United States Environmental Research EPA-600/9-78-024
Environmental Protection Laboratory September 1978
Agency Gulf Breeze FL 32561
Research and Development
&EPA American-
Symposium on
Use of Mathematical
Models to Optimize
Water Quality
Management
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AMERICAN-SOVIET SYMPOSIUM
ON
USE OF MATHEMATICAL MODELS TO OPTIMIZE
WATER QUALITY MANAGEMENT
KHARKOV and ROSTOV-ON-DON, USSR
December 9-16, 1975
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
GULF BREEZE, FLORIDA 32561
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DISCLAIMER
This report has been reviewed by the Gulf Breeze Environmental Research
Laboratory, U.S. Environmental Protection Agency, and approved for publica-
tion. Mention of trade names or commercial products does not constitute
endorsement or recommendation for use.
11
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FOREWORD
The Joint American-Soviet Committee on Cooperation in the Field of
Environmental Protection, established by an agreement signed May 23, 1972,
in Moscow, identified eleven ecological problem areas for cooperative
investigation and exchange of information. In accordance with these objec-
tives, the Joint Symposium on Mathematical Models to Optimize Water Quality
Management was convened December 9-16, 1975, in Kharkov and Rostov-on-Don in
the USSR.
During the Kharkov portion of the symposium, participants exchanged
experience on methodological questions related to planning in accordance with
legal requirements and organizational units existing in each country.
Discussions focused on technical processes connected primarily with questions
of simulation and optimization modeling of processes that determine the
dynamics of water quality within river basins.
The symposium noted a specific process in the work organization and in
the development of mathematical models, and established grounds for further
scientific-technical cooperation in the analysis of the actual water protec-
tion planning process and realization of these plans.
Papers presented in Rostov-on-Don described a wide >range of methods for
mathematical modeling and many specific models. These papers can be used to
compare and evaluate techniques of mathematical modeling developed in the
cooperating countries and will serve as the basis for further progress in this
field.
Symposium participants revealed two main problems confronting scientists
from both countries. The first difficulty is to determine various constants
(coefficients) that describe water ecosystems. The second difficulty is the
selection of water quality criteria and the complexity of defining boundary
conditions of the model. Participants agreed that both questions cannot be
resolved at the present time.
The symposium demonstrated great mutual interest in the exchange of
scientific ideas and information. It actively involved the participants and
confirmed the usefulness of joint discussions of actual water protection
problems.
The symposium showed that the problems of modeling the ongoing processes
in water bodies, and the problem of surface water quality management, are
very complex. At present, the solution (development of a universal approach
to the model) are a long way off. We hope that the joint efforts of the
scientists from the USSR and USA will attain this goal.
iii
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The published materials reflect the .original_positions of both sides
before the_discussion developed at the Kharkov part of the symposium.
Publication of the proceedings by the Environmental Research Laboratory, U.S.
Environmental Protection Agency, Gulf Breeze, Florida, fulfills the protocol
agreement for Project II.1 requiring simultaneous and independent publication
in both countries after coordination of manuscripts.
T.T. Davies
Acting Director, EPA Environmental
Research Laboratory
Gulf Breeze, Florida USA
USA Chairman of Project II-l
V.R. Lozanskiy
Director, All Union Scientific Research
Institute for Water Protection
Kharkov, USSR
USSR Chairman of Project II-l
iv
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ABSTRACT
The American-Soviet Symposium on Use of Mathematical Models to Optimize
Water Quality Management examines methodological questions related to simula-
tion and optimization modeling of processes that determine water quality of
river basins. Discussants describe the general state of development and
application of mathematical models designed to predict and optimize water
quality management in the USA and USSR. Subjects of papers presented by
American and Soviet specialists include: planning comprehensive water quality
protection systems; modeling water quality in river basins; geographic-econ-
omic aspects of pollution control systems; identification of ecosystem models
by field data; management decisions for lake systems on a survey of trophic
status, limiting nutrients, and nutrient loadings; a descriptive simulation
model for forecasting the condition of a water system; mathematical ecosystem
models and a description of the water quality in water bodies; mathematical
modeling strategies applied to Saginaw Bay, Lake Huron, to eutrophical pro-
cesses in Lake Ontario, to hydrodynamics and dispersion of contaminants in
the nearshore, and to coastal currents in large lakes; and the construction
of a model of Lake Baikal on the principles of self-organization. Publication
of the proceedings held December 9-16, 1975, in Kharkov and Rostov-on-Don,
USSR, is in compliance with the Memorandum from the Fourth Session of the
Joint American-Soviet Committee on Cooperation in the Field of Environmental
Research.
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CONTENTS
Foreword ill
Abstract v
Participants in the Symposium ix
Planning Comprehensive Water Quality Protection Systems—D.P. Loucks . . i
Optimization of Water Quality Control in a River Basin—A.K. Kuzin ... 36
Modeling Water Quality in River Basins—Ye.V. Yeremenko 55
Water Quality Management Models: Specific Cases and Some Broader
Observations—R.J. DeLucia and T. Chi 92
Construction of Models for Forecasting and Controlling Water Quality in
a River Basin on Principles of Self-Organization—A.G. Ivakhnenko . . 127
Geographic-Economic Aspects of Pollution Control Systems—
C. ZumBrunnen 155
Modeling and Decision-Making Aspects of Water Quality Management—
M.A. Pisano 180
Determining Water Quality Criteria for Water Flow in Solving the
Problems of Controlling Water Pollution—Kh.A. Velner, V.I. Gurariy,
and A.S. Shayn 193
Mathematical Modeling of Eutrophication Processes in Lake Ontario—
R.V. Thomann 208
Modeling the Dispersion of Pollution and Thermal Pollution in Water
Bodies—O.F. Vasilyev 246
Identification of Ecosystem Models by Field Data (Inverse Problems of
Ecology)—V.B. Georgiyevskiy 263
Mathematical Modeling of the Hydrodynamics and Dispersion of
Contaminants in the Nearshore—W. Lick , 282
Modeling and Verification : The Two Indispensable Legs for Progress
toward Understanding and Management of Aquatic Systems as Demonstrated
by a Study of Coastal Currents in Large Lakes—C.H. Mortimer 315
VII
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Construction of a Model of Lake Baikal on Principles of Self-
Organization—A.G. Ivakhnenko 340
Management Decisions for Lake Systems on a Survey of Trophic Status,
Limiting Nutrients, and Nutrient Loadings—A.F. Bartsch and J.H.
Gakstatter 372
Mathematical Modeling Strategies Applied to Saginaw Bay, Lake Huron—
V. Bierman, W. Richardson, and T.T. Davies 397
Descriptive Simulation Model for Forecasting the Condition of a Water
Ecosystem—A.B. Gortsko 433
Mathematical Ecosystem Models and Description of the Water Quality
in Water Bodies—R.A. Poluektov 442
Final Report 451
Vlll
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PARTICIPANTS
AMERICAN SIDE
Tudor T. Davies
Co-chairman of the Symposium
Acting Director, Environmental Research Laboratory
U.S. Environmental Protection Agency
Gulf Breeze, Florida
Mark A. Pisano
Co-chairman of the Symposium
Director, Water Planning Division
U.S. Environmental Protection Agency
Washington, 'D . C .
Russell J. DeLucia
Meta Systems Inc.
Cambridge, Massachusetts
Daniel P. Loucks
Professor, Department of Environmental Engineering
Cornell University
Ithaca, New York
Craig ZumBrunnen
Professor, Department of Geography
Ohio State University
Columbus, Ohio
A.F. Bartsch
Director, Environmental Research Laboratory
U.S. Environmental Protection Agency
Corvallis, Oregon
Wilbert Lick
Professor, Department of Geology
Case Western Reserve University
Cleveland, Ohio
C.H. Mortimer
Director, Center for Great Lakes Studies
University of Wisconsin, Milwaukee
Milwaukee, Wisconsin
IX
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Robert V- Thomann
Professor, Environmental Engineering and Science
Manhatten College
Bronx, New York
Victor Bierman
Scientist, Environmental Research Laboratory
U.S. Environmental Protection Agency
Grosse lie, Michigan
SOVIET SIDE
V.P. Lozanskiy
Co-chairman of the Symposium
Director, All Union Scientific Research Institute for Water Protection
Kharkov
A.A. Zenin
Co-chairman of the Symposium
Director, Hydrochemical Institute
Novocherkassk
E.V. Yeremenko
Laboratory Chief, All Union Scientific Research Institute for Water Protection
Kharkov
R.A. Galich
Scientific Secretary
All Union Scientific Research Institute for Water Protection
Kharkov
G.A. Sukhorukov
Laboratory Chief, All Union Scientific Research Institute for Water Protection
Kharkov
Yu.V. Yurkov
Doctor, All Union Scientific Research Institute for Water Protection
Kharkov
P.N. Matveyev
Department Chief, All Union Scientific Research Institute for Water Protection
Kharkov
Yu.M. Lugovskoy
Department Deputy Chief
All Union Scientific Research Institute for Water Protection
Kharkov
V.V. Piotrovsky
Senior Engineer, All Union Scientific Research Institute for Water Protection
Kharkov
x
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V.B. Stradomsky
Deputy Director, Hydrochemical Institute
Novocherkassk
M.N. Tarasov
Laboratory Chief, Hydrochemical Institute
Novocherkassk
A.A. Matveyev
Laboratory Chief, Hydrochemical Institute
Novocherkassk
V.T. Kaplin
Laboratory Chief, Hydrochemical Institute
Novocherkassk
Yu.V. Fliegelman
Interpreter, Hydrochemical Institute
Novocherkassk
T.Ch. Kolesnikova
Laboratory Chief, Hydrochemical Institute
Novocherkassk
A.B. Gorstko
Department Chief
Scientific Research Institute of Engineering and Applied Mathematics
Ro s t ov-on-D on
V.A. Znamenskiy
Department Chief, State Hydrological Institute
Leningrad
A.G. Ivahknenko
Department Chief
Institute of Cybernetics of the Ukrainian Academy of Sciences
Kiev
I.A. Koloskov
Laboratory Chief, Main Administration of Hydrometeorological Services
Moscow
A.Ch. Ostromogilskiy
Senior Scientist, Main Administration of Hydrometeorological Service
Moscow
A.C. Litvinov
Senior Scientist
Institute of Inland Water Biology of the USSR Academy of Sciences
Borok
xi
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I.D. Rodziller
Sector Chief, All Union Scientific Research Institute
Moscow
V.L. S imonov
Senior Engineer, Ministry for Reclamation and Water Management
Moscow
Yu.M. Tulupchuk
Senior Scientist
Central Scientific Research Institute of Complex Utilization of Water Resources
Kiev
L.M. Boychuk
Senior Scientist
Institute of Cybernetics of the Ukrainian Academy of Sciences
Kiev
N.A. Lobzovskiy
Senior Scientist, Limnological Institute of the USSR Academy of Sciences
Leningrad
V.A. Vavilin
Scientific Secretary
Institute of Water Problems of the USSR Academy of Sciences
Moscow
V.A. Rozhkov
Sector Chief, State Oceanographic Institute
Leningrad :
I.I. Gavrilov
Senior Scientist, Moscow State University
Moscow
Yu.M. Lebedev
Laboratory Chief
Pacific Ocean Institute of Oceanography of the USSR Academy of Sciences
Vladivostok
V.B. Georgiyevskiy
Department Chief, Institute of Hydrobiology of the USSR Academy of Sciences
Kiev
O.F. Vasilyev
Department Chief, Institute of Hydrodynamics of the USSR Academy of Sciences
Novosibirsk
xii
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PLANNING COMPREHENSIVE WATER QUALITY PROTECTION SYSTEMS
Danjel P. Loucks
INTRODUCTION
This paper will review a number of mathematical modeling components that
can be combined and solved for the preliminary planning of economically effi-
cient .comprehensive water quality management programs for river systems. The
selection of this topic . for this symposium s.tems from the discussions with
VNIIVO specialists that took place in the spring and summer of 1975 in the
USSR and in the U.S. The modeling techniques to be reviewed in this paper
hopefully reflect an approach to long-run water quality protection planning
that might be appropriate for various rivers (.e.g. the Severski Donets) in
the USSR.
Water quality management models based on optimization techniques have
been developed and used for the preliminary definition and evaluation of al-
ternative designs and .operating policies of ...a variety of water quality control
options. These include, but are not limited to, (1) wastewater treatment
facilities, (2) wastewater effluent storage lagoons, (3) land disposal sites
for wastewater disposal and additional treatment, (4) flow augmentation re-
servoirs, and (5) instream aeration devices.. Each of t.hese optipns can in-
fluence the quality of water in river systems. While other engineering alter-
natives are available (such as bypass piping of wastewater effluent and re-
gional treatment facilities), it is this set of five water quality protection
alternatives that will be considered and modeled in this paper.
The simultaneous examination of all appropriate engineering measures
available for water quality protection permits the identification of a more
economically efficient or cost-effective solution to water quality protection
problems than would be pq.ssible if only one or .two such measures were con-
sidered. The purpose of developing and using optimization and simulation
models for water quality planning, of course, is to identify those particular
measures, their design specifications, and their operating policies, that
will .satisfy water .quality, economic, .and possibly other objectives.
In the USSR, one of the more important objectives is,cost minimization.
Therefore, the discussion in this paper will be oriented towards that single
objective of cost minimization, while, meeting .various stream quality standards
involving dissolved oxygen (DO), biochemical oxygen demand (BOD), conservative
substances [e.g., chlorides (CI)], and nutrients (e.g., nitrates).
Before beginning this review of some optimization modeling techniques
for water quality protection, it is important to understand how people in the
U.S. view the role of water quality management optimization models, as opposed
1
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to ecosystem simulation models, and. even the meaning of optimization in this
context. In spite of the relatively powerful solution procedures (algorithms)
and the computer speed and capacity available today for solving large complex
optimization problems, these procedures and computers are inadequate for
solving any reasonably complex water quality protection problem. Thus, some
simplifying assumptions are necessary regarding the biological-chemical and
physical phenomena that occur in a water system and also regarding the poli-
tical or management objectives.
Even if one could be certain of all of the values of various model para-
meters, there is still considerable uncertainty concerning the most correct
form of a model of a particular aquatic system. Because of limitations in our
knowledge of aquatic ecosystems, even the relatively complex multispecies eco-
system models that are currently being used for predictions of the biological,
chemical, and physical quality of water bodies are limited (Russell, 1975).
These ecosystem models are very data-demanding, and hence very costly. They
are also extremely non-linear and multidimensional, making it difficult to
incorporate them into an optimization program for estimating least-cost man-
agement solutions.
Given these limitations of existing ecosystem simulation models, one
could question the value of developing optimization models that require many
more simplifying assumptions and approximations. The answer is that such
models are useful if through their use one is able to reduce a large set of
possible engineering design and operating policy alternatives to a relatively
small number for more detailed (and expensive) examination with simulation
procedures. Our experience suggests that optimization models of water quality
protection systems cannot find the best or optimal solution, but, if applied
correctly, they can be used to eliminate clearly inferior alternatives from
further consideration. When there are many engineering or management alter-
natives available, the need for a preliminary screening technique is substan-
tial. The cost and time required to simulate even a fraction of all alterna-
tives could be prohibitive.
Water quality optimization models are also less data-demanding than the
ecosystem simulation models. Thus, with the speed of their solution, these
models are a potentially useful tool for preliminary screening of alternative
designs and operating policies. The specific applications of water quality
optimization models discussed by DeLucia and Chi (1975) at this symposium are
examples that could be cited to illustrate this point.
The models to be outlined in this paper will be static models. They will;
not be able to answer such questions as when additional wastewater treatment
or storage lagoon capacities should be made available, or to permit prediction
of the quality profile in a water body over a daily cycle. These models will
simply enable "snapshots" to be taken at particular future periods, corre-
sponding to the exogenous values of the wastewater flows at each industrial or
municipal site. They will only be able to estimate the average water quality
values in rivers with specified flow characteristics. Such models are for
long-term planning, not short-term control. For greater precision, simulation
models are required, and in fact, such models should be used to check and to
improve upon the results of preliminary screening models (such as those out-
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lined in this paper) prior to the actual implementation of a particular water
quality protection plan.
This paper will first discuss each component of a comprehensive water
quality model. These modeling components will include a flow augmentation
reservoir, a wastewater treatment facility, an effluent storage lagoon, a land
disposal system, and instream aeration. These components can then be combined
into a single comprehensive model of any particular river system, such as the
Severski Donets. Finally, the paper will discuss procedures that can be used
to solve any resulting comprehensive model.
MODEL COMPONENTS
Low Flow Augmentation
Changes in streamflow can influence the water quality that occurs at any
particular location in a river. Augmenting the flow with relatively high
quality water dilutes the waste load, increases the stream velocity, spreads
the degradation reaction further downstream from the waste sources, and af-
fects the atmospheric reaeration rate. Temperature changes, if any, may alter
the biochemical processes. Colder augmenting flows tend to decrease the de-
oxygenation and reaeration rates, thereby extending oxygen deficits and BOD
concentrations for longer distances downstream. Lower temperatures also in-
crease the saturation concentrations of dissolved oxygen. Increased flows
also result in increased scouring of bottom or benthal deposits, which in turn
increases the waste load of the stream.
The net effect of all changes resulting from augmented streamflows may
increase or decrease the concentrations of oxygen, BOD, conservative sub-
stances, and nutrients, depending upon the location of the waste sources and
the particular sites where quality is to be maintained. Hence, to put it
simply, flow augmentation may be good or bad, or good and bad simultaneously,
depending on a particular location in the river basin, the extent of augmenta-
tion, and other circumstances. The only way to evaluate the advantages and
disadvantages of this alternative for water quality protection is to include
it with the overall comprehensive water quality modeling effort. Just how
this might be done is the subject of this section of the paper.
Water quality standards are usually based on a design streamflow condi-
tion. For example, in the more humid portions of the U.S., this design flow
is the particular yearly minimum average 7-day flow that is equalled or ex-
ceeded 90% of the time (9 years in 10 on the average). This is commonly
called the 7-day, 10-year flow. In much of Eastern Europe and in the USSR,
it is my understanding that this design flow is the minimum average monthly
flow exceeded 19 years out of 20, or 95% of the time. Both flows are minimum
flow conditions that are exceeded most of the time. They are based on the
assumption, generally but not always correct, that low-flow, stream-flow
conditions result in poorer stream qualities than those that exist during
higher streamflow conditions.
Upstream reservoirs that are used, in part, to supply waters for low flow
augmentation, must be designed and operated in such a way as to increase the
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design flow condition, i.e., to increase the flows that will be equalled or
exceeded by the same probabilities as the original unaugmented design flow
conditions. Hence, models that are appropriate for the definition and evalua-
tion of flow augmentation policies must be able to predict the reservoir
storage requirements and operating policies that will maintain the probabili-
ties of exceedence of augmented design streamflows, i.e., those streamflows
that are used as a reference for water quality standards.
Low-flow augmentation models may differ depending on how the design
streamflows are defined (Loucks and Jacoby, 1972). However, each low-flow
augmentation model is developed to estimate reservoir storage capacity re-
quirements and operating policies associated with any particular set of aug-
mented streamflows. These flows will later be used in a water quality model
for estimating the appropriate capacity of wastewater treatment facilities,
lagoon storage basins, and other water quality protection options needed to
meet various stream quality standards.
Water allocated for low-flow augmentation is not usually available for
other uses, and hence some opportunity cost or loss is incurred. Additional
reservoir storage capacity may also be required to meet the low-flow augmenta-
tion requirements, and this too costs money. Thus, a portion of the overall
economic efficiency or cost minimization objective for water quality protec-
tion will be related to the total cost of flow augmentation, C^(Qr) + C^(Kr),
provided by each reservoir r having an annual flow augmentation release of at
least Qr at least 95% of the time, and a storage capacity of Kr- That portion
of the overall least-cost objective related to low-flow augmentation can be
written as:
minimize I Cr(Qr) + Cr(Kr) (1)
r 1 2
To determine the storage capacity, Kr, and operating policy needed to
provide an annual augmenting release of Qr or equivalently a series of monthly
flows Q£ downstream of each reservoir site r that are equalled or exceeded 95%
of the time, one can examine overyear and within-year storage capacity re-
requirements separately. First, the overyear capacity requirements can be
defined by a series of continuity equations for each year y of record, equa-
ting the initial storage volume Sy plus the annual inflow l£, less the flow
augmentation release Qr, less any excess release E^, to the final storage
volume, S£+i.
Sj + i£ - Qr _ Er = sr+i for each year y (Vy) (2)
To estimate the probability that a release of at least Qr can be made
from reservoir r each year, the concept of a mean probability of exceedence
can be introduced. The mean probability that any particular streamflow will
be equalled or exceeded is based on the assumption that any future flow has
an equal probability of falling within any interval defined by the ordered
sequence of historical (and/or synthetic) streamflows. Suppose, for example,
that there exists a record of n unregulated annual streamflows, hence n+1
streamflow intervals. Ranking these flows so that the largest streamflow has
the lowest rank, m=l, and the lowest streamflow has the highest rank, m-n,
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the mean probability that a future unregulated annual streamflow will equal
or exceed a flow of rank m is m/n+1. The expected recurrence interval of an
annual flow of rank m is n+l/m years, the reciprocal of the mean probability.
An estimate of the mean probability of a given unregulated streamflow
makes it .possible to define the mean probability of any particular reservoir
release Qr. If the release variable Qr is contained in each of the n equa-
tions (2), then the probability of its being equalled or exceeded is n/n+1.
If this Qr variable is omitted from the equation 2 having the lowest annual
inflow I , i.e., the year having an inflow of rank m=n, then the probability
of its being equalled or exceeded is reduced to n-l/n+1.
Including the Qr term in equation 2 only when inflows Iy have a rank of
m less than or equal to n+1 times the desired mean probability of exceedence
p (say 0.95) will insure that the release Qr will be equalled or exceeded
lOOp percent of the time. If, for example, there exists an annual unregu-
lated streamflow record of 19 years, the inclusion of a release variable Qr in
each of the 19 equations (2) will ensure that the flow augmentation release
will be equalled or exceeded with a probability of 19/20 or 95% of the time.
Overyear reservoir storage capacity, SJ, required for any flow augmenta-
tion release Qr is obtained from the maximum value of all the storage volumes
sy i so yy (3)
To ensure a particular distribution of the annual flow augmentation re-
lease Qr in each month t, within-year reservoir storage capacity may also be
required. This can be obtained by another mass balance, equating the sum of
the initial within-year storage volume, s^, plus that average proportion 6?
of the annual flow release Qr that flows into the reservoir during month t,
less the monthly flow augmentation release Q£, to the final storage volume
sr , for each month t, (when t=12, t+l=l).
a^ j- R n'- — n^ — c^- v> f/i}
b_. T Pi_L^ V+- — t+1 \^J
The fraction gt can be determined from the average monthly uncontrolled
inflows to the reservoir, I£ = 1/n ^ I t
y
g = I^/E Ir Vt (5)
The total reservoir capacity Kr required for each monthly flow augmenta-
tion release Of will be the smallest capacity, Kr, that contains each overyear
storage, s£, and within-year storage, s£.
Kr >_ S£ + s£ Vt (6)
The model just developed, equations 1 through 6, can be solved for vari-
ous chosen values of Q£. Each set of augmented flows Q£ can then be incor-
porated into a water quality management model, to be discussed shortly. Once
particular vectors of Q£ have been selected as optimal, reservoir rule curves
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defining the releases that can be made as a function of calendar time and
existing storage volumes can be established. Such rules enable the operators
of reservoirs to meet the flow augmentation quality and reliability require-
ments. The formulation and use of these types of reservoir design -and opera-
ting models, and how they can be used to derive reservoir rule curves, has
been described (Loucks, 1976).
Wastewater Treatment
The concentration, CW?, of any potential pollutant p, (e.g. , BOD, chlo-
ride, nitrogen) in the wastewater produced at site i, can be reduced by vari-
ous unit treatment processes, k. The quantity;of wastewater, QW^, that is
routed through each unit treatment process k will determine the cost and
removal efficiency of the overall wastewater treatment facility. To illus-
trate how this overall efficiency and cost can be estimated, assume there are
K^ possible unit treatment operations at site i (e.g., grit chamber, primary
and secondary sedimentation basins, activated sludge unit, chlorination,
etc.), each costing CT^(QW^), depending on the hydraulic capacity, QW^, and
each removing a known fraction, fg, of pollutant p. The portion of the cost
minimization objective related to wastewater treatment is
Ki
minimize I I CT^(QW^) (7)
i k
The concentration, CP, of each pollutant in the effluent of the treat-
ment facility depends on the quantity routed through each unit treatment pro-
cess, and the configuration of these processes. Mass balance equations for
both flows and pollutant masses are easily written for any proposed configura-
tion of unit treatment processes, and these will define the mass and concen-
tration of each pollutant in the effluent. An example of how these conserva-
tion equations can be written is illustrated below in Figure 1. In Figure 1,
process 3 might be considered an alternative to 2 and 4; the only difference
perhaps would be the recycling of a fraction p of the flow between 2 and 4.
Note that each of the expressions in Figure 1 is linear with respect to
the unknown flows, QW, in each link of the treatment system. Since the final
effluent flow equals the known influent flow, the concentrations, CP, are the
unknowns. If certain alternative wastewater unit process configurations in-
clude a portion of the flow being returned to a previously considered unit
process, e.g., a return flow as shown in unit process k=3 in Figure 1, the
resulting expressions for masses of pollutants may be nonlinear unless the
recycled portion p is known. These looped components (e.g., a and b in
Figure 1) can be considered as a single component whose hydraulic capacity
must include the quantity of the recycled flow.
The total fraction, X?, of each pollutant p removed by the entire treat-
ment facility can be determined by comparing the known influent concentration
to the effluent concentration in the model solution.
X£ = C£/CW£ Vp (8)
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Influent
Effluent
i—
(1+p) Of
a I H b
.j
Figure 1. Flows and pollutant concentrations throughout a potential waste-
water treatment system.
Process k Influent Flow Effluent Concentration CP.
1
2
l qw±
Q _< Q - Q
CWP(l-fP) - C^
;± CWP(l-fP)(l-fP) = CP,
4
SYSTEM
QjlQj
QW.
i
CWP(l-fp)(l-fP) = C?
i) + c^ (qi -
J - C?3 *
- Q?)
P£ (Q? - Qj)
*0btained by equating the recycled concentration to the effluent concentration.
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Defining some constants, y?> such that for a variable X.,
X. = y?xP Vp (9)
and comparing the costs derived from equation 7 for various values of X^, the
cost of wastewater treatment at each site i can be expressed as a function
CT.(X.). This procedure is appropriate at each site i only where a particular
pollutant p is dominant, i.e., where it will dictate the design of the treat-
ment facility. If this is not the case, then each unit process must be in-
cluded in the overall model. However, when a single pollutant p is dominant,
the detailed analysis of each wastewater treatment component k can be omitted
from the more comprehensive river basin model. The objective component for
wastewater treatment,
minimize Y CT (X ) (10)
I i i
and constraint equations
CW?(1 - X±/Y?) = C? Vp (11)
are all that are needed in the more comprehensive model of a river basin
water quality protection system.
What is needed for the remainder of the overall water quality protection
model are the quantities or masses of pollutants in the wastewater effluent
of the treatment facility. This effluent may either be stored temporarily in
a lagoon or discharged directly into a water body or disposed of directly on
land. Assuming lagoons are reasonable alternatives for the temporary storage
of treated wastewater effluent during periods of low stream, or land assimi-
lative capacity, they too can be modeled for the purpose of determining cost-
effective lagoon capacities and operating policies.
Effluent Storage Lagoons
The ability to temporarily store treated wastewater effluent when the
assimilative capacity of the receiving land or water is low, and release it
when the assimilative capacity is high, may permit some reduction in the total
cost of water quality protection. Effluent storage lagoons can be modeled and
included in the overall planning model to determine whether or not any cost
savings can be achieved by their installation and operation. The objective
component of these storage lagoon submodels will be to minimize the total
annual cost, CL.(KL.), of constructing and operating each lagoon of capacity
KL..
i
minimize Y CL (KL ) (12)
i i 1
The lagoon itself can achieve some removal of certain nonconservative pollu-
tants. Relationships between pollutant removal efficiencies and effluent
detention times in the storage lagoons are hard to obtain, especially in view
of the possibility of relatively long detention times, incomplete mixing, and
the varying temperatures and storage volumes. It appears, however, that
removal rate constants can be estimated from the literature for long storage
-------
periods in temperate climates (Fisher, 1968; Renn, 1973). Assuming first
order reactions for the removal of each pollutant p, a mass balance for each
month can be written relating storage volumes, S^, influent and effluent
concentrations, CP and UP., and flows QWti and Wti> in eacn month t at each
site i. t:L
p — -p p p
- QWtiC? - QWtiC?i - k?Cti( - 2 - (13)
t = 1,2,..., 12 (if t = 12, t+1 = 1)
where :
c? = concentration of pollutant p in the treated effluent entering the
storage lagoon in month t.
CTP . = average concentration of pollutant p in the effluent released from
the storage lagoon during month t.
k.P = pollutant p removal rate constant for the effluent in storage
lagoon during month t (temperature effects included), month .
QW . = volume of wastewater influent to storage lagoon in month t.
TJW . = volume of wastewater effluent discharged from the storage lagoon
to the land during month t.
St^ = volume of wastewater in the storage lagoon at beginning of month t.
Often there exists a maximum limit of pollutant removal that can occur
in a lagoon, i.e., after long detention times no more removal occurs. This
can be stated, for example, as
TJP > some fraction of the influent concentration, C? Vt , i (14)
ti - !
C _1_ C
The term , t+l.i ti-\ i-s t^16 average volume of treated wastewater ef-
fluent in the storage, lagoon at site i during time period t. The St.'s are
obtained from the continuity equations for storage and may vary from period
to period. The effects of evaporation in each month t from the surface of
the lagoon can be approximated by the fraction et^
£tiSti + ^Wti - ^ti - St+l,l ^ <15>
where the lagoon capacity, KL^, must be no less than each storage volume,
str
KLi 1 sti *t,± (16)
In order to provide adequate detention times for the settling of suspend-
ed solids in the stored effluent, a minimum detention time, 6m^n, may be re-
quired.
s "-1
It is also important to ensure that the storage lagoon does not become
anaerobic, especially during late winter and spring. This limits the depth
-------
of the lagoon and the BOD loading per unit area-day. The lagoon area, LA^, is
determined by the actual and maximum allowable daily BOD loadings.
D QW c!
LA. _>_ maximum BOD load ing/day = max, t ti L (18)
al 1 r>T.TaK1 e ROTl/nn-it- 01-03— r\ a-vj t ^ T D '
allowable BOD/unit area-day t LRn
h
where:
LR, = allowable daily BOD loading per unit area for storage lagoon of
depth h.
D = the number of days in month t.
The area, LA^, of the storage lagoon must also be no less than the maxi-
mum storage volume divided by the depth, h-^, at site i.
LAi 1 Sti/hi vt (19)
The maximum of the areas, LA^. found by equations 18 and 19, will be the area
required for the storage lagoon. If the land area required for a lagoon costs
money, then this annual cost should be included in objective function, equa-
tion 12.
The wastewater effluent in a storage lagoon can be either discharged into
water or onto land, depending in part on the type of pollutants contained in
the effluent, the availability of suitable land and its costs, and on the
capacities of water and land to assimilate the wastes remaining in the efflu-
ent.
If land is available, then one alternative for additional wastewater
treatment and stream quality protection is to discharge the wastes onto land.
The extent to which this is'possible can be estimated by including this alter-
native, where feasible, within the comprehensive water quality protection
planning model.
Wasjtewater Disposal on Land
Land disposal of wastewater effluents can be an effective means of
further removing various pollutants, such as BOD, nitrogen, phosphorus, patho-
gens (bacteria and viruses), heavy metals, and the like. This additional
treatment of wastewater effluent is sometimes referred to as effluent polish-
ing. In the U.S., land application of wastewater effluent is increasingly
being considered as an important option in the protection of water quality.
This is only true, of course, in situations where suitable land exists and
when the continued application of effluent would not adversely affect the
physical characteristics and productivity of the land.
If land disposal is to be considered, a model is needed to estimate the
quantity of treated wastewater effluent that can be disposed on land without
violating various maximum allowable concentrations of nutrients in the
drainage water. Since any soil system has a limited capacity to "polish"
secondary effluents, restrictions must be placed on the amount of concentra-
10
-------
tion of pretreated wastewater applied per unit area.
The capacity of most soils (especially those found near the Severski
Donets) to remove essentially all of the BOD, phosphorus, and heavy metals is
quite satisfactory for average wastewaters. Hence, modeling the interaction
of these pollutants in the soil is not necessary. While the accumulation of
phosphorus and heavy metals cannot be dismissed in the long-run, especially
when applying secondary effluents over a considerable period of time, the
primary focus is usually on nitrogen. The nitrogen in the wastewater effluent
is mostly of the N03~N form because of the storage detention times in the
lagoon which allow almost complete nitrification (Renn, 1973).
Nitrogen is highly mobile in soils, is often associated with eutrophica-
tion, and can also be a public health problem. If the nitrogen concentration
exceeds the drinking water standard (in the U.S. this equals 10 mg/£ N03-N)
in recharged groundwater, its use for water supplies may be severely limited
due to the lack of an economical treatment technology that can remove nitrate.
Organics are not desired in water supplies because they often impair their
taste and odor. By addition of organics (e.g., from sludge and wastewater
effluent), the slow depletion of organic matter in the soil is reduced or
prevented, but care is also needed to prevent excessive concentrations of
organics in the runoff (Renn, 1973). Hence a method is needed to assist the
environmental engineer in designing and operating economically efficient
wastewater treatment and land disposal systems.
In this section a simple model is proposed which can assist in evaluating
land disposal schemes of pretreated wastewater effluent on well-drained flat
grassland for a variety of wastes and climatic conditions. The model can be
used to estimate the maximum hydraulic and assimilative capacity of the soil
and the capacity and operating policy of an effluent storage lagoon, if re-
quired, due to seasonal or monthly changes in the soil assimilative capacity.
Since the costs, Cl£(IA^), for irrigation and drainage systems can be expressed'
as a function of the irrigation area, IA-j_. required, the objective function of
this portion of the overall comprehensive model can be written
minimize £ CIi(IAi) .(20)
i
The land which receives the lagoon effluent from the treatment plant or
the storage lagoon must be relatively level and well-drained. It may or may
not be underlain by an impermeable layer. The drainage capacity of such soils
that are underlain by an impermeable layer depends on the hydraulic conductiv-
ity coefficient of the soil and the spacing of the tilage drains.
The soil drainage capacity, d, is dependent on the horizontal distance
between tiles, s, the vertical distance between the impermeable layer and the
tile drain, hQ, the depth of the water table above the impermeable layer, H,
and the hydraulic conductivity of the soil, k.
?k o o
d = * (H2_h2) (n)
11
-------
The upper layer of the soil is- kept aerobic by maintaining the water
table below the root zone. This permits normal plant growth. Hence, the
application rate of wastewater effluent must be limited so as not to exceed
the drainage capacity of the soil. Application rates can be calculated from
a soil-water balance which includes precipitation and evapotranspiration.
Mt+l,i - Mti + IRti + Pti - Eti - dti Vt'i (")
where at each site i,
Mt± £ M± Vt,i (23)
dti - di Vt»i (24)
and
d^ = drainage capacity of soil (equation 21)
dti = drainage rate during month t
Etj_ = average evapotranspiration during month t
IR^i = application rate of wastewater effluent during month t
M^ = moisture content in the soil at saturation
^ti = moisture content in the soil at the beginning of month t
Pj.£ = average precipitation during month t
The relationship between that portion, LQt^, of the lagoon ef fluent, QW"ti
discharged on land, the application) rate, IRj..^, and the land area, IA^, used
for disposal of wastewater effluent, is given by
LQtj. £ IR^ClAi) (25)
For most locations, average monthly values for precipitation, Pt^, and
evapotranspiration, E^, can be obtained from climatological summaries. If
E£ is not readily available, it may be estimated for certain crops from hand-
books (Agricultural Engineers Handbook, 1961) , or with the aid of some empir-
ical formula (Hamon, 1961).
If the average monthly temperature falls below 0° Celsius, the soil may
be frozen. When this occurs, no wastewater effluent mayl be discharged.
Where snow accumulates, soils may be frozen even after the average monthly
temperature rises above 0° Celsius.
The application rate of wastewater decreases if soil clogging occurs due
to the suspended solids content in the wastewater effluent. By ensuring that
the application" rate, IR^, does not exceed a safe limit;, IRmax ^, favorable
infiltration characteristics can be maintained over long periods of time.
IRti 1 IRmax,i Vt (26)
Having a model of the hydraulic and physical aspects of the wastewater
land disposal system, it is now necessary to model the soil nitrogen balance.
Haith (1973) has presented a concise summary of monthly inventory equations
describing the soil organic and inorganic nitrogen. For convenience, the site
12
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subscript i will be dropped here, but would be needed in the overall model if
more than one land site was being considered for wastewater disposal.
where:
organic N : Ot+1 = (l-mt)0t + X0t
inorganic N : I = I +
Vt
+ XIt -
Vt
(27)
(28)
It = soil inorganic nitrogen at beginning of month t
L£ = inorganic nitrogen lost due to drainage during month t
™.(- = fraction of soil organic nitrogen converted (mineralized) to
inorganic nitrogen during month t
Nt = inorganic nitrogen removed from the soil by plant growth during
month t
0 = soil organic nitrogen at beginning of month t
XI t = inorganic nitrogen added to the soil due to wastewater irrigation
during month t
XOfc-= organic nitrogen added to the soil due to wastewater irrigation
during month t
The quantity of organic nitrogen, XO , and inorganic nitrogen, XI t, added
to the soil can be found from the wastewater application rate, IR^ , and the
concentration of nitrogen,
j,
in the wastewater effluent.
X0t =
XIt =
IR
t
IR
Vt
Vt
(29)
(30)
In the above equations, the parameter a is the fraction of organic nitrogen in
the total nitrogen of the applied wastewater effluent, and the pollutant p
represents nitrogen.
The mineralization rate of nitrogen depends, in part, on soil tempera-
ture. If the temperature drops below 0° Celsius, no mineralization is assumed
to take place. The yearly mineralization rate can be estimated from a know-
ledge of -the fraction of organic matter,
m
Q ,
annually mineralized in the soil.
The monthly fraction mineralized, mt, can be estimated
tionship.
with a linear rela-
mt = m0Tt/
Vt
(31)
where T is average temperature during month t in °C. The sum includes only
those months t whose average temperature is greater than 0°C.
If no nitrate adsorption in the soil is assumed for month t, the N03~N
concentration, nt, in the soil solution is simply the average quantity of
inorganic nitrogen in the soil, (It + It+j_)/2, divided by the average "soil
moisture content, (Mfc + Mt+1)/2, in month t.
13
-------
nt = K
M
Vt
(32)
t+l
Drainage occurs when the soil is saturated, i.e., when Mt = M. Hence, the
N03-N concentration, nt, in the drainage water is
n. =
2M
Vt
(33)
The amount of NOo-N lost, L,., in the drainage water is equal to the average
concentration of N03-N in the drainage water, n times the quantity,
water drained.
Vt
Substituting equation 33 into 34 yields
= dt(It
+ I
t+1
)/2M
Vt
of
(34)
(35)
Grass or pasture land has a high uptake of nitrogen compared to crops
such as corn or wheat. Therefore, grass appears to be an ideal cover crop for
wastewater disposal sites, at least for nitrogen control. Among other ad-
vantages are reduced erosion and runoff, maintenance of satisfactory infiltra-
tion capacity, fairly even nitrogen removal over the whole land area in con-
trast to "point uptake" by corn, for example, and fairly uniform growth rates
for most of the year. Moreover, grasses grow well in wet soils, a common
characteristic of many potential land sites used for wastewater disposal.
Cooke (1967) estimates that grass takes up about 2/3 to 3/4 of the applied
nitrogen. Sopper (1973) found an uptake efficiency over 97.3% when canary
grass was irrigated with secondary wastewater effluent. Thus a reasonable,
and conservative, estimate of the maximum nitrogen uptake, N for grasses
could be assumed to equal 70% of the available nitrogen in the soil.
N <_ 0.7(m Ot
XI
Vt
(36)
On the other hand, the uptake of nitrogen usually does not exceed some
maximum value, Nmax, depending on the plant type (Overman, 1975).
_
Vt
(37)
If the yearly potential nitrogen uptake is known, monthly uptake rates
for the growing months (e.g., months with average temperatures, T > 0) can
be estimated. For example*, a nitrogen removal of 420 kg/ha per year is not
unusual when grass is heavily fertilized. If we assume a maximum annual up-
take of 420 kg/ha and a growing season of April to October, with April and
October removing only one-half the amount of the other months, the maximum
monthly uptakes, N
max
for May through September.
would be 35 kg/ha for April and October and 70 kg/ha
14
-------
Constraints limiting the nitrogen content of the drainage water can be
formulated in at least two ways. One way is to limit the total annual
quantity of nitrogen loss in the drainage water per unit area, ^L to a cer-
tain maximum amount, Nmax, t
r max /oo\
I Lfc <_ N (38)
Another way is to restrict the concentration of nitrogen in the drainage
water. For example, constraints could be written ensuring that the NOg-N con-
centration of the drainage water, nfc, does not exceed the permissible nitrate-
N concentration for drinking water, e.g., 10 mg/£.
n < 10 Vt (39)
t
This constraint can be modified to require that only the average N03-N con-
centration of the drainage water be no greater than the permissible nitrate
standard.
The total organic carbon applied per unit area should not be in excess of
the assimilative capacity of the soil. Overloading will cause anaerobic con-
ditions and insufficient removal efficiencies for COB. About 4.5 to 11.2
metric tons/ha-year of organic carbon are needed to maintain a static organic
carbon content in the soil. Most of the carbon will come from plant residues.
Sandy soil can assimilate loadings of up to 67 metric tons/ha-year of organic
carbon without detrimental effects (Thomas and Bendixen, 1969). Such high
loading rates are not likely to be achieved by applying storage lagoon ef-
fluents; yet they may be obtained by spraying sewage sludge. Defining C as
the maximum permissible organic carbon load, the constraint limiting the
application of organic carbon
£ Cp IR £ C p = organic carbon (40)
will rarely be limiting for wastewater effluent disposal on land.
Stream Water Quality
So far various land based wastewater treatment and disposal alternatives
have been modeled. The resulting effluent from these facilities will be dis-
charged into the stream and affect its quality. The extent to which all dis-
charges affect water quality will determine, along with each facility's cost,
the extent to which such facilities will be built and operated.
Recall that the effluent from the storage lagoon, Wt-p is divided into
that which is discharged into land, LQt^, and that which is discharged into
the river or stream, RQt-p at each site i and month t.
3Wti = LQti + RQti Vi,t (41)
If land is used for wastewater disposal, essentially all the pollutants are
removed except nitrogen, and the quantity, Lt^IA^. or concentration, nt^, of
15
-------
nitrogen may be limited by constraint equation 38 or 39 if these are included
in the model. The quantity of drainage water that eventually enters the
stream from the land application site is difficult to predict and depends con
siderably on the particular site and its relation to the stream. Whatever it
is, it will usually be negligible in comparison to the total streamflow and
hence can be ignored.
The quality of each pollutant p discharged to the stream directly from
the storage lagoon at each site i, as obtained from equations 13 and 14,
equals the lagoon effluent concentration C"^. times the effluent flow, RQti-
As previously indicated, nitrogen is perhaps the only significant pollutant
contained in the drainage water of many land application sites, and its
quantity, from equation 35, is Lti. Hence, the total mass, m^, of each pol-
lutant entering the stream equals
mP. =
ti
Lti P = nitrogen (42)
Up RQ. . Vp + nitrogen (43)
ti C1
The total mass and concentration of each pollutant p at any particular
site in a stream will depend on the quantities of waste discharged upstream
and on numerous factors such as the stream flow, velocity, temperature, depth,
decay rates, and the like. A variety of equations have been developed and
used to predict, for given stream flow and wastewater effluent conditions, the
average mass of pollutant p that will exist at a site j in the stream in month
t resulting from a discharge of one unit mass of pollutant p at site i up-
stream. These coefficients can be denoted as d^jt, anc^ are functions of the
design stream flow conditions in period t. Hence for each discrete flow aug-
mentation level [£ Q£, (equation 4), plus the appropriate interflow between
upstream reservoirs r and site j], there will be a set of coefficients, df^...
o J
The total mass, M£•, of pollutant p at each site j will equal that from un-
controlled sources, U?•, plus that resulting from the controlled discharges.
For wastewater effluents having a biochemical oxygen demand, BOD, the coeffi-
cients d? for estimating both BOD and dissolved oxygen deficits are de-
scribed ill the symposium paper by DeLucia and Chi (1975). These are just some
of the types of equations developed to predict, in a very simplified manner,
stream water quality.
Constraints limiting the concentration of pollutants at any particular
site j would have the general form:
M /F . j£ Maximum allowable concentration
J at site j in month t Vt,j (45)
where F is the design streamflow at site j in month t.
16
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Artificial Instream Aeration
Artificial instream aeration is another method of increasing the dis-
solved oxygen concentration of streams. This is usually accomplished by in-
jecting oxygen or air into water through a network of perforated pipes or by
rotating devices that cause surface turbulence, thereby increasing the area
over which oxygen transfer can occur. These methods may be particularly
efficient for the temporary improvement of near anaerobic conditions, i.e.,
at sites where the dissolved oxygen deficits are relatively high.
The oxygen transfer rate due to aeration devices varies directly with the
oxygen deficit, the water quality, temperature, and with the flow. Denoting
£i as *-fre ra-te of oxygen transfer per unit of power input under actual condi-
tions, H^ as the rate of oxygen transfer under standard conditions (i.e., tap
water quality at 20°C, and at one atmosphere of'pressure), DODti as the dis-
solved oxygen deficit concentration at site i in period t (i.e., a particular
value of the left-hand side of one of equation 44), and Kfci a parameter re-
lated to the actual streamflow conditions at site i in period t, then
Ht± = HsKti (DODti) Vi,t (46)
The oxygen transfer rate, ORti, at site i from artificial aeration is the
product of the weight of oxygen per power input unit, Hti, the total power
input per unit of time, Wti, and a coefficient g needed for unit conversion.
ORt± = 3Ht± Wt± (47)
The oxygen transferred per unit of time, ORt_p at each site i cannot exceed
the aerator capacity KA^ times the total weight transferred per unit power
input, Hti,
ORt± £ BHtl KA± (48)
which is equivalent to specifying that the power input per unit of time, W^.
cannot exceed the capacity, A-^.
Assuming that the oxygen transfer per unit of time, Ht£, is an average
value throughout period t and that the aerators are operating at a constant
power input W^, the estimation of the minimum cost combination of capaci'^^s,
KA-;, and power inputs, Wt^, can be simplified. Since there is no need for
(and higher costs associated with) extra capacity, power input per unit of
time can be equated to capacity KA-. Hence from equations 46 and 47
ORti = BHS Kti (DODtl)(KA±) (49)
in which both DOD . and KA^ are unknowns. The total annual cost CAi(KAi) is
now a function of only the capacity, KA., and this portion of the overall
objective can be written
minimize I CA^KA^ (50)
i
17
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MODEL SYNTHESIS
To illustrate how these model components can be combined into a more com-
prehensive model, it will be sufficient to consider the simple river basin
system shown in Figure 2. This system includes two reservoirs (sites 1 and 2),
a source of BOD which can be treated and then stored in a lagoon for later
discharge onto land and/or into the water (site 3), the river system itself,
which contains one point at which artificial aeration may be feasible (site 5),
and a water quality site (site 6) in which standards exist restricting the
chloride, BOD, DO, nitrogen, and phosphorus concentrations.
LEGEND
/\ Reservoir
f~\ Treatment Facility
Q Effluent Lagoon
Q Wastewater Irrigation Land
<^> Aerator
Quality Standards
Figure 2. A hypothetical water quality management problem.
Flow Augmentation from Sites 1 and 2
At reservoir sites 1 and 2, reservoir 1 has a capacity of K1. The
existing capacity K2 at reservoir 2 can be increased, K2, at a cost C2(K2).
The costs for providing a 95% firm flow of Qr at each reservoir site (r=l,2)
are the loss of benefits or opportunity costs C^Q1) and C2(Q2) of using that
firm flow elsewhere. Hence, the total reservoir costs equal
CR =
C2(QZ) + C2(K2)
(51)
18
-------
where for each reservoir site r, overyear continuity requires
Sr + Ir - a Qr - Er = Sr Vy (52)
y y y y y+1
tor each year y of record and
fl if rank m of I is <_ (n+l)(.95)
ay =T<1 otherwise, depending on what is considered
[reasonable in years,that a failure occurs (53)
(If y = the last year of record, then y+1 = the first year of record.)
Within-year continuity requires
r ij:
sr + _ Qr - Qr = S; Vt.r (54)
if t = 12, t+1 = 1
The storage capacity, Kr, necessary to release a ,95% firm yield of of is
estimated from
so 1 Sy yy>r (55)
and
/*" -v
r = 1
,r
+ s.l Vt (56)
Kr r = 2
Waste Treatment at Site 3
Since carbonaceous BOD is the dominant waste at site 3, a cost effective
analysis can be made of potential wastewater treatment facilities at site 3,
each associated with a particular removal fraction, X3 . Once this has been
done, by using procedures already discussed, the treatment cost CT3 can be
expressed as a 'function of the removal fraction X3 .
CT3 = CT3(.X3) (57)
The effluent concentration of carbonaceous BOD equals
' • CW^ (1-X3) = CP p = carbonaceous BOD (58)
The effluent concentration of all other pollutants, including nitrogenous BOD,
nitrogen, and phosphorus, is
CWP(1-X3/YP) = CP (59)
19
-------
Storage Lagoon at Site 3
The equations for the lagoon at site 3 are those required for the con-
tinuity of storage and pollutant concentration.
3 Vt (60)
Vt (61)
-
Vt (62)
Additional constraints may be required to define appropriate detention times,
loading rates, and lower bounds on concentration reductions in the storage
lagoon.
S3 + S , , o
_JLr » _ minimum detention time Vt (63)
2 QWt3
o
KL^ ^—(QW^ CP)/LR^ p = BOD Vt (64)
^t
0 — ^Q ,
j - D tJ 3 h
C^ 2i some minimum fraction of influent
concentration, C? Vt (65)
The cost of the storage lagoon is
CL3 = CL3(KL3) (66)
The lagoon effluent can be released to either the land or directly into
the river.
QWt3 = LQt3 + RQt3 (67)
Wastewater Disposal on Land at Site 3
Constraint equations are needed to define the soil moisture balance and
ensure tt,,..ii the application of wastewater does not exceed the soil drainage
capacity.
Mt+l,3 =Mt3 + IRt3 +Pt3 - Et3 - dt3 Vt
Mt3 <_ M3 Vt (69)
20
-------
£ D3 Vt (70)
The land area required depends on the application rate and the quantity re-
leased to the land
LQt3 = IRt3 IA3 Vt
The application rate in each month should not exceed some maximum safe sus-
tained value.
IRI] Vt
t3lI]Wx,3
Also required are the inventory equations for soil organic and inorganic
nitrogen.
3 = (l-mt3) Ot3 + X0t3 Vt (73)
t3 + mt3°t3 + XIt3 - Lt3 - Nt3 vt (74)
where
X0t3 = a "cP3 IRt3 p = nitrogen Vt (75)
XIfc3 = (1-a) "c£3 IRt3 p = nitrogen Vt (76)
The quantity of nitrogen in the drainage water equals
Lt3 = dt3(It3 + It+l,3)/2M3 Vt (77)
The cost of the irrigation land area required for wastewater disposal is
CI3 = CI3(IA3) (78)
Finally, the total quantity of each pollutant p entering the stream at
site 3 equals that from the irrigation area, if the pollutant is nitrogen,
and that discharged directly from the lagoon, if any.
/ Lf-3 if p = nitrogen
mt3 - ^t3 RQt3 +
-------
where
KL >_ St4 Vt (81)
The chloride concentration C"t4 in the effluent, QWt4, discharged into the
stream, is obtained from the mass balance:
4 - Ct4) = QWfc4 C4 - QWt4 Ct4 Vt (82)
The above equation assumes no loss of chloride in storage. The last term
represents the mass of chloride, m?4 (p = chloride), released to the stream at
site 4 in each period t.
The cost of the lagoon equals
CL4 = CL4(KL4) (83)
Dissolved Oxygen Deficit at Site 5
The dissolved oxygen deficit concentration at site 5 results from the
controlled BOD source at site 3 and all uncontrolled sources. For any parti-
cular design flow condition in each month t, the dissolved oxygen deficit at
site 3 equals
DODt5 = mPj ' ^ct + mP^ • dP^ + ^ . Vt (84)
where pc and pn are the carbonaceous and nitrogenous BOD indicators and y'
refers to the DO deficit from uncontrolled upstream sources.
Artification Aeration at Site 5
The reduction of dissolved oxygen deficit at site 5 is a function of the
existing deficit, the capacity of the aerator 'and some known coefficients for
each design flow condition.
ORt5 = BHsK±(DODt5) (KA5) Vt (85)
The dissolved oxygen deficit immediately downstream of the aerator is
DODt5 - ORt5 = DOD+5 Vt (86)
The cost of the aerator is
CA5 = CA5(KA5) (87)
Stream Quality -standards 'at Site 6
For any particular design flow condition, the dissolved oxygen deficit,
DODj-g, at site 6 is simply a constant [Sty^ = exp (-reaeration rate constant
times time of flow from site 5 to site 6)Jtimes the deficit at site 5, plus
22
-------
the additional deficit, u , caused by uncontrolled BOD sources between sites
5 and 6. t56'
DODt6 = (DODt5)(656t)
Vt (88)
The dissolved oxygen at site 6 will equal the saturation quantity of oxygen
for the design flow condition, DOSt6, less the deficit quantity DODt6. This
quantity of DO can be converted to a concentration which may be required to be
no less than some minimum allowable concentration. Letting F , be the design
flow at site 6 in month t,
= Qt + Qt + (interflow between sites 1, 2, and 6
exceeded 95% of the time) Vt (89)
The dissolved oxygen standard can be written:
6 - DODt6)/Ft6 L minimum allowable DO con-
centration in month t at
site 6. Vt (90)
Constraints ensuring that other pollutant concentrations are no greater than
their maximum allowable concentrations can be expressed as:
(/, dj_-jt mE • + ^tG^ ^ ^t6 — maxi-mum allowable concentration of
i pollutant p at site 6 in month t Vt,p (91)
Model Objective
The overall objective of this cost-effective model is to minimize the
total annual cost, i.e., the sum of equations El, E7, E16, E28, E33, and E37.
minimize CR + CT3 + CL3 + CI3 + CL4 + CA5 (92)
MODEL SOLUTION
There are, no doubt, numerous ways to solve the model developed in the
proceeding section, or any similar water quality protection model. Since
there are some nonlinear terms in the objective and constraints, it is cer-
tainly possible to obtain at least locally optimal solutions with a nonlinear
programming algorithm, e.g., the sequential unconstrained minimization tech-
nique (SUMT) developed by Fiacco and McCormick (1968) . While this and other
nonlinear programming techniques are capable of solving water quality pro-
tection models similar to those described in this paper, they are usually not
as readily available or operational in the U.S. as are linear, separable,
mixed integer, or dynamic programming algorithms, some of which can result in
globally optimal solutions.
Note that the objective function, while nonlinear, is separable. Hence
each separate function can be approximated by linear segments if desired.
Figure 3 reviews three procedures for doing this. Linear programming, using
approximation methods (a) or (b) , can be employed if the function F(X) in
Figure 3 is to be maximized. Note method (b) requires no additional con-
23
-------
straints for increased accuracy, only an increase in the number of variables
w.. Yet method (b) cannot be used if the concave function F(X) is to be
minimized. For minimization of concave functions (or maximization of convex
functions) separable programming applied to method (a) will ensure that each
allocation x^ equals its upper bound, a^-a^^, before x^+1 > 0, but there is
no guarantee that a global minimum will be obtained. To guarantee a global
minimum, the mixed integer programming method (c) can be used as a solution
technique. Note that any function of the form /.nFjXj using method (a) is
and
+
using
equivalent to £^£ . F . (a.j-aj_1)wp using method
method (c) . Details of these linearization and solution procedures can be
found in most texts in operations research.
F(X)
Method Figure 3. Piecewise linearization procedures.
(a) Optimize F(X) = F x + F2x2 + F3x3 + ...
Subject to X = Xj + x2 + x, + ...
xl 1 al
< a3 - a2
Method
(b) Maximize F(X)
Subject to X
F(a1)w1 + F(a2)w2 + F(a3)w3
+ a2w2 + a3w3 + ...
Method
(c) Minimize F(X)
Subject to X
blzl + Flxl + b2z2 + F2x2
xl 1 alzl x2
zl + Z2 + Z3 +
all z. integer
24
. . . < 1
b3z3
_ <^ a3x3 •
-------
The products of quotients of two or more decision variables found in the
constraint set pose a more difficult problem in model solution. At each
wastewater treatment'site these nonlinear expressions result from a considera-
tion of effluent storage and land application of wastewater effluent. If only
one, or perhaps two, pollutants are of concern at each site, then each treat-
ment/disposal site can be examined separately, by dynamic programming or by
gradient techniques, to develop a cost function for the mass of pollutant
discharged into the stream at that site (Koenig and Loucks, 1975). This per-
mits much of the detail of each treatment/disposal site to be omitted from
the overall'river'basin model.
If a variety of pollutants are discharged into the receiving water at
each wastewater treatment site, then it is necessary to include the model
components of each treatment site in the overall river basin model. This may
pose a problem if computer capacity is limited, especially if any linear pro-
gramming algorithm or its extensions are used.
Models similar to, but larger than that constructed in the previous sec-
tion have been solved at Cornell University by using separable and mixed
integer algorithms.' Separation of product terms has been accomplished by use
of logarithms (to the base two) and piecewise'linearization procedures. The
separation approach is illustrated below for a product of two nonnegative un-
knowns, X and Y having maximum values of Xmax and Ymax.
Let cz = XY
Assume X, Y >_ 1 and ax =,X, by = Y.
The constants a, b, and c are chosen so as to ensure that, x, y, and z do
not exceed 10. It is in the range from 1 to 10 that the piecewise lineariza-
tion of the Iog2 function can be relatively accurate.
Next, define variables log^x, Iog2y, and Iog2z such that
Iog2c + Iog2z = Iog2a + Iog2x + Iog2b + Iog2y
or
Iog2z - Iog2x - Iog2y = log (—)
Now all that is needed is to relate each variable x, y, and z to its Iog2,
e.g.,
. n
X = y x.
1-1 X
0 < x. < IP. Vi
log x = I ' a.x. - constant
x X
25
-------
where a. is the average slope of the logarithmic function in interval i. For
example, if the interval of each x. is 1 (n=10), then the values of ot^, begin-
ning at i=2 (i.e., X _> 1) are 1.0, 0.58, 0.42, 0.32, 0.26, 0.22, 0.18, 0.17V-
and 0.15.
Assuming X^ < 1, and that c^ = 2 (instead of °°) , then the constant in the
above equation is 2.
If the ct.'.s associated with each variable (e.g., x, y, and z) are the
same, then separable programming will yield global solutions, albeit approxi-
mate ones to the original nonlinear problem. If a small value is assigned to
any variable in the solution of the model, it is usually a signal that the
value should .be 0, and the model can be resolved after setting it equal to 0.
Admittedly this logarithmic separation is a relatively crude approach to
solving the model and it takes a few iterations to determine the appropriate
constants a, b, and c and zero values variables that are contained in non-
linear expressions. But the method works and the solutions obtained so far
have been sufficiently accurate for the preliminary screening of design and
operating policy alternatives for water quality protection.
Undoubtedly the most difficult nonlinearity encountered in any water
quality protection model that includes both the river flows and wastewater ef-
fluent discharges as endogenous or unknown variables is the dependence of the
pollutant transfer functions, d?. , on streamflow. From a practical point of
view, it seems wise to separate trie analysis of design streamflows, Qr, from
the remainder of the problem. Before solving- the water quality portion of the
problem, a variety of reasonable flow vectors Q = {Qr} can be assumed, and
their minimum cost, CR, can be determined by solving equations El through E6.
Then for each Of, the appropriate transfer coefficient, d?. , can be derived
for use in the water quality model, equations E7 through Enz, where the cost,
CR, is now known.' The vector Qr that minimizes equation E42 will be the most
cost effective of all those assumed flow vectors Qr. This approach is dis-
cussed in considerable detail elsewhere (Loucks and Jacoby, 1972).
CONCLUSIONS
Rather than a comprehensive state-of-the-art report on water quality pro-
tection modeling, this paper has attempted to briefly outline several ap-
proaches to water quality protection modeling. Emphasis has been placed on
problem definition and model development, not on model solution techniques.
The models presented are not intended to be definitive; they only illustrate
how various water quality control subsystems can be modeled and analyzed.
Problem definition, modeling, and analysis involve considerable judgment
in addition to some mathematical and computational skills. How this is accom-
plished in any particular situation depends not only on the problem itself,
but also on the skill of the systems analyst(s), and on the available data,
programming algorithms, and computational facilities. Clearly, the systems
models and techniques illustrated in this paper are not substitutes for judg-
ment, but are only a means of enhancing this judgment by providing information
and opportunities that otherwise might not be available.
26
-------
Experience to date* has 'demonstrated that some of the more obvious types
of information and opportunities provided by systems approaches to water
quality problems include: •1) an increased capability for defining and eval-
uating possible alternatives and for keeping a wider range>bf options open
and available for analysis at each level of decisionmaking; 2) an improved
capacity for testing assumptions and data to estimate the effects of economic,
hydrologic, political, and technological uncertainties; 3) a method whereby
all assumptions and judgments', and the consequences of these assumptions and
judgments, are made explicit and available for all to see and question, if so
desired* (This 'feature is viewed'as a distinct disadvantage by some observers.);
and 4) a means of communication between all the participants in the planning
and evaluation stages.
The analysis of water pollution control problems, like any public policy
planning process in a dynamic environment, is never completed. It is an
iterative process, one that involves continual updating of information and
techniques of analysis. Since this activity costs money, the extent of any
increase in the types and precision of. the data collected or any increase in
modeling sophistication depends on the additional benefits to be-achieved.
These costs and benefits are often difficult to1 assess,; especial-ly when envi-
ronmental quality problems are involved. Needless to say, ;however, if the
systems analysis studies are done well at each level of decisionmaking, the
increased data base should enhance the content>;of-the debate>'over what deci-
sion to make—a debate that can center on which assumptions are best rather
than on whether the ?best solution is given a particular set of assumptions.
As the art and science of environmental systems analysis mature and as
more qualified analysts-become available and involved in environmental quality
problems, the contributions from quantitative systems analyses, together with
the more traditional' planning and economic-engineering studies, should, have a
substantial influence on environmental quality, policymaking processes.
NOTATION
The following notation is listed in the order presented in the paper.
Physical units are expressed in terms of length, L, mass, M, and time, T.
Depending on the actual units used, many of the equations in the paper may
require a 'constant for unit conversion, e.g., from mg to kg or from days to
months.
Variables, Functions, and Parameters
Cq(Qr) = annual opportunity cost ofj a release Qr from reservoir r, $.
C£(Kr) = annual cost of a capacity Kr at, a-reservoir site r, $.
f 3
S = initial storage volume in year at reservoir r, L .
Ir = net inflow to reservoir r in year y, L3T-1.
Er = excess release from reservoir r in year y, L3T-1.
27
-------
S0 = overyear storage capacity of reservoir r, L3.
^t = proportion of average inflow in month t to total annual average
inflow.
I t = inflow in month t in year y, L3!"1.
If = average inflow in month t at reservoir r, L3T-1-
st = initial within-year storage in month t in reservoir r, L3.
CWP = concentration of pollutant p in wastewater influent at site i,
ML"3.
U
QWjL = quantity of wastewater routed through treatment process k at
site i, L3!"1.
K1 = total number of treatment processes k at site i.
CT^(QW.) = annual cost of treatment process k at site i having a hydraulic
capacity of QW., $.
fP = fraction of concentration of pollutant p removed by process k.
C? = concentration of pollutant p in the wastewater effluent of the
treatment facility at site i, ML~3.
X? = fraction of pollutant p removed by treatment facility at site i.
CT.(X.) = annual cost of treatment facility at site i as a function of its
"design" efficiency X., $.
CL.(KL.) = annual cost of effluent storage lagoon of capacity KL^ at site
i, $.
KLj = capacity of effluent storage lagoon at site i, L3.
o
Stj[ = initial storage volume in lagoon in month t at site i, L .
"C?. = average concentration of pollutant p in the effluent lagoon
during month t at site i.
kP = removal rate constant for pollutant p in month t, T .
"QW . = volume of wastewater effluent discharged in month t from storage
lagoon at site i, L3!"1.
e . = evaporation rate fraction.
= minimum detention time, T.
= lagoon area at site i, L2.
28
-------
Dt = number of days in month t.
LR, = allowable pollutant loading rate for storage lagoon of depth h,
ML~2T-1.
^i = maximum depth of storage lagoon volume at site i.
GI-^(IAi) = annual cost of land area, LA^, used for wastewater disposal at
site i, $.
d = soil drainage capacity, LT"1.
n0 = vertical distance between impermeable layer and tile drain, L.
s = horizontal distance between tiles, L.
H = depth of water table above impermeable layer, L.
k = hydraulic conductivity of soil, LT"1.
Mt^ = moisture content of soil at beginning of month t at site i, L.
d . = drainage rate during month t at site i, LT"1-
Et. = average evapotranspiration during month t at site i, LT"1 .
Pti = average precipitation during month t at site i, LT
IRti = application rate of wastewater effluent in month t at site i,
LT-1.
LQt• = quantity of wastewater effluent discharged to land in month t
at site i, L3!"1.
IRmc,^ • = maximum wastewater application rate at site i, L,T-1.
itlci^v 1
It = soil inorganic nitrogen at beginning of month t, ML"2.
mt = fraction of soil organic nitrogen converted to inorganic
nitrogen during month t.
N = inorganic nitrogen removed from the soil by plant growth during
t month t, ML~2-
0 = soil organic nitrogen at beginning of month t, ML~2-
XI = inorganic nitrogen added to soil by wastewater application
*" during month t, ML~2 .
XO = organic nitrogen added to soil by wastewater application during
t month t, ML"2.
29
-------
max
RQti
d?.
IJt
Hti
H
DODti
OR
ti
W.
ti
KA±
CAi(KA1)
Indices
n
m
y
r
i
t
P
= fraction of organic nitrogen in the total nitrogen content of
the wastewater applied to land.
= concentration of N03~N in the drainage water, ML~3.
= maximum permissible organic carbon loading rate, ML"2!"1.
= quantity of wastewater effluent discharged to the stream at
site i during month t, L3T-1.
= mass of pollutant p entering stream at site i during month t,
MI"1.
= mass of pollutant p at a downstream site j resulting from the
discharge of one unit of pollutant p at site i upstream, in
month t, M.
= total mass of pollutant p at site j in month t, M.
= quantity of pollutant p at site ] in month t resulting from
uncontrolled sources, M.
= design streamflow at site j in month t, L .
= rate of oxygen transfer per unit of power input under actual
conditions L .
= rate of oxygen transfer per unit of power input under standard
conditions, L"1.
= dissolved oxygen deficit concentration in month t at site i,
ML~3.
= oxygen transfer rate, MT"1.
= power input per unit of time, LMT~2.
= aerator capacity, LMT"1.
= annual cost of aerator of capacity KA^ at site i, $.
number of years (or segments)
rank of an annual flow
index for year
index for reservoir site
index for wastewater source or treatment site (or for segment of
a nonlinear function)
index for month of year
index for a type of pollutant
30
-------
Operators
2. = sum
V = for all
3 = such that
31
-------
REFERENCES
Agricultural Engineers' Handbook, Rickey, C.B., P. Jacobson, and C.W. Hall
(eds.) 1961. McGraw-Hill, New York.
Cooke, G.W. 1967. The Control of Soil Fertility. Crosby Lockwood and Son,
Ltd., London.
DeLucia, R.J. and T.W. C•?. 1975. Water quality management models: Specific
cases and some broader observations. Paper prepared for the US-USSR
Symposium on the "Use of Mathematical Models to Optimize Water Quality
Management," Kharkov, USSR.
Fiacco, A.V. and G.P. McCormick. 1968. Non-Linear Programming: Sequential
Unconstrained Minimization Techniques. John Wiley and Sons, New York.
Fisher, C.P., W.R. Drynan, and G.L. Vanfleet. 1968. Waste stabilization pond
practices in Canada. In: Advances in Water Quality Improvement. E.F. Gloyna
and W.W. Eckenfelder (eds.), University of Texas Press, Austin and London.
Haith, D. 1973. Optimal control of nitrogen losses from land disposal areas.
ASCE Proceedings. Journal of Sanitary Engineering Division, Vol. 99, No.
EE6, p. 923.
Hamon, W.R. 1961. Estimating potential evapotranspiration. ASCE, Journal of
the Hydraulics Division, Vol. 87, No. HY3, Proc. Paper 2817.
Koenig, A. and D.P. Loucks. 1975. A management model for wastewater disposal
on land. Second Annual National ASCE Conference on Environmental Engineer-
ing Research, Development, and Design, University of Florida, Gainesville.
Loucks, D.P. and H.D. Jacoby. 1972. Flow regulation for water quality manage-
ment. In: Models for Managing Regional Water Quality. R. Dorfman, H.D.
Jacoby, and H.A. Thomas (eds.), Harvard University Press, Cambridge, Mass.
Loucks, D.P. 1976. Water Quality Management Models. In: Systems Approach to
Water Management. A.K. Biswas (ed.), McGraw-Hill Publishing Co., Inc.,
New York.
Overman, A.R. 1975. Effluent irrigation of pearl millet. ASCE, Journal of the
Environmental Engineering Division, Vol. 101, No. EE2, Proc. Paper 1123,
pp. 193-199.
Renn, C.E. 1973. Management of recycled waste-process water ponds. U.S.
Environmental Protection Agency, Washington, D.C.
32
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Russell, C.S. 1975. Ecological modeling in a resource management framework.
Resources for the Future Working Paper QE-1, Washington, D.C.
Sopper, W.E. 1973. Crop selection and management alternatives—perennials.
In: Recycling Municipal Sludges and Effluents on the Land. National
Association of State Universities and Land-Grant Colleges, Washington, D.C
Thomas, R.E. and T.W. Bendixen. 1969. Degradation of wastewater organics in
the soil. Journal of the Water Pollution Control Federation. Vol. 41, No.
5, Part 1, p. 808.
33
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DISCUSSION
KUZIN: In this paper, the current approach to solving complex problems of
planning technical water protection measures for water basins has been
described. The principles for creating an economic-mathematical model for
optimizing a system of activities has been reported in detail.
1. The proposed model for determining minimum cost for a system of
measures is static and is intended for a fixed planning period. Therefore,
with each essential change in source conditions, the model must be recon-
structed, and the earlier accepted solutions are corrected. This procedure
can be considered as a process of uninterrupted planning.
2. A description of non-point source pollution control in cost-effect
terms is considerably more complex than measures for point sources of water
pollution. Therefore, when the influence-of surface flow on water quality
must be taken into consideration, it is necessary to consider the essential
complication of the global model.
3. It is not clear what type of model is being used to calculate pol-
lutant transformation in the water body. Simplified models based on instan-
taneous mixing of wastewaters with river water and numerical solutions by a
computer do not present any particular difficulties. However, dilution in
rivers is calculated with a model based on the turbulent diffusion differen-
tial equation, the complexity of solving optimization problems increased,
especially in the case of a subdivided grid system.
4. When the purpose function for solving optimization problems for the
system of water protection measures presented generally in the report are
minimized, the parameters for each water body are determined.
The approach to the development of models is also followed in the USSR,
and offers a perspective method for solving the complex problems of planning
water protection. It is necessary to show that the application of methods
for optimizing and planning water protection in the USSR can be broader be-
cause of the possibility for manipulating the treatment level-of wastewaters
from industrial establishments. As you know, stringent limits have been
placed on industrial discharges by a 1972 law enacted in the US.
STANISHEVSKIY: Dr. Louck's report contains information on economic-mathe-
matical modeling of the following water protection measures: (1) flow
augmentation; (2) wastewater treatment; (3) construction of wastewater lagoons
for preliminary treatment; (4) irrigation by wastewaters for preliminary
treatment; and (5) artificial aeration. These measures can be used jointly
or examined as alternatives for water protection.
34
-------
It is necessary to note that in most previous U.S. efforts related to
optimization problems, water quality was monitored only for one parameter—
dissolved oxygen concentration. In this report, four indicators were used—
dissolved oxygen concentration, biochemical demand in oxygen, content of
chlorides, and nitrates.
As a whole, the material presented in the report is of scientific
interest and confirms that research approaches for development of models for
water protection measures and their optimization being carried out in the
U.S. and USSR have been chosen wisely. An exchange of scientific information
on the problem will be beneficial to both countries.
THOMANN: I would just like to indicate that it is no more difficult to in-
clude diffusion in any of these optimzation models. In fact, the inclusion
of a turbulent and tidal dispersion and diffusion in estuaries in our country
goes back ten years to the Delaware estuary. Optimization includes multi-
dimensional, (two- or three-dimensional steady-state diffusional problem
structures), as well as multivariable feedback and feed-forward systems.
These extensions do not necessarily contribute to the complexity of the
problem providing all extensions are linear.
Also it is getting easier to write equations but not so easy to apply
those equations to the real world. It seems that once the door was opened,
our ability to write equations increased dramatically, but I really believe
the problems we face are the successful application of those equations to
actual planning problems and situations.
35
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OPTIMIZATION OF WATER QUALITY CONTROL IN A RIVER BASIN
A.K. Kuzin
INTRODUCTION
In the USSR, a comprehensive approach to the development of water pollu-
tion control is represented in the general plan for use and protection of wa-
ter resources, especially when capital investments for the development of
sewer systems under the plan have been determined. The structure, composi-
tion, and effectiveness of water protection measures are determined during the
development of long-term regional and river basin utilization plans. Broad
use and protection of water resources should be in accord with the long-term
plans for the national economy. The methodology for the development of such
schemes, however, has not been perfected. In the majority of cases, only a
few sites for effluent treatment plants have been evaluated. The application
of methods of optimization of technical solutions is limited.
The implementation of these schemes within the long-term plans of nation-
al economic development can lead to rigid requirements for water quality pro-
tection construction projects which, at present costs, would mean an annual
increase in expenditures of millions of rubles. Evaluation of the numbers of
variants of future water pollution control systems, and decisions on optimum
solutions to problems by a specified evaluation criterion, will be possible
only through a wide application of methods of mathematical modeling with mo-
dern computers for analyzing very complex models.
One main difficulty in developing an acceptable technique for optimizing
anti-pollution measures within the limits of a river basin is the exceptional
complexity of analyzing the interrelationships of water quality, expenditures
for protection, and economic profits. Even for a small section of a river,
under the conditions of a probabilistic river flow availability, the complex-
ity of chemical and biological processes of degradation, discharge of ef-
fluents into the water body, and the connection between the foregoing, can be
only approximately established. Another serious difficulty is the multiplic-
ity of possible combinations of control actions on water quality. It is ob-
vious that a special scientific approach, such as systems analysis, is needed
to define the optimum combination of questions related to the complex problem.
The process of developing a model for optimizing water protection measures
over the entire basin, classifying optimization problems, selecting basic
principles to use for solution, and considering certain types of problems in
detail is described for systems analysis.
36
-------
Constructing Large-Scale Models of Water_QualityControl
In a broad sense, water quality control is considered a system of effects
on the basin and its environment. These effects are projected both for the
distant future and for perturbation that may be caused by accidents. In the
first case, long-term strategy can be developed by considering statistical ob-
jectives and corresponding capital investment policies. In the second case,
it is only possible to prepare a response plan because it is impossible to
forecast breakdowns at effluent treatment installations and oil pipelines,
leaks from gas storage tanks, tanker accidents, and so forth. The combination
of problems connected with the functioning of the river basin and its environ-
ment are considered in developing a strategy for water quality control.
First, we shall consider the presence of cause and effect links between
the input, as an object of effect on the environment, and output, as a trans-
formation resulting from different effects. The input is determined as a
measure of the influence of controllable or non-controllable impulses that
can be evaluated quantitatively. The output is determined as the state of
water quality at a given point of the river basin at a corresponding moment of
time. If we represent the water quality as a vector U-^Ct) where U± is the
water quality in i-th part of the river ba.sin at the period of time t, and the
state of factors acting upon the system as a vector Tf-^Ct), where I?i are the
factors acting upon the i-th part of the river basin, then the relationship
between the input and output can be determined with the help of a certain
transition matrix F-^:
F^Ct) = ^(t) (1)
Equation 1 is the most general form that can be applied over a wide range
of conditions. Indeed, the water state vector allows us to characterize its
quality at any point in the basin at the required moment of time. The vector
of effect factors can reflect all elements of man's economic activity affect-
ing the state of the basin. The transition matrix, for different specific
applications of this equation, show the change in the vector of the first
type, depending on the value of the second one. In water quality control,
during long-term changes, we are not interested in the state of water quality
at each point or at any moment, but rather the average monthly or average
annual indices for separate cross-sections of the basin. Therefore, detailed
description of processes of transformation of water quality are not necessary.
In practice, the periods of the effective functioning of water protection
measures are assumed to be indefinitely long [H^Ct)]. This is due to the long
period of construction, operation of water protection installations, and the
considerable inertia of their effect.
In addition to the consideration of the limits of a river basin as a
single hydrographical system (in a physical sense), socio-economic considera-
tions also exist. These are revealed, in particular, when basin water quality
goals, as well as technical water protection measures, are planned. Indeed,
when water quality goals are established for the limits of the basin, the
specific conditions affecting water quality can be fully evaluated, and the
economic profits and losses, corresponding to different water quality goals,
37
-------
can be statistically evaluated. In planning for technical water quality pro-
tection through a complex study of the main sources of water pollution in the
given basin, a wide application of methods for optimization and mathematical
modeling is possible with modern computers to provide the required water goals
for minimum resource expenditure.
Optimization of water protection measures over the entire basin is diffi-
cult because of the complexity of mathematical description, the diversity of
environmental effects on the water quality state of the basin, and the volumi-
nous initial information requiring preliminary mathematical treatment. Two
approaches exist for modeling complex processes: to construct a gigantic
"super-model" with the development of a complex algorithm of optimization, or
to create a system of several small separately optimized models. The main
advantage of the so-called "super-model" is the possibility of a totally inte-
grated and centralized control of the system. Such models, however, have very
serious disadvantages which make them inefficient when modeling complex manu-
facturing and/or economic processes.
For solving optimization problems in the field of water protection, the
"super-model" cannot be recommended for the following reasons: Such a model
would not be reliable, because admissible simplifications in small-scale
systems (accepted during the construction of physical-biological-mathematical
models of effluent transformation in a water medium relationship), character-
istics of the existing and forecasting states of the basin, and the quality
of effluents can essentially distort^both the input and output of the global
model of the basin and its environment. It is also known that the pollutant
transformation equations are suitable for describing local processes and can-
not be modeled for the entire basin with any degree of authenticity.
The modification and updating of "super-models" is complex. The realiza-
tion of basin water protection programs is affected by a number of economic,
technical, and social factors; therefore, the once-developed program would
undergo constant specifications and modifications. In addition, it is practi-
cally impossible to simultaneously complete the whole complex of planned
measures. Their realization, as a rule, is piecemeal. The global model re-
quires time-consuming analyses, because obstacles can interfere with the pro-
cess of perfecting programs for river basin water protection.
Thus, the construction of the "super-model" does not allow us to bal-
ance the effects and efforts in separate parts of the program or to in-
vestigate their influence on social transformations, which is necessary when
ranking program components and the initial allocation of resources for com-
pletion is increased or decreased.
Further, the use of the global model in planning of water protection
measures hinders the "man-machine" dialogue, which limits creative interven-
tion in the decision-making process. The links between exact algorithmic
methods of problem solution and non-mathematical ones and, in particular
heuristic methods, are excluded. Their application allows us to reject un-
necessary calculations and obtain results with less expenditure of machine
time.
38
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Finally, the computational abilities of general-purpose computers are
limited both in memory size and time required for calculations. Experience in
solving water protection optimization problems within the limits of a river
basin, by using the "M-222" computer, has shown that such problems can be pro-
grammed only if subdivided into less complex subproblems.
These considerations favor an alternative approach to the economic-mathe-
matical modeling of river basins—a system of mutually coordinated optimiza-
tion models. Effluent discharges, concentrated in small stretches of water
bodies (within the limits of large towns or industrial centers), facilitate
the creation of separate water protection installations, definable by optimi-
zation models. Within the limits of a river basin, hundreds of water protec-
tion installations can be organized. These are intended to maintain the re-
quired water quality in the adjacent sections with minimum capital outlay.
It is not obligatory to match the optimization models throughout the
entire basin. It is frequently more expedient to match only certain models
chosen by heuristic methods. In this case, a relationship between the optimi-
zation models in the evaluated stretches is established: The models of two
neighboring stretches are connected in sequence through a transformation model
which has a feedback link through an analyzer (Figure 1). The role of the
transformation model is to transform the-outputs from one model into the in-
puts of the other by an algorithm. The analyzer (or a group of individuals)
creates the model on the basis of engineering intuition and determines if the
input-output of joined models should be changed. Since the feedback process
between models can be repeated a number of times, the analyzer must decide
when to stop the iterative process of matching models. The water quality at
the inflow and outflow of each evaluation stretch is thus established.
Classes of Optimization Models
Optimization problems can be solved by three classes of models: 1) opti-
mization models with functionally different technical methods; 2) models with
differing interactions of technical methods and types; 3) optimization models
with operatively different technical solutions.
The optimum structure of water protection measures is determined by
Class I models. The measures are divided into three groups depending on the
stage of the water pollution process. The first group is related to the pro-
duction process of the main product, such as transition to new raw materials
or equipment that reduce the volume of effluents or improve its qualitative
composition by recycling water or utilizing effluents, etc. The second group
acts on effluents prior to discharge into water bodies: effluent treatment,
or accumulation, burning, pumping into underground voids, etc. Finally, the
third group utilizes measures that are implemented in the water bodies: dilu-
tion of effluents by water releases, water aeration, biological reclamation,
and so forth.
The first group of pollution preventative measures is the most effective
and reliable. Plans for water pollution management are developed within
possibilities of advanced technology, as specified in the fundamentals of the
water laws of the USSR (1971) and by Cherkinskiy (1971). Currently, rates
39
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and concentration of effluent discharge are defined relative, to the units of
production for particular branches of industry with some allowance for ad-
vanced technology developed within the framework of the CMEA (1973). Measures
in the second and third group are alternatives only.
Figure 1. Diagram of links of models of one type when water protection meas-
ures are otpimized within the limits of a basin. C - number of
models of one type, T - transformation model, A - analyzer.
The degree of efficiency of the measures with differing functions is es-
timated by the decrease in pollutant concentrations at the end of the river
section under sutyd. When evaluating this type of problem, it is impossible
to evaluate all the available technological solutions; generalized relation-
ships of cost effectiveness can be derived from optimization models of class
2 and 3 solutions.
The class 2 models allow us to determine the optimum siting capacity and
the number of waste treatment systems on the basis of information from class
1 models related to the amount of treatment required. Characteristics of the
technical efficiency of alternative treatment methods, which can be derived
from class 3 model output, must be known from an exact solution. Approximate
40
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solutions require only the generalized characteristics of alternative tech-
nologies.
The links existing among the models of the above classes are shown in
Figure 2. The investigation of these links allows us to expose the character-
istic features of a multilevel hierarchical system of models by a successive
vertical arrangement of subsystems, interdependence of actions of subsystems
of different levels, and the ability of top level subsystems to interfere at
lower levels (Mesarovich et al., 1973).
The successive vertical arrangement of models results from the level of
their abstraction. Higher level models are less structured and are difficult
to formalize quantitatively. Their behavior is related to large subsystems
which have broad strategic possibilities. Decision-making on the top level is
the most complex. Models of a lower level describe the specific aspects of
the system's behavior and are easily formalized. These problems have numeri-
cal solutions.
The interdependence of actions of the subsystems is the existence of two
kinds of links between them: downward, to successively define the problem,
and upward to specify the characteristics of technical and economical effec-
tiveness of the solutions, i.e., the higher level defines the problems at the
lower level. The higher level subsystems are assigned priorities so that
they can interfere with and control subsystem actions at the lower level.
The interference takes place both prior to the moment of decision-making
at the lower levels, and afterwards. In the first case, the action regions of
the lower subsystems are specified (e.g., the degree of technical efficiency
of the measures) on the condition that the efficiency of the whole system
would be ensured by the chosen criterion. In the second case, previous de-
cisions are corrected if the basic assumptions were erroneous.
As shown, the problem of optimizing water protection measures within the
evaluation stretch has all the distinctive features of a complex multilevel
problem. For a solution, it is expedient to use a multilevel approach. The
experience of realizing complex large-scale problems shows that with the mul-
tilevel approach, the available resources are used more efficiently. The
problem is considerably simplified by organizing the hierarchical structure.
Construction of a multipurpose system of models begins with development
of a higher level of hierarchy. Information on the lower level can be ob-
tained by aggregating the variables of this level, since this effects the
efficiency of decision-making on the higher level. Aggregation allows the
production of a simplified description of the lower-level system. Thus, for
the first level subsystems of a hierarchical system of models for optimizing
water protection measures over a river basin stretch, the information is re-
presented as a simplified "costs-effectiveness" relationship for each measure.
The information on each separate object in the particular measure is not con-
sidered.
The development of subsequent levels of a model's system is also based on
a simplified description of the lower level by aggregating the variables. A
41
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certain mutual autonomy of the decision-making process exists on each level in
relation to other levels. This makes it possible to pass a considerable a-
mount of the work of finding an optimum plan for the entire system to the
lower levels. The optimixation of subsystems of each level is carried out by
using appropriate mathematical methods and algorithms.
Figure 2. Diagram of choosing the type of measure and its parameters.
I - choice of the type of measure; II - choice of capacity of ob-
jects entering the measure; III - choice of technical effectiveness
of objects; k - number of possible measures; n - number of objects
entering the measure; z - number of variants of the technical ef-
fectiveness of the object.
42
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The construction of a three-level hierarchical system of models for op-
timizing water protection measures over a section of a basin is preferable
in the most complex cases, particularly for evaluating sections with many
water pollution sources that can be ameliorated by a variety of pollution
control methods. A one or two-level system can be Used successfully. The in-
vestigator decides on the number of hierarchical levels required. The deci-
sion is governed by the necessity to match model complexity with the complex-
ity of the problem considered.
The Stages of Solving Optimization Problems
In addition to different classes and types of optimization problems for
water protection measures, there are general methods for simulation. The
development of an optimization model to protect water quality requires such
stages as: defining the objective function of the problem, establishing
limitations, choosing a suitable hydrological model and a model to simulate
mixing and self-purification processes, analyzing and statistically treating
technical-economical indices, choosing the numerical method, and developing a
computer program framework.
Problem Formulization—
The- initial step in a problem of any class or type is the formulation of
objective function. One fundamental requirement is to minimize resources re-
quired for desired results.
The objective of the optimization procedures is to maintain the water
quality standards for a specific body of water. These standards are defined
as the maximum permissible concentrations (M.P.C.) of pollutants in water.
The M.P.C. standards are defined in terms of the protection of human health
(drinking water standards), protection of fishing resources, and the preser-
vation of ecological systems. Water quality standards for municipal, recrea-
tional, and fishing waters have been established; additional standards exist
to cover unique water bodies. Standards for industrial and irrigation water
sources are being developed.
The effectiveness of the pollution control system is evaluated in terms
of the total cost for the pollution control complex to achieve the required
water quality standards.
For the most general case, this can be formulated as
k
min S = min £ S, (P.), (1 = 1,2,...,k) (2)
p-| 1=1 ' '
B F Y ., ^on
provided U (U , I P ,11 ,n ) <_ U (3)
1=1 ]
where S,S-, are the total costs of the complex and 1-th measure, correspond-
ingly;
P-i - efficiency of the 1-th measure;
u8on - M.P.C. of the pollutant in a water body;
U^ - calculated pollutant concentration, depending on the background
43
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concentration;
UF - efficiency of water protection measures, and also the intensity
of mixing 3 and self-purification n processes.
Modeling Processes in a Water Body—
The generalized linkage between water quality objectives and available re-
sources is described in equations 2 and 3 for a water body. It is possible to
link "process" models of self-purification, hydrology, and hydrography into
economic mathematical models.
The discharge model is required to determine the hydraulic flow in the
river system, particularly to determine the minimum mean monthly discharge per
year with 95% confidence. This rated discharge is derived from the projected
annual discharge curve, which should be based on long-term observational data
(at least a 60-year period). If the discharge records are for shorter periods
than derived statistically with defined confidence, limits can be developed.
Binomial distribution curves are the most suitable for the annual discharge
values. To establish the theoretical probability curve from observed dis-
charge, the coefficient of variation, which is the standard deviation, is ex-
pressed as a percentage of the arithmetic mean discharge. The values for the
coefficients and the computational fo'rmulas are given in the Hydrographic
Manual (1972).
The variation within annual discharge patterns can be defined by develop-
ing discharge graphs (hydrographs) q=f(t). The discharge hydrograph for a
river with regulated flows in particular stretches is controlled by the flow
control strategy. The discharges can be determined from a model.
The so-called "mixing model" allows us to determine, first, the distance
at which a complete mixing of the effluents with the water of the receiving
body takes place, and second, the maximum pollutant concentration at any dis-
tance from the place of its discharge into the water body. The values of
pollutant concentration in natural water and in effluents, the discharge flow
and the discharge conditions of the effluents, and the various characteristics
of the water body are required initial inputs to the models.
Selection of the different mixing models are outlined in the manual de-
scribing rated hydrographic characteristics (G.G.I., Leningrad, 1973). Finite
difference models, in which the finite increments are substituted for the dif-
ferentials in the steady turbulent diffusion differential equation, are re-
commended as the most suitable method for evaluating steady dilution in riv-
ers. The equation in the finite difference form is reduced to an evaluation
relationship to calculate the concentrations cross-section of the stream. A
plane and spatial solution scheme exists for solving this problem; the spatial
scheme provides a more exact solution. The accuracy of the calculation also
increases if the transverse circulation and the irregularity of depth along
the stream are considered.
If it is necessary to include self-purification processes, the one-dimen-
sional water quality model based on the following equations is accepted as the
most suitable solution by Vasilyev, Temnoyeva, and Shugrin (1965)
44
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*^T ^T 1 !H ^1T
O i * | OJ_i — -L O * o I i
v ~~— i D^W ) ^ k n T ~4~ f / r~ \
3t at W ox 3x 1 T (j)
where v is the flow velocity;
W - flow cross-sectional area;
- dissolved oxygen at saturation;
ki - BOD rate coefficient;
kz - reaeration coefficient;
z - free surface water level;
fQ , f - mass of oxygen and organic substances delivered per unit of time,
respectively, characterized by BOD.
The integration of equations 4 and 5 is performed with numerical methods
of implicit and explicit difference schemes.
The appropriate type of pollutant transformation model for a water body
depends on many factors (availability of initial data, required accuracy of
calculation, computer resources, etc.). Therefore, the choice of model is
significant for solving optimization problems.
Building "Cost Effectiveness" Relationships—
To^develop and complete economic-mathematical models, it is necessary to
describe water pollution prevention measures in terms of costs and effects.
To build the relationship S(P), a statistical treatment of the technical-
economic effectiveness of advanced technology is needed. Stanishevskiy and
Kuzin (1974) have described a least square method which allowed the develop-
ment of empirical formulae that have a high correlation coefficient.
In constructing relationships between costs and specific water protection
parameters of a technological process, several variables must be analyzed
simultaneously by multiple correlation statistics. For effluent treatment
sites, a formula has been developed which relates costs for effluent treatment
Sor to the treatment efficiency P, pollution concentration in untreated ef-
fluent Ucm and the capacity of the treatment process q.
Sor = n(P,Ucm,q) (6)
For the majority of effluent categories, the relationship between costs
and untreated effluent quality indices has a discrete character and cannot be
described analytically. A considerable increase in pollutant concentration
in effluents leads to a change in the treatment technology, since the "costs-
quality" relation within the framework of existing technology is rather weak.
For example, to decrease BOD2Q in municipal effluents, where the index is
within 100 <_ BOD2Q < 250 mg/£, a common biological treatment is used, for
which the cost indices differ by 10 to 15%; at BOD2Q of effluents exceeding
250 mg/£, preaerators are additionally provided, and instead of the common
aerotanks, two-stage aerotanks with regenerators are used. This immediately
increases the cost by 30 to 40%. Therefore, it is expedient to average
45
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technical-economic indices for separate gradations of effluent quality.
When this concept is incorporated into equation 6, it then becomes
S°r = r^P,q), (rKL, 2, ...y) (7)
where y is the number of water quality levels for which the cost indices are
averaged.
In particular, given a BOD of untreated effluents up to 250 mg/£ and
their discharge within 10 £ q ^°200,000 m3/day, the following formula for
determining the expenditures for a decrease in oxidizability S°y (kopeks/m )
(Stanishevskiy and Kuzin, 1974) is derived:
sor =
(15.103 P3 + 25.881 + 4.74)q(0.015P-0.4122) given 0.2 £ P £ 0.85
(406.80 P - 308.i3)q(0.684P-0.9847) glven 0.85 £ P £ 0.99 (8)
The form of the cost-effectiveness relationship in conjunction with other
factors, determines the choice of the numerical method for -the optimization
problem; therefore, the problems of constructing the relationships and the
choice of numerical methods are directly interrelated.
Choice of Numerical Optimization Methods—
There is no formalized method for selecting numerical methods to solve
optimization problems. However, the requirements for an efficient method are
known: its complexity should not exceed the complexity of the formulated
problem; the accuracy of the solution must be adequate; and the computer pro-
gram should run on general purpose machines with minimum computing time.
Optimization problems of various classes and types can be solved by inte-
ger, linear, dynamic programming, and steepest gradient descent methods and
some of their combinations. These methods have no essential advantage, how-
ever, for solving simple problems when the measures within a non-ramified
stretch of the river are optimized and a small number of water protection in-
stallations (up to ten) is examined. However, relatively greater calculation
accuracy can be derived from integer or dynamic programming.
When a medium class computer such as the M-222 is used for calculations
and the number of installations in a pollution control system is increased by
50 or more, then efficacy of the various numerical methods described differs.
The steepest gradient descent method is the most efficient for defining an
approximate solution to the extremes of a function with several variables.
More accuracy can be obtained from the combined method which is a synthesis
of dynamic programming and relaxation research. Integer linear programming
and dynamic programming are limited by computer memory.
During the formulation of optimization problems for small ramified
stretches of a river basin, dynamic programming and the steepest gradient
descent method are accepted methods. The choice of the numerical method de-
pends on the character of the problem solved, the type of model of pollutant
transformation in water bodies, the form of cost effectiveness relationships
46
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and numerous other factors. Therefore, the choice of the numerical optimiza-
tion method requires an individual approach which considers specific features
of the formulated problem.
Typical Optimization Problems for Water Protection Measures
This section describes the most common types of optimization problems for
water protection measures solved individually and examined in the structure of
complex optimization models. The problems are related to different classes of
the technical solution optimization problems in the field of water protection.
Optimizing Effluent Treatment —
The treatment of effluent discharged into water bodies is one of the most
important measures preventing water pollution. Considering the treatment
plants as a complex, a considerable economy between 10 to 30% can be achieved
by optimizing the degree of effluent treatment. It is assumed that the role
of treatment in reducing the water body pollution has been determined, and the
treatment installation sites and, consequently, their capacity, have been
established. Therefore, the analysis becomes a comparison by the optimum
criterion, of operationally different alternatives, i.e., in determining such
levels of effluent P treatment for which the total costs of treatment S°r at
the n effluent treatment plant would be minimum, i.e.,
£ or
min Sor = min I S± (P±) , (i=l,2, . . . ,n) , (9)
pi i=l
given the limitations
n
I D^CP) >. Uor, (10)
Pf n < P. < P (11)
where U is the specified decrease of measured pollutant concentration in the
water body as a result of effluent treatments;
U. is the decrease of measured pollutant concentration by the i-th ef-
fluent treatment plant;
P. n5p?ax is the minimum permissible and maximum possible effluent treatment
level, respectively.
Further model development requires interaction with the hydrological model
of the water basin, with the model of the process of mixing effluents with
river water, and also with the models of biochemical demand for dissolved
oxygen and reaeration of river water. This is required if the pollutants de-
livered into the water body are capable of biochemical oxidation.
Depending on the availability of initial data and corresponding accuracy of
calculations, the solution of equations 9-11 can be achieved with the help of
various models and numerical methods. It is during the forecast of necessary
water pollution control measures and long-term planning that the hydraulic
characteristics of the river are only approximately determined and the possi-
ble technological schemes of treatment cannot be exactly specified. As a
47
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consequence, the number of river water quality parameters taken into account
in the calculation is limited, simplified process models are constructed, and
numerical methods yielding approximate results are chosen. During short-term
planning of water protection measures, when the characteristics necessary for
evaluation are specified, it is possible to use models and calculation methods
with relatively exact solutions. For example, the use of models of pollutant
transformation in river water, constructed on the basis of the turbulent dif-
fusion equation and dynamic programming, can be used (Kuzin and Stanishevskiy,
1974).
Optimizing the Dimensions of Effluent Lagoons—
In the absence of economically substantiated methods for effluents treat-
ment, the accumulation and discharge of effluents during periods of high flow
are permitted, providing the dilution capacity of the rivers is adequate.
Where effluent lagoons are permitted, their efficiency is determined by
the decrease in pollutant concentration at water intake during effluent dis-
charge periods. If the discharge from the lagoon is well organized, then
their efficiency depends on the capacity W which is related to their total
cost SH by the relationship SH = f(WH).
In the case where several lagoons are constructed within a river stretch,
and the effluents are characterized by the same limiting pollution index, the
problem of choosing the optimum lagoon dimensions arises, i.e.,
min SH = min I S? (W?), (12)
i=l
given the limitations n „
I U» (W?) < UH8°n, (13)
where S is the total cost of all the effluent lagoons considered as a com-
H ?leX'
U^ is the increase of measured pollutant concentration in a water body
due to functioning of the i-th lagoon;
U on is the maximum permissible increase of measured pollutant concentra-
tion in a water body as a result of effluents discharge from all
„ . lagoons in the system.
W^ is the minimum capacity of the i-th lagoon depending on the sanitary
rules for effluents discharged into water bodies;
wHmax is tjie maximum capacity of the i-th lagoon depending on the availa-
bility of sites suitable for construction.
The quantity UH on is found when the problems of the higher levels of the
hierarchical system are solved; that is, when part of the various measures
for protecting the water body from pollution are determined.
In contrast to the hydrological model used for determining the level of
effluent treatment, when the minimum average monthly river flow of 95%
48
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frequency is taken as the rated value, models of the intraannual and long-term
flow distribution are constructed when the lagoon dimensions are determined.
The discrete dynamic programming method is most effective as a numerical
model for solving equations 12 to 14. The solution begins with effluents
discharge at the existing hydrological regime of the river. Then the effluent
discharge from the lagoons is determined, and then the regulating volume, a
sum of the seasonal and long-term components, is found for each lagoon. This
problem is considered in detail by Mukhopad (1975) .
Optimizing the Volume of Acceptance Releases —
As a result of the fast growth in the volume of industrial and municipal
effluents in the last decades, the natural flow of rivers in many water basins
is insufficient to dilute thoroughly treated effluents. In such cases, either
special releases to increase the dilution capacity of a water body or further
decreases in the quantity of pollutants in discharged effluents are required.
Decisions on the level of effluent treatment, and the volumes of effluent
discharge, should consider these measures by methods of mathematical simula-
tion and economic optimization. Parameters which yield the required water
quality indices in the monitoring sections at a minimum total cost should be
chosen.
An important feature of flow augmentation as a pollution prevention meas-
ure is the ability to dilute the effluents of all point sources located in the
release zone, and so reduce concentrations of different kinds of pollutants.
Therefore, when developing an objective function for cost minimization of a
system of water protection measures by using flow augmentation and effluent
treatment, it is necessary to provide for the possibility of optimizing ex-
penditures for effluent treatment of differentiated categories, i.e., the
objective function of the problem can be presented as:
n m
min S = ruin [wnsn(Wn) + 365 I I q^S?? (P±1)] (15)
W.P-t i=l i=l
given limitations on:
the maximum rated river flow Qr providing for flow augmentation
max
Qr < Qr , (16)
Wn is the maximum size of the water reservoir and the volume allocated
-for flow augmentation
Wn < wnmax, (17)
the concentration of limiting substances in the monitoring section of
the water body
B Son
Uj 1 U . , (j=l,2,...,m), (18)
the degree of effluents treatment P
min max
49
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where S is the total cost for flow augmentation and treatment of effluents
from point sources located in the zone effected by flow augmentation;
Sn,S.. is the total specific costs for flow augmentation and effluent treat-
ment of the i-th source, containing the j pollutant correspondingly;
m
is the number of kinds of limiting pollutants taken into account.
The equations 15-19 present a multistep problem. After choosing the
river flow with some provision for flow augmentation, depending on the accu-
racy of the calculations required and the limitations 16 and 17, each step of
the corresponding capacity of the river reservoir Wn is calculated, and the
problem of optimizing the level of effluent treatment is solved. As a numeri-
cal method for solving the problem, it is expedient to use the steepest gra-
dient descent, since this method provides a maximally quick determination of
the extreme function from several variables. This is necessary during the
multistep process to determine optimum solutions (Kuzin and Osmachko, 1975).
Problems of determining optimal capacities of treatment plants, optimizing
the distribution of aeration structures on rivers and reservoirs, optimizing
measures for prevention pollution of point sources by surface flow from cities
and agricultural areas, and several other factors are not examined in this
report. Mutually matched solutions to various optimization problems during
the development of basin water pollution prevention programs will significant-
ly decrease economical costs.
CONCLUSION
In the context of this paper, water quality control has been considered in
the development of a long-term optimum strategy determining capital investment
policy. The basin water protection programs, part of the nation's overall
water protection program, are based on strategic objectives for 10 to 15
years.
Weighty socio-economic prerequisites favor the basin approach to the con-
trol of water quality-effect factors in developing water quality goals and
water protection measures.
There are at least two ways of constructing an economic-mathematical model
for optimization of anti-pollution measures on a drainage basin scale: con-
struction of a united "super-model" or a system of small mutually matched
models. Although a totally integrated and centralized control of the system
is possible, a number of serious shortcomings of a practical nature limit the
use of "super-models."
Optimization problems in the field of water protection are divided into
three classes: Class 1 - problems of optimizing functionally different tech-
nical facilities; Class 2 - problems of interaction of differing facilities;
and Class 3 - problems of optimizing operationally different technical solu-
tions. The investigation of relations among them has shown that during the
optimization of water protection measures within the limits of a basin or
basin section, a multilevel system of models can be developed. This system
requires a multilevel approach.
50
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The solution of optimization problems follows the sequence of formaliza-
tion. This involves modeling processes in a water body, working out the re-
lationships reflecting the link of cost indices to parameters of the techno-
logical processes, choosing numerical solution methods, and developing algo-
rithms for computerization.
The most widespread types of optimization problems are the problems of
optimizing the level of effluent treatment, the value of sanitary releases,
dimensions of effluent treatment installations, and parameters of aeration
devices for rivers and water reservoirs.
The decision-making methodology outlined here, based on modeling and
optimization, is a powerful analytical tool for complex problems. Its wide-
spread use in the development of basin water protection programs (given the
present dimension of the problem) will yield high economic profits to the
national economy.
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BIBLIOGRAPHY
Cherkinskiy, S.N. 1971. Sanitary conditions of effluent discharge into water
bodies. Moscow, IZD-VO LIT-RY PO STR-VY, 208 p.
Consolidated rates of water discharge and quantity of effluents per unit of
production for various branches of industry. 1973. Moscow, Stroyizdat,
368 p.
Fundamentals of the water laws of the USSR and the Union Republics. 1971.
Moscow, 32 p.
Kuzin, A.K., and L.P. Osmachko. 1975. Optimization of the volume of sanitary
releases. In: "Vodnye Resursy," No. 2, pp. 90-102.
Kuzin, A.K. and S.A. Stanishevskiy. 1974. Problems and principles of optimiz-
ing water protection measures. In: "Vodnye Resursy," No. 5, pp. 111-124.
Manual for determining rated hydrographic characteristics. 1972. Leningrad,
Gidrometeoizdat, pp. 435-472.
Mesarovich, M., D. Maks, and I. Takakhara. 1973. The theory of hierarchical
multilevel systems. Moscow, IZD-VO "MIR," 334 p.
Mukhopad, V.I. 1975. The choice of optimum capacities of a system of two
regulating lagoons. In: "Problemy Okhrany Vod," Issue 5, Kharkov, pp.185-
190.
Practical recommendations for evaluating effluent dilution in rivers, lakes,
and water reservoirs. 1973. GGI, Leningrad, 98 p.
Stanishevskiy, S.A. and A.K. Kuzin. 1974. Mathematical description of rela-
tionships reflecting the link between expenditures and the parameters of
treatment processes. In: "Problemy Okhrany I Ispolzovaniya Vod," Issue 4,
Kharkov, pp. 121-127.
Typical technique for determining the economical effectiveness of capital
investments. 1969. Moscow, IZD-VO "Ekonomika,", 16 p.
Vasilyev, O.F., T.A. Temnoyeva, and S.M. Shugrin. 1965. Numerical method for
evaluating unsteady flows in open channels. IZD. Academy of Sciences, USSR,
Mekhanika, No. 2.
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DISCUSSION
DeLUCIA: First, I want to comment on Dr. Kuzin's system using smaller
interrelated models rather than super models and to point out the analogy to
work being done in the U.S. Recently partially decoupled systems have been
used by the U.S. In these decoupling or decomposition systems, total decom-
position and solution take place, either by heuristics or specific algorithms
My second general observation pertains to Dr. Kuzin's suggestion that
smaller interrelated models allow easier intervention in the man-machine
interface. I would agree, but caution not to forget there are many means for
intervention when we use large optimization models. First, it is most
important that our younger generation of applied mathematicians who work with
these problems do not ignore that experience and intuition of our older
colleagues. Second is the use of heuristics to do site analysis that you
then incorporate into a larger model; namely, for considering the cost of
treatment or removal at a particular point. One can enter into a model, a
super model, a functional form that represents prior analysis, looking at a
whole range of technologies—whether it be treatment, land disposal, etc.
My third comment concerns whether super models necessarily require
assumptions in the transformation equations relevant to the physical system
that question their validity. My comments are twofold: I believe Dr.
Loucks wanted to point out that one never suggests that super models
(mathematical programming models) be used without correlation with simulation
models and linear assumptions.
GORTSKO: In my opinion, optimization models are not the only instruments for
solving questions of water resource management. Optimization is, without
doubt, necessary, but if we are talking about optimum water quality within
the limits of a river basin, then the question of an optimal goal arises.
If we optimize, developing a system not based on the best alternative, the
results will not be good enough.
The approach for developing an analogous model is different. First, it
is necessary to construct a simulation model of water resource use within the
limits of the region. This model can be less refined, but serves as the
base for optimization models. A simulation model is necessary to forecast
what will occur within a specific time period. A simulation model of an
aquaculture complex is being applied in Rostov, USSR.
KUZIN: By "global model," I imply a model of a whole river basin system. I
have not selected all indices, only those accepted for modeling a given
river basin. When verifying the "inflow-outflow" model, we compared the
economic indices to confirm that the expenditures decreased by 10-30%.
53
-------
Joint examination of our models will allow us to enrich the methodology
being used in both countries. Creation of models for specific objectives
should be the final stage of cooperation.
54
-------
MODELING WATER QUALITY IN RIVER BASINS
Ye.V. Yeremenko
INTRODUCTION
The suitability of specific waters for particular uses is determined by
the combination of physical, chemical, and biologic criteria; however, current
mathematical models of water quality use mainly physical criteria. The bases
for mathematical models are hydrodynamic or hydraulic equations, and the tur-
bulent diffusion equations for calculating pollutant concentrations with some
allowance for non-conservative properties.
Depending on whether the size of a river basin section or the whole river
basin is considered, different data needs on pollutant concentration in the
river are required. If a long stretch of watercourse is considered, it is
expedient to use one-dimensional solutions; otherwise a three-dimensional or
two-dimensional solution should be used. The zone within which the pollutant
concentration is not completely mixed in the cross-section is called the
three-dimensional diffusion zone. We shall consider the calculation of the
pollutant concentration in this zone.
Calculation of Pollutant Concentrations in the Three-Dimensional Diffusion Zone
We shall consider whether the concentration in this zone can change not
only in the three coordinate directions, but in a relatively shallow flow—
only in the plan. Since the three-dimensional diffusion zone is small, the
pollutant is considered to act as a conservative substance within the zone.
In the USSR, it is common practice to follow the method described by
Karaushev (1969) to discuss three-dimensional mixing zones. This method
describes two and three-dimensional fields of pollutant concentration in pris-
matic channels using the difference method based on the turbulent diffusion
equation.
^(i+§
By using the explicit difference scheme for a square grid, this equation is
reduced to a simple formula for calculating the concentration
Ck+l,n,m = -4(Ck,n+l,m + Ck,n-l,m + Ck,n,m+l + Ck,n,m-l)
(2)
55
-------
in the next section. The distance between the sections is
AX, - (3)
4Dm
There is no diffusion of the pollutant through the boundary surfaces 3C/3n=0.
Similar relations were obtained for solving the two-dimensional plan problem,
where instead of equation 2, we have
and instead of equation 3, the relation
=
2Dm
In the above relations, V is the mean cross-section velocity. Dm the
turbulent diffusion coefficient.
The above formulas make it possible to calculate the concentrations in
steady-state flows. The diffusion coefficient Dm is assumed to be identical
in each direction and equal to the coefficient in the vertical direction
D =
2 MC
where H is the mean depth of the channel,
C is the Chezy coefficient, and
M is the Chezy coefficient functions.
However, in equation 2, the difference between diffusion coefficients in
different directions in real channels has been indicated. The essential in-
fluence of this circumstance is described by Ye.V. Yeremenko (1975).
On the basis of the solution of equation
D4t + D4 + f
with the initial condition C(X,0) = n (X) and the boundary conditions 3C/3X2 =
3C/3Xo =0 (applied to a rectangular channel with a steady power source) of the
form
(X2 + X^ + n)
exp[-
56
-------
(X
+ exp[-
K)
3s Jf 3; CF=Q/VBH is the mean
cross-section concentration. B is the width of the channel, and Q is
the water discharge.
The dimensions of a three-dimensional diffusion zone were established, given
different relations D3/D2. The distance between the source and the section,
where C*max=Cinax/Cp=l.05, was accepted as the length of the three-dimensional
diffusion zone 1.
Figure 1 illustrates the results of the evaluation of zone 1 with differ-
ent D3/D2 and Chezy coefficients C, which indicate the essential influence of
the relation D3/D2. Given the known relation between coefficients D3/D2=3.4,
obtained by Elder (1959), 1 decreases in comparison with the case D3/D2=1 by
a factor of more than three.
1/2B 300
250
200
150
100
90
Figure 1. Influence of the ratio D3/D2 on the length of the three-dimensional
diffusion zone (location of the source is indicated in the figure).
57
-------
Yeremenko, Kolpak, and Selyuk (1975), using this and the solution of
equation 8 for a rectangular channel, the difference method, the diffusion
equation, calculations, and generalization of results, constructed nomograms
to determine 1(C*max=l.05) in channels of rectangular and parabolic cross-
sections of a sector type, with different positions of the source in the
cross-section and relations B/H.. An example of such a nomogram for a channel
of a rectangular cross-section is given in Figure 2. Since at C*->1 the slopes
of the concentration variation curves are very steep, then the values of zone
1 essentially decrease at C*<1.05. This can be described with diagrams that
show the decrease of the maximum value (C*max) along^the channel. Such dia-
grams, generalized for different Chezy coefficients C, are also obtained for
three different kinds of cross-sections, depending on the relation H/B and
the location of the source (Figure 3).
B/H = TOO
20 30 40 50 60 70-0 250 500 750 1000 1250 1500 1750 L = I/2Q
v>
Figure 2. Nomogram for evaluating the length of the three-dimensional dif-
fusion zone.
The dimensions of the three-dimensional diffusion zone can be essentially
affected by the transverse circulation that, in a number of cases, exceeds the
coefficient anisotropy effect, and so special methods of calculation are being
developed.
Karaushev (1969) , in the case when the transverse velocity components
are considerably smaller than the longitudinal component, suggests evaluations
by calculating the partical trajectories determined by the equation
dx
dz
Vl(Xi,X2,X3) V2(Xi,X2,X3) V3(Xi,X2,X3)
(9)
58
-------
B
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 x =
MC2B
Figure 3.
Variation of the maximum pollutant concentration along the flow.
Conventional designations 1-H/B=1/10; 2-H/B=l/20; 3-H/B=l/40; 4-H/B
=1/70; (the location of the source is indicated in the figure).
Therefore, it isT understood that changes in the pollutant concentration
at any point in the grid area during transition from one section to another
are due not only to transfer and diffusion according to equation 1, but also
to displacement from transverse velocities. The evaluation is also carried
out by formula 3, and the displacement, according to equation 9.
The appearance of the transverse circulation is usually accounted for by
the curvilinear plan contour of the channel. In this case, a method of eval-
uating the three-dimensional pollutant concentration field in the channel of
a rectangular cross-section is considered by Fedorov, Lapshev, and Bezobrazov
(1975). They proceed from the equation in cylindrical coordinates
Urv 9C +
r 9n
j 1C + n
r9r z
9C
= Dz(-
92C
_!
r
_9C
9r
(10)
where Un, Ur, and Uz are the velocity components in the coordinate directions.
Dz=gHV/MC (g is the acceleration due to gravity).
The boundary and initial conditions of the solution of equation 10 are
1C
9r|r=R+B/2
3. 1C =0; c|. =. = C0(r,z),
where R is the radius of curvature of the channel axis.
(11)
59
-------
The velocity components were determined from work by Rozovski}* (1957) in
which the equation 10 was reduced to the form
ai 1C + _3C = _gH (92C + J^__ _9C + 82C
R+y 9-n 3y MC "By"2" T;
where a1 = 1.17 - 0.5(z/H)2;
a2 = H[2205(|)Ti _ 3363^) + 680]
JVT y hi rl -D
with 0 5 z £ H, -B/2 £ y £ B/2, f)>0.
To integrate equation 12, the explicit scheme of the finite-difference
method was used, thus the main recurrence formula is
i k = TJd i k[- - 2Ai (— + \)
JjK- L i> J JKLT 1 \, yz
+ c . . , , Ai + AL(C, , k+1 - c . . ,_-,) } ,
ijJ-J-j^ Y2 2 ijJs1^^-1- i>JjK i
Y2
where i,j,k is the numbering of the grid nodes correspondingly along n;y,z;
T is the grid spacing along the coordinate n ;
Y and y are grid parameters along y and z, correspondingly;
_ gH
BI -
The stability of the solution by the explicit scheme is achieved by
choosing the parameters T, Y> and y according to the conditions
AL+BL> o, I - 2Ai(i + V -!L> o
Y2 Y T Y2 y^ Y
Thus, the choice of the spacing and the grid parameters T, Y> and y depend
essentially on the initial parameters of the channel specified by the rela-
tions R/B and B/H.
The influence of the boundary conditions is achieved by extending the
matrix (C-j^j^) by the two last rows and the two last columns, setting their
elements equal to the corresponding neighboring elements of the original
matrix.
The computation is usually carried out for relative or conditional con-
centrations obtained by assuming the area of the constant value of concentra-
tion equal to the area of one segment. To pass from these conditional con-
centrations, it is necessary to multiply the results by the coefficient r.
depending on the discharges of effluents and the flow, and the dimensions of
the accepted grid.
e = Qn(J-D(k-l)
VBH (13)
60
-------
According to the algorithm constructed on the basis of these expressions,
it is possible to compute the concentration distribution in any cross-section
located at the angular distance n from the place of discharge.
In the course of preliminary estimates, it is sometimes important to
evaluate only the distribution of maximum concentrations along the channel.
This problem is solved by the method of multiple evaluations of conditional
variants, which are characterized by relative parameters R/B and B/H, and by
subsequent statistical treatment of the series of concentration determined as
functions of angular length. As a result, an estimation formula was obtained:
Cmax = (C0 - CF) exp(-3| n ° ' 25) + Cp, (14)
where g = f (R/B; B/H) .
By solving the plan problems, it is also possible to determine the
equalization zone of the concentration in the flow section in channels of
great width. Sherenkov (1975) considered pollutant distribution in a mean-
dering channel, using a two-dimensional model. The equations for calculating
flow are written in the curvilinear system of orthogonal coordinates parallel
to the dynamic axis of the flow. These steady-state motion equations, after
discarding the terms of the highest order of smallness, can be presented in
the form
8Vo , rl 9V§ , 3 TT „ o" T7 TT g 3H , 9 T g XV2 , Ts
" -
3S ^ sn 2h & (15)
aav2 = _e il + In.
K s B 9n ph' (16)
9h 1 3 T, , , 9 _ TT v a TT i, _ n
i¥ + K ^S ^ + to Vnh ' K ^ " ° (17)
Here Vs , Vn are the values of the mean depth longitudinal S, where S and
transverse n are the coordinate velocity components;
a is the coefficient for the irregularity of velocity distribution over
the depth of the channel;
H is the free surface water level;
h is the depth of the channel; curvilinear system (metric) coefficients
K=l-an, Kjj=Kz=l;
Ts,Tn are the tangential stresses of the wind acting on the free surface
of the channel;
Tsn are the mean depth turbulent tangential stresses caused by the
horizontal turbulent momentum transfer;
A is the coefficient of hydraulic friction on the bottom; and
p is the density of water.
The system of equations is closed using the Reichardt-Konovalov hypo-
thesis 2
VnSn -
P o
61
-------
which gives good correspondence with the experiment, with A being determined
for uniform flow conditions at a specified water discharge Q and slope T
from equations
The calculation of the distribution of velocities and free surface levels
of steady flow is carried out on the basis of equations 15-17 with boundary
conditions at
S = 0; Vs = Vso(n); Vn=Vno(n); H=H0(n), (20)
and at S>0; Vs=Vn=0 along the bank line by successive approximations with
transition to new variables
where Byy(S) is the coordinate of the left bank, B(S) is the width of the
river at the section; the solution region is transformed into a semi-strip of
unit width, and equations 15-16 are written in the general form
.s^.^-^S.,,, (33)
where a,(S*,i) =--A* ; a9(S*,i) = 1(3BA _ tJB) _ 2aaA; a,(S*,i) =
1 aB^ z B 3S 3S ^ *
f2(S*,i) = a. (24)
Ph
The character of the boundary conditions (equation 20) in the new coordinates
does not change. The free surface H(S*,i) is presented as a sum of the mean
section surface Hp(S) and the deviation from the mean surface
H = HF + £(S*,i); = . (25)
0181
In solving the first approximation, it is assumed that
9HF 9? __j 3?
- ~
- - — • —
3S* 9S 9S (26)
and equation 26, after substituting into it equation 23 , is written in the
form
3iY|+a 3V|+ (a3-as-afi)v| + a, ™Z = f . (SA t) _ a_f . (27)
3S* x 81? 23i 3 5 6 s 49S^ r^b*^; a5r
62
-------
The quantity 9Hp/8S* enters equation 27 as a parameter and is unknown before-
hand .
The parabolic equation 27 is solved numerically by the standard algorithm
at,different values 9Hp/9S* and by the calculated values Vs, the integral con-
dition of continuity is checked:
Q = B}vs(i)h(t)di (28)
where Q is the specified discharge in the river channel.
The value of 3HF/3S* and Vs(i), when the equation (28) is valid, is the
required result. Substituting the solution of the first approximation VsW
into equation 23, we obtain the value £(•"-) for the first approximation.
i 1
? = B{/[a6(Vs1)2 - f2(S*,i)] di-/di [agCVg1)2 - f2(S*,i)] di} (29)
Substitution into equation 22, using equation 25 allows the values Vs of the
second approximation to be determined.
Calculations have shown that the second approximation is necessary only
when
oB - 1 and crB > 0.3 (30)
The determination of values Vs and H makes it possible to find Vn from
the continuity equation 17.
The semi-empirical pollutant transfer equation in the plan in the curvi-
linear orthogonal coordinate system, parallel to the flow dynamic axis, is
written in the form:
M + Is JO + Vn 19 = _1 [_3 h Dss 39 + _3_ hDsn ii + -3 hDns 1° + _i hKDnn 36]
9t K 3S 3n Kh 3S K 3S 3S sn 3n 3n nfa 3S 3n nn 3n
+ Gn ~ GO + f,
(31)
where 0 is the mean depth pollutant concentration value, Gn and Go are the
pollutant flow through the free surface and into the bottom, and f is the rate
of generation or decay of the pollutant (due to biochemical transformations)
at a given point of the flow plan.
The value D forms the matrix of values of effective diffusion coefficients
that take into account both turbulent and convective diffusion caused by the
heterogeneity of the velocity field and by secondary flows. To determine.D,
the approach used by Maron (1971) to solve the eigenvalue problem, is general-
ized by Sherenkov (1975) for a complex flow. Sherenkov describes the rela-
tionships for determining D; equation 31 is integrated by the difference
method. An evaluation of the distribution of velocities and concentrations
in dissipative discharge of effluents on one of our rivers is illustrated in
Figure 4.
63
-------
0.8 M/day
0 0.4 0.8U M/day
0 0.4 0.8 U M/day
.x,
^V-.. 0 0.4 0.8 1.2U M/day
-P-
0
50
0 0.01 0.02 0.03
100M
Figure 4. Results of the evaluation of the flow plan and the pollutant concentration in a river channel:
a is the contour; b is the contour of the relative concentration 0/VO according to the evalua-
tion; dots show the values of Q/VO according to the results of measurements; arrows show the
values of V according to field measurements; I and II are dissipative discharges through which
the pollutants are delivered; V0 is the initial pollutant concentration.
-------
Paal (1975) describes a pollutant diffusion problem in which the pollu-
tant is a non-conservative substance. The initial equation used is:
(32,
In the solution, the limitation of the function is assumed at X -*».
If there is a discharge of effluents, the solution of equation 32 for
determining BOD is presented in the form
L = LCTE - 1 - exp (-(xa +|.)v _ XL (33)
~
whpre LQ^ and g are correspondingly BOD and the discharge of a liquid effluent
To determine the distribution of the oxygen deficit, the following for-
mula has been obtained :
LCT - [expC-Kl.) _ exp (-K) ] } '
-+5)2v
The coefficient ~U^ is determined from experimental data (Paal) for smooth
channels by the formula
Evaluation of Pollution Concentration in One-Dimensional Problems
One-dimensional solutions are necessary for long channels when the three-
dimensional diffusion zone only occupies a small section of the channel. The
evaluation of concentrations of non-conservative pollutants in non-steady-
state stream flows is of general interest. If we assume that water bodies are
vertically well mixed (no density stratification) , the most complete system
of equations for evaluating the pollution concentrations affecting water
quality is described by Yeremenko (1975). The equations are:
I + vl = £!x (WDi} + K2(srs) - K1L -
+ V = _ (WDM) - K N + f •
n N
= I
9t 8X W gX SX
65
-------
fc;
9t 3X W 3X 3X
II + v.3T. = 1 !_ (wn9T) _ B JA [e (T ) _ e] - BiT - AR + AT} + f
3t 3X W 3X 3X W l m a
IQ + 2V-^ + B(C2-V2) 12 = gW[(i + 1 ^)Fr
3t 3X 9X B 3X r
,,3Z + 3Q = g
I> - - C> * / n f \
9t 3X (36)
where S, L, and N are the respective concentrations of dissolved oxygen, BOD is
due to carbonic and nitrogenous organic compounds; C^ is the concentration
of the i-th ingredient present in waste water; T is the water temperature; f
is the capacity of the sources of pollutant; Q is the water discharge; W is
the cross-sectional area of the channel; V=Q/W is the mean flow velocity; Z
is the free surface water level; Fr is the Froude number; B is the width of
the channel at water level Z; g is the influx of water per unit length of
channel; K} ,Kn, K^ are the coefficients of ingredient decay rates; D is the
longitudinal dispersion coefficient; P is the photosynthetic production of
oxygen; Pp is the oxygen consumption by water plants; PM> the oxygen consump-
tion by bottom sediments; AI and BI are the coefficients of heat transfer by
evaporation and convection, respectively; Tg is the air temperature; all
other designations which characterize temperature balance are contained in
the regulations operating in the USSR (Fedorpv, Lapshev, and Bezobrazov) .
The first five equations are the transfer equations, and the last two are
Saint-Venant equations evaluating the non-steady flow motion.
A similar system of equations for evaluating pollutant concentration was
used by Vasiliyev and Voyevodin (1975) .
The values of the coefficients D, KI , K , K-^, K2, and also the values P,
Pp, P^ are required to integrate the transfer equations. Almost all the
quantities can be determined by investigations of specific water bodies or
from published values. In the papers presented, special attention is given
to the determination of the longitudinal dispersion coefficient.
In Paal's paper (1975), the following relations were obtained from labor^
atory investigations of regularities of variation of coefficients D for dif-
ferent regions of resistance (Figure 5) .
= 3400Re-0.5, (37)
RU*
(38)
= 16 (I,)1-25 (39)
RU* u*
66
-------
3.0
D
lgRU
2.5
3.2
4.4
4.6
2.0
3.8 * 4.0
4.8
.5.0
1.5
••
-------
Vasiliyev and Voyevodin (1975), using Harlimann's work, determined the
longitudinal dispersion coefficient for an open prismatic channel by the
formula
D = 20.2 /glvlR/C. (40)
For an arbitrary channel, however, the longitudinal dispersion coefficient can
be an order of magnitude higher than for a prismatic channel.
The determination of the dispersion coefficient in a non-steady-state
flow in channels of great width is considered in Yeremenko ' s paper (1975).
Taylor's assumption that the effect of the longitudinal diffusion is
negligible, when compared with the diffusion in directions transverse to the
direction of flow, is used as the initial three-dimensional diffusion equation
^C + \T, SC • v 5C + w CF_T\ 9C = __3 /n 8C x , _9
V V ^ ° +
' (41)
where F(X2,t) = Vi(X2,t)/Vh, X2 = ^,
Vi^ is the mean depth velocity; D2 £3 are constant in the cross-section diffusion
coefficients .
By averaging equation 41 with depth in respect to a rectangular prismatic
channel
3~C 3C , 1 8 hv 1r(F-l)"cdx2 =
+ V + h
3t h8Xl h 3XX o ' (42)
_ , h
is obtained where C = — /cdx-
h 0
By inserting the trivial result 1 ^_ hv }(F_1} ~d- = Q
h 9X1 o
into equation 42, we obtain, after subtracting 41 from 42:
Vh /(F-l) (C-C) dx2 - Vh(F-l)
n
h 8X1 n o 3X1
(D
v
J 3X2 x /9X2" (43)
Using Taylor's approximation (O"C) from equation 43, we obtain the equa-
tion for determining C = f(D2, V2 , U)
3C
3X2 3X2
If C is known, it is possible to determine the longitudinal dispersion coef-
ficient E12 in a vertical plane, because
68
-------
hVh/(F-
=
Then substituting V^ = VH(X ,t), where X3=3-; V is the average velocity by
active section, equation 42 can be written as
9C , T7 9C , TT/YI_-I\ 9C
V(n 1} = " ' (45)
Averaging equation 45 by width and assuming no loss of the pollutant into the
banks of the channel, we obtain
ft + Hr + 55 - ""V9-1' Sx3 - £ O*) , (46)
B
where G=1/B /rdx . Subtracting as before, equation 45 from 46, and fixing
C-H5, we obtain an equation for defining U.
(hD3||-) =
h 8X3 ^38X3' ~ VW"" 3X! (47)
The solution for this equation, as is generally known, offers the possibility
for finding the horizontal longitudinal dispersion coefficient of E13 because
By substituting equation 48 into equation 46, it is possible to substitute
longitudinal dispersion coefficients in horizontal and vertical planes, be-
cause
It + V~9X^ ~ W W^ l2 13 "ax^ (49)
Yeremenko (1975) shows that, given variable motion in wide channels, the
characteristic time for the change of the flow velocity structure across the
channel is essentially greater than the characteristic time change of the
velocity field in the vertical plane. Consequently, in a number of cases, it
is possible to determine the coefficient E-io by the relationships determined
for the steady-state solution, while the non-steady-state effect can be re-
solved by the coefficient E-^J the value of which is affected by the velocity
component V"2- The extent to which the presence of the velocity component V~2
affects the coefficient of longitudinal dispersion is established by calcula-
tions for a flow with a parabolic velocity distribution
Vh(F-l) = !l (X2 - li. - 1),
These were used for solving equation 44, presented in the form
69
-------
~ a
9C
9X
a a
£ (X2 -
8C
(50)
where a = V2/aU&; a = 0.067 is the Boussinesq coefficient; U5v is the dynamic
velocity. The velocity component Vo was assumed to be constant over the
depth.
Since the generally accepted form of the dispersion coefficient is the
expression E12 = 3hU*, coefficient 3 had to be determined by expression
Vh(F-l)~cdx9
= n
hU*9C/9X
1
The results of evaluation of
following table
3, depending on the ratio V2/U*, are given in the
V2/U*
1.00
0.67
0.335
0.067
0
-0.067
-0.335
-0.67
-1.0
141,980.0
2,799.0
91,
11.
7,
5.
1.4
0.52
0.25
,4
,0
.0
,3
From the table, it follows that coefficient 3 changes non-symmetrically .
Given V2/U*>0, this coefficient changes by several orders of magnitude, where-
as given V2/U*<0, the coefficient changes by only one order of magnitude. The
qualitative verification of the result was carried out experimentally in a
hydraulic tray 29 cm wide and 4.0 meters long. The concentration distribution
has been measured after a single instantaneous discharge from a point source
into the experimental channel. The distributions were obtained in the steady
and variable flows with the hydrograph shown in Figure 6a. The results
in Figure 6 show the increase in the longitudinal dispersion coefficient in a
variable flow.
The integration of equations for determining the BOD and the dissolved
oxygen deficit in a steady-state flow is described by Paal (1967) .
The original equations are as follows:
9t
IS
9t
+
9S
+ K,L = 0,
+ K2S = 0
(51)
with the initial conditions L=0 and "5=0, given t=0.
70
-------
20
16
12
7 day
Q= 15.7 I/day
7 day
-Q= 10 I/day
Q = 4.3 I/day
VN.
1.6
3.2
4.8
6.4
t day
Figure 6. Influence of the unsteadiness of the flow on the dispersion of the
pollutant concentration in time: is steady motion; is
unsteady motion.
Diagrams of the introduction of effluents are presented in the form of a
step function determined for values t>0 in the following manner
71
-------
LQ in the interval (Xt 0, given X + °° (53)
The Laplace transformation method was used to solve equation 51; the
boundary conditions in the initial section were determined by the Heaviside
unit function. Given the indicated initial boundary conditions, the solution
for determining BOD takes the form
i v —n n v.-TTf'i-—-i T "^
L = 1 _a_ Jexp — (V- /Vz + 4^0) £ SCTi • [erfc(^_l5r 1T;
2 Q+q T 2D i=0 2/D(t-ir)
where q is the effluent discharge; L^xi is the effluent BOD concentration _^e
within the time interval t . . . (i+1) ; erfc(Z)=l-erf (Z) , while erf(Z)=2/Aj e d
For a conservative substance, it should be taken that K^=0. 0
The dissolved oxygen regime of the channel, with the above-mentioned con
ditions, is determined from the solution obtained in the form
= - ID
rT
^ -L / i; / i i Z.\]
Kl__ ____^ _
(55)
Equation 55 is a calculating formula for determining the river oxygen regime,
if the effluent oxygen deficit remains constant with time, as in real situa-
tions .
To evaluate the pollutant concentration in non-steady-state flow, it is
first necessary to integrate the Saint-Venant equations. This is described
by Vasiliyev and Voyevodin (1975). The solution of the equations, after ap-
proximation by difference methods, is carried out by a numerical method using
an implicit difference scheme.
A scheme of open channels of canals is naturally represented in the form
of a system of intersecting segments and nodes (vertices), each at the end of
some segment. On each segment, the required functions satisfy a system of
Saint-Venant differential equations with initial conditions, and, in each
72
-------
vertex, they satisfy the boundary conditions or the conjugation conditions.
The Saint-Venant equations are written in characteristic form, which in the
vector form can be presented as
AU9U
AR8l = F> (56)
_ gW
P _ /I - B(V+C)\ , /V-C 0 \ v TTr,/^ ^ _ 7 70 T,- 3W
R - I , ' I A =1 I K = WCXR h = Z - Z , B= ——
VI - B(V-C)/. V 0 V+cI 8h ,
\ x 7/ N /•» '
B is the width of watercourse that has a depth h; Z° is the bottom of the
channel. Equations 56 are approximated by using an implicit difference of
the first order accuracy by t and of the second order by X.
K n ,
F , (57)
n
Here T is the interval along the t axis common for all segments; A is the in
terval along the X axis, generally different for various segments.
K+1
The second order calculation for the vector Fn follows
As a result, each segment contains a system of linear algebraic equations
of unknowns U§ °f the following form
= Dn, (58)
where
n = 1,2,..., N-l; An = -Cn = (AR); Bn = (R-r) Dn = Bnl + T.
For the extreme points of the segment n=0 and n=N, the difference equations
are obtained by approximating the first equation for n=0, and the second
equation of system 56, n=N. As a result, we obtain
B12 ZK+1 + all QK+1 + al2 ZK+1 = dl
00 00 01 01 0
R110K+1 + B127K+1 , C110K+1 , a!27K+l _ ,2
BN ^N ^N ^N LN qN-l + aN ZN-1 ~ dN
The system of equations 58-60, together with the conjugation conditions
and the boundary conditions, form a closed system of linear algebraic equa-
tions in the unknowns Q^ , Zn -1- (n=0,l, . . . ,Nm; m=l,l,...,m) within the upper
73
-------
temporal layer tK+^. Given this conjugation, there will be conditions where
channels merge in the intermediate points causing an influx of water; for
example, the case of a concentrated influx or outflow of water at the boundary
of two sections of the channel (Figure 7).
I
Figure 7. Variants of flow conjugation: a - concentrated inflow (outflow);
b - division of flow; c - confluence of channels.
Qz = Ql + (^; Z2 = Zl
(61)
At the separation and juncture of flows in a node (Figure 7), there are con-
jugation conditions
Q3 = Q! + Q2; Z1 = Z2 = Z3 (62)
Without calculating the structure of difference equnl; irm.s and boundary condi-
tions (which in some cases can be even non-linear), the direct solution of
the system 58-60, together with the boundary conditions and the conjugation
conditions, is practically impossible. Vasiliyev and Voyevodin (1975) suggest
two algorithms for the solution of such systems of equations, which consider,
first, the type of system of channels, and second, the tridiagonal structure'
of the matrix of difference equations. The conjugation conditions are pre-
sented in the form of the following group of equations:
74
-------
a) balance relations; b) relations between the parameters on the ends of the
segment and the parameters in the vertex, which is the left or right end of
this segment; and c) relations between the parameters in the vertex.
For example, during the calculation of hydraulic characteristics, it is
convenient to choose as the parameters in the vertex: QJ as additional in-
flux, and Z^ as the water level in the vertex. Let j be the number of the
vertex, m the number of the segment, and Mj the combination of numbers of
segments contiguous to the given vertex; then the boundary conditions or con-
ditions in the vertex can be written in the form
a) ^ I on the left end of the
mEM1 lnAn = <& lm = segment
-I on the right end of the
segment (63)
b)
Zm = Zl, rnEM-j (64)
c) .
<£ = Q*(t) (65)
The following two cases are considered:
1. Ramified system of channels. In this case, it is assumed that the
system of vertices and segments constitutes a tree-type graph. The basic pro-
perty of the graph is that the number of its vertices exceeds by a unit the
number of segments. It is not difficult to demonstrate the existence of at
least two vertices to which only one segment is contiguous. To one of these
vertices is assigned the number 1, to the other (M+l) (where M is the total
number of segments in the system) . Then the numbered vertex and the segment
with the same number are excluded from consideration. As a result, the new
scheme of vertices and segments possess the same property as the preceding
one; i.e., except for the vertex with number (M+l), at least one vertex con-
tinues to exist in the new system to which only one segment is contiguous.
The vertex and segment are assigned the following number and the process of
numbering is continued until the last segment is numbered. It must be noted
that if mEM- , then m
-------
Y = (£ + C)'1 ^ ~ C-^ (68)
n
Assuming that XQ and YO are known, given n=N-l, we obtain
%-l = XN-1UN + YN (69)
With the help of relations 69, unknowns QN-1 and ZN-I in equation 60 are
eliminated, and, as a result, the following relation for the last point of the
m-th segment is obtained
and, using condition 64 which is linearized if necessary, we obtain
(%v = #i + £• (7o)
Here, j is the number of the vertex which corresponds to the right end of the
segment .
If m
-------
The solution algorithm is divided into the following stages (Rozovskiy
1957). At first, only the segments are considered in any sequence. The
system of equations 58 on each segment is reduced to the form
U = XU + Y + WU0, n = 1,2,...,N-1, (73)
ZK+1
Here U0=(0° ), coefficients Xn,Yn,Wn for n=l are calculated from the first
equation of the system 58, by eliminating the unknown QQ using equation 59,
while for the remaining n>_2 by the recurrence formulas
Yn = (Bn + CnXn_1)-1(Dn - CnYn_!),
Wn = -(Bn + CnXn_1)-1CnWn_i. (74)
Only when n=N-l will this relationship be obtained:
UN-1 = XN-1% + YN-1 + %-lV (75>
which is used to eliminate the unknowns l%_i in equation 60. As a result., on
the right end of each segment, a relation of the following form is obtained
(QN>m = (vnZN)m + (U12zo)m + (y13)m. (76)
In order to obtain a similar relationship on the left end of the segment,
equation 59 is rewritten in the form
Q0 = -m + ^03^' (77>
Equations 76 and 77, given m=l,2,..., together with the conjugation conditions
61-65 in each vertex, form a_closed system of equations relative to the un-
known parameters in the Q^ Z^ vertices and Q, Z on the ends of the segments.
The derived system is efficiently solved by the iteration method (Rozovskiy
1957).
After the unknowns on the ends of the segments have been calculated, the
solution in all the internal points for all the segments is determined with
the help of equation 75.
A computational framework developed on the basis of these algorithms
allows calculation of a wide range of hydraulic problems for open channels.
The most efficient method of program construction for multipurpose programs
is the combination of separate functional elements (modules). The calculation
of various and different problems require various combinations of such
elements. A series of computational modules which contain the variety of re-
quired algorithms is organized and can be applied to specific situations.
77
-------
Some of the required coefficients (V,W), necessary for solving the
transfer equations, are derived from integration of Saint-Venant non-steady-
state motion equations. Yeremenko, Kolpak, and Selyuk (1975) integrate the
transfer equations by numerical methods using implicit and explicit differ-
ence schemes. The conditions differ, depending on the difference scheme.
The initial equation is of the equation 36 type and can be closed under
special conditions.
f + vf-|;l-KC + f(x'':) (78)
The implicit two-layer difference scheme is approximated by the following
differences
and is solved by the sweeping method
W-jT^) (
(79)
where the sweeping coefficients are determined from recurrence relations
- -*li - > 31+1/2 =
B± - Aiai_1/2
in which
AI = -^ (v± + wipi + wi+iDi+i 2Bi = i + KC + T (w±+1D1+1 +
AA W±AX ^' 2W±(AX)Z
+ 2W±Di + B^D^i, ; M± . -^ C"i°i + "i-l°i-l _ v , , G .
-------
The results of the investigation of various difference schemes by
Yeremenko, Kolpak,and Selyuk (1975) suggest that the Dufour-Frankel three-
layer explicit difference scheme, which does not require limitations on the
spatial step, is the best available method. The initial equation in this case
is presented in the difference form as
OJ+1 = CJ'1 + 2T[-V- + ±- (DW>i+l ~
-I -i L 1 T.T •
2AX 2AX
CJ ,-, - CJ+1 - d"1 + CJ ,
"-1-1 J - -"JL +2T[fi - KCJ]
T 1
The stability criteria of this difference scheme are
rAX2. AX i
(82)
L2D ' |V
When the explicit difference scheme is used, the diffusion effects in the
closing section are not excluded, and equation 78 is replaced by a finite-
difference two-layer pattern
-2 - - D
- - N - - (83)
for which the following stability conditions are derived
D < .AX < 2D 2 (AX)2
V - ~ V T - 3VAX - 2D
2D < AX T > 2 (AX)2
V - ~ 3VAX - 2D (84)
These stability conditions do not contradict the stability conditions of
the Dufour-Frankel scheme. Yeremenko et al. (1975) used the following formula
to determine the longitudinal dispersion coefficient
D = 1.5BV, (85)
where B is the flow width.
Two methods are used for calculating the conjugation conditions when the
influx into the river at the junctions E of tributary confluence is considered
(Technical Guide, 1963). The following condition is valid if diffusion is not
neglected at the point of confluence of flows 1 and 2 into stream 3.
CE(3)(QE(1)+ QE(2)> = QE(I)CE(I) + QE(2)CE(2)
The concentrations Cg/^and C ,„•) are determined by algorithms described
above for explicit or implicit schemes.
For cases when the influence of diffusion cannot be neglected, the con-
jugation condition takes the form
79
-------
(87)
Various difference approximations of condition 87 are possible. Numerous cal-
culations showed that it is best to use the continuity condition CE(]_) =
CE(2) = CE(3)> in which case the obvious equality Q3 = QI + Q2> instead of 87
gives
+
9X
(88)
By using conditions 86 and 87, it can be shown that the influence of diffusion
is important only at points adjacent to node E (Figure 8).
28 N
Figure 8. Calculated pollutant concentrations in two confluent flows at times
ti and t2 . Curves 1 are results obtained without calculating dif-
fusion effects; curves 2 calculate diffusion effects; 0, the value
of concentration in conjugation node, wiLh calculation of diffusion.
80
-------
Similar conjugation conditions are considered by Vasiliyev and Voyevodin
(1975). Here, as in the calculation of variable flows, the case of concen-
trated influx or outflow of water, confluence, or disjunction of flows is con-
sidered. For instance, in the case of confluence of flows, the condition
^E(l) = ^E(2) = ^E(3) i-s equivalent to equation 87. Also, two forms of cal-
culating the flow conjugation, corresponding to cases 86 and 87, are consider
ed. The transfer equation is presented in the form
V_. () + F(C) (89)
9t 8X V 8X
where
and is approximated with the help of the ordinary implicit first order differ-
ence scheme in t, and of the second order in X (Maron 1971).
AX2
WK+1 CK+ - Cg _ ^K+l [(WD}K+1
T * ri~l~-L
* ri~l~-L - *\ .- ___ . -- . —
AX2
(90)
Here the coefficients and the right hand side of the expression are calculated
approximately as follows (Sherenkov 1975) :
•
As a result, for each segment the following system of implicit equations
relative to the unknowns is obtained, C+l, n=l,2, . . . ,N-1
= dn n = 1,2, ... ,N-1 , - (91)
where an = _J (Q - /2; 1 n = (Q +
% ^ dn = BnCn + ^
In addition to equations 91, at the points n=0 and n=N, the following equa-
tions are developed
K+1
_ _ , fr
Fo ~ ^o C0 l£XJo
K+1 _ K+1 K+1 ,WD,K+lr K+1 K+1
Fo ~ % co - ) (CN - C_) (93)
Together with the boundary conditions and the conjugation conditions, equa-
tions 91 to 93 form a closed system relative to the unknowns
81
-------
in the. upper temporal layer. The solution sequence of the equations is
similar to the variable motion calculation. This solution algorithm has a
disadvantage in the implicit difference scheme which is described by Yeremenko
et al. (1975), but in spite of the transformation, the solution of equation 92
remains sensitive to the condition AX <_ 2D/V. They also describe the mathe-
matical model which evaluates the effect of surface flow on concentration
distribution in a channel, using the turbulent diffusion equation. With an
essentially non-uniform concentration distribution in the actual flow section,
the transfer equation is
iC + V1C = 1 ^L (DW 1C 1 iT 75 + f , (94)
3t 3X W 3X 3X' W 3X
where 1/W(3T/3X) is an additional term which describes the unevenness of the
dispersion in the flow section, and
T = TV CdW - DW — , "C = 77 /CdW, (95)
1 9X' W
0 -> _ 0 ->
where C = C(X,t) - C, y = Vx(X,t) - V
The estimation of the term containing T is derived for the case when the
concentration of pollutant delivered by the flow does not change with time,
while the discharge of influx water is negligible. The influence of the
term containing T is investigated by comparing the solution of the three-
dimensional problem, in the specified velocity field, with the one-dimensional
problem, using equation 95 without calculating the term using the following
conditions :
c:(x,o) = o, "c(o,t) = o. (96)
The three-dimensional problem
with the boundary conditions
sr
C(0,X2,X3) = 0, |^ = 0 (98)
where ?> is the contour of the flow and n is the normal to this contour,
integrated with the help of the known locally unidimensional variable direc-
tion method (Samarskiy, 1971). The lateral pollutant influx is specified by
the source function F(X1,X2,X3). Calculations were carried out for a
trapezoidal channel with a specified cross-sectional distribution of longi-
tudinal velocity. In the calculations, various distributions of sources along
the channel were specified as (a) of constant strength, (b) constant for
specific periods, and (c,d) sinusoidal. The results of calculating the three-
dimensional problem, after the concentration C has been averaged in the cross-
section, are shown in Figure 9. The results of the solution to the one-
dimensional problem are also displayed in Figure 9; curves 1, corresponding to
the solution of the one-dimensional equation, are located lower than curves 2
82
-------
the plots of the averaged function C.
B
8
6
4
2
10
8
1.5 3.0 4.5 6 X, KM
1.5 3.0 4.5 6 X, KM
1.5 3.0 4.5 6 X, KM
Figure 9. Diagrams of variation in concentrations along the X axis given dif-
fered specification of function F(X ,X2,X3) (curves 1 are the solu-
tions of the three-dimensional problem averaged over the cross-
section; curves 2 are the solution of the one-dimensional problem
without calculating the term; the dots indicate the solution of a
one-dimensional problem, with a calculation of the term F(X,,X ,X ).
The error fluctuates within 10% to 25%. In some cases therefore, the
additional term in equation 94 should be taken into account and its approxi-
mating expression introduced. For this, the pollutant transfer equation is
considered without taking into account the diffusion effects
Vx— = F(X15X2,X3).
83
-------
Integrating this equation by X}_, we have
where K is introduced from the condition
— ~f = — /— dW
If the quantiuies introduced are considered, equation 91 assumes the form
ar . __ ar 'I a an -i 3fx _ (101)
Here f = ? = i/FdW
W
These calculations, given different specifications of function F and ic=1.05,
indicate that calculation of the additional term in equation 101 decreases
the error.
Solution of Some Problems of Water Quality Management
The algorithms given above allow us to define water quality control
problems. Yeremenko (1975) considered some problems of controlling the
quality of effluent discharges. Preparation for discharging from storage
lakes for different purposes should include determination of the impact of
effluent discharge into a variable flow to obtain maximum use of the assimila-
tive capacity of the receiving water body, to determine the minimum discharge
hydrograph, and determine an acceptable discharge. It is assumed that there
are existing standards that limit the pollutant concentration in the river
at the points of effluent discharge.
The solution to these problems can be attained with the help of the
mathematical models described in this paper by using the explicit difference
scheme 81.
Determination of the effluent discharge regime required for discharge
into a variable flow, can be reduced to the calculation of the source capacity
given a limitation on BOD concentration equal to L; from equation 81 we find
= KL - [-V. + - DW)j+l - (DW)^! . Lj+i - Lj-j Ln+i - 2L
J - - + -
2A - 2A
(102)
An actual example of this type was solved for a channel having a width of
B=150m, initial depth hQ=4.0m and specified QO(X, t)=227m3/sec, Qo(0,t)=SQo
sinflt/T (T is the specific given period of time), Q(N)=f(Z); L(X,0)=0; L(0,t)=0;
L(4000m,t)=4.2m/m3=r=constant; the concentration of dissolved oxygen S(X,0)=
6.0g/m3, S(0,t)=6.0g/m3 was given. In the calculations, Snp=9.0g/m3, K^O.2
I/day, K2=0.4 I/ day, were accepted. The results of the calculation are shown
in Figure 10(a,b) , where the time intervals are ti=3360 sec, t2=10860 sec,
84
-------
t3=18060 sec, ti4=25260 sec.
The feature of pollutant dispersion in a variable velocity flow, as in
Figure 10(a), is the relatively small velocity of pollutant dispersion, in
comparison with the velocity of propagation of water. Figure lOb indicates
a calculated change of assimilative capacity with time.
Q
O
Q (M3/sec)
,
(gr/m3) (gr/m3)
8.4
6.2
4.0
24.0 X(KM)
<3
fi (gr/m° sec)
6
4,800
14,400
24,000 t (sec)
Figure 10. Results of calculating pollutant concentrations and source
capacity in an unsteady flow.
The determination of the minimum discharge hydrograph that will permit
an acceptable release of effluent into a watercourse was also considered. A
85
-------
typical problem is the release hydrograph during daily control of the load at
a hydropower station. At a fixed capacity of pollutant delivery, the minimum
discharge should maintain BOD within the required standards. This problem is
solved by matching. To speed up the calculations, it is possible to apply
standard programs to find the roots of functions on a finite segment. A
typical solution for a variable flow in a 500m-wide channel is described in
the section X=0 by the hydrograph shown in Figure lla. The problem is
solved when a minimum release discharge is found, which maintains maxL
-------
M3/sec
5000
4000
3000
2000
1000
Lgr/M3
8-
6
Q2=483
= 451
= 507
12
a
16
20
24
t hour
L
gr/M3
20
40 60 80 100 x, KM
b
Figure 11. Results of calculating minimum water discharge given water
releases for a guarantee of required water quality in the flow.
87
-------
BIBLIOGRAPHY
Elder, J.W. 1959. The dispersion of marked fluid in turbulent shear flow.
J. Fluid Mech. 5, No. 4.
Fedorov, N.F., N.N. Lapshev, and Yu.B. Bezobrazov. Formulation and solution of
several problems of conservative pollutant dispersion in water flows.
Paper presented at symposium.
Karaushev, A.V. 1969. River hydraulics. Gidrometizdat, Leningrad.
Maron, V.I. 1971. Mixing mutually soluble liquids in a turbulent flow in a
pipe. PMTF, No. 5.
Monin, A.S. and A.M. Yaglom. 1965. Statistical hydromechanics. Moscow,
"Nauka," 4.1.
Paal, L.L. 1967. On the longitudinal diffusion of pollutants in small flows.
Paper and reports on the problems of self-purification of water bodies and
the displacement of effluents. Tallin.
Paal, L.L. Mathematical models and their application for calculating water
quality in flows. Paper presented at the symposium.
Rozovskiy, I.L. 1957. The flow of water at the bend of an open channel.
Academy of Science, Ukr. SSR, Kiev.
Samarskiy, A.A. 1971. Introduction to the theory of difference schemes.
"Nauka," Moscow.
Sherenkov, I.A. Two-dimensional model of flow and pollutant dispersion in a
meandering river channel. Paper presented at symposium.
Technical guide for the calculation of cooling ponds. 1963. Gosenergoizd,
104 p.
Vasiliyev, O.F., T.A. Temnoyeva, and S.M. Shugrin. 1965. Numerical method of
evaluating unsteady flows in open channels. Academy of Science, USSR,
Mekhanika, No. 2.
Vasiliyev, O.F. and A.F. Voyevodin. Mathematical modeling of water quality in
open channel systems. Paper presented at symposium.
Voyevodin, A.F. 1973. Sweeping method for difference equations determined in
a complex. ZhVM and MF, Vol. 13, No. 2.
-------
Yeremenko, Ye.V. Several problems on modeling water quality in streams.
Paper presented at symposium.
Yeremenko, Ye.V. and V.Z. Kolpak. 1973. Evaluation of the concentration of
a passive pollutant in a river with tributaries. In: Problemy Okhrany
Vod, Issue 4, Kharkov.
Yeremenko, Ye.V., V.Z. Kolpak, and N.I. Selyuk. Calculation of pollutant
concentrations in a three-dimensional diffusion zone when solving one-
dimensional problems. Paper presented at symposium.
89
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DISCUSSION
THOMANN: I am slightly acquainted with the state-of-the-art in mcdsling
water quality in the USSR. The material presented by Dr. Yeremenko illus-
trates a considerable step forward for Soviet scientists. I congratulate him.
On the basis of practical testing of river water quality modeling in the
U.S., it is possible to analytically indicate that results with equations
that consider dispersion do not significantly differ from solutions received
without this consideration. Can you comment on this situation?
YEREMENKO: The situation mentioned by Di. Thomann is a known fact. It is
correct, for the most part, in stationary flows and steady flows when waste-
waters are discharged. In a nonstationary case, the calculation of diffusion
processes can play an important role. Therefore, it is necessary to conduct
an analysis and to give a basis for the necessity or expediency of cal-
culating diffusion.
YENETSIANOV: I would like to add to this report, namely to the part related
to the equation that describes the distribution of the pollutant in a river
flow. When describing the physical processes of a river, a linear diffusion
equation is used as given, without critical analysis. From our point of view,
it is possible to use this equation in the manner indicated only when
describing the behavior of nonconservative substances. It reflects rather
accurately the physical processes occurring with nonconservative components.
Nonconservative pollutants can be divided into two large groups: 1) "hetero-
phased" pollutants that interact physically and chemically with sediments,
and 2) essentially nonconservative pollutants for which the law of conserva-
tion of mass is not fulfilled.
For the first group we proposed to supplement the equation with a term
which describes the interaction with the sediment phase of the pollutant.
In both phases, the pollutant must be in a thermodynamic equilibrium. Time
is required to establish this equilibrium. If the equilibrium is shifted to
the nonliquid phase, then the concentration in solution can essentially
differ from the real content of the component on a given stretch of a river.
When the equilibrium is shifted to the aqueous phase, a large quantity of the
component is extracted from the nonliquid phase.
The second modification concerns the linear diffusion coefficient. The
criterion is written in the following manner: D/vL where D is the diffusion
coefficient, v is velocity, and L characteristic size. Depending on the
value of this criterion, both Dr. Thomann and Dr. Yeremenko are correct.
90
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BEZOBRAZOV: The impression has been created that research on diffusion
processes is gradually yielding to wider, more problematic questions of
constructing water quality control models. However, you must realize that
most of these models at one stage or another must be based on solutions to
diffusion problems. Many questions, connected with the diffusion processes,
have been solved. Some of these questions received exhaustive analysis in
Dr. Yeremenko's report. However, as more questions are solved, more
controversial and complex questions arise, in particular, problems of
evaluating the relationships of the turbulent diffusion coefficient, or, to
be more exact, the relationships of its various components. It is necessary
to consider the influence of allowing isotropy or anisotropy of the diffusion
coefficient on the dispersion of a passive pollutant.
RODZILLER: I agree with Dr. Thomann's evaluation of Yeremenko's paper. He
gave us an exhaustive summary of the material presented for the symposium.
However this is not an exhaustive summary of all the work in progress in
this country. Mathematical models must influence decision-making which will
maintain clean water bodies. I fully agree with Dr. Thomann that it is easy
to write a complex equation, but that it is difficult to solve it, and like-
wise difficult to put it into practice. Therefore, we must heed Dr. DeLucia's
words: a simple and accessible solution is better than a mathematically
elaborate one.
For practical goals, I consider it necessary to concentrate our atten-
tion on solving the water quality question for water bodies depending on the
arbitrary quantity of the discharge and the arbitrary composition of the
wastewaters. In the models being developed, it is necessary to consider
substance transformation processes. I assume that you have such models,
perhaps not absolutely exact, but which allow you to solve engineering
problems.
SELYUK: When planning water economy activities and regulating quality of
water sources, it is necessary to consider self-purification processes.
Questions of the detailed study of physical processes in water bodies, and
the development and perfection of mathematical models of these processes
deserve particular attention.
YEREMENKO: I must agree with Venetsianov and Bezobrjazov1s remarks on the
necessity to continue research on the behavior of nonconservative pollutants
in a water body. This work is complex. In the report presented, a deter-
mined relationship between the diffusion coefficient's components is actually
shown. We understand that this is not a rule for all circumstances. There-
fore, we have tried in the report to illustrate the influence of the choice
of this value and used its determined meaning to obtain a concrete solution.
We thank you for all the valuable and useful comments, which we will try to
take into consideration in further work.
THOMANN: By U.S. experience, diffusion processes are not as important as the
kinetics of photosynthesis processes, oxygen consumption, transformation of
various forms of nitrogen, carbon, etc., for calculations and river basin
management. These factors are taken into consideration in the practical
models being developed in the U.S.
91
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WATER QUALITY MANAGEMENT MODELS:
SPECIFIC CASES AND SOME BROADER OBSERVATIONS
Russell J. DeLucia and Tze-wen Chi
INTRODUCTION
Some Issues Associated with the Planning Process and Model Development
In the process of planning investment in water pollution control of a
river basin, a great deal of analysis is needed to examine many pertinent
facets of the problem. There are interacting physical, economic, social, and
political subsystems in a basin. Each subsystem can be complex and involve
time-consuming study. Models are often used to ensure and facilitate consist-
ent analysis. However, there is no one prescribed approach for model build-
ing.
A model can be comprehensive enough to include all subsystems. A model
of this type in general will be coarse and simplified for computational con-
venience and, therefore, will seldom provide sufficient details of the sys-
tem's response. To remedy the shortcoming, other models are used to generate
refined information. Some of these models will be subsystem specific while
others will include more than one subsystem. They can be designed to be used
conjunctively in that output of some models may be input to others. For
example, a highly aggregated optimization model generates plans which can be
further analyzed by a detailed simulation model. In addition, models can be
designed for feedback iteration. The appropriateness of a single model versus
many models and how to structure it (or them) depends upon the analyst's
experience, preference, and the particular system under consideration. In
practice, one would start with relatively simple and robust model(s) and re-
fine toward more reliable model(s).
In the broadest sense, the development of a water quality model is depen-
dent on the characteristics of the problem being examined. We x^ill not at-
tempt, in this paper, to provide an exhaustive list of the issues that arise
in this process, but we do wish to present four areas of concern that in-
fluence the development. These are:
1. the relationship of planning activity to the problems of variable
specifications;
2. the nature of the institutional or client setting;
3. the type and quality of data available for model development; and
4. the availability of resources for model development and implem-nta-
tion.
In the first area, the issue of the specificity with which the model's
endogenous variables must be defined is the primary consideration. For
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example, although a river basin may have an important nonpoint waste load,
this pollution in-p-u-t can be accounted for in the model as an equivalent point
source because specific control alternatives for nonpoint loads are not being
considered. Another example is a case in which non-treatment alternatives,
such as flow augmentation or bypass piping and regional treatment, are under
consideration, but they need not be included in the model specification.
Rather, such alternatives are dealt with by running the model under an alter-
native flow regime (to account for flow augmentation) or by developing treat-
ment cost functions which reflect transport costs (to account for piping and
regional treatment). There are other issues, of course, having to do with the
nature of the pollution problem (e.g., the importance of nutrients) and the
nature of the preference function (e.g., the problem of equity versus effi-
ciency) .
The second area of concern is associated with the institutional or client
setting of the planning problem. Here the issues influence decisions related
to the structure of the mathematical model and its level of documentation.
Is the model for general use or is it to be employed in the analysis of a
specific problem? Is the planning problem strictly a water quality management
problem or is the model to be part of a larger planning effort and will it
have to be made compatible with other quantitative models?
The more general the model use will be, the stronger the argument for
detailed documentation and the inclusion of a variable structure that explic-
itly deals with many of the issues suggested in the area discussed above. On
the other hand, if the model is being developed for a very specific planning
problem or is strictly a water quality management model, then both the struc-
ture and ultimate documentation of the model can be limited. Strict water
quality management problems argue for less complex models. The availability
of complementary models suggests that the model's variable structure may be
limited and these variables or characteristics should be considered in a sec-
ond or complementary model.
The last two areas (availability of information and resources) jointly
influence both the level of complexity of model variable structure and ulti-
mately its implementing computer code. The more complex the model in general,
the more the data, computing, and professional resources necessary. In some
cases significant prior information allows the use of a more complex model
than would be warranted by the study budget. Consider, for example, the
situation in which there is adequate existing field data to allow extensive
calibration of parameters. Additional data from the present study could then
be developed slowly for the ultimate verification exercise.
The development of the Saint John River model provides an example in
which many of the issues in the four areas arise. As jthe model is presented
in the following section, we will highlight the issues we considered most
important and discuss the important features of the discussions that led to
the development of the model in its present form.
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The Saint John Model
The model discussed in this paper was developed for initial use in the
Saint John River Basin described briefly below. However, the characteristics
of the model reflect not only the particulars of the specific Saint John plan-
ning problem for which it is being used. In addition, its characteristics
reflect the planning requirements and constraints implicit in the 1972
Amendments to the Federal Water Pollution Control Act (Public Law 92-500)-
The model was developed and was documented so that it would be useful in
examining basin or area-wide planning efforts required or suggested by various
sections of the Amendments.
The model was developed for use in a planning study conducted by a team
of consultants, under a U.S. Environmental Protection Agency contract with the
Northern Maine Regional Planning Commission in a joint venture of E.G. Jordan
Co., Inc., and Meta Systems, Inc.
The Northern Maine Regional Planning Commission in cooperation with the
state agency is undertaking aspects of water quality planning mandated by
P.L. 92-500. The initial model users are members of the consultants' staff
who have been involved in the development of this and other models. However,
the model development was funded by EPA and is being documented at a level
sufficient to allow ready application by other users.
Prior to discussing particular characteristics of the model, it is useful
to describe some of the particulars of P.L. 92-500, the nature of the basin
for which the planning is being done, and complementary planning efforts that
have been done.
The P.L. 92-500 calls for the attainment of "secondary" treatment or bet-
ter by 1977 for publicly owned treatment works and best-practicable control
technology for industries provided that these levels of treatment/control will
result in meeting water quality standards. In cases where these levels of
technology are insufficient to meet the standards, more stringent technology
would be called for. P.L. 92-500 further states that when it is necessary to
move beyond secondary and best-practicable, the targeted reductions should
reflect economic considerations. More stringent 1983 goals, namely the appli-
cation of best-practicable by publicly owned treatment works and best-avail-
able technology economically achievable by industries, are also required. The
explicit two-time period-treatment targets are incorporated into this model
that considers both capacity expansion and level of treatment questions.
The Saint John River drains the northern third of the State of Maine in
the U.S.A. and portions of the Provinces of Quebec and New Brunswick in
Canada. The planning effort which sponsored the development of the model(s)
discussed focused on the Maine portions of the basin and the international
boundary waters between New Brunswick and Maine. Another planning study was
carried out by a Canadian team and there was a series of both formal and in-
formal exchanges of information. In addition, the United States planning
study and the development of the Saint John model benefited significantly
from the results of an earlier methodological study sponsored by the Depart-
ment of the Environment of Canada. This earlier study used the Saint John
94
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Basin as an example and included the development of a prescriptive model; a
nonsteady-state stochastic simulation model developed in this earlier study
of the Saint John Basin was also available for use.
The watershed is sparsely populated, with large areas of timberland and
agricultural land both of which contribute significant nonpoint pollution
loads. There are, however, a number of large volumes of either untreated or
partially treated industrial (pulp and paper, potato processing) and domestic
effluents which have created severe water quality problems.
Model(s) Overview
The model(s) developed actually comprise a set of complementary models.
There is often no unique mathematical model for a particular problem (Meta
Systems Inc, 1971). Rather, a set of linked, sometimes hierarchical models,
is most useful. In this study, we have developed both a descriptive and a
prescriptive stream/river model and a descriptive lake model. Only the
stream/river models are discussed in this paper.
A descriptive model relates pollutant loads with stream quality indica-
tors. This model can be run independently as a steady-state, quality simula-
tion model. It also produces the matrix of descriptive relationships that
become part of a prescriptive non-linear (separable) programming model which
minimizes present value of costs over a multiple-period, time horizon. Capi-
tal costs are distinguished from operating and maintenance costs and appro-
priate relative multipliers may be introduced when useful. The impact of
reimbursement funds (both federal and state) and cost sharing of joint muni-
cipal industrial treatment works may be included in this manner. Provision is
made in the model for inclusion of various equity and budgetary constraints.
The physical systems relationships are based on stream hydraulics and
kinetics with some modification for reservoirs and lakes that may be integral
to the system. The dissolved oxygen/biochemical oxygen demand (DO/BOD) re-
gime is modeled by using a modification of the empirical Streeter-Phelps (1925)
relationship. The carbonaceous and nitrogenous BOD loads are handled individ-
ually (the latter including an appropriate lag effect). The combination of
long travel time and pulp and paper industry loads present in the Saint John
River, with their resultant significant nitrogenous loads, argued for this
inclusion. Point and nonurban nonpoint pollution sources are significant.
They are explicitly included with the options of including cost of controls
for both point and nonpoint sources since source control options differ.
The kinetic characteristics of the major pollutants (pulp and paper waste,
potato processing waste, and municipal wastes) can vary considerably. The
inclusion of polluter-specific kinetics allows explicit consideration of dif-
ferent control levels for different polluters. The nutrient regime is exam-
ined by considering nitrogen and phosphorus as total species and therefore
essentially as conservative elements. The analysis of this model is based on
the assumption of steady-state regimes (although, for example, variation of
the hydrologic regime is easily accomplished).
The structuring of a prescriptive water quality management model which
incorporates relationships between pollution loads and stream quality based
95
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on "steady-state" solutions of stream and/or estuary equations is not new.
The uniqueness of the prescriptive model presented herein lies in: a) its
structure reflecting planning problems arising from P.L. 92-500; b) the use of
a synthetic pollutant to express the cost removal function; and c) a series of
heuristics adopted to overcome possible non convex difficulties.
The first of these will be obvious from the' structure presented below.
The second and third are somewhat interconnected.
The Model —
The basin is divided into activity points or reaches labeled j, j=l,
...,J, and quality points or reaches i, i=l,...,I. For particular i and j,
the reach or point may be identical.
The descriptive approach entails the development of a set of transfer
functions, T-^, relating the quality at point i with the pollution at point or
reach j. T-H is a vector transfer coefficient in that this relationship is
developed for each of the H\ pollutants and quality indicator couples we are
examining (DO, BOD, N, P) . Letting T-H& represent the scalar element for
pollutant £ in the vector TJ-J_, we note that T-H£ is a function of geometry,
flow, kinetics, etc. T-JJ_£ is a function of system physics.
The descriptive river model uses stream hydraulics which are overwhelm-
ingly advective with little or no dispersion, and are based on a steady-state
analysis. The pollutants (instream) that are discussed below are treated with
zero or first-order kinetics dependent on their conservative or nonconserva-
tive nature. ;
The calculation of the transfer coefficient for the nonconservative
organic BOD and DO interrelationships is programmed using variations of the
Streeter-Phelps (1925) equations as follows: Extensions to the Streeter-
Phelps equation given by Camp (1963) and Dobbins (1974) are for a BOD level L,
§ = -(kl + VL (1)
where the terms k-^ and kg represent constants respectively for deoxygenation
and sedimentation, and:
— = k,L - k,D - A
dt * 2 (2)
where k2 is the oxygen reaeration rate, D is the oxygen deficit, or saturation
concentration less\ actual concentration, and A is the photosynthetic production
rate. Integrating (1) for Lfc , the concentration at a point corresponding to
time t downstream from an initial concentration L is:
Lt = V
k3)t
Utilizing equation 3 it is possible to integrate equation 2 for D the oxygen
deficit at any time t downstream from an initial deficit DQ , thus:
96
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Dt =
- (kj + kg)
Lo -
K3)t
-k2t
- e
-k9t -k9t
i 2 + D e 2
o
(4)
Equation 4 thus provides the means for determining transfer coefficients for
BOD and dissolved oxygen in the models.
If F-• denotes the flow between upstream activity point j and downstream
point i, then stream velocity and depth are functions of flow—V(Fj^), H(F^) .
Velocity and depth in turn are determinants of the reaeration.
Using the Thackston and Krenkel (1969) formulation, the reaeration coef-
ficient may be written as:
k2 =
-------
BOD nitrogenous/DO, DO/DO, Nitrogen/Nitrogen, and Phosphorus/Phosphorus. DO
is not a pollutant, but a transfer coefficient must be developed for this
couple since upstream DO influences downstream DO as indicated in equation 4.
The T-J describe the transfer of a pollutant concentration lumped at
point j to point i downstream. We have also included in the model provision
for pollutant loadings arising by runoff from the area drained by each river
section. By assuming a constant increment of flow/length and unit uniform
loading along the reach, we can derive a transfer coefficient for the propaga-
tion downstream to reach i of a pollutant load linearly distributed along
reach j. In centra-distinction to the situation discussed above, the flow in
such a reach is not constant, i.e., I
and an F-.- such that
does not exist; rather, one has an F.
+
M4
(8)
where F is the increment of flow/length in reach j. Then
I
V (\r + V 1 "C1 .
K2VK1 K3j 1
1 e
k,F^ V(F-;, F-)
1 1 -^ j
(k +
x + kg)] F±
-k^WF^Fj)
+ k3)
-e
(9)
To a good approximation the velocity can be regarded as constant:
V(F1,Fj) - V(Fj) = V(Fj;L)
(10)
Clearly, in reaches downstream of j , the pollutant, so introduced propagates
simply according to the T^.
The transfer coefficients then constitute the necessary relationships for
descriptive modeling. The downstream quality can be calculated by multiplying
the upstream load times the transfer coefficient, and summing over the up-
stream loads and appropriate couples. For example, if Q^ is the DO at point
i, LN the nitrogenous BOD load at LC-s the carbonaceous BOD load at then
LNj the nitrogenous BOD load at j, LC-s the carbonaceous BOD load at j
Qi=
jeJ
LN T
(ii)
J is the set of pollutant points upstream from i, j* the uppermost point in
the system, and ^ indicates the BOD nitrogenous/DO couple, £ the BOD car-
bonaceous/DO couple, and &3 the DO/DO couple. The prescriptive model will
have constraints of the form 0^ <_ "Q^ where "0^ is the quality target (stan-
dards) .
In addition to the pollutants and quality indicators mentioned above,
the model utilizes an additional parameter, biomass potential that indicates
98
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the extent to which substances in the waste stream distort the biological
activity of streams-beyond natural levels. The biomass potential concept was
developed as a part of another study (Meta Systems, Inc., 1973). Certain pol-
lutants (or certain levels and intensities of pollutant loadings) cause irre-
versible ecosystem disruptions; we do not deal with such regimes. The pollu-
tants in which we are interested are those which have reversible effects at
pertinent loadings. These may cause overfeeding and overpopulation, or mal-
nutrition and loss of species diversity; they are responsible for excess pro-
ductivity which causes only such perturbations to the natural system from
which recovery is possible through simple physical and biological activities.
For given hydraulic conditions, a measure of the relative extent of excess
productivity in a water body is the biomass potential, a parameter that may
be estimated from conventional water quality criteria.
[Biomass Potential (BP) =1.47 BOD5 +4.57 TRN + yP, where yP ~ 30]
When this parameter is combined with stream size, dilution ratio, and deten-
tion time parameters, a measure of the degree of reversible distortion of an
aquatic ecosystem is obtained. Biomass potential is an appropriate water
quality criterion to be used in the post-1977 period. At that time, standards
will be met and DO criteria will no longer be as relevant. As written above,
the biomass potential is expressed in terms of an equivalent estimate of ul-
timate BOD (NH+ + 202 + H20 -> NOg + 2H30+; 2 x MWQ /I x MWN = 64/14 = 4.57)
that follows from the oxidation of nitrogen in the2reduced stage (as in
ammonia NHg to nitrate NO^). The basis of the estimate y - 30 is discussed
in the Meta Systems report (1973). A predecessor to the concept of biomass
potential (or equivalent) as a water quality criterion was contained in
Standards of the Royal Commission on Sewage Disposal in Great Britain (Fifth
and Eighth Reports of the 1898 Commission). The basis of the standards,
which were developed from extensive surveys of small streams draining densely
populated areas in the United Kingdom, was that the 65°F, 5-day BOD of the
receiving water should be less than 4.0 mg/£ during the low-flow warm weather
season. This was perhaps the first example of a true BOD or biomass standard
for streams; similar standards were adopted subsequently in the United States.
Most stream standards, however, related to dissolved oxygen concentration; BOD
concentration in streams was regarded as significant only with regard to the
effect on dissolved oxygen. Dissolved oxygen reserves are directly important
only in connection with safety factors pertaining to maintenance of aerobic
stream conditions. The dissolved oxygen parameter has little direct correla-
tion with water quality for recreation and aesthetic uses, because stream
standards in many regions of the United States today (e.g., New York and the
New England states) do not use the oxygen criterion to distinguish among the
classes of good quality waters—Classes A, B, and C.
In addition to serving as a useful water quality criterion especially for
the post-1977 period, biomass potential proves a useful synthetic pollutant
to utilize in costs of pollutant control expressions.
The Prescriptive Model—
Most early water quality management models only utilized the BOD carbona-
ceous/DO relationship. Many were single period annual cost models (Loucks and
99
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Jacoby, 1972). Cost functions were often expressed as total annual cost as a
function of BOD^ removal efficiency.
Complete treatment plants consist of a series of processes and/or mecha-
nisms acting simultaneously or in series. Although one or more of the pro-
cesses may often be present primarily for the removal of a particular pollu-
tant (i.e., biological treatment processes for the removal of BOD) treatment
plants are in fact "joint product" plants removing BOD, suspended solids, nit-
rogen, etc., subject to all the difficulties associated with allocating the
costs to the joint products. Thus, expressing total costs in terms of a single
constituent such as BOD was an artificial mechanism but one that proved useful
in models that focused primarily on the BOD^/DO link.
Programming models which incorporate the nitrogenous BOD in the quality
relationships (constraints) while expressing costs in terms'of carbonaceous
BOD often result in convexity difficulties. The problems stem from the fact
that in the range of BODp removals for which the marginal costs rise with
higher degrees of removal—the same is not true for one of the other "joint
products," typically BOD..,.
Planning for capacity expansion problems of models involves nonconvex
issues even when dealing only with the BOD^/DO couple. These follow directly
from the existence of economies of scale in treatment plant costs.
In the model, costs are expressed as a function of biomass potential
removal and capacity. The specific planning requirements explicit in the
P.L. 92-500 allow the use of certain heuristics to examine both degree of treat-
ment and capacity expansion questions in a computationally efficient manner.
A convex separation programming algorithm is utilized by allowing the use of
readily available computer codes (IBM, 1971).
The model is formulated with two decision periods; one to reflect the
decisions necessary to meet the 1977 water quality goals of the P.L.92-500, the
other reflecting the 1983-1985 goals.
The following paragraphs present the cost structure utilized in the model,
generalized for the nth point-source polluter. From the discussion above
we suggest that cost functions exist in the form shown in Figure 1. For
the purposes of discussion, capital cost functions only are shown, and the
curves are schematic. Annual operation and maintenance costs are assumed to
have the same general shape.
The model requires that, in the neighborhood of a particular design capa-
city flow, X, D1 <_ X <_ D2 for a specific waste source the cost function can be
approximated in the following form:
Capital Costs = K = aX^ + f (r) D <_ X <_ Di+1 (12)
rl £ r
where X is the design capacity, and r the design removal efficiency, and
where ^ , D}, and D.^ are tae bounds over which equation 12 is valid. Note
100
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that in general f(r) is convex, and aX obviously concave for X > 0, a > 0,
and 0 < 0 < 1.
The rationale for this approach is to select a particular design load
(capacity) X, and approximate the concave function with a linear term. This
linearization is schematically depicted in Figure 1.
Concave region
approximated by
a linear segment
/ Convex region approximated
by piecewise linear segments
Figure 1. Linearization of concave segment of design flow cost function.
As discussed below, the size of the "neighborhood" in which the lineari-
zation is valid, and hence the adequacy of the approximation, is a function
primarily of the flow growth rates. Tc consider this method in greater detail,
examine a two-period capacity expansion problem. Let D-^ and Do represent the
design loads for the wastewater flow in periods 1 and 2. They are a function
of the flow growth rates for the particular point sources and hence a function
of population growth. The smaller the growth rate, the smaller will be the
difference between D^ and D2, and hence the smaller the neighborhood of ap-
proximation.
, = total capacity to be built by start of the first time period;
The decision variables include:
X,
X2 = incremental capacity to be built by start of the second time period;
TQ = percent removal in existing facility;
r^ = percent removal in first period;
^2 = percent removal in second period.
The parameters include:
DI = design flow for first period;
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D~ = design flow for second period;
A = D2 - DI
rml = minimum percent removal for first period (secondary) (assume r
is greater than or equal to rQ);
r 2 = minimum percent removal for second period (best practicable);
TI = length of first period (years);
T2 = length of second period (years);
(Ti + 12 is the length of the planning horizon).
There are three sets of constraints; namely, those that apply to each
treatment site, those that apply to the stream quality, and those that are
budgetary.
Treatment Facility Cost Constraints—
The following constraints apply to the definition of the costs of capa-
city and removal efficiencies for each wastewater treatment plant:
DI <_ Xi <_ D2 (in the cast of no existing plant) (13)
Qo £ Xi ^< D2 (where Qo is an existing capacity) (14)
0 _<_ X2 (the non-negativity condition) (15)
TI >_ rml (minimum level of treatment for the first period) (16)
r2 _> rm2 (minimum level of treatment for the second period) (17)
The explicit assumption made in the model is that the cost functions, for
a particular concentration and flow, may be written over a limited range of
capacity and removal efficiencies as a separate function of capacity and re-
moval .
A mechanism for expressing the cost functions for each period remains to
be developed. If K, represents the capital costs prior to the first period,
and K2 the capital costs during the first period, assuming XQ to be zero,
(For purposes of simplicity of presentation, XQ will be assumed to be zero
henceforth. For situations where it is non-zero, simply substitute X-^ - Xg in
the applicable cost functions and budget constraints.), then the expenditure
prior to the first period must involve three separate terms. The first term
is the capital cost associated with a design flow of D^ and a removal effi-
ciency, TQ. The second term reflects the change in cost resulting from the
difference between Xj_, at the beginning of period 1, and the design flow DI,
at the beginning of period 1. As explained above, for small differences X^ -
Dl, the change in cost can be approximated as a linear function, a(Xi - DI) .
The coefficient "a" is the slope between the capital costs associated with
increasing the removal efficiency from rQ to rj_, assuming that the flow of the
treatment is DI. The removal cost is a convex function of the form depicted
in Figure 2. The capital costs to be incurred by the start of the first time
period can therefore be written as:
ki(X:, TI) = KI + a(Xi - DI) + fi(ri) (18)
where KI is defined by equation 12 when D± + DI and X± = XQ.
The capital costs incurred during the first period, T-j_, depend on whether
incremental capacity, X2, will be constructed during this period, i.e.,
whether X2 = 0 or X2 > 0. If X2 = 0, then the capital costs of the first
102
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period equal only the costs of increasing the removal efficiency from X-^ to
k2(X2, r2) = fi(r2) -
if X = 0
(19)
as shown in Figure 3. If X~ > 0, the capital cost includes the cost of con-
structing additional flow capacity, X2, and increasing the removal efficiency
of the total capacity flow, X^ + X2, to X2. The cost of additional removal
efficiency, i.e., from r^ at the beginning of period 1 to r2 during period 1,
for a flow capacity of X]^, is defined in equation 19. The unit cost of re-
moving a fraction of the waste, r2, from an additional flow X2 can be esti-
mated by a linear approximation, b, of the initial portion of the cost func-
tion for capacity X given r^, k(X r^), as illustrated by Figure 4.
k(D -
- k(0 Xi)
b =
D, -
(20)
The removal efficiency costs for X2 are approximated by the function f2(r2)
which is based on an X2 of D2 - D]^. The capital costs may now be written as
k2(X2, r2) = bX2 + f2(r2) + Mr2) - Mrl> if X2 1
(21)
This equation is only an approximation of the actual costs associated with X
and r2, based on an assumed D, and D2. If the values obtained for X2 signi-
ficantly differ from either 0 or D2 - D]^, then the b and f2(r2) should be
changed to more accurately estimate the total capital cost before resolving
the model.
Figure 2. Capital costs incurred by the start of the first time period as a
separate function of capacity, X and removal, r.
103
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Figure 3. Capital costs incurred during the first time period when X = 0.
Figure 4. Capital costs incurred during the first time period when X9 > 0
104
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Now there are at least two approaches to handling the expression k^(X
r~) : a) a mixed integer formulation; or b) a heuristic utilizing a "loss
function on X2-" Both have been investigated in the Saint John study.
The first approach suggests that TI be a dichotomous variable and
0 1 X2 1 A
then:
k2(X2, r2) = bX2
fi(r2) -
(22)
(23)
The second approach suggests a linear loss function in X_ as a replacement for
fo(r2). It should be noted in general that f2(r2) will not be a major term
of k2(X2» ic2) in the cases under consideration because the capacity expansions
during the second time period will be small.
Consider
f2(r2)
Let r~ be the maximum expected removal for the second period and f~ =
f(rm1J; then let
ml
f2(r2) - (X2 - XT) —
hence
k2(X2, r2) = b(X2 - Xx) + (X2 - Xx
- f1(r1 - rQ)
Collecting terms, etc., we can write
kj (X1 , rj) = ajXj + f^rj) + constant terms
k2(X2, r2) = a2X2 - a3X1 + f].(r2) - f^r^ + constant terms
(24)
(25)
(25a)
(25b)
where a-i, a^, and a^ are positive coefficients and f-^ is a convex function. A
similar argument leads to the following for operation and maintenance costs:
02(X2, r2) = b2X2 - b3X3 + g1(r2) - g^r^ (26b)
where g.. is a convex function and b,, b2, and b-^ are positive coefficients.
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In the objective function both the capital and operating cost functions
will be multiplied by discount factors &l, 02» and ei> 92'
The objective function then is
min Z = T
jeN
1
JXlj+flj
+
JEN
2j
X
2.
j (r2j)
(27)
where the second subscript j indicates a particular pollution source, of which
there are N. The N pollution source points j are a subset of the J activity
points defined earlier.
There are four types of constraints:
1. local constraints;
2. removal and load relationships;
3. instream quality constraints;
4. budgetary and equity constraints.
The first of these are equations 13 through 17 associated with each of
the cost functions. The second set of constraints relates the removals and
loads of the various pollutants to that of biomass potential. To facilitate
this, we define the following sets of variables and parameters:
Parameters :
C. = the concentration (#/mgd) at waste point j before treatment, of
the pth pollution in the tth period. p=l,...Np. p=l for biomass
potential
L. = the daily load (///day)...
= C. pt-Dt, where Dt is the design flow defined above.
Variables:
r. = removal percentage of the pth pollution at the jth waste point in
period t
jPtrjlt
(28)
relates the removal of the other pollutants to that of biomass potential.
Within the range of removals considered, the linear relationship is adequate
in most cases
Ljpt = tne daily load (///day) at waste point i after removal of pollutant
p after treatment in the tth period
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jltrjlt Ajlt= 1' (29)
The third class of constraints are instream quality/waste load relation-
ships and the limits (or targets) placed on instream quality. We let Q^nt be
the nth quality measure at the i-th stream point in period t. Recalling the
structure of equation 11 the instream quality/waste load relationships are of
the following form
Q-tnr = y y T „ L + Q. T „ (30)
JrK jl\ ^ J Ji£
JO t-l\. tx. IS-
and the quality targets are of the following form
QintlQint t=l, 2 l-l... (31)
where the transfer coefficient T... is based on an appropriate low flow
regime, usually the 1-year, 7-day low flow.
Here J is the set of pollutant points upstream from i, and j* is the
uppermost point in the system. K is the set of pollutant/quality measure
couples appropriate and H is the appropriate quality measure/quality measure
couple (e.g., BOD nitrogenous/DO couple).
The last set of constraints that may be of interest are budgetary and
equity constraints. These constraints may take the form of limitations on
capital and/or operating expenditures for a particular polluter or for sets
of polluters. Let K -(rt-j) and Ot^(rt^) represent the cost functions of
equation 25 and 26 for time period t and pollution source j; then such con-
straints might be of the form
Ktj(rtj) <.Ktj (32)
Q*.I(T^I) < "0~tJ for some or all of j, t (33)
where the righthand sides of equations 32 and 33 reflect capital and operation
and maintenance budget limitations. Possible equity constraints might take
the form
Ktjl(rt.i) < a. I Kt.(rtj), a., < 1 (34)
reflecting the fact that irrespective of marginal costs, one polluter should
not bear more than some percentage of the basin costs. There are a number of
other equity or political constraints that may be introduced.
In summary, the model selects treatment levels and capacity expansion
sizes and timing based on a criterion of minimization of weighted present
value of capital and operating costs. The weight can be varied to reflect
either federal resource cost criteria or local/regional (after subsidy) costs
or other alternative weights.
107
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The selection is constrained by a) minimum treatment and capacity con-
straints which reflect federal legislation (these are augmented with local
constraints to facilitate model solution); b) waste removal and load relation-
ship; c) instream quality constraints which reflect stream standards and/or
targets; and d) various budgetary and equity constraints.
Concluding Comment';;- -
The Saint John Basin planning study which sponsored the development of
these model(s) is just being completed. Still, some observations can be made
regarding the usefulness of the model(s). As has been the case in the past,
the models for prescriptive purpose will be of more limited use than the de-
scriptive purpose. There are two reasons for this. First, although knowledge
of least-cost solutions of control alternatives of both treatment level and
capacity expansion are important in the planning process, the number of pollu-
ters in this case is small and an ad hoc analysis with a descriptive model is
sufficient to investigate a reasonable range of options. In some measure this
was foreseen early in the study and it was the potential for wider application
and the availability of EPA sponsorship that led to the prescriptive model
development.
Second, there is the need to examine control under a variety of criteria
in addition to least cost. For example, these criteria are associated with
a concern for such things as: a) cash flow implications for municipalities;
b) energy requirements of alternatives; and c) administrative requirements of
control alternatives, both during construction phase and continuing operation.
All of these are of some concern in the Saint John Basin and the investigation
is amenable to use of the descriptive model with some complementary ad hoc
prescriptive investigations rather than the prescriptive formulation which is
limited in its criteria structure. This issue is briefly discussed in the
following pages since the implications for the usefulness of prescriptive
models is more general.
The Need to Calibrate/Verify the Des_crjLgtive_ Rel_a^tionships^
This section discusses an indispensable step, namely validation, in water
quality modeling. In specifying the functional (or structural) relationship
among variables in the previous sections, we have made assumptions, many of
which are for convenience, while others are founded on the observational data
of similar systems. For example, we have postulated that the Saint John River
behaves like the Ohio River and some laboratory systems in that the BOD degra-
dation is proportional to the BOD remaining. We also assume that the rate of
degradation in our systems may be different from that of the Ohio River* and
is to be determined by the pertinent data. Therefore, it is important to
realize that a water quality model is, at best, an approximation of the actual
system, and that results obtained are only as reliable as the input informa-
tion and the accuracy of the operations performed on the data.
Recognizing that a model is only an approximation of the real system, it
appears logical to visualize that water quality modeling is merely a curve
*The Ohio is the river that the "Streeter-Phelps" formulation first
utilized (Camp et al., 1969).
108
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fitting in a river system. This line of recognition motivates one to apply
fruitfully the available curve-fitting techniques in water quality modeling.
For instance, statistical theory of estimation and hypothesis testing are all
useful in model building. In the following, we present an example to illus-
trate the application of hypothesis testing. Failure to recognize the quality
modeling as curve-fitting has in part contributed in making water quality
modeling a field of ambiguity and mystery. Too often the calibration/verifi-
cation procedure is described as a distinct and creative step in water quality
modeling. However, it is merely, in fact, an ad hoc procedure to fill par-
tially the role of estimation and h-ypothes-is-testing.
Our example*is a river reach receiving nitrogenous wastes. Travel time
between upstream and downstream points is about three days. On the basis of
data collected in the summer of 1973, the rate constant of nitrogenous BOD
decay is estimated to be 0.04 per day in the reach. In the degradation of
nitrogenous BOD, nitrogen is transformed into nitrate. With an intent of
proving (or disproving) the value of 0.04 per day, samples were taken during
the summer of 1974. Sampling times at up and downstream points were properly
spaced to account for the traveling time. Both nitrate concentration and
stream flow were measured to compute the nitrate flux at both points. These
samples are tabulated as:
Sample Upstream Downstream Difference
Number in 1000 Ibs. in 1000 Ibs. in 1000 Ibs.
1 28.3 24.2 -4.10
2 18.0 21.3 3.3
3 13.0 17.7 4.7
4 18.2 20.6 2.4
5 18.1 20.8 2.7
6 17.4 24.5 7.1
7 16.4 24.5 8.1
8 19.6 23.1 3.5
9 23.0 43.9 20.9
10 24.3 56.3 32.0
11 24.3 32.1 7.8
12 44.7 27.2 -17.5
13 55.0 36.9 -18.1
14 26.2 61.0 34.8
15 22.4 59.5 37.1
16 34.0 39.2 5.2
17 33.1 25.0 -8.1
18 23.2 23.9 0.7
19 12.8 19.7 6.9
20 19.2 18.3 -0.9
21 18.8 22.6 3.8
22 20.6 26.2 5.6
23 16.5 29.5 13.0
*This example is derived from actual data and analysis undertaken in the
provision of technical assistance to EPA Region III under Contract 68-01-2223,
109
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Sample
Number
24
25
26
27
28
29
30
31
32
MEAN
SD(X)
SDQO
Upstream
in 1000 lb_s_;
18.9
21.1
21.8
22.3
23.6
46.2
57.2
31.4
35.5
25.8
11.1
2.0
Downstream
in 1000 Ibs.
23.6
23.9
40.8
57.8
32.8
42.0
58.
68.
,2
,3
41.8
34.0
14.6
2.6
Difference
in 1000 Ibs.
4.7
2.8
19.0
35.5
9.2
-4.2
1.0
36.9
6.3
8.2
14.0
2.5
Simple statistics are also computed in the last lines of the table. Our
problem is to prove that the increase in nitrate flux downstream is signifi-
cant enough to justify a rate constant of 0.04 per day. For this rate con-
stant, an increase of 3.3 x 103 Ibs. of nitrate will be observed if an average
flux of 25.8 x 103 Ibs. is carrying across the upstream boundary. According
to the theory of statistical technique, (Miller and Freund, 1965), one can set
HO : ]i - U2 = °
to test against the alternative
= 3.3
Appropriate statistic for tests of this type is
t =
(X1 - X2) - & 8.2-0
2.5
= 3.28
The stochastic variate, t, is distributed in accordance with a student's fre-
quency with 31 degrees of freedom. The value, 3.28, is within critical value
tfl. 025,31 * 2.02 of 95% significant level; therefore, HQ (no significant in-
crease in nitrate downstream) cannot be accepted. While type I error is 5%,
type II error of the test can also be calculated as 9.0% (=Pr{lt/> 0.06}).
Since 9.0% of type II error is fairly convincing, to accept H^ (i.e., 0.04 per
day, 0.06 - 2.02 - 8'2 ~ 3'3) is reasonable.
2.5
Admittedly, on many river basins the rigorous estimation and hypothesis
testing cannot be carried out due to paucity of data. However, it appears
that some less precise and rigorous procedures must be adopted to justify the
Bodel before its use. But our point here is that, whatever procedure is used,
a clear description of the steps taken, and concurrent adjustment on parame-
ter's values must be provided. In the following we present the procedure used
to justify the Saint John Basin model.
110
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Although in the full Saint John study, calibration/verification proce-
dures were undertaken for all of the sub-basins, the discussion below will
focus on only a portion of one sub-basin, namely the Grand Falls headpond
region. This region is highlighted since it contains interesting hydraulic
characteristics and is also the region receiving the largest pollution inputs.
A schematic of this segment of the basin is contained in Figure 5.
Edmunds ton
f ^Madawaska
^/Ste.Basile
Grand Isle
river
hydraulics
Quisibis
Leyland
reservoir
impoundment area
Buren
Figure 5. Spatial location of point sources in example application.
Recall from equation 11 that the quality indicator concentration instream
at a point i is a complicated summation of products of the upstream loadings
and the appropriate transfer coefficients. Within the reliability of sampling
theory, the quality indicator level is known from the instream measurements.
The problem is then one of examining the system-with respect to those unknown
values of the loads and transfer coefficients. The large number of unknown
parameters (e.g., loadings, instream decay, and sedimentation rates, etc.)
necessitate numerous initial assumptions which in turn affect the computed
profile. To create a better comparison between measured and computed profiles,
the "art" of calibration becomes important in determining which characteris-
tics of the computed profile are incorrectly being models or are in need of
incremental adjustment. During the calibration exercise it is essential to
maintain an understanding of the physical system being modeled since apparent
calibration can be obtained by incorrect adjustment of parameters; however,
this would create incorrect forecasts of the water quality levels if the model
was subsequently used in a predictive or verification-type mode.
Ill
-------
In this example we discuss only the calibration exercises associated with
the non-conservative pollutants since their transfer coefficients are more
complex.
A series of line schematics identifying the spatial location of sampling
stations along the river sections provides a medium of comparison between the
measured and computed water quality profiles. On these schematics, the DO-BOD
stream sample information collected during a particular sampling is plotted.
The ranges of the sample measurements are denoted by the wide vertical bar at
each of the sampling stations, the length of the vertical bar indicating the
spread of the measurements. The short cross-bar on the vertical bar denotes
the mean value of the measurements. It is important to point out that the
plotted values for BOD profiles represent the ultimate total BOD values. The
need for considerations of ultimate (as opposed to 5-day) BOD values arises
because of the extensive time of travel for transit through the river system.
The transformation of the instream samples of 5-day BOD is complicated because
the ratios of ultimate carbonaceous (BOD^) and nitrogenous (BOD^) values to
the 5-day values are dependent on the nature of the source. A method for com-
puting the transformation was developed by utilizing the best estimate of the
transfer coefficient and the pollution release levels to create a weighted
coefficient for the ratio of 5-day to ultimate BOD conditions for the instreara-
sampled BOD^ levels.
Manipulations of the sampling results in accordance with the procedures,
such as those indicated above, provide the appropriate data to compare with
the mathematically computed results. Prior to the mathematical computation
phase, however, there were many considerations that received attention, the
more important being:
1. Checking the one-dimensionality assumptions—the assumption of one-
dimensionality (i.e., cross-sectional uniformity) implicit in the mathematical
model is an important approximation-that must be critically examined parti-
cularly for regions of considerable river width, such as those found in im-
poundments like the Grand Falls headpond. Experimental evidence of the cross-
sectional homogeneity was provided by sampling at intervals across the cross-
section. The results identified no strong channeling effects, indicating that
one-dimensional assumption is appropriate.
2. Selection of reaeration levels—the reaeration coefficient, k2, is
one of the rate constants incorporated within the mathematical model computa-
tion. Since the rate constant cannot be measured accurately in a polluted
stream, and is a function of the characteristics of the stream (e.g., velocity,
bottom slope, etc.), as a first approximation, the appropriate reaeration
levels must be derived from literature equations. Efforts to match sampled
and computed profiles were complicated because a considerable instability of
the flow regime occurred during the sampling periods. Sensitivity analysis
of reaeration coefficient levels was undertaken for both the river regime and
the headpond to determine the magnitude of the changes in the velocity and
reaeration that probably occurred during the sampling program. This informa-
tion was useful in suggesting the level of uncertainty and therefore the differ-
ence between the experimental and calculated profiles that might be attributed
to the selection of the reaeration coefficient level.
3. Since the Grand Falls headpond creates an impoundment for a distance
of about 30 miles upstream of the dam, it was important to establish the con-
112
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consequent effects on the pollutant transport processes (e.g., Does the water
stratify?). Use of a technique based on a reservoir Froude number (Water
Resources Engineers, Inc., 1969) indicated that the reservoir does not strati-
fy. This finding was later supported by experimental evidence. The absence
of stratification, however, did not preclude a considerable decrease in velo-
city in the wide portions of the reservoir and a consequent difficult task of
establishing~levels for the sedimentation coefficient.
The calibration computations are necessarily an iterative process, each
trial refining appropriate parameters and hopefully improving the comparison
between measured and calculated water quality profiles (subject to the pro-
viso that no parameter be adjusted without valid reason). A synthesis of the
calibration exercises on the Basin is documented in Table I. The information
has been tabulated by a) verification run number, b) general comments, c) flow
regime, d) point source, e) background contribution, and f) rate coefficients.
The second heading describes the general emphasis of the calibration run. The
last four columns denote general sub-sectors of the problem and entries in the
columns describe the specific refinements made in each run. The successive
runs of the calibration generally indicate activity in sectors or columns from
left to right (e.g., It is necessary to establish the hydrologic regime prior
to refinement of background loading contributions).
Several general comments will clarify elements of the table:
1. The flow regime of the computer program is developed by summing suc-
cessive contributions along the river system. Therefore, in order to match
the hydrologic regime of the sampling program, one must iterate, trying flow
additions and then modifying, if necessary, to create a match of hydrology at
many points of the system.
2. The literature estimates of background pollutant contributions are
necessarily given as a range of values. Successive trials were necessary to
establish the general area within the literature range which was appropriate
for the Saint John River. (This range factor was particularly important for
the conservative pollutants since the principal N and P contributions develop
from background sources.)
3. As illustrated in the example results presented in Figure 6, the
series of minor adjustments eventually provided good comparisons to the in-
stream measured values upstream of Edmundston. Successive adjustments to
parameters for the region downstream of Edmundston, however, failed to create
the comparison with the instream measured parameters that was expected. ihis
divergence indicated an important contribution or activity was being omitted.
Adjustments to coefficients and loadings produced an improvement in measured
and computed profile, but the profiles still demonstrated a wide divergence
in behavior. Subsequent bottom sampling indicated a significant benthal
activity in the area. This finding represented an important affirmation of
the value of the calibration exercise.
The comments included on Figures 6 through 9 indicate, in addition to
Table I, the sequence of trials taken prior to a reasonable calibration. Too
frequently in discussion of water quality models, it is only the equivalent of
Figure 9 and a brief discussion which is presented.
113
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TABLE I. DESCRIPTION OF:SEQUENTIAL CALIBRATION EXERCISES
Run
1
General
Testing for configura-
tion of data deck, e.g.
connectivity of node
points
Flow
regime
First trial at
flow contribu-
tions to create
hydrologic regime
Point
source
First trial at
loading contri-
butions
Background
contribution
First trial at
background con-
tributions
Reaction
rates
First trial
at decay
rates
•c-
Results—connectivity
is okay
Evaluating flow con-
tributions
Attention focused on
point source contri-
butions
Requires adjust-
ment
Refinement of
contributions
Minor adjust-
ments
Modification of
levels to1 re-
flect new data,
new BOD5/BODU
ratios
Refinement
of travel
times in
Grand Falls
headpond
Results—point source
contributions much too
high to agree with in-
stream measurements
Hydrologic re-
gime in agree-
ment with values
from sampling set
#1
Reduction in
levels needed—
checking of as-
sumed conditions
Attention focused on
point source contri-
butions
Decrease in
point source con-
tributions
Results—movement in
correct direction but
insufficient—back-
ground loadings too high
Checking of as-
sumed conditions
(Continued)
-------
Run
TABLE I. Continued
General
Flow'
regime
Foint
source
Background
contribution
Reaction
rates
Attention focused
on background
levels
Lowering of
values in ac-
cordance with
some litera-
ture values
Adjustment
of ki level
for Emund-
ston Eraser's
to reflect
revised es-
timates
Results—concentra-
tion levels still
too high in compu-
ted levels
Attention focused on
background levels
throughout river
basin
Results—large dis-
agreement still pre-
valent in Grand Falls
headpond
Attention focused on
Grand Falls headpond
Results—agreement bet-
ter between computed
and measured but still
significantly different
Contributions
too high
Lowering of
background con-
tributions in
particular areas
Adjustment of area
vs depth relation-
ships to test sen-
sitivity in Grand
Falls areas
Computed profiles
not very sensitive
to adjustments
Adjustment
of sedimen-
tation rates
in Grand
Falls head-
pond
(Continued)
-------
TABLE I. Continued
Run
General
Attention focused on
Grand Falls headpond
Results—as in (7)
Attention focused on
Grand Falls headpond
Flow
regime
Point
source
Background
contribution
Reaction
rates
Reduction in
reaeration
rate G.F.
headpont; up-
ward adjust-
ment of rate
Similar to
(8)
k - .001
Results—same as in
(7) and (8)—must be
factor causing diver-
gence in addition
10 Attention focused on
Grand Falls headpond
Results—General agree-
ment between computed &
measured profiles but
further adjustment of
benthic & decay rates
required
11 Attention focused on
Grand Falls headpond
Addition of es-
timated benthic
demand
Incremental adjust-
ments in travel
times between
points in headpond
Decrease in Ed-
munds ton load-
ing
(Continued)
-------
TABLE I. Continued
Run
General
Flow
regime
Point
source
Background
contribution
Reaction
rates
Results—improvement in
agreement but sedimenta-
tion and therefore BOD
rate of change is too
high in headpond
12 Attention focused on
Grand Falls headpond
Results—sedimentation
rate still too high,
magnitude of load must
still be too great
13 Attention focused on
Grand Falls headpond
Results—agreement very
good between experimen-
tal '& -computed profiles
Decrease in Ed-
mundston load-
ing
Decrease in Ed-
mundston load-
ing
Edmundston contri-
bution apparently
significantly less
than originally
thought
Adjustments
in lf-2 to re-
flect O'Con-
nor formula-
tion; adjust-
ment of k3
from .8 to
.6 in head-
pond
Adjustments
of k2 to .2
in headpond
-------
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Q.
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8-
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I
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h*»_
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Legend
I
s
\
"" — "-.
^~^.
::
T denotes variation in
J- variable measurement
T denotes mean of
I measurement
•^>.
Nj
T
Decrease in point source load and
sedimentation rate level
/
^
'
\
F 10 1 20 301 40 150 160 70
a — K o ?
2 o _ PC
3 I § £ -1 o
Q < "- Distance in miles |
5j (measured from Dickey ) -g
UJ
\
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T
i
-A-
80 1 90 100
§ -o
C C C irt
«> D 3 0 =
tn > UJ >- p
Grand Falls Heodpond ,
Figure 8. DO and BOD profiles, mainstream St. John River verification runs,
I10
c 8
S c
>. 6
o
•a 4
5
1 2
V)
O Q
20
16
12
I 8
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S4
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ease
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ource
Legend
T denotes variation in
1 variable measurement
I denotes mean of
^ measurement
•~-^
mag
1
nitut
e
and decrease in sedimentation level
— .
-.
/
,/
— -
/
— • —
' .
0 ' 10 I 20 301 40 150 '60 70
III * IE
Q J | ^ -gO
< ^ Distance in miles |
f (measured from Dickey) -6
UJ
"***^.
— -
T
L
\l
_ 80 1 90 100
s §| "
« >« t
Grand Falls Headpond
\=0
3u_
•i
Figure 9. DO and BOD profiles, mainstream St. John River verification runs,
119
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Concluding Comments
We believe that there are two types of application or implementation prob-
lems associated with the use of water quality planning models. The first prob-
lem is one of acceptance of a descriptive or prescriptive mathematical model
for use in an actual water quality planning study. The second is associated
with the implementation of a water quality plan that is based at least partial
ly on a study in which a mathematical model has been employed.
Currently there is little impediment to the use of descriptive models in
water quality planning in the United States. Model building as a water qual-
ity management analysis procedure has been gaining popularity. Advancement
in the modeling skill has improved the ability of a model to account for rele-
vant elements of the problem examined. As a consequence, the image of model-
ing has been enhanced. However, one of the most important stimulants to the
application of a model for analysis may be attributed to a shift in the feder-
al legal procedures in dealing with pollution control in the United States.
In a traditional procedure of constitutional governments, an individual
awaiting trial in a court is regarded as innocent until he (or she) is proven
guilty. The burden of proof that he (or she) is guilty lies with the plain-
tiff. A similar procedure has been applied in the litigation of pollution
control in the U.S. The court (or environmental regulatory agency) is re-
quired to assume the burden of proof of damage due to a waste discharge. If
the evidence is insufficient to prove damage, then waste producers (private
or public) are freed of the charge. Pollution control efforts following this
procedure have not experienced a great deal of success in improving environ-
mental quality. Since the burden of proof is on the regulatory agencies, in-
formation is valuable to them but not to waste dischargers. However, data
collection and analysis efforts assumed by the agencies are inadequate to cope
with the problem. Moreover, little cooperation can be expected from waste
dischargers, since lack of information acts in their favor. The universal
shortage of waste discharge information throughout the U.S. (and an unwilling-
ness to accept results obtained on the basis of a model) may be partially at-
tributable to the reluctance of waste producers to cooperate. The apparent
measure to remedy the situation is to reverse the burden of proof; namely,
a waste discharger would be required to provide evidence to show his innocence,
otherwise any discharge is regarded as harmful to the environment.
The National Environmental Policy Act requires that for any new construc-
tion larger than a certain size, an assessment must be carried out to evaluate
the environmental impact. The National Pollution Discharge Elimination System
(NPDES) specifies more precisely that a waste producer must obtain a permit
for waste discharge. Public Law 92-500 also states that any polluter must
have a minimum treatment prior to discharge and if the water quality standard
is still not met, the level of treatment must be increased to reduce the
amount of discharge. In these pieces of legislation, legal procedure appears
to shift the burden of proving non-damage to the waste producers. Because of
this change, data collection and analysis are essential for polluters. Many
large waste producers have built up personnel capable of using some sort of
analytical techniques to perform pertinent studies. There is evidence that
120
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these staffs are susceptible to a relatively new method—model building—in
their continuous efforts to ensure consistent analysis.
The application or use of prescriptive models has not been nearly as
widespread. We believe that this results from the inability of the models to
produce acceptable control plans to inform those involved in the decision-
making process. At fault are the criteria functions employed. The functions
are based on cost minimization or benefit maximization. The former criterion
is often inadequate because it does not encompass issues associated with
distribution or equity. The latter criterion, while suffering from some of
these same shortcomings, is of little use in guiding the selection of accept-
able plans because water quality benefits are usually not measurable.
On the other hand, there are cases, when the situation is complex and in
which it is useful to have the solution alternatives generated by prescriptive
models as the basis for comparison, where one finds such models implemented.
The Saint John is such a case. In this we sought an implementation of the
model but not an implementation of the plan based on the solutions of the
prescriptive model. The eventual plan implementation will be based on a
number of issues integrated into plan formulation process, only some of which
stem from results of the prescriptive model.
121
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REFERENCES
Camp, T.R. 1963. Water and its Impurities. Reinhold Publishing Co., New York.
Camp, Dresser, and McKee. 1969. Report on the development of a mathematical
model for minimizing construction costs in water pollution control. Report
to the United States General Accounting Officer.
Chi, Tze-Wen. 1972. Wastewater conveyance models, Models for Managing Re-
gional Water Quality. Dorfman, Robert, Henry D. Jacoby, and Harold A.
Thomas, Jr., eds., Harvard University Press, Cambridge, Mass.
Deininger, R.A. 1965. Water quality management: The planning of economically
optimal pollution control systems. Ph.D. Thesis, Northwestern University,
Chicago, Illinois.
Dobbins, W.E. 1964. BOD and oxygen relationships in streams. ASCE, JSED, Vol.
90, No. 3.
Graves, G., G.B. Hatfield, and A. Whinston. 1969. Water pollution control
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124
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DISCUSSION
UBERMAN: Describe the decision-making process developed from data received
from the automatic monitoring system on the Delaware River Basin?
THOMANN: The Delaware River has seven water quality monitoring stations,
from the upper river areas to the estuary. The data collected by these
stations have been used for at least the past 10 years in making decisions
about water quality on that river. Just one example: in 1965, we had a very
severe drought and there was a great fear that salt water would penetrate into
the estuary and affect the water supply. The mathematical model time variable
for salt water in the estuary was used, together with the data from the
monitoring stations, to demand releases from upstream reservoirs to control
the salt-water intrusion. Each day a two-week forecast of salt-water intru-
sion was made from monitoring data and the mathematical model.
YEREMENKO: It seems very valuable to me that Dr. DeLucia's report tries to
systematize American work on mathematical modeling. This will help us
understand the directions in which modeling is headed. In this report, models
are divided into descriptive and prescriptive. The descriptive models can be
based on two approaches: an enlarged parametric approach and an approach
with distributed parameters. The first is characterized by a separation of
sections and by an examination of all the processes within the section. This
is considered less accurate. However, I would like to hear something about
the quantitative comparison of these approaches. I familiarized myself with
the existing approaches. An enlarged first approach which was developed by
Dr. Thomann, and which is widely used in the U.S., is not evaluated
quantitatively in comparison with the second approach. If we use the second
approach, then we come up against a whole series of limitations. I am of the
opinion that the first approach does not have any limitations. The first
model, in the report, was used for the River Tiber, the second for the St.
John River. The first model, which is considered a less refined model, is
connected with linear dispersion. The second model does not consider linear
dispersion. Why is a little-known effect considered in the first less-refined
method, and not considered in the second?
KOLOSKOV: Today we heard two very interesting reports by Dr. Yeremenko and
Dr. DeLucia. I get the impression that the U.S. and USSR differ in their
approach to the development of mathematical models for decision-making. How
should further research be carried out? In my view, these directions supple-
ment one another, and we should develop them.
If well-studied rivers are modeled, we possess a sufficient volume of
information and it is necessary to use the models which Dr. Yeremenko has
talked about. At present, we have rivers for which it is necessary to develop
125
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water quality management mathematical models, and the volume of information
which we can use and put into the model is very limited. In this case, these
approaches which our American colleagues have developed can be used.
We should note that one or another of the models accepted obviously will
be determined by strategy and tactics for its use for optimizing water quality
management and decision-making and by the time interval which is accepted
when solving a given problem. If we have problems with accident situations,
and speed and accuracy are required to achieve a decision, then Yeremenko's
method allows us to obtain good results. On the other hand, if we create a
mathematical model to forecast for a long time period and to develop manage-
ment systems, then the American system is applicable.
We should note that in our country, methods similar to the American
methods are being developed, in particular, in the Moscow River where a
model was used to forecast water quality.
DAVIES: The subject of our symposium is "The Use of Mathematical Models to
Optimize Water Quality Management," and we are here to discuss the best
method for achieving goals. I think that everyone understands that many
approaches can be used for a determined set of problems and that often the
nature of the problem requires the application of a less complex model.
However, there are cases when very complex models are necessary. One of the
basic goals of models is to provide the decision-maker with the best avail-
able analysis of the problem and the ability to consider alternative strate-
gies. We must be convinced that the models are scientifically credible to
have any confidence in their use. We have spoken only very briefly about the
verification of a complete model. The easiest and probably the cheapest part
of modeling is writing the equations. The usefulness of these equations
depends on how closely they are able to replicate real processes. We have
heard many papers in which efforts to model river basins are described, but
we do not know how well verified these models are. I think that we should
turn particular attention to that question.
USACHEV: It seems to me that the problem of water quality management is first
of all a monitoring problem. The second problem is joining the scales of the
model with the results of the measurements. An actual river is a stratified
three-dimensional environment. If we describe the model by means of a one-
dimensional equation, we must find a way to average the results of the
measurements that will agree with our model. The third; problem is accuracy
and timing for measurements. I believe that more detailed mathematical
research is required on attaching models to specific water bodies.
DeLUCIA: I support Dr. Davies' position on the question of the necessity
for strict verification. The evident difference between verification and
calibration is absent. In my report, physical phenomena are examined. We
should pay attention to analyses of the physical, economic, or demographic
aspects of the problem, and likewise to how the decisions will influence
people who carry out water management. When discussing the St. John River
model, I was not implying that the optimized model can be used only when
there is a non-stationary hydraulic system. It can be used with different
hydraulic systems.
126
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CONSTRUCTION OF MODELS FOR FORECASTING AND CONTROLLING WATER QUALITY IN
A RIVER BASIN ON PRINCIPLES OF SELF-ORGANIZATION
A.G. Ivakhnenko
INTRODUCTION
The purpose of this research is to create mathematical models that ade-
quately describe the dynamics of river basins and are applicable both for
forecasting and for deliberately changing and controlling the basins.
Considering the complexity of controlled systems (i.e., river basins),
and the incompleteness of information obtainable both with existing monitoring
methods and from other factors, self-organization methods are considered the
most promising mathematical models for identifying and controlling such
systems.
Work in this field from 1971 to 1975 was conducted at the Institute of
Cybernetics of the Academy of Sciences, of the Ukraine. Construction of
forecasting models for river basins developed for management purposes, was
also obtained at the VNIIVO Institute of the USSR Ministry of Land Reclamation
and Water Use.
Synthesis of Models of the Principle of Self-Organization
The problem was called direct modeling using experimental data. Pre-
viously, similar problems were solved by regression, analysis with the method
of least squares (r.m.s.). The same type of the regression equation was
indicated (its power and number of terms); the result of modeling is subjec-
tive. With regression analysis, it is impossible in principle to establish
the structure of an equation (model) of optimum complexity, since the rule is:
"the more complex the model, the more accurate it is (up to the zero-order
error)." This serves as the basis for the theory of model multiplicity (Byr,
1963). In reality, overfitting a model is as bad as oversimplification.
The principle of model self-organization using a computer offers the only
model of optimum complexity. Therefore, it is sufficient to indicate, in
accordance with Godel's incompleteness theorem (Byr, 1963), a certain external
criterion. The self-organization principle is formulated as follows: given
gradual complication of the model, the number of criteria (called the selec-
tion criteria), with properties of external addition (Byr, 1963), passes
through the minimum, determining the model of optimum complexity for each
criterion. It is always possible to select such an order of model complexity
and sophistication (e.g., increasing the number of terms in the power of the
polynomial) at which the: minimum will be the only cine. The theory of self-
organization as a selection criterion mainly uses ijhe criterion of regularity,
127
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the value of r.m.s. error as measured at a separate sequence of experimental
points:
mm,
K=l
where K is the point number;
Nnp is the number of points in the test sequence;
n is the actual value of the function in a given point;
(n-n*) is the interpolation error.
Recently, several more expedient criteria have been proposed: the
criterion of non-displacement, the criterion of the balance variables, the
variation criterion, and others (Ivakhnenko, 1973). For the synthesis of a
model of optimum complexity and sophistication, a method of sequential testing
for models (scanning) with gradual increase in complexity, is used. The value
of the selection criterion is calculated for each model, and the model with
the smallest value is chosen.
Scanning of models offer a non-parametric method and do not require any data
on statistical or other properties of the object (steadiness, non-linearity,
non-Markovian, etc.). It is important to note the objective character of the
optimum complexity model synthesis. According to the principle of self-organ-
ization, the technician needs only to feed the experimental data into the
computer and indicate the selection criterion. Then the computer, without
human help, can synthetize the model and indicate its authenticity. The
depth of the selection criterion minimum (Ivakhnenko, et al., 1967) is a meas-
ure of this authenticity. Only when an adequate minimum value is obtained,
can it be certain that the problem of model synthesis is solved. If the
minimum is not small enough o 1 7 n
2°, 2Z1, 2/u, respectively. From this, one may infer that complete model
scanning (Table I or II) is possible only if the number of variables is no
greater than three or four. The number of variables in real problems reaches
one thousand. For problems of such large dimensions, various algorithms to
rationalize scanning are used instead of complete model scanning. The goal
is to find the optimum complexity model. These algorithms are called the
128
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TABLE I. EXAMPLE OF THE GRADUAL COMPLICATION OF-A POLYNOMIAL (FOR SINGLE VARIABLE)
_ IlLe__T_erms^_Qf__.theJ&ponen,tial Kplmogprpy-Gabpr Power Polynomial (.__-_
Y1=QQ+Q1t Y2=Q0+Qlt2 Y4=Q0+Qlt3 V^l^ Y16=Q0+Qlt3
Y3=Q0+Q1t+Q2t2 Y5=Q0+Qlt3+Q2t Y9=Q0+Qlt^+Q2!t Y17=Q0+Qlt5+92t
Y22=Q0+Q1t5+Q2t3+Q3t2
-------
TABLE II. EXAMPLE OF THE GRADUAL COMPLICATION .QFAPOLYNOMIAL (FOR TWO VARIABLES)
OJ
o
The Terms-jof the Exponeutial^J^imfil£t£.--KQla£LgQrQ.v-,Gabor.-PQHer .. Polynomial 2
Xf | _ X2 __ | _ XlX2 _ 1 _ X2
Y16=Q0+Q1X2
Yg=Q0Q1X1+Q2X1X2
Y15=Q0+Q1X1+Q2X1+Q3X2+ Y2
+QitX1X2 Y2l+=Q0+Q1X1X2+Q2X|
Y25=Q0+Q1X1+Q2X1X2+Q3X|
-------
group method of data handling (GMDH). The method of scanning rationalization
:is of no importance. It depends on the character of the problem and the
talent of the researcher. The number of newly proposed GMDH algorithms grows
continuously. By now,, about 'a hundred different versions of the GMDH algo-
rithms have been published (see "Automatika" journals). The purpose of all the
algorithms is to obtain a result by complete scanning of all models in se-
quence. Regarding the selection of a set of the model arguments, the GMDH
algorithms provide for two methods: algorithms with protection of variables
(all arguments introduced by the technician), and GMDH algorithms without pro-
tection of the variables. (An excess of possible arguments are given.) The
computer selects some arguments to ensure the minimum of the main selection
criterion. Examples of successful computer self-organization of optimum
complexity models for forecasting and management by the GMDH algorithm without
protection of the variables follow.
Mathematical Models for Forecasting River Flow
The self-organization principle of model scanning with gradual increasing
complexity, allows determination of river flow forecast models that contain
diverse sets of arguments and types of base functions.
Synthetized models can be presented as the sum of the so-called "trend"
and "remainder."
Q = fx(t) + f2(t,Xl5X2,...,Xm),
where f,(t) is a trend, the sum of the optimum number of harmonics with
non-multiple periods;
f2(t,Xj,X2,.Xm) is the remainder, the optimum complexity exponential polyno-
mial, which considers the influence of different environmental
factors on the river flow.
In the results of modeling the flows of the Dnieper and the Severski
Donets rivers, the value of the remainder is small. The inference is that the
river runoff is basically determined by the trend ordinates in the time
functions. The GMDH algorithm intended for separation of the harmonics trend
at non-multiple frequencies is described.
Insufficient attention has been given in applied mathematics to the
harmonics analysis of fluctuating processes with non-multiple harmonic fre-
quencies. The Fourier series expansion pertains to the case of multiple fre-
quencies (Ivakhnenko, 1973). A special criterion, called harmonics ordinates
balance criterion, is used as the selection criterion. It consists of a set
of linear equations that express a specific determined relation among the
neighboring process ordinates. Figure 1 gives an example of frequency deter-
mination, when separating one or two harmonics. If frequencies are known, it
is easy to determine their amplitudes by the r.m.s. method, and then determine
the imbalance of the regression equation (Ivakhnenko et al., 1975). In ac-
cordance with the GMDH algorithm, the sum of imbalance in points of the veri-
fication test succession becomes more accurate as the number of separated
harmonics in the trend separation program increases.
131
-------
a.
Figure 1. Determination of the W and W^, W2 harmonics frequencies by the har-
monics ordinate balance criteria.
a. Harmonics W
0 TT f(i+D + f(i-l)
2cosW = FTTT .
W=?
b. Harmonics Wi and W2
al +
,
= cos2Wz;
w2=?
132
-------
The simplest algorithm is one with sequential separation of one result in
each set (F=l). However, the version of the algorithm in which several (F>1)
harmonics are separated for each set is often more accurate. In defining the
first set of all given points, possible trends are selected with one, two,
three, etc., harmonics. The maximum number of harmonics is Mmax <_ N/3. From
these, no single one is selected, but rather a F number of trends of each type
that most completely satisfy the ordinate balance criterion requirements.
Afterwards, F remainders are calculated. (The remainder is the difference
between the process ordinates and each trend of the first set.) In selecting
the second set of remaining given points, the Mmax of trends is separated with
a number of frequencies from 1 to Mmax, respectively. From the whole set F-
Mmax of trends obtained for the second set, F of the best trends of this set
is selected by the same criterion. The trends are separated for all points,
but selection of the best F trends are carried out by the points of a separate
verification test succession. The described procedure is repeated in the fol-
lowing sets. Complexity of the model, depending on the number of sets to be
selected, increases until the "imbalance" value of the ordinate balance cri-
terion, measured by means of the test sequence of points, decreases. In the
last set, a single solution is selected corresponding to the minimum deviation
from the ordinate balance criterion. Thus, selection has optimized both the
number of harmonics and the number of separated remainders. The method of
problem solving is rational sequential scanning by the ordinate balance cri-
terion with a gradual complication of the model (the number of harmonics to
be summed up).
Long-Term Forecast of the Average Annual Flow of_ the^Dnieper River
The harmonic components frequencies may be selected by using the new
GMDH algorithm described above, and also the sequential scanning of a discrete
set of frequencies by the GMDH algorithm with the regularity criterion
(Ivakhnenko et al., 1973). The frequencies are analytically determined by the
new GMDH algorithm with the ordinate balance criterion. This criterion re-
quires that a determined ratio between the advanced and delayed ordinates of
the process be expressed through the sum of the harmonics with non-multiple
frequencies (Figure 1)- The forecast calculation time on the BESM-6 computer
is reduced from three hours to five minutes; the accuracy of forecasting for
the Dnieper River flow is generally not impaired.
The initial data (Ivakhnenko et al., 1973) were used to forecast average
annual flow of the Dnieper at the extrapolation interval of Ty=10 years (from
1971 to 1980 inclusive) by the ordinate balance criterion. Data on Dnieper
flow for 100 years (from 1871 to 1970 inclusive) were used for creation of the
training (95 points) and verifying test successions. In the second verifying
sequence, a part of the forecasting interval, data for 1971, 1972, 1973, were
used.
The results of the forecast and the best forecast obtained by Ivakhnenko
et al. (1973) are given in Table III. The error in the examination sequence
for forecasting using the proposed GMDH algorithm with F=l:6(2)=20.2%, with
F=5:6(2)=11.0%. The error in the sequential scanning algorithm of the dis-
crete values of frequencies, proposed by Ivakhnenko et al. (1973), is equal to
6(2)=17.2%. The new algorithm is actually almost as accurate as the previous
133
-------
one, but calculation time is much shorter. In Table IV, the parameters of
the harmonic trend of the optimum complexity, which produced the smallest
value of A(2) in the examination sequence, are included.
TABLE III. FORECASTS OF THE DNIEPER RIVER FLOW AT KIEV BASED ON DATA FOR
1871 to 1970 ; ^^^——^
j
!
[Year
Forecast of Harmonic Trend
Regularity
criterion
Criterion of the
balance of variables
freedom of selection
F=l F=5
Actual
flow
Spread of
forecasts
by GMDH
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1635
1061
909
925
1218
1328
1145
1197
1413
1500
1560
1266
795
1018
1326
1284
1219
1480
1787
1641
1600
1095
1128
1156
1344
1480
1132
1000
1586
1435
1830
1030
1060
1300
—
—
—
—
—
—
1460-1900
1050-1400
800-1050
8003-1300
1050-1350
1150-1550
1000-1300
1000-1300
1300-1600
1300-1700
Forecasting the Average Monthly Values of River Flows —
Studies of the river flow data showed that the flow of water can be
determined for each month with sufficient accuracy by regression equations:
Q± = fi(Qyear,
1=1,2,3, ... ,12
where Qi is the average monthly water flow for the given month;
Qyear is the value (or forecast) of the annual flow; and
Qi-1 is the mean flow for the previous month.
Properties of the river flows are such that the regression equation of
Qi = fj. (Qyear), gives a certain accuracy only for April during the annual
flooding period. For other months, sufficient accuracy of regression is not
obtained unless the annual rainfall is indicated (low, medium, high water) or
if the regression contains both Qyear and Qi-l. This is clearly shown in
Figures 2a, 2b, and 2c .
The regression equation using only Qj_=f (Qyear) for the Dnieper River
shows too large a dispersion. The scatter of points, given as an example in
Figure 2a, cannot be expressed with sufficient accuracy by a line. Only
April provides an exception; here the points have comparatively small dis-
persion around the line of regression (Figure 2b) .
Introduction of one more axis, Q±_I, is sufficient to show that all the
points are sufficiently close to a certain surface (Figure 2c) , corresponding
to the equation with two indices: Q^ = f (Qyear, Q-_i).
134
-------
Q year
Q year
a.
Qi '
Qi-1
• .•
/•'•'•• .' V'-'. -v
'
c.
Figure 2. Reduction in the dispersion of points at the transition from the
regression equation Q^=f(year) to the equation Qj_=f (Qyear Qi_]_) :
a. Scatter of points for all months, except April;
b. Regression curve for April;
c. Regression surface for all months including April.
Figure 2a may be considered a projection of the points of Figure 2c on to
the Q-L_]_-Qyear plane. These figures define the accuracy of the equations:
the regression surface in Figure 2c is at a small angle to the axis of Qyear,
so that even a considerable error in the forecast of average annual flow
shows little influence on the results of the average monthly flow forecast.
A close approximation to the average annual flow may be obtained. For example,
it is sufficient to point out the type of rainfall projected for the year,
high, average, or low, to considerably reduce the dispersion of the points
from the regression surface.
135
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TABLE IV. OPTIMAL HARMONIC_TREMD JTRUCTURE_
Set #
1
Quality of j
harmonic Frequency i Periods
component in Radians j (years)
2 3 | 4
Coefficient
A
5
B
6
1 4 0.434
1.084
1.886
2.716
2 5 0.351
0.877
0.480
2.124
2.752
3 8 0.219
0.621
0.907
1.293
1.747
2.149
2.596
2.922
4 7 0.263
0.574
1.147
1.507
2.011
2.477
2.880
14.48
5.80
3.33
2.31
17.90
7.16
4.25
2.96
2.28
28.69
10.12
6.93
4.86
3.60
2.92
2.43
2.15
23.45
9.32
5.48
4.17
3.13
2.54
2.18
26.9
34.3
-72.6
6.6
5.8
0.8
-48.2
6.3
-93.7
95.7
-72.9
35.9
-90.1
-100.1
26.9
50.3
-7.8
-51.4
-132.4
92.8
53.3
-3.0
-1.7
-13.2
2.8
2.7
-11.7
48.5
-79.3
45.4
-46.9
46.1
6.9
-10.4
-8.4
5.5
45.6
3.0
1.2
-69.5
-19.0
-9.6
45.1
-40.4
31.7
22.9
-22.6
46.0
It is possible to select the f± function of optimum complexity by complete
scanning of all possible regression equations since the number of arguments is
small (two). Complete scanning is also used in the so-called combination GMDH
algorithm (Ivakhnenko et al., 1973). One equation of the optimum complexity
is selected for each month.
Forecasting Average Monthly Flow in the Dnieper River—
Forecasts of the average annual flow in the Dnieper have been taken from
the work of Ivakhnenko et al. (1973). Average monthly flow values were based
on data from 1878 through 1967 inclusive. Ninety experimental points were
divided into the training and test succession. The points were arranged by
the value of dispersion. The points with lesser dispersion (40 points) were
included into the test succession, and the points with larger dispersion (50
points} in the training sequence (the third method of regularization). The
values of the regression equations coefficients were determined initially
only in the training sequence. The complexity of the equations increased
136
-------
until the error measures in the test sequence (regularity criterion) were
reduced. After determining the equations of optimum structure (separately
for each month), their coefficients were recalculated for all the points
("coefficient adaptation stage").
By scanning all the possible polynomial models with two arguments up to
the third power inclusively, we were able, by the regularity criterion, to
select twelve polynomials of optimum complexity of the following type:
Qi = Qo,i + alj:LQyear + &2,±Q±-1. + a35iQ2year + a^Qyear- Q±_i + a5>±q2i_1,
1=1,2,3,...,12.
The values of coefficients obtained are given in Table V. Results for fore-
casting are given in Table VI and in Figure 3. The error of the forecast in
the examination sequence (Table VII) is equal to A(2)=25%. This points to the
considerable accuracy. The error does not increase with the increase of pre-
diction time; the 1977 forecast is as accurate as that for 1974. A separate
study (carried out by Stepashko) showed the stability of the forecast from
calculation errors; the 1974 error occurs only in the results closest to 1975
and has almost no influence upon the accuracy of further forecasts.
TABLE V. VALUE OF COEFFICIENTS FOR AN EQUATION FORECASTING AVERAGE MONTHLY
FLOW IN THE DNTF.PER RIVER
i i
i
1
2
3
4
5
6
7
8
9
10
11
12
1 ao,i
43.03
633.10
-16.80
-1236.00
-799.80
692.90
275.80
21.49
492.90
525.60
137.20
935.50
j al,±
0.186
-0.298
0
4.253
3.945
0
0
0
-0.599
-0.260
0
-0.570
| a2,i a3,
0.966 0
0 0
1.312 0
i au,i
0.000288
0.000526
0.000284
0 -0.000348 0
-0.212 0
0.0668 0
0
0.000103
0.308 0.0000897 0
0.943 0
0
0.717 0.000227 0
0 0
1.017 0
0 0
0.00054
0
0.000719
a5,i
0
0
0
0
0
0
-0.
-0.
0
0
0
-0.
00000262
000112
000339
Mathematical Models for Forecasting_the Quality_of^ R.iyer Water
It is not necessary to discuss the importance of forecasting reliable
river water quality indices. Predictions of the content of chlorides and
dissolved oxygen in the water of the Severski Donets River were compared by
methods based on regressive analysis, on Lagrange interpolation polynomials,
and group method of data handling (GMDH).
The self-organization principle is most fully utilized by the GMDH. One
algorithm was described by Chukin (1973) . It is designjed to forecast non-
stationary cycle processes. The initial information is1 provided by the time
series, from which the following matrix is formed:
Yi5j(i=l,l; 1=1,n),
137
-------
where 1 is the length of a separated cycle; and
n is the number of cycles in the training sequence.
The following polynomial is accepted as the approximating function:
The algorithm allows the power of the polynomial to be changed by accepting
the assignment for N.
N=10-complete cubic polynomial;
N=8-polynomial without "particular" cubes;
N=6-full square polynomial;
N=3-linear polynomial.
The polynomial coefficient are detenuin&l by the r.m.s. method. The selec-
tion of coefficients is determined by the GMDH.
7000 - -
1000 --
1971
1972
1973
1974
1975
1976
1977 t years
Figure 3. Average monthly Dnieper River flow forecast graph.
1. is the actual flow; 2. is the forecast.
An ALGOL-60 language program was compiled for this calculation and the
values of 1=6, N=3 were selected. The experimental data (analyzed every 4
hours) served as the basis for selecting the length of the cycle to be sepa-
rated. Accuracy was maintained by the use of linear regression, and selection
(by the computer) of an optimum Lagrangian polynomial of the first power that
served as the basis for selecting the linear polynomial.
138
-------
TABLE VI. FORECAST OF DNIEPER FLOW, M3/Second
u>
I
JYear
1971
1972
1973
1974
1975
1976
1977
r .... .__
Average
Annual
Flow
1814
1398
930
909
1180
1217
1283
I i II
1129 1168
806 809
726 711
713 703
708 721
706 722
705 727
III
2119
1368
1104
1087
1171
1180
1202
Average Monthly Flow
IV | V i VI | VII
5332 5229 2016 1182
4035 3870 1507 911
2418 2358 1075 682
2342 2291 1060 674
3227 3158 1286 793
3424 3278 1321 812
3647 3491 1386 846
VIII
979
787
612 .
606
698
713
739
IX | X i XI
856 892 1045
664 663 812
571 571 718
570 569 716
603 603 751
612 611 759
628 627 775
XII
693
730
710
711
708
710
715
-------
TABLE VII. RESULTS OF THE FORECAST OF AVERAGE MONTHLY FLOW IN THE DNIEPER IN THE EXAMINATION
SERIES (a); GIVEN ACTUAL VALUES OF ANNUAL FLOW (b); AND VERIFICATION OF FORECAST
STABILITY (c), Md/Second.
Year
1
Average
annual
flow
2
Average monthly flow
I I II j III i IV T V
.... 3__ |__ 4--j— y -)-- g -| - ?- -
VI
8
VII
9
VIII
10
IX ! X 1 XI
11 1 12 1 13
XII
14
r .m. s . f
for the !
Year !
15 !
.p-
o
1968a 1200 850 616 830
b 1200 705 720 1174
c 2000 6.91 764 141-9
407 2870 1070 628 610
3366 3223 1305 803 706
587-6 5848 2284 1325 1074
561 814 797 706
608 607 755 709 0.173
974 1057 1213 1039 0.705
1969a
b
c
1970a
b
c
1971a
b
c
1972a
b
c
1973a
b
c
1340
1340
3000
2490
2490
4000
1830
1830
1830
1030
1030
1030
1060
1060
1060
772
704
707
1290
686
295
1380
1123
1687
1200
840
840
600
708
708
674
729
855
1290
789
610
2230
1168
1711
806
781
781
522
712
712
666
1219
1835
1360
1578
132
1950
2124
3119
772
1237
1237
1031
1131
1131
2910
3837
8387
10000
7193
10200
4900
5380
5380
2310
2775
2775
2624
2880
2880
3760
3675
9263
5970
7503
12820
3150
5282
5282
1950
2677
2677
1979
2773
2773
1660
1444
4164
2310
3111
6814
1530
2038
2038
900
1155
1155
1159
1180
1180
1320
877
2322
1160
1766
8691
1340
1194
1194
798
724
724
852
731
737
882
762
1606
908
1337
1973
951
987
987
684
645
645
895
655
655
767
645
1893
784
1369
3149
801
866
866
556
579
579
668
583
583
694
643
2812
943
1719
6286
931
905
005
568
580
580
667
584
584
828
792
2997
1670
1885
6532
1290
1058
1058
647
111
727
968
731
731
1220
721
2641
1630
1684
2969
1470
903
903
1110
707
707
859
707
707
—
0.237
1.590
—
0.317
0.909
—
0.348
0.361
—
0.288
0.288
—
0.226
0.226
-------
Comparative analysis of the results (Table VIII) indicates that the
methods of the regression analysis, the modified method of the Lagrangian in-
terpolation polynomials, and the GMDH can be used for water quality fore-
casting. The GMDH method gives the most accurate results and should be con-
sidered as the most practicable method for water quality forecasting.
TABLE VIII. FORECAST OF THE CONCENTRATION OF CHLORIDES AND DISSOLVED OXYGEN
1
Experimental
i data
ici~
482
489
482
468
468
440
Average
°2
7.20
7.90
8.00
7.30
6.90
6.80
quad-
Method
Linear
regression
ci-
453
453
452
452
451
450
°2
7.40
7.40
7.40
7.30
7.30
7.30
Quadratic
regression
Cl
459
458
457
456
455
454
°2
7.40
7.40
7.30
7.30
7.30
7.30
Lagrangian
polynomials
Cl~
454
454
433
425
418
411
°2
6.50
6.20
5.60
5.70
6.50
7.10
Lagrangian
polynomials"
ci-
469
462
474
418
420
411
°2
6.80
7.50
7.80
6.30
7.60
7.10
GMDH
ci-
474
473
471
470
480
465
°2
7.40
8.00
7.80
7.00
7.10
6.80
ratic error
of forecast
Relative
error, %
25
5.3
0.42
5.60
24
5.1
0.45
6.00
38
8.1
1.40
18.60
35
7.4
0.49
6.70
14
3.0
0.19
2.50
In "Possibilities of Applying Perceptions to the Problems of Evaluating
Conditions of Water Objects" (VNIIVO, Kharkov), Belogurov and Uberman proposed
that the Kolmogorov formula be used in a continuous network of non-linear
transformers. The Kolmogorov formula presents the continuous function of
several variables by super-position of the continuous functions of one •varia-
ble, and the equation:
2n+l n
Q = f(aia2...an) = |=1 q^^ XijCX^]
This provides a mechanism for accepting individual water quality measurements
from a monitor. Output from 30-35 water quality sensors in a monitor can be
fed into this non-linear transformer to determine Q. Belogurov and Uberman
associated this idea with the theory of perceptrons and called the proposed
non-linear functional transformer an analog continuous perceptron (ACP).
Since accuracy of the ACP has not been studied, and no experimental verifica-
tions have been made, the concept is still preliminary.
Methods of Managing the Discharge of Power _PlLant_R^s_ervpj.r_is
A plan to regulate power plant discharge must consider several objectives:
1. Power to provide the planned electrical energy;
2. Water transport to maintain the navigation channel depth;
3. Irrigation to maintain water for irrigation; and
4. Fish farming to maintain conditions for aqua-culture.
141
-------
This task is a multiobjective management project. For brevity, an
example is cited: the Dnieper reservoir cascade managed only by two criteria:
a) power and b) fish farming. The problem is solved with the equations for
water body dynamics and the equations of the loss functions. Application of
the principle of self-organization is most effective for the synthesis of
these equations on the basis of a small set of observations, especially when
synthesizing the loss functions.
Synthesis of Equations for the Discharge in Dnieper Reservoirs—
The dynamics of the Dnieper discharge of water reservoirs may be de-
scribed by the following system of difference equations. (In this system, the
water flow lag time is not taken'into consideration, since the time intervals
are considerably larger than water flow time in the reservoirs. Therefore,
the Kanev reservoir dynamics are not considered.)
,
<3>
.(5) (f)
't-1' Vt ' Vt
T.t '
(5).
7Zt •
where Z1, i=l,5 is the water level at the dams at the end of the t-time inter-
val respectively for the Kiev, Kremenchug, Dnieprodzerzhinsk,
Dnieper, and Kakhovka water reservoirs (state variables);
V^, i=l,5 is the volume of water passed through the hydraulic units of
the Dnieper power stations at the t-time interval (variables
of control);
V^ i=l,5 is the total volume of water flowing into the Dnieper water
reservoirs at the t-time interval (variables of perturbations).
Water balance equation:
+ 1
V£t - Vlt +
2t
3t
+
+ v|t + v
6t
V7t
V£t>
1=1,5,
where ^ , ±=1,5 is the basic inflow into the i-th reservoir during the t-
time interval;
V^t, ±=1,5 is the base flow into the i-th reservoir during the t-time
interval;
v|t, ±=1,5 is the volume of effluents discharged into the i-th reser-
voir during the t-time interval;
v^t» i=l> 5 is the volume of atmospheric precipitation into the i-th
reservoir during the t-time interval;
V^t, ±=1,5 is the volume of water evaporated from the i-th reservoir
during the t-time interval;
V^t, 1=1,5 is the volume of water for municipal needs taken from the
i-th reservoir during the t-time interval;
Vib. , i=l,. 5 is the volume of water discharge in the i-th reservoir
142
-------
during the t-time interval;
g 1=1,5 is the percolation losses in the-i-th reservoir during the
t-time interval. ">
Water volumes V^ , ...,Vg , i=l> 5 are determined. £rom an additional fore-
cast for the given number of control intervals t=l,N.
The problem of identifying the equation lies in determining the f func-
tion. These functions can be found either by analyzing the cause and effect
links (determinate method) or by the GMDH algorithms (self-organization method)
for a given series of observation points. In the- latter, the complexity .of
f functions generally increases, lowering the regularity criterion.
Identification of the Equations of Dynamics of the Dnieper Reservoirs —
Equations of the Dnieper reservoirs dynamics were identified on the BESM-
6 computer on the basis of statistical data (Table IX) from the hydrologic
annual report, according to the program given by Ivakhnenko (1971), and are
as follows:
A. Kiev reservoir
z£ = 2.71 • (0.76 • YI + Y2 - 3.1 • YI • Y2) + 2.71;
= 0.09 - 0.24 • (v£ - 2.54)/2.54 + 0.42 • ~(vit - 2. 68.) /2. 68 - 0.12 •
• - 2.54)/2.54] • (vt - 2.68)/2.68;
Y2 = 0.03 + 0.96 • (ZjL-L - 2.7)/2.7 + 0.15 • (v£ - 2.54)/2.54 + 0.19 •
• [(Z^_! - 2.7)72.7] • (v£ - 2.54)/2.54
B. Kremenchug reservoir
Z^ = 4.22(1.34 ' YI - 0.35-Y2 - 0.02 • YI • Y2 ) + 4.22;
YI = -0.0173 + 0.7544 • X2 + 0.2902 • X3 + 0.1168 • X2 • X3 ;
Y2 = -0.0231 - 0.0302 • Xj + 1.0682 • X2 + 0.2486 • X], • X2 ;
Xi = -0.37 - 1.05(v2 - 3.46)/3.46 - 0.08(v|t - 0.98)/0.98 + 1.37 •
• [(v£ - 3.46)73.46] • (V|t - 0.98)70.98;
X2 = 1.15(Z2_1 - 4.22)74.22 + 0.35 • (v£ - 2.5)/2.5 - 0.12 • (Z^_x - 4.22)7
4.22 - (v£ - 2.5)72.5;
X3 = -0.09 + 1.32 • (Z^_! - 4.22)74.22 + 0.26 • [(V? - 0.98/0.98]- 0.3 •
• [(Z^ - 4.22)74.22] • (v|t - 0.98)/0.98
C. Dnieprodzerzhinsk reservoir ••
l = 3.77 • (0.02 + 0.74 • YI - 1.29 • Y2 + 20.22 • YI • Y2 ) + 3.77;
YI = 0.06(y2 - 3.45)73.45 - 0.02(v|t - 0.17)/0.17 + 0.06 - [ (v| - 3.45)73.45]
- 0.17)70.17;
143
-------
Y2 = 0.01 + 0.25 • (ZJL-L - 3.78)/3.78 + 0.04 • (vjl - 3.57)/3.57 - 0.8 •
• [(Z|_! - 3.78)73.78] • (v| - 3.57)/3.57
D. Dnieper reservoir
z£ = 7.42 • (-0.003 + Y! +0.53 • Y2 +25.66 • Yj • Y2) +7.42;
YI = 0.01 • (v£ - 3.7)/3.7 - 0.002 • (v£ - 0.03)/0.03 + 0.001 • [(v£ - 3.7)/
3.7] • (v| - 0.03)/0.03;
Y2 = -0.007 + 0.387 • (Z^ - 7.41)77.41 - 0.01 • (VJ? - 3.59)73.59 - 1.84 •
• t(zt-l - 7.41)77.41] • (VJj - 3.59)73.59
E. Kakhovka reservoir
Z^ = 3.63 ' (0.017 + 0.39 ' Y! + 0.55 • Y2 - 2.17 • Yx • Y2) + 3.63;
Y! =-0.3 • (v£ - 3.23)73.23 + 0.01 • (v£ + 0.42)/(-0.42) + 0.31 • [(v| - 3.23)
73.23] • (v£ + 0.42)/(-0.42);
Y2 = -0.05 + 1.62 • (ZjL! - 3.62)73.62 + 0.16 • (v£ - 3.71)73.71-2.22 • [(^_1
- 3.62)73.62] • (vjt - 3.71)73.71
TABLE IX. INITIAL DATA FOR IDENTIFYING THE DYNAMICS EQUATIONS FOR THE DNIEPER
RESERVOIR*
ariable
Month
II III IV V VI VII VIII IX X XI XI
I
,m 3.17 2.76 2,26 3.47 4.00 3.90 3.70 3.76 3.84 4.09 3.88 3.95
,m 3.53 3.17 2.76 2.26 3.47 4.00 3.90 3.70 3.76 3.84 4.09 3.88
,km3 3.44 3.97 4.67 5.87 5.90 2.99 1.91 1.83 2.34 3.09 3.16 3.58
,km3 4.08 4.57 5.11 2.88 4.09 2.18 1.51 1.29 1.72 2.32 3.39 3.48
,km3 3.21 4.49 2.80 2.52 1.82 1.60 3.90 5.33 -0.11 -0.13 -0.36 -0.29
*For the sake of brevity, only data on the Kakhovka reservoir for 1968 are
presented. Analogous data were used for all other reservoirs. Levels are
presented with respect to the elevation of 12 m.b.c.
Synthesis of Loss Functions—
Reservoir dynamics equations may also be obtained by determinate reason-
ing. To obtain the loss functions, however, self-organization appears to be
the only method possible. The synthesis of two loss functions,(for power
generation and for fish farming), is given here as an example of a two-crite-
ria problem of power station discharge management.
144
-------
The power generation by the Dnieper discharge is considered the first
control criterion. Electric power generation depends on water levels and
volumes of discharged water as follows:
where E?: is the consumption of electric energy of the i-th hydroelectric
power station during the t-time interval; and
N is the number of [control intervals.
i
The forecast of fish catch in the T+l year at the Kremenchug and Kakhovka
water reservoirs is based on the mathematical dependence of fish catch for the
Kremenchug and Kakhovka reservoirs on the water level regimes of these re-
servoirs (Ovchinnikov, 1972; Pirozhnikov, 1969):
T+1
ZVIIIT'ZVIIIT-1'
VIIIT-2'
where
YT-I>
J ,
ZVIIIT'
3=2 ,.5 is the forecast fish catch in tons in the year T+l in the
j-th reservoir;
J=2'5 is the fish catch in the T-l year in the j-th reservoir;
j=2,5 is the level of j-th reservoir decrease in March of the
current year T, characterizing the winter decrease in the
reservoir;
ZVIIIT-1' ZVIHT-2' J=2'5 ls the level of 1~th reservoir decrease in
August of the present and previous year, two years, char-
acterizing the summer decrease in the reservoir.
The self-organization GMDH algorithms allow us to determine the function
type and evaluate the coefficients in the criteria equations.
Synthesis of the Energy Loss Function for the Dnieper Hydropower Stations —
Synthesis of the control criteria for each separate Dnieper hydroelectric
power station included in the total power criterion was carried out on the
BESM-6 computer on the basis of the statistical data presented by the United
Monitoring Office of the South, according to Todua et al. ;(1973) . The fol-
lowing criterion dependencies were obtained. The variables E^ ( i=l ,5) are
measured in relative units of electric energy generation, ZJjL (i=l ,5) in meters,
V- (1=1,5) in kilometers.)
= (60.7
18
28.
2 =
(30 - 2
(129.3 - 38Z3. - 15V3 - 1.3(v£3>)2+
[-44.6
[66.5 +
+ 12.2Z-!vj!)
Synthesis of Fish Farming Loss Functions for the Dnieper Water Reservoirs —
Synthesis of the forecasting criterion dependencies of fish catch for the
Kremenchug and Kakhovka water reservoirs were carried out on the BESM-6
145
-------
computer on the basis of the statistical data (Table X) presented by the
Piscicultural Scientific Research Institute of the Ministry of Fish Husbandry,
according to Todua et al. (1973).
= 22402 -
- 1.5Y
T-l'
= 15816 -
Although only a small number of observations are presented in Table X,
the synthesis could be performed by using the GMDH loss function.
TABLE X. INITIAL DATA FOR THE SYNTHESIS OF LOSS FUNCTIONS IN FISH FARMING
i Variables
I
v(5),
T
z(5)
z<5)
^VIII
z(5)
z(5)
VIII
ton
T,m
,T'm
T_l,m
,T-2'm
Year
1967
8440
3.83
3.62
2.68
3.42
1968
1969
Kakhovka
9395 8814
3.76
3.63
3.62
2.68
2.76
3.72
3.63
3.62
Kremenchug
Y(2)
T '
z(2)
(2) '
7
^VIII
z(2)
AVIII
z(2)
VIII
ton
T'™
T,m
» A
,T-l'm
m 1 Jm
,1-2'
7000
5.79
4.98
6.12
4.84
6454
1.47
5.38
4.98
6.12
Synthesis of the Multicriteria
The principle of
tion
and fish
farming
vations, which may be
ZEE
i t
t t—
1 Zt Vt>
7619
0.87
5.54
5.38
4.98
Op t imum
self -organization
1970
Reservoir
7501
3
3
3
3
.41
.76
.72
.63
1971
7367
3
3
3
3
.46
.95
.76
.72
1972
7002
3.
3.
3.
3.
02
84
95
76
1973
9332
3.62
4.40
3.84
3.95
Reservoir
7591
0
5
5
5
.87
.82
.54
.38
Management
allowed us
loss functions on the
wr i 1 1 en
, 1=1,5,
basis
8671
0
5
5
5
of
to
of
.96
.84
.82
.54
Water
find
7778
3.
5.
5.
5.
06
64
84
82
Reservoir
the
power
a small series
in the following generalized
t=l,12,
9399
2.31
—
5.64
5.84
Discharge
genera-
of obser-
form:
146
-------
In each point of sixty-dimensional space (five parameters and twenty
time intervals), it is possible to calculate electric energy generation and
fish catch. In the indicated hyperspace, there are points where, within the
limits, electrical power generation is maximum (point Q^max, Figure 4), as
well as .a point which corresponds to a maximum fish catch (point Qvmax,
Figure 4). In accordance with the existing theory of multicriteria manage-
ment, this problem of two-criteria management should be solved on the Q^max -
Qymax line of tangency for isoclines of two extreme hills, i.e., the so-called
Pareto line (space).
The idea of vector optimization consists of an option by the expert of
any point on this line (from the maximum energy to the maximum fish catch).
N = 60
Boundary of the optimization region
Figure 4. Conditional description of 60-dimensional hyperspace for two-
criteria optimization:
Qgmax is the point of maximum generation of electric power;
Qymax is the point of maximum fish catch;
Qj?max - Q max - Pareto line, on which the compromise optimum
solutions should be searched;
1,2,3,4,5, and 6 - points of vector optimization (see Table IV);
Synthesis of Two-Criteria Optimum Control of the Dnieper Water Reservoirs
Discharge—
The dynamics equations of the Dnieper water reservoirs, the energy loss
and the fish farming loss functions, were synthesized. The values of the
total inflows are indicated in Table XI. At certain limitations in the
optimization space (Ivakhnenko and Ovchinnikov, 1975), six of its points on
the Pareto line were obtained:
I " ~ • 1 1
i 2 T
3 | 4
5 6 |
, Relative
Units Electro
Energy Generated
Yrp., -
T+l,m
0.77
15431
0.76
15792
0.75
16354
0.73
17812
0.72
18340
0.70
18942
147
-------
The above mentioned Pareto points were determined by the Ovchinnikov (1974)
method. The vector optimization consists of selecting any one of these
points.
Synthesis of Discharge Management with Forecast Optimization—
The above example illustrates how certain optimum management for a cur-
rent year is obtained. It will be more reasonable to direct optimum manage-
ment toward obtaining the long-term optimum, rather than the short-term
optimum situation. The goal of optimization is continuously displaced forward
for the given preceding interval. The difference between the synthesis pro-
blem of managing the discharge from hydroelectrical power reservoirs by two
criteria given a projected management period, as opposed to a solution given
a fixed period, is that the table of effective solutions (in Pareto sense)
must be recalculated at each new management cycle. The management period,
i.e., the number of preceding cycles, remains constant, but the object, the
first value of an effective problem solution management sequence, is realized.
The investigations have shown an extraordinary stability of management
with forecast optimization: at a sufficiently long projecting interval,
management becomes stable even when given an unstable object [the regime of
"artificial stability" (Ivakhnenko, 1971)].
CONCLUSION
Complete scanning of models with their gradual complication, or ration-
alized scanning according to the GMDH algorithm based on the principle of
self-organization, allow us to objectively solve important problems of fore-
casting, identifying, and managing water ecosystems.
Examples have been given of the methods application, including forecast-
ing of river flow (with accuracy around + 20%). This figure indicates the
relationship of determinative (80%) and stochastic (20%) components of river
flow. Synthesis of the dynamics equations of the hydroelectrical power
station discharge water reservoirs and the loss functions (loss for energy
generation and fish farming according to short tables of initial observation
data) have been shown.
The equations obtained (mathematical models) were used for synthesis of
two-criteria management of the hydroelectric power station discharge in ac-
cordance with the ideas of multicriteria optimization.
148
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TABLE XI. VALUES OF TOTAL INFLOWS (PERTURBATION VARIABLES)
•Total j
I inflow I Month
! km3 ! I | II I III j IV | V j VI I VII | VIII | IX I X | XI | XlT
V(l) 1.96 1.41 1.05 1.08 1.88 1.75 1.40 1.17 0.81 0.95 8.15 6.80
Zj I—
(2)
V ' 0.48 0.41 0.25 0.85 0.58 0.70 0.55 0.43 0.50 0.82 2.32 3.44
£t
V*-3-1 0.05 0.02 0.01 0.01 0.20 0.17 0.12 0.21 0.07 0.12 0.70 0.28
-0.06 -0.06 -0.07 -0.07 -0.01 -0.01 0.01 0.02 -0.03 0.01 0.01 0.01
-0.80 -0.87 -0.63 -0.40 -0.16 -0.11 -0.02 -0.12 -0.44 -0.49 -0.32 -0.58
-------
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DISCUSSION
OSTROMOGILSKIY: Have you compared harmonic analysis with spectral analysis?
IVAKHNENKO: Analytical determination of frequencies was conducted by a
colleague from the Institute of Cybernetics of the Ukrainian Academy of
Sciencies, Vysotskiy. Interestingly, it establishes the physical frequencies
in the water body not revealed by rapid furie transformation. Thus, the
furie transformation, by not revealing frequencies, gives only an approxima-
tion.
GLUTSYUK: How accurate are your predictions of yearly flow with the GMDH
method?
IVAKHNENKO: We believe that we can project 2 to 3 times the time span of
available data. Using data for a period of 100 ye-ars, we can obtain data for
200 to 300 years ahead.
GLUTSYUK: Is it possible to unify the process of forecasting yearly flow for
various rivers?
IVAKHNENKO: The algorithm is universal. We are trying to heighten the
accuracy of forecasting, taking into consideration the discharge and flow rate
of a series of rivers, for example, the discharge of the Nieman for fore-
casting the Dnieper, and vice versa. This reflects, to a certain degree,
Professor Kalinin's ideas. He affirms that one river repeats the discharge of
another river with a displacement of 2 to 3 years.
GLUTSYUK: Have you compared the data from your analysis and then produced a
proposition using your equation? Have you compared the forecast -with your
data? Have you compared the results of the forecast with individual points?
IVAKHNENKO: For river flow, data for a large number of past years, e.g., for
the Dnieper since 1912, is available for several rivers. We can verify the
accuracy of forecasting on the data for that past time period. Still one
more verification method exists "to change the direction of the time flow"
to calculate the path of the process in a reverse direction. This method is
used when a small number of points are present.
OSTROMOGILSKIY: Is it possible with the help of the given method to forecast
anomalous climatic phenomena and their consequences, for example their
influence on river flow?
IVAKHNENKO: The algorithm of monthly river flow was used by one of the Kiev
organizations for forecasting soil humidity. This algorithm was verified.
153
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Droughts for example are random occurrences, and do not enter into the fore-
cast. Average forecast values for a decade are performed with a 10 to 15%
accuracy.
OSTROMOGILSKIY: In order to construct your system for forecasting, you need
information on a river. How do you take into consideration morphometric
river changes such as thresholds disappearing on the Dnieper and construction
of dams and water reservoir cascades? What, in this case, is the value of
long-term forecasts for 100 to 200 years?
IVAKHNENKO: Several scientists confirm that the Dnieper flow is the same now
as it was 1,000 years ago. We forecast average yearly flow. Each year, in
the month of February; the water reservoir is emptied in order to collect
the fresh spring water. The Ukraine gets in an average spring 50 km of
water from the atmosphere. The basic function of water reservoirs is to hold
this water. It is necessary to increase the volume of the reservoirs. The
yearly cycle takes place exactly on time. The yearly flow does not depend on
the presence of water reservoirs. An average monthly discharge of flow is
made for the upper Dnieper where there are no reservoirs.
154
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GEOGRAPHIC-ECONOMIC ASPECTS OF POLLUTION CONTROL SYSTEMS
Craig ZumBrunnen
INTRODUCTION
The purpose of this paper is to examine four questions involved with
water pollution. The first question deals with the geographic-economic dis-
tribution or equity problems arising from either water pollution generating
activities or water pollution control measures. Essentially this short ini-
tial section attempts to apply benefit-cost concepts within a spatial context.
The second section examines the issue as to whether the mere territorial size
or jurisdictional scale of a pollution control system enhances or impedes pro--
gress in achieving its abatement objectives. Regardless of the specifics, any
pollution control system must have a geographical component. In other words,
what are the inherent advantages and disadvantages associated with pollution
control regions at various geographical scales? This paper will investigate
the utility and disutility of the various scales of two types of water quality
control regions, namely, political-administrative jurisdictions and natural
regions such as river and lake basins. The third part briefly evaluates four
broad strategies for water resource management: a) moral suasion, b) bar-
gaining, c) government regulation, and d) economic inducements. Both of the
last two water pollution control strategies are predicated upon governmental
legislative action, but the latter is presumed to involve more decentralized
decision making. Finally, current U.S. pollution control policies are re-
viewed .
Rather than focusing upon technical approaches to water pollution con-
trol, this paper deliberately addresses itself to some of the legal, territo-
rial, and economic strategies of water pollution control systems because the
author believes that the specific institutional arrangements play a primary
role in determining the successfulness of any pollution control measures re-
gardless of the technical components of the scheme. For instance, even a
technical process or apparatus which cheaply and effectively intercepts a
given pollutant or renders it innocuous will prove to be an ineffective safe-
guard of water quality if the legal, organizational, and/or economic components
of the scheme do not provide sufficient incentives or inducements to ensure
that the technical apparatus is installed, operated, and maintained efficient-
ly and properly.
Although the paper refers primarily to the institutional context of
American, Canadian, or West European water quality management, most of its
arguments are applicable as well to Soviet water quality management despite
certain differences in the institutional environments.
155
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Finally, two assumptions are implicit in all of the following discus-
sions. First, the true price or value of pure water, while certainly not
infinite, may be greater than both the U.S. and the USSR have assumed thus
far. Second, and following from the above assumption, the objective of any
pollution control system should be to minimize the total costs of residual
disposal (i.e., the sum of total damage and treatment costs) which is achieved
by equating the marginal costs (marginal costs are equivalent to the first
derivative of a total cost curve) of abatement and the marginal costs of
pollution damages.
While the costs of treatment can be ascertained because of the costs of
resources utilized in such abatement measures, a price tag on pollution
damages is exceedingly difficult to determine. In both the U.S. and the USSR,
technical as well as political approaches are used to arrive at surrogate
damage costs. The former refers to the use of knowledge about the biological
and physical effects of pollutants on the environment to arrive at acceptable
or minimum water quality standards (Freeman et al., 1973). The latter refers
to the political process whereby the costs of achieving a certain minimum
water quality are judged to be either too high or acceptable (Freeman, et al.,
1973).
In terms of economic efficiency in a systems context, the key constraints
for ensuring a least-cost solution require: a) the equating of marginal
abatement costs among all pollution control options and b) among all effluent
dischargers. The economic and legal institutions of the United States essen-
tially preclude such an optional solution as undoubtedly such measures would
necessitate changes in the competitive advantage of privately-owned, competing
enterprises. Although technical change and innovations do alter the competi-
tive advantages of such firms continually, voluntary actions favoring such
systems optimization seem remote and such solutions promoted by local, state,
or the federal government seem very difficult to implement as is suggested
later in the paper. In this respect, the Severskiy Donets project undertaken
by VNIIVO appears to possess a solid theoretical foundation.
Spatial Impact of Water Pollution
The question immediately posed is whether it is possible for generaliza-
tions to be made concerning the spatial distribution of positive and negative
externalities that ensue from activities generating water pollution. A second
question arises regarding the spatial distribution of benefits and costs ac-
cruing from pollution control efforts.
Positive externalities refer generally to direct or indirect pecuniary
benefits that individuals or firms derive from any given pollution-creating
activity, that is to say, direct returns in the sense of profits and wages and
indirect in the sense of lower real resource costs or production costs. Nega-
tive externalities refer to such items as increased expenditures for municipal
water treatment, reduced amenity values, and bioecological disruption. A._-
though these first two types of losses can be measured or estimated in mone-
tary terms, it is almost impossible to assess monetarily the deleterious bio-
ecological impacts of water pollution
156
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The benefits of pollution control are manifested chiefly as improved
recreational opportunities and amenities (Freeman, 1973), as well as reduced
outlays for purification of urban water supplies. The costs of pollution
abatement can be grouped into two broad categories, namely real resource costs
and factor income costs. The former pertain to labor, land, and capital that
must be devoted by various means to curtailing the quantity (and harmful
quality) of pollutants otherwise discharged, whereas the latter represent al-
terations in labor costs and in capital incomes resulting from pollution con-
trol measures (Freeman, et al., 1973). Certainly there are inverse relation-
ships between positive pollution externalities and control costs, and between
negative pollution externalities and control benefits. The point at issue is
thus to determine whether or not these aspects of pollution and pollution con-
trol generate a coherent geographical pattern.
Dales (1972) has argued that American literature on pollution exhibits a
very strong "uni-directional" bias. He has claimed that, because most Ameri-
cans live on or near river systems (as opposed to lakes), it appears to be
fairly easy to identify geographically who pollutes whom. (The bathroom
graffito in Pittsburgh admonishes us to "flush the toilet because Cincinnati
needs the water," whereas Cincinnati claims Memphis needs it, and in Memphis,
that New Orleans does.)
Lake pollution, on the other hand, often displays no strong directional
bias. Water exchange and flushing may be slow, even in lakes with large out-
flows. Hence, some lakes function in a manner similar to the oceans in acting
as pollution sinks. The geographical problem of culprit—victim identifica-
tion thus appears to be much more complex in the case of lake pollution than
in river pollution.
However, even with river pollution, can the situation be accurately de-
picted as one of upstream culprits and downstream victims? Maybe with respect
to negative bioecological externalities, the answer is yes (assuming, of
course, that biological self-purification is less than 100 percent effective).
However, the situation as regards the geographical pattern of pecuniary ef-
fects is much less clear. For the positive pecuniary externalities to be
localized upstream and the negative ones downstream, the assumption of no
inter-community movement of goods and residents must be made (Dales, 1972).
The foregoing rather simplified assumptions, at least, would have to be
satisfied before a distinct "polluter—pollutee" spatial pattern could be
drawn. Obviously, the assumption of immobile goods and people is untenable.
Hence, no easy generalization is possible concerning the spatial distribution
of positive and negative externalities of water pollution for either river or
lake.
With regard to the second question, that of the spatial impact of the
benefits and costs of pollution control measures, no simple generalizations
can be made, except that amenity values downstream are improved as river
pollution is abated. The issue is further complicated by the fact that there
is no difference in economic efficiency whether the polluter or the pollutee
bears the financial burden of pollution control costs and/or damages (Coase,
1972; Turvey, 1972). Hence, the spatial benefit-cost distribution is at
157
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least partially dependent upon the specific, legal delimitation of rights.
Such questions really deal with equity considerations, or the impact of
control measures upon the distribution of income and welfare. Although im-
proved air quality in the U.S. would probably benefit the inner city poor and
black populations more than it would the more affluent suburbanites, improved
water quality would probably accrue disproportionately to the more affluent,
as they alone have the financial resources to avail themselves of many water-
based recreational activities and riverfront lots. If the costs of control
measures are passed on to the consumer, they will, in all likelihood, consti-
tute regressive solutions. Freeman and coworkers (1973) have discussed two
broad categories of cost-shifting policies, namely cost subsidy and adjustment
assistance. These policies could do a great deal to ameliorate the inequita-
ble features of pollution control measures.
Pursuing this direction of thought, it is difficult to make any simple
generalizations concerning the spatial impact of water pollution on the one
hand and pollution control measures on the other. An investigation of the
relative merits of various types and scales of territorial organization with
respect to effective abatement of water pollution may, however, contribute to
more efficient resolution of the different interests involved.
Geographic Scale of Control Systems
Criteria for Optimal Control Scale—
In discussing the optimal scale for control measures, the first criterion
should be that net social welfare including ecological considerations is maxi-
mized. This can be accomplished by selecting that pollution control level
where the marginal costs of pollution abatement are equal to the marginal ben-
efits, as depicted schematically in Figure 1 (Koleda, 1971). There are two
criteria of optimal coordination scale: jointness efficiency and distribution
efficiency. Cox (1974) has argued that jointness efficiency depends upon two
factors: the degree of internalization of externalities and the exploitation
of scale economies. Tullock (1969) has linked these two factors to the juris-
dictional area by assuming a) a negative exponential function relating the
costs of uninternalized externalities to the size of jurisdiction (see Figure
2) and b) a U-shaped function relating the marginal (pure water) production
cost to the size of jurisdiction (see Figure 3). By itself the first assump-
tion means that jointness efficiency increases with jurisdictional area,
whereas the second clearly specifies the optimal size as being the one at the
bottom of the U-shaped, marginal-cost curve.
Distribution efficiency is achieved._if_ individuals can consume their
perceived optimal amounts of the public good (in the form of pure water) at
the given price such that their marginal private benefits are equal to their
marginal private costs. Distribution inefficiency can occur under two dichot-
omous situations. On the one hand, non-uniform provision of the good may
exist within a context of homogeneous demand curves. Ori the other hand,
heterogeneous demand curves may exist with uniform provision of the good
(Cox, 1974). However, in the case of water pollution control, the demand
curves for the pure water would probably be heterogeneous and, because of
158
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MARGINAL COST
MARGINAL BENEFIT
0
Pollutants removed
100%
Figure 1. Optimal Pollution Control (Q = optimal abatement level)
Figure 2. Uninternalized costs and iurisdictional size.
159
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d)
-t-J
CD
c
CD
—
O
o
O
CD
C
'cn
L_
CD
Size of Jurisdiction
Figure 3. Marginal production of clean water.
geographical and accessibility factors, the provision of pure water would be
non-uniform.
There are tradeoff problems between the criteria for jointness efficiency
and distribution efficiency. As the jurisdictional size is enlarged to inter-
nalize a greater proportion of the externalities and to reap the benefits of
scale economies, the economic preferences, and hence the demand curves, of the
residents in a given region become more disparate. This situation is likely
to lead to a decline in the distribution efficiency. However, in the case of
water pollution control, where non-uniform provision of the public good is
the rule, the tradeoff is not so simple. Cox (1974) has argued plausibly that
the theoretical problem of optimal territorial scale becomes one of selecting
the scale," ...at which the ratio of the prices of jointness efficiency and
distribution efficiency is proportionate to their marginal rate of substitu-
tion" (Cox, 1974).
This theoretical discussion of optimal criteria for territorial scale has
one glaring drawback and that is the lack of any satisfactory way of measuring
and utilizing these criteria. There are probably many empirical factors that
affect jointness and distribution efficiencies. Hence, for some conflicts in
territorial partition, Cox (1974) has suggested the use of empirical movement
networks to delimit de jure territories. One of the major caveats that he
attaches to his procedure is that it reflects only the significance of human
movements, and hence would be of little utility in trying to obtain the op-
timal jurisdictional scale for flood control or pollution control districts.
(This supposed limitation will be mentioned again later in the discussion.)
160
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Despite the limitations mentioned in the previous paragraph, it .still
seems possible to develop some useful rules of thumb concerning the relation-
ships between the size of control districts and the effectiveness of any
abatement system. Thus, this analysis will first focus upon the advantages
and disadvantages of various political jurisdictions.
Political Jurisdictions—
As we know, in the United States, most of the readily used political
tools for environmental control are firmly attached to the lowest levels of
government because they involve the ownership and use of the land and associ-
ated waterways.
Nevertheless, the disadvantages of local jurisdictions (such as a munici-
pality or county) seem to outnumber their advantages. Projects developed at
the local level tend to be single-purpose ones (such as an urban water supply
or a sewage treatment plant). Accordingly, unless forced to do so, project
planners rarely consider the results of their actions upon other water users
(Davis, 1968). In fact, intramural jealousies among neighboring communities
often act to promote inefficiencies in the system and stifle possible econo-
mies of scale in the construction of regional projects. If the pollution
control strategy were to involve effluent charges, a local political unit
could not readily extract payment from polluters located outside its territo-
rial authority, even if their effluent contaminates its waterways. Therefore,
the application of charges to polluters located within the control district
might not result in any appreciable improvement of local water quality unless
pertinent agreements between regions can be made. A similar jointness inef-
ficiency can occur if water quality or effluent standards are the primary
tools for control. This is true especially when the volume and quality of ef-
fluent discharged locally is insignificant in comparison to the total waste
load entering the local waterbodies.
Local revenue from any charges might not be sufficient to improve water
conditions. Even though large-scale, multi-stage, waste treatment facilities
may be the most efficient and effective, a local governmental unit probably
would not be able to afford them. The dynamics of the situation call for
political integration or, at least, financial assistance from higher level
political units.
On the other hand, a strategy predicated upon effluent standards might
be even less satisfactory because it would have at best a weak revenue genera-
ting capacity. The only logical mechanism for producing revenue would be the
levying of fines upon enterprises that violate the established standards.
However, local jurisdictions are very susceptible to pressure applied by large
industrial enterprises, especially those that employ a large percentage of
the local labor force. Bowen's (1970) account of the 1948 air pollution
disaster in Donora, Pennsylvania is an all too poignant example of the inef-
fectiveness of local political units in situations of monopsony. If the fines
are low, offending firms may simply pay them and pollute as usual. Large
fines levied by local governmental bodies are more likely to produce protrac-
ted and losing battles in court than augmented revenues or improved water
quality. This has been demonstrated with Lake Erie problems and also the
residuals discharged into Lake Superior by the Reserve Mining Company of
161
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Silver Bay, Minnesota. Furthermore, local governmental bodies may even be
pressured by surrounding local jurisdictions into ignoring unpopular pollution
controls in order to attract industry rather than to chase it away.
On the positive side, local political units possess, in principle, a
better knowledge of local conditions, both hydrologic and economic, than do
the larger scale political units. They may be in the best position to esti-
mate the importance of different levels of water quality to the local popula-
tion. In addition, they may be able to devise a plan for land use that would
minimize the effects of pollution or maximize the efficiency of an abatement
system. Unfortunately, these advantages would probably be offset, at least
partially- by difficulties in attracting and retaining a competent profes-
sional staff. Similarly, local agencies are likely to be the least competent
in the latest and most appropriate field, laboratory, and planning techniques.
Some potential improvement in th^ effectiveness of water pollution con-
trol may occur at the state level. In the first place, states are usually
willing to attempt the formation of coordinated agencies dealing with outdoor
recreation, water quality control, fish and wildlife management, irrigation,
and water supply. They can probably get a reasonable gauge on what demands
exist, and exercise considerable control over these activities. Secondly,
states are in a better position to control water quality of major parts of
river basins (by internalizing externalities) through comprehensive investi-
gation of water conditions, and the institution of differential methods such
as zoning, charges, or standards. Thirdly, state revenues might be sufficient
to undertake large-scale projects, if needed.
On the other hand, there usually is strong competition for available
state tax funds, especially from highway and education agencies. If federal
funds are accepted, they often have strings attached. . Richer states that are
not dependent on dirty industries may institute tighter controls. Poorer
states might then be tempted to establish lenient control systems to attract
industry and thus increase their income base. Such a situation may even be
indirectly beneficial in reducing per capita income disparities between
states. However, any economic benefits to the residents of the poorer states
would be at least partially diminished by losses in amenities and by increased
hazards to health. Furthermore, as state boundaries are not coterminous with
river basins, this differential control might conceivably occur in a non-opti-
mal manner within particular basins. For instance, the dirtier part of a
river might be in a rich state that institutes stringent controls, even though
the water is less susceptible to further pollution damage than is clean water
in a poor state. In other words, the marginal loss from additional contamina-
tion is less in the rich state than in the poor one (see Figure 4). An alter-
native scenario of interstate "standards cutting warfare" suggests that sub-
optimal pollution control levels (namely too little) may be reached as firms
either move, or threaten to move, to states having more lenient controls.
Efficiency of a state control system for a particular river or lake basin pro-
bably varies directly with the fraction of the basin within the state bound-
aries. Obviously, the inclusion of the headwaters territory is crucial. For
example, all else being equal, pollution control over the Mississippi River
would be more effective within the state of Minnesota than within the state
of Louisiana.
162
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MCpp
MCpr
LEGEND
State boundary
Clean stream
Polluted stream
Marginal cost of pollution in a poor state
Marginal cost of pollution in a rich state
Figure 4. Marginal pollution costs and political boundaries (MC > MC )
163
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These limitations of control at the local and state levels may lead to
the conclusion that a federal pollution control system would be preferable.
Indeed, control at the federal level does have some factors to commend it.
First, it can avail itself of greater monetary, technical, and human re-
sources. Second, it has more political leverage to deal with monopsony prob-
lems at lower jurisdictional levels. Finally, at the federal scale, a larger
share of water pollution externalities would be internalized. However, the
emergence of multinational corporations raises the potential specter of inter-
national pollution "standards cutting warfare" whereby the threat to the
entire global ecosystem could be dangerously increased.
In addition, control at the federal level has at least three drawbacks.
First, the larger the scale of the control system, the greater the bureaucrat-
ic maze and tendency to fracture responsibilities for water management into
agencies concerned with specific types of programs (Davis, 1968). Also, in
the specific case of the United States, lobbying interests are very powerful,
and therefore, certain types of solutions may be looked upon with more favor
than others. Large-scale, capital-intensive, public projects, such as sewage
treatment plants, may be favored because industry is not affected directly,
and manufacturers involved in selling pollution abatement equipment will
stand to gain, and such projects will be "visible" benefits from a political
viewpoint (Davis, 1968).
A program at the federal level may simply promulgate minimum standards
designed to be uniformly applicable. However, such a policy might merely
translate itself into spatially homogeneous levels of mediocre pollution con-
trol. Furthermore, information gathered from a federal viewpoint might not
possess a sufficient degree of differentiation in terms of the characteristics
of, and influences upon, different river and lake basins, nor appropriate
awareness of the various demands and needs of the surrounding populations.
For example, waters already polluted to a certain extent are less susceptible
to additional degradation than clear, clean water. A more rational division
of uses in such a situation might be to zone the former as a repository for
industrial waste and the latter for recreation. An optimal solution within a
basin, or between nearby basins, may be to superimpose a strategy for manage-
ment of venous-arterial water quality, utilizing currently differentiated
conditions of water quality as one criterion for territorial partition. Un-
fortunately, with the exception of interior drainage basins (such as the Cas-
pian Sea) the world's oceans seem to serve as the ultimate sinks for indus-
trial, agricultural, and human wastes. Hence, the ultimate control over water
quality necessitates an international scale of control (that is to say, the
complete internalization of externalities). Because collective decision
making, even at the international scale, is plagued by free-rider problems,
there is great need for altruism.
Two tenative generalizations about the optimal scale of political juris-
dictions for effective pollution control may be enunciated. First, the
theoretical effectiveness of the control system seems to increase as exter-
nalities are internalized. Second, with the possible exception of the global
scale, all political jurisdictions seem not only sub-optimal but more impor-
tantly inappropriate units for the task envisioned. An interesting alterna-
tive which may help resolve some of the problems of political jurisdictions,
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is the use of river and lake basins as the most suitable divisions for water
pollution control systems.
Natural Jurisdictions—
The fact that both air and water pollution ignore political boundaries,
and the existence of a plethora of inter-jurisdictional conflicts over water
management authority in the United States, is a convincing argument for the
judicious conversion to the basin concept of water quality control. With
slight modification, Cox's (1974) theoretical arguments, linking movement
networks to territorial organization, can be used to support the adoption of
river and lake basins as the most appropriate territorial units for any system
of water pollution control. Instead of focusing upon networks of human move-
ment, it is simply necessary to concentrate upon networks of water movement as
the basic organizing principle for selecting the optimal geographical scale of
pollution control districts. Obviously, control districts conterminous with
drainage basin boundaries are ideal for maximizing jointness efficiency.
In terms of the optimal scale for a natural jurisdiction, the dynamics
seem to call for nothing less than integral river or lake basins, regardless
of whether they are interior drainage systems or part of an oceanic drainage
system. Specific basin control measures could be added to enhance distribu-
tion efficiency by seeking to collect and utilize both differential hydraulic
and socio-economic data. Where a particular basin encompasses two or more
nations, the optimal solution calls for international, basin-wide control.
Again, the ideal solution seems to be some form of effective, global, collec-
tive bargaining agreement to control ocean pollution.
The appropriateness of river or lake basins as the territorial units
used for pollution control systems seems intuitively obvious. Yet in the
United States such natural planning and management regions are few in number,
have evolved sluggishly, and in all instances known to this author, have very
circumscribed authority over multipurpose or multi-use water resource manage-
ment. Accordingly, the universal adoption of the basin management concept
seems to be a logical requisite for substantive improvement in the efficient
and ecologically prudent management of America's water resources. Unfortu-
nately- the multitude of entrenched political interests and governmental
agencies involved with water resource management do not augur well for the
widespread adoption of the comprehensive basin approach to water resource
management in the U.S.
The People's Voice
Fisher (1970) has pointed out that a network of knowledgeable voluntary
associations and private technical associations can be a useful link to relate
public and private agencies with jurisditional decision making. In the usual
cost-benefit analyses conducted on behalf of agencies, benefits are generally
overstated, and costs fail to take into account the more remote damages and,
more particularly, the social contexts of water quality control. The Brandy-
wine study (1968) attempted to alleviate this problem of single purpose plan-
ning by specifically incorporating the people's view of the role that the
natural environment plays in their lives.
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Unless some means is established to combine water basin with political
region, quality will be pursued in a piecemeal fashion. Political jurisdic-
tions will continue to be involved in tradeoffs, and appointed bodies will
continue to appease those they represent. More than ever, we must focus on
the political democratic process as well as on economic theory. In doing so,
we accept the responsibility for maintaining a dynamic dimension, and deter-
mining its receptiveness to infusions of new information and mechanisms of
feedback.
Strategies of Control
In a summary fashion four pollution control strategies will be evaluated
in terms of four criteria: a) economic efficiency, b) incentive structure,
c) transactional or information costs, and d) enforcement burden.
Moral Suasion—
A policy or strategy of water pollution control based upon the use of
moral persuasion to convince polluters to voluntarily refrain from polluting
or reduce their water-borne residual discharges as an obligation to society
can be summarily dismissed as an ineffective policy doomed to failure within
the context of a socialist as well as market economy. First of all, without
accompanying subsidies or payments, no strong economic incentive exists to
implement pollution abatement measures because of price competition in market
economies and free-rider problems in both forms of economic organizations.
The private firm that altruistically and unilaterally takes positive actions
to either reduce or treat its wasteload incurs increased production costs (in
many instances) which can either manifest themselves as lower profit margins
(and even outright losses) or higher-priced, less competitive products. Thus,
firms have an economic incentive to continue to pollute.
To make matters worse, there is no guarantee that the altruistic firm's
abatement measures will be economically as efficient as the potential ones
a more "business-like" enterprise which decides against control measures
could have adopted.
While transactional costs may be rather modest, moral persuasion has
little effective means of enforcing compliance. One could argue that environ-
mental groups in America, through the skillful use of the mass media, could
tarnish the public image of polluting enterprises which choose to ignore the
moral urgings of the citizenry for cleaner water with the hope of creating a
consumer boycott of the offending firm's products. Such boycotts are also
susceptible to free-rider problems, especially if the polluting firm's goods
are priced lower (as could be expected) than competing businesses who yield
to moral force. Then, too, a firm may discover that it is cheaper to hire
a public relations firm to resurrect its public image than to reduce the
environmental damages which its activities generate.
In summary, a policy based on moral persuasion leads not to the control
of pollution, but rather to the continuation of pollution thus reducing both
net GNP (Gross National Product) and the welfare of society (Dorfman, et al.,
1972).
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Bargaining Solutions or the Tort System—
Bargaining solutions represent an improvement over moral exhortations.
Such a strategy is predicated upon certain prescribed legal rights and obliga-
tions with regard to water use. In certain limited situations, negotiations
leading to either bribes or mergers may produce economically efficient solu-
tions if bargaining is perfect. For example, if A and B are (profit-maximi-
zing) firms with the former representing the polluting actor and the latter
the polluted or harmed actor, then negotiations between them can yield three
economically optimal outcomes: a) Pollution costs could be internalized by
a merger, b) B might agree to pay A to modify the nature or scale of its
polluting activities, or c) If the law allows B to be compensated for losses,
then A might pay B to accept the optimal amount of loss imposed by A. In the
last two scenarios, the optimum is defined as A's type and scale of activities
and B's reaction to them which maximize the algebraic sum of A's gain and B's
loss as compared to the situation where A imposes no diseconomy on B. As
was noted previously, the fact that compensation liability is reversed in the
last two bargaining outcomes has no impact whatsoever on economic efficiency
(Turvey, 1972). Thus, when negotiations are possible, governmental inter-
vention in the United States is not for the sake of efficiency, but for a
societal sense of justice. However, as Turvey (1972) illustrates graphically,
for optimality to exist it is necessary that any levy assessed against A be
paid to B, otherwise sub-optimal conditions will obtain.
In the example above, both protagonists have effective incentive to bar-
gain with each other. However, real world situations are rarely that simple.
Water is a flow resource as opposed to a stock resource and as such has many
of the attributes of a free good or a public good. It is exceedingly diffi-
cult from a physical point of view to assign property rights to a fluid re-
source. Accordingly, except for certain limited situations, water resources
in the U.S., as in the USSR, are publicly owned. Furthermore, in both socie-
ties water has a multitude of different users (and uses), many of which have
conflicting if not incompatible interests in water resource management.
The key limiting factor to the applicability of a bargaining or tort
system of water pollution control is the sheer number of potential partici-
pants. In brief, transactional costs of bargaining solutions increase expo-
nentially as the number of actors increases. Finally, bargaining solutions
are plagued with free-rider problems which aggravate enforcement problems.
Government Regulation—
For the purposes here government regulations can be divided into three
general categories: a) prohibiting certain uses, b) limiting certain uses,
and c) prescribing protective measures.
The most stringent policy, prohibition, is needed in certain cases such
as bans on the use of persistent biocides and polychlorinated biphenol (PCB).
The cumulative effect or biological magnification of certain deleterious re-
siduals may yet prove that the policy of prohibition has not been invoked
frequently enough. In general, however, optimality does not require the com-
plete elimination of pollutants. Provided the penalties established for non-
compliance are high, the enforcement burden is likely to be low as are trans-
actional costs. The chief problem with a prohibition policy is that it
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represents an extreme position with limited applicability. However, because
of ecological response time lags, prohibition has probably not been invoked
often nor expeditiously enough in the U.S. and elsewhere.
A policy of limiting certain uses is of more general applicability and
may take two common forms: a) instream water quality standards (in which
case it is often exceedingly difficult to assign responsibility for violated
standards in situations x^here there are several dischargers), and b) discharge
standards based upon some technical-political-economic estimate of the total
allowable discharge (the assimilative capacity) for a given region. The lat-
ter system would probably include a licensing of polluters. Both forms are
difficult to enforce (especially with regard to occasional infractions),
require costly monitoring, are likely to be inefficient, and have few self-
regulating properties or incentives to conform to governmental directions.
While creating lower transactional costs, and being politically expedient,
discharge standards which are geographically and sectorally uniform are sub-
optimal.
The third regulating device involves prescribing protective measures,
such as uniform secondary treatment processes or specific types of pollution
control apparatus. This solution by action approach has limited applicability
as polluters usually have better knowledge as to the technical and cost effec-
tiveness of various technical options. Thus, specific means of pollution
control ordered by government are likely to be both technically and economi-
cally inefficient. On the other hand, such an approach has low transactional
costs and perhaps an acceptable enforcement burden.
In a more systematic fashion, Dorfman and Dorfman (1922) enumerate four
general difficulties that beset governmental attempts to regulate water
quality: a) inadequacy (and cost of information, b) crudeness of regulatory
instruments, c) practical difficulties of enforcement, and d) haphazardness of
burdens. The section discussing current U.S. water quality management sup-
ports their general dismal thesis.
Economic Inducements—
Economic inducement policies may be divided into two broad categories:
a) subsidies or payments and b) taxes or effluent charges.
Subsidies, tax incentives and the like may effectively discourage or en-
courage different (socially and ecologically more desirable) ways of water
resource use. They appear appropriate where capital costs are the only
stumbling block to the improvement of water quality. Transactional costs and
enforcement burdens would likely be modest and certainly lower than the just
cited examples of more direct forms of government control and regulation. An
effective positive incentive structure can be created as part of either a
subsidy or a tax incentive program of water pollution control. The chief
liability of these "carrot" approaches is that there is little likehood of the
policy being even remotely geographically, ecologically, or economically ef-
ficient. Then, too, there are some blackmail or coercion dangers in some pay-
ment policies. For example, Shkatov (1968) suggests a scheme in which a
Soviet enterprise's incentive fund would receive a two-kopeck bonus from the
state for every cubic meter of completely purified sewage which it discharged.
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In the U.S. such a proposal, if adopted, would create a host of new and pro-
fitable firms engaged in "the business of not creating pollution!" These and
other problems with subsidies are discussed in the section dealing with cur-
rent U.S. water quality.
Tax and effluent charge policies seem far more encouraging for a number
of reasons. A tax policy here refers to a tax on pollution generating pro-
ducts while an effluent charge policy refers to a charge (or tax) on the ef-
fluent discharged (differentiated according to the quantity, quality, time,
and place of the discharge). Both approaches would internalize pollution ex-
ternalities and significantly reduce enforcement burdens, compared with other
possible policies.
A tax on pollution generating commodities would likely have both low
information costs and enforcement burdens provided the tax was uniform ac-
cording to commodity. However, such a strategy would be sub-optimal because
it is improbable that damage functions for the manufacture of a given commod-
ity are homogeneous for all production sites. The tradeoff for increased ef-
ficiency would be increased information costs as to commodity-specific and
location-specific estimates of damage functions.
A charge scheme could be established similar to the federal income tax
system in which firms would have to monitor (record) their emissions, report
their composition and quantity (file tax forms) to a public agency and make
the required payments (taxes) to the agency. Audits and unannounced inspec-
tions would, of course, be necessary. While involving considerable expense
(information costs), an effluent standards policy would require similar ex-
penditures. On the other hand, for optimality to obtain the governmental
agency only needs to know the quantity and quality of the effluent discharges
and not marginal cost curves of individual emitters.—the latter being informa-
tion essentially unobtainable as costs are often regarded as enterprise
secrets.
Not only would a charge system create a permanent incentive system to
reduce residuals, but furthermore, individual firms have the best knowledge
as to the most efficient option to employ to equate marginal pollution con-
trol costs with marginal damage costs (effluent charges). Thus, a residuals
charge system would tend toward optimality.
The "correct" charge, unfortunately, could not be ascertained easily by
trial and error without violating parato-optimality. For instance, while
charges could be changed readily, fixed capital in pollution control facili-
ties are far less flexible (Dorfman et al., 1972). Thus, considerable bene-
fit-cost analysis should precede the adoption of a charge policy. It seems
that some of the linked econometric-biologic-hydrologic simulation models dis-
cussed at this symposium could greatly facilitate such types of analysis and
guide decision makers involved in the elaboration of the specifics of a
residuals charges policy. However, in most cases there might well be far
greater risks of initial underinvestment in "effluent charge-avoiding" treat-
ment facilities and altered production processes than the converse simply as
a consequence of national economic expansion.
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It appears that a carefully conceived residuals charge policy has more
to recommend it on both theoretical and practical grounds than any of the
other major strategies thus far proposed or implemented.
Current U.S. Pollution Control Policies—
In the United States, the failure of moral suasion, the limited applica-
bility of bargaining solutions, and the public good aspects of water resources
in the face of deteriorating national water quality conditions culminated in
a public demand for government regulation of water quality.
The potentially effective federal Refuse Act of 1899 which prohibits the
discharge into any waters of "any refuse matter of any kind or description
whatsoever" without the polluter having first obtained a permit from the U.S.
Army Corps of Engineers has been, in effect, repealed by the U.S. Justice
Department's policy of selective non-enforcement (Freeman et al., 1973).
The Water Pollution Control Act of 1956 established a federal subsidy
program for the construction of municipal waste treatment facilities. How-
ever, the subsidy program'pertains only to the initial capital construction
costs and not to maintenance and operating expenditures. Therefore, the
initial incentive to build treatment plants is not coupled with an incentive
to operate them efficiently. Such an incentive, however, could be created
fairly easily.
Since many industrial and commercial waste generators are connected to
municipal sewage lines, the federal grants program results in reduced sewer
charges and hence, a substantial subsidy to business. Furthermore, it under-
cuts the incentives for all effluent dischargers to seek alternative pollution
control measures. Recent federal water pollution control legislation stipu-
lates that all municipalities must use secondary waste treatment processes
(85% organic removal) and that all discharges from all sources of pollution
are to cease by a target date of 1985. The first regulation leads to a mis-
allocation of resources as some regions will undoubtedly experience insuffi-
cient treatment costs and others excessive ones. On the other hand, such a
uniform treatment regulation greatly conserves on transactional or information
costs. The second stipulation> albeit laudable, will probably be counter-
productive as it sets unattainable expectations (and implicitly assumes that
the price of pure water is infinite). The enforcement burden of such a water
quality goal (zero discharge) would indubitably precipitate a massive break-
down in the system of water quality management.
The second major U.S. policy instrument intended to encourage investment
in waste treatment facilities is contained in the Tax Reform Act of 1969
which allows accelerated depreciation of such investments for tax purposes.
(Really a tax subsidy program should be categorized as an economic inducement
strategy.) This tax subsidy reduces the after-tax costs of pollution control
investments (Freeman et al., 1973). However, both the federal grants program
and the tax subsidy, in effect, allow dischargers to create and dispose of
large quantities of residuals without incurring the full costs of their ac-
tions and at the same time use taxpayers' revenue to cleanse the waters they
have dispoiled.
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This last statement is predicated upon the author's support—with certain
critical qualifications—of the polluter pay principle. Subscribing to such
a principle appeals both to a sense of justice and economic principles. With
regard to the latter, whenever someone does not bear,the full costs of one's
actions, there is a natural tendency to squander und^rpriced resources. The
crucial qualifications entail the development of a pollution pay strategy
which is not regressive in its functioning. For example, in any polluter pay
system, the costs of compliance will unavoidably be passed on to the consumer.
Such an arrangement is economically desirable, but may be socially inequitable
for consumer necessities having low price elasticities of demand. Thus, with-
out an accompanying income adjustment component (such as a guaranteed income),
a polluter pay principle will lead to socially regressive resolutions of
environmental quality problems.
Although U.S. water pollution control statutes provide for various legal
sanctions for non-compliance such as fines and the legal authority to halt
the operations of polluters, in practice compliance has been more voluntary
than compulsory. In some cases various governmental enforcement agencies
have resorted to fines, court action and the like, but they have not been very
effective. Overall the effectiveness of existing U.S. pollution control poli-
cies have been characterized more by failure than success. Accordingly new
policies and strategies need to be tried, especially ones using a combination
of residual charges and income adjustment programs.
Conclusions
A number of tentative conclusions can be enumerated at this time. First,
it is difficult to state any simple generalizations about the human-geographic
manifestations of water pollution on the one hand, and pollution abatement
measures on the other. Second, political jurisdictions are far less appro-
priate than river or lake basins as the territorial organizing principle be-
hind any pollution control system. In this regard, the USSR may be ahead of
the U.S. Third, current U.S. water pollution control policy leaves much to
be desired. Fourth, and most importantly, a plea is made for the acceptance
of the polluter pay principle and the establishment of charges or fees for
the use of nature's freshwater assimilative capacity.
Effluent charges currently exist in a number of both East and West European
countries. It seems more than a little ironic to this author that many of the
socialist countries of Eastern Europe have accepted the concept of residuals
pricing while the so-called price-oriented market economies of England and the
United States have stubbornly resisted such a strategy. Any number of Soviet
academicians and economists have suggested an effluent charge which is:
a) proportional to the quantity and quality of unpurified effluent discharged
and b) differentiated by industry and product. Brown and coworkers (in press)
add the additional economically rational requirements that: a) charges be
differentiated by production techniques within industries, b) charges be
differentiated geographically to reflect local conditions, c) all polluters
regardless of size should pay the same unit charge provided the opportunity
costs of their residual emissions are the same, and d) charges ought to vary
through time to reflect changes in the opportunity costs of pollution.
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As Freeman and coworkers (1973) similarly conclude, a residuals charge
strategy has four strong factors to recommend it. First, a charge strategy
will produce a given level of pollution abatement at a lower total economic
cost than alternative strategies because of the economic incentives it creates
and the decentralized decision making as to the scale of emission reduction
and the specific measures. Second, not only is the incentive to control
residuals discharge direct and powerful, it is durable. Third, governmental
policy makers utilizing a charge system need less detailed information about
individual polluters to achieve optimality than if they depend on an alterna-
tive system such as direct regulation. Fourth, a residuals charges strategy
will create less of an enforcement burden on the existing judicial system
because collecting the charge is similar to the paying of taxes. To be
economically, geographically, and socially desirable, the residuals charges
strategy should be superimposed on a basin-wide water resources management
system and combined with an income adjustment program to ameliorate the
possible regressive aspects of the policy.
Finally, the first part of this paper pointed to the need for global
cooperation with regard to environmental pollution control. This joint
Soviet-American symposium on water pollution control is not only in the spirit
of this need, but also is a tangible example of such needed international
cooperation.
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pp. 27-34.
Brown, Gardner, and Ralph W. Johnson. Cleaning Up Europe's Waters: Econo-
mics, Management, Policies, (chapters 9 and 12 of preliminary draft of
book forthcoming by Praeger Publishing Co.).
Cherniavsky, I. 1970. We must protect water. Current Digest of the Soviet
Press, Vol. XXII, No. 46, pg. 17. Translated from Izvestiia, July 1, 1970,
Pg- 3.
Coase, Ronald. 1972. The problem of social cost. In: Economics of the Envi-
ronment, Eds. Robert and Nancy S. Dorfman,(W.W. Norton & Co., New York)
pp. 100-129.
Cox, Kevin R. 1974. Territorial organization, optimal scale and conflict.
In: Locational Approaches to Power and Conflict. Eds. Kevin R. Cox,
David, R. Reynolds, and Stein Rokkan, (Sage Publications, New York)
pp. 124-137-
Dales, J.H. 1972. Land, water and ownership. In: Economics of the Environ-
ment. Eds. Robert and Nancy S. Dorfman (W.W. Norton & Co., New York)
pp. 171-186.
Davis, Robert K. 1968. The Range of Choice in Water Management. (The Johns
Hopkins Press, Baltimore, Md.) pp. 8-10.
Dorfman, Robert and Nancy S. Dorfman. 1972. Economics of the Environment.
(W.W. Norton & Co., New York) pp. xiii-XI.
Fisher, Joseph L. 1970. The several contexts of water. In: America's
Changing Environment. Eds. Roger Revelle and Hans H. Landsberg. (Houghton
Mifflin Co., Boston, Mass.) pp. 31-38.
Freeman, A. Myrick III, Robert H. Haveman, and Allen V. Kneese. 1973. The
Economics of Environmental Policy. (John Wiley & Sons, New York), pp. 94-
148.
Khvostenko, N, August 19, 1970. Vzysklvat"podorzhe. Izvestiia, pg. 3.
Koleda, M.S. 1971. A public good model of governmental consolidation. Urban
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Loiter, M. 1968.' Economic measures in the rational utilization of water
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Mazanova, M. 1969. Territorial'noe proportsii razvitiia ekonomiki. Planove
khoziaistvo. No. 2, pp. 10-19.
Oziranskii, S.L. 1968. Plata za vodnye resursy. Planovoe khoziaistvo. No. 9,
pp. 67-75.
Shkatov, V. 1968. Prices for natural riches and the perfecting of planned
price formation. Current Abstracts of the Soviet Press, Vol. 1, No. 6,
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Sukhotin, lu. 1968. Exchange of opinions: Evaluations of natural resources.
Current Abstracts of the Soviet Press, Vol. 1, pp. 3-4.
The Brandywine Plan. 1968. Chester County Water Resources Authority, West-
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Environmental Studies, University of Pennsylvania, Philadelphia, Pa.
Tullock, G. 1969. Federalism: Problems o-f scale. Public Choice. Vol. 6,
pp. 19-29.
Turvey, Ralph. 1972. On divergences between social cost and private cost.
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DISCUSSION
DeLUCIA: I would like to make a few comments that in some cases will
amplify and reinforce, and in some cases disagree with Dr. Zumbrunnen's
remarks. First, with respect to his comments on regulatory approaches, one
very important thing to remember is that frequent discussions of regulatory
approaches implicitly assume zero or little transaction time, but the
empirical evidence in the United States is that this is not so. Hence, the
resulting transaction costs due to delays and to the resulting regulatory
process are significant. Second, with respect to market mechanisms and ef-
fluent charges, I would suggest that there has been recent work that explic-
itly estimates mechanisms which, when used with a combination of regulatory
and effluent charges, minimize the risks of underinvestment, deal with the
issues of uncertainty of response, and estimate the implications of single
charges versus commodity or specific charges. In fact, this work is an
example of a Zumbrunnen proposal—the use of some of the models, such as those
we have been discussing, to examine effluent charges and other mechanisms.
This work is referenced in my paper. Lastly, I would like to take very strong
exception to Dr. Zumbrunnen's strong positive comment on the "polluter pays
principle." In my opinion, speaking solely of the situation in the United
States, the 1972 amendments to the Water Pollution Control Act are markedly
different from earlier water resource legislation in the United States in
that, in both the preamble and in the text, the Congress makes it clear that
it views environmental quality as a national goal and objective. It is quite
clear from reading the legislative history that the Congress's intent is that
a significant portion of the financial burden should be borne nationally.
Moreover, and perhaps most importantly, the legislation mandates a behavior,
but gives no option or flexibility to the local or regional institution. In
contrast, earlier water resources legislation in the United States gave
options to states and municipalities. Earlier legislation was the basis for
the economic literature of cost-sharing in proportion to benefits received.
LANDYZHENSKIY: When I was familiarizing myself with Dr. Zumbrunnen's paper,
I was struck by the fact that so many questions presented in this report
relate to conditions in our country. Our countries are very large and have
a high concentration of industrial enterprises in specific regions—fre-
quently in regions with limited water resources. In the U.S. and USSR, there
are so-called "hot spots," where the controlling and the executive agencies
are late in passing preventative measures. As a result, difficult situations
arise. In order to improve these situations, it is urgently necessary to take
cardinal measures, often with considerable expense or material loss to stop
the activity of the polluter. All this affects the interests of many agencies
and many people. These interests are often contradictory, and the search for
the best decision is exceptionally difficult.
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The report shows that when solving a question of water protection, even
the most effective measures will not produce the desired result, if the
jurisdicial, organizational, and economic acts will not create the necessary
conditions for this. The problem of so-called "jurisdicial territory" in the
U.S., on which the jurisdiction of the controlling agencies is propagated,
also exists in our country. In the Soviet Union, government control for
protecting of water bodies is vested in local agencies. Determining the best
sizes for jurisdicial territory remains an unsolved problem. Research and
practical experience show the expedience of creating an administration for
river basins. Using the Kuzin terminology, such an administration should
be called "global." In our country, only one such administration has been
created for protecting the Caspian Sea. The whole water body enters into its
jurisdiction. Other similar organizations—of which there are slightly more
than 100—have a limited territorial sphere of influence which affects the
coordination of decisions for complex water protection programs for the
whole river basin. The Volga River is controlled by four basin inspectorates.
We should agree with the conclusion that the most suitable administrative
division for systems for controlling water pollution is at the river basin
level. It is evident that areas of control that coincide with the boundaries
of river basins are ideal for maximizing the effectiveness of consolidation.
When a given water body includes the territory of two or more governments, the
best decisions require international control within the boundaries of the
whole basin. The scheme for complex use and protection of the Tisa River
requires the participation of Czechoslovakia, Hungary, Rumania, the Soviet
Union, and Yugoslavia.
YAVOROVSKIY: I will only touch upon one point—the application of the
"polluter pays principle." Solving technical, mathematical, and technological
problems is possible under identical conditions, on an identical basis under
socialist economic conditions, as well as under capitalist conditions.
Solving organizational problems is a different matter; and still more
differences arise when examining economic questions. Significant differences
exist in the approach to the economics of water protection on the strength of
differences in the original conditions. As Dr. Zumbrunnen noted in his
report, the problem of Water protection in the U.S. is very specific. It is
solved under conditions by which the subjective economic interests of the
enterprises take precedence over the objective economic interests of society.
The fact that a single water body under U.S. conditions is administratively
separate is the second essential difference. This arrangement leads to the
expediency of applying the "polluter pays principle"; i.e., the system
takes into consideration the subjective harmony of production and the
interests of individual establishments, not the objective differences of the
functioning of the water bodies. In this case, the clear advantage of this
method is its simplicity. However, we feel that its basic flaw lies in its
inability to fully consider the natural differences in conditions where
various water bodies and separate parts of a single water body are used.
Water protection measures under conditions in a socialist system have
their own economic specifics. Our problems are solved under conditions in
which differentiated economic requirements to protect a water body from
pollution are basic to water protection. The fact that expenditures for
treatment should be reduced not to a common level but to objective, socially
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necessary costs is another characteristic. At present, we are faced with a
task which our colleagues are solving. For various enterprises, equal
economic conditions should be created for cost accounting that does not
depend on the location of the enterprise and the water protection factor.
We have not solved all economic questions of water protection, nor have
you in the U.S. However, the "polluter pays principle" obviously cannot be
used under our conditions, since it does not produce the best solutions within
regions or basins. Dr. Zumbrunnen said that this principle guarantees the
introduction of the necessary means. However, it must be said that excess
financing is possible in this case, but when there is an accurate economic
solution to a problem, this is not the best variant. Therefore, at present,
we feel that an economic basis to water protection must be found on the
following premises:
1. Costs for water protection measures should be differentiated. This
type of cost should be taken into consideration in the formation of production
net cost and should influence the economic results of the industrial enter-
prise.
2. Cost accounting should provide for differentiated revenue payments
based on the location of the industrial production's water bodies. This type
of cost should equalize the original conditions of the enterprise'-3 production
activity.
Such an economic position in the approach to water protection questions
is formulated in the work of the Corresponding Member of the USSR Academy of
Sciences Bunich and Academician Fedorenko, in particular the latter's work,
"The Mechanism for Economic Stimulation under Socialism."
A method that requires payment for water resources and considers the
cost of water in projecting the net cost of industrial production would be
most effective for solving economic problems related to water protection.
In conclusion, I must say that the problems confronting us and our
colleagues are the same. However, the paths for solving these problems, in
particular the economics, are different.
LOZANSKIY: I agree with Dr. Yavorovskiy and must add that water legislation
in our country forbids water pollution. Therefore, it is not a case of
payment for pollution, but of assessing guilt or criminal responsibility. It
is prohibited to pay money for breaking laws.
The question of payment for discharging waste in our country is based on
this premise: the mass of pollutants accepted by a water body should be
determined within the framework of its ability to assimilate so that actual
water quality standards are observed.
THOMANN: I have three questions and one preparatory remark. It is fashion-
able to say in our country that the present system is not working, yet there
is much evidence to the contrary; evidence that the present system of
government control has produced improvements in water quality. On the other
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hand, T personally am very sympathetic toward the concept of effluent
charges. I believe they can and should be experimented with under our
present legislation.
My questions, however, relate to my belief that effluent charge problems
have not been sufficiently analyzed. First, what should be done with
municipal waste discharges? All discussion of effluent charges has been
directed toward industrial firms. Second, water quality problems are
associated with the discharge of many types of wastes, not just BOD. Previous
work not withstanding, I have not yet seen a convincing analysis that relates
effluent charges to the many variables in water quality of interest (for
example phosphorus, nitrogen, BOD); how do effluent charges account for
different water quality variables? Third, the proponents of effluent charges
talk about a trial and error method. But how do you measure the success of
this program? For example, if I established a fee system for a river during
years of high flow, I may observe good water quality for that system; then
the drought comes, bringing low flows and water quality deterioration. I
find out my charge system was not adequate; but by .then it is too late—we
may have money, but the fish have died.
DeLUCIA: I would like to clarify my last comments. When I said I disagree
with the "polluter pays principle," I disagree only with respect to public
institutions. What do you do with the municipal polluter? You build into
your fee system both a charge and a subsidy so that you can design a charge
that "keeps the system intact" by forcing a municipality to pay at a parti-
cular level based on the marginal unit of waste discharged, but you allow
a significant subsidy based on what is removed. This subsidy is not given
to industry.
As for other pollutants, we have argued strongly that the appropriate
regulatory mechanism is a combination of regulatory and charge systems. In
the latter case, problems associated with a trial and error mechanism are
largely eliminated, since there are direct upper bounds on the discharge, and
lower bounds on the treatment or removal. The effluent charge is utilized
to efficiently allocate resources in the range of the cost removal curve
where the marginal costs are highest. I would further suggest that flow-
dependent, administrative requirements, although feasible, may outweigh the
benefits. Additional alternative market mechanisms have been suggested for
eliminating the trial and error method.
BARTSCH: I want to inject a different and perhaps simplistic point of view.
I personally disagree with the "polluter pays principle." I believe that
this approach gives industry the opportunity to do what it has always done,
and that is to attempt to get by with the very least. From my own observa-
tions on industrial performance, I doubt very much that matching pollution
control to assimulation capacity is a potentially realistic objective. I
call attention to the fact that in the U.S. (and I suspect that in this
regard we are no different from the USSR), we have been especially efficient
in exploiting the total environment. I call attention to the manner in which
we have cut trees while speaking about a sustained yield. To exploit the
land, we have destroyed the buffalo. There are many examples of ways in
which we have degraded the air. We have also been most efficient in
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insulting the total environment. I believe we need only look at the history
of litigation between industry and official bodies in connection with water
pollution control to raise serious questions about the efficacy of this
approach. The issue and the goal is not how much can we get by with; the
issue ought to be how clean can we make our environment.
ZUMBRUNNEN: I see no problem for instituting effluent charges that are
differentiated both according to the quantity and the quality characteristics
of the effluent. One possible use cf mathematical models would be to make
some connection between the charges and simulated capacity of the water
resource, thus resulting in a type of collected social decision-making.
Accordingly, we should come up with an absolute limit cf allowable abuse or
misuse for a particular body of water, and accordingly, correlate our
charges, perhaps through an auctioning system. It seems that effluent
charges would be difficult to apply, except to point sources of pollution.
I am not suggesting that effluent charges should be the only water pollution
strategy; but rather one tool which has not been used in the U.S. but
might be used. With regard to Dr. Bartsch's comments, I am in complete
agreement with him. However, it seems to me that by using an effluent charge,
we can build in an economic incentive for industrial firms to reduce their
discharges of wastes. Indeed, it seems to me that the Soviet Union does have
an advantage over the U.S., at least in theory, because of central planning
of industrial locations, industrial expansion, and so forth. At this point
I would like to ask a question. What in fact are the inputs in terms of the
location of industry, the production of industry, and the estimated effluent?
1 am curious as to whether some of the outputs of VNIIVO's efforts here
would flow back to say, GOSPLUN, so that some of the modeling work here might
suggest that industrial expansion should not take place on this river basin
but elsewhere. I understand that your law states that any pollution is
essentially illegal; we have the same law in the United States—the 1899
Refuse Act. I understand the legal implications, but I see no difference in
introducing the charge for water use and introducing a charge for effluent.
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MODELING AND DECISION-MAKING ASPECTS
OF WATER QUALITY MANAGEMENT
Mark A. Pisano
INTRODUCTION
Several of the papers presented at this symposium discuss the manner in
which mathematical modeling components and approaches may be utilized in the
solution of complex water quality problems. The papers also discuss several
key economic and political concepts that are used in the water quality manage-
ment planning process of the United States. Finally, the papers discuss some
specific case study applications of these modeling approaches, as well as how
some of the key planning concepts were employed in a particular area.
This paper will present an overview of how the above approaches and con-
cepts have been employed in the United States in implementing the Federal
Water Pollution Control Program. Modeling, water quality analysis, and com-
prehensive planning approaches have played a very fundamental role in the
implementation of our national program. This paper will describe that role
and provide an assessment of modeling as it has contributed to our water
quality management decisions.
Historical Background
To properly understand the key role of modeling in the decision-making
process established under the Federal Water Pollution Control Act of 1972,
it is necessary to understand the basic approach embodied in this statute.
The central thrust of the Water Program is that control measures them-
selves, that is, the application of technology, will establish the basic level
of water pollution control. Through successive improvements in technology,
eventually approaching a target of zero discharge, wastes to the environment
would be progressively eliminated. To a large extent the raison d'etre of
the program is a reliance on continuous advancement in abatement technology.
The program is based on the assumption that any waste discharge to water would
potentially upset the natural integrity of water; therefore, any increase in
the removal of pollutants is desirable.
The ;tatute also recognizes that the uniform application of technology at
any one time cannot guarantee the appropriate level of control necessary to
ensure that every water body would be adequately protected. Similarly, this
statute also recognizes that over time the application of higher levels of
technology might not be necessary to achieve the desired level of health and
environmental protection. Thus, the concept of a control level dependent upon
the desired in-stream beneficiary uses of a water body and criteria needed to
180
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support those uses is still maintained. Furthermore, the statute stipulates
that if this beneficial use cannot be protected through the application of the
technological standard established for that period, a higher level of treat-
ment technology can be required. On the other hand, if over time it is found
that it is not technologically feasible and/or economically practical to move
to zero discharge, then the water quality standards, in conjunction with
beneficial uses, can be used to determine a less stringent level of control.
There are two basic approaches for establishing levels of control in the
United States. The first is effluent guidelines which function like perform-
ance specifications defining what treatment technology can do and when the
technology can be implemented. The second is the intended water uses for a
specific water body. Models and water quality analysis are the vehicles em-
ployed to determine the needed control levels to protect these uses. Thus,
models and water quality analysis become the critical factors in determining
the levels of control to be established on industrial and municipal discharg-
ers.
There are two basic approaches contained in our statute for establishing
levels of control, i.e., technology established limitations and water quality
established limitations. There are different rationales and basic assumptions
employed in each approach. The technology driven approach is based on the
assumption that the problem cannot be solved without technology and, moreover,
that a technological solution is administratively easier to implement. Cause
and effect analysis, or water quality modeling analysis, is not needed in this
case, rather a simple demonstration of the existence of technology that is
adequate to establish a control level. The water quality analysis approach is
predicated on the basis that in some instances technology may not be adequate
to protect beneficial uses, while in other cases the application of technology
might not be cost effective. A key weakness of this approach has been the
historical difficulty in demonstrating cause and effect analysis.
Prior to the passage of the 1972 statutes, an analysis was performed of
the total municipal and industrial control costs that would result if in-
creasing levels of technology were applied. The figures in Table I indicate
that a five-fold cost increase could result from going from secondary treat-
ment to a zero discharge. An assessment of the cost of doing individual water
quality analysis and tailoring the control level to the level needed to pro-
tect beneficial uses was not made. The concluding section of this paper does
provide some insights into this issue.
Water Quality Analysis Approach
The first step in determining control levels using water quality analysis
is to establish the goals or intended water uses for a specific water body.
These goals, embodied in the term "water quality standards," are established
jointly by the states and the federal government. An underlying objective
employed in the establishment of these standards is the achievement of fish-
able and swimmable waters where attainable. There are several key tasks to
be performed in determining the standard. The first involves an evaluation
of the biological criteria needed to support a fishable ecosystem—a balanced
system of fish, shellfish, and wildlife—and an analysis of the health
181
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criteria needed to permit swimming in that particular water body. This eval-
uation seeks to make a determination of specific biological and chemical
criteria needed to protect the intended uses. This evaluation includes an
assessment of the natural background conditions for this given water body.
If there are higher natural levels of pollutant constituents that would pre-
clude an intended use of fishable or swimmable waters, then a lower use can be
established. Additionally, if there are technological, economic, or institu-
tional factors that would preclude the attainment of a fishable or swimmable
level, then a lower use can also be established. In order to evaluate the
above relationship between implementation, feasibility, and intended water
use, it is necessary to be able to first determine a control strategy, and
secondly to be able to relate the effects of this control strategy with the
level of water quality. Water quality analysis, including modeling considera-
tion of alternatives, is the tool employed to provide the decision-makers
(local, state, and federal officials) with the information needed to determine
the attainable beneficial use and the necessary control program.
TABLE I. TOTAL NATIONAL COSTS* (Dollars in Billions)
Dischargers'
level of
removal
100%
95-99%
85-90%
10-Year
capital
expenditures
94.5
35.3
17.6
20-25 Year
operating
costs
220.0
83.5
43.2
Total
expenditures
316.5
118.8
60.8
Annual i zed
costs in
1981
21.1
8.4
4.1
-'Excludes $12.0 billion costs for intercepting sewers.
Water Quality Analysis
The starting point for the analysis is water quality standard generally
predicated on the level of fishable/swimmable water, unless natural background
conditions precluded this use. The analysis would be performed for each pol-
lution parameter which is considered important in order to protect the in-
tended uses. The analytical procedure for performing this analysis is out-
lined in Figure 1. Each source contributing that certain pollutant to the
water body should be identified and alternative structural and nonstructural
corrective measures identified. The final treatment control strategy for
this segment should reflect a combination of control methods which will meet
water quality standards for all water quality parameters.
As noted in Figure 1, modeling is generally the appropriate method of
ascertaining the effects of proposed alternative abatement strategies. To
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Determine DO
Standard
Select
Background
DO Deficit
Water Quality i
Model f"
Input "Secondary"
Treatment Level
@ Each Point Source
Meets DO
Standards
(Including Background)
YES
NO
Allocation as
Given by Best
Practical Treat-
ment
Increase Dis-
charge to Level
Before Standard
is Violated
Increment Dis-
charger by Dis-
crete Level for
Total Segment Loads
Water
Quality
Model
_y
"Equivalent"
Reserve Capacity
Reserve Capacity =
(R) Max. Allowable
Check for
Standard
Attainment
Maximum Allowable
Discharger Load
Select Reserve
Factor (R)
Allocation = R x
~I
Maximum Allowable I
\
f
Check for
Upper Bound
Constraint
Figure 1. Suggested Allocation Procedures (BOD-DO Example).
183
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choose the proper analysis, modelers consider the following factors:
Physical Sytems: Rivers, lakes, and estuaries obviously have unique
characteristics which must be addressed. Each system is to be represented as
either one, two, or three-dimensional and in either dynamic or static time
frames.
Complexity of Problem: Water quality concerns arising from dissolved
oxygen depression, temperature, eutrophication, toxics, or biological contam-
ination will each dictate a different approach.
Available Resources: A basic practical concern is the amount of time and
money which is available for data gathering and analysis. The dependence of
planning programs on modeling results requires time-phasing, adherence to
schedules, and initial accurate estimates of what can be accomplished in the
available time period.
Data: Acquiring a sufficient amount of data that is required to make the
decision at hand must be determined. Collecting more data than is needed in
order to make a decision is often a pitfall in the analysis. The problem that
needs to be solved should be defined and limited before the data collection
effort is initiated.
Environmental Risks/Potential Investment: In choosing the level of
analysis, the consequences of making a wrong decision and the expenditure
resulting from this decision should be considered in determining the actual
level of analysis.
The above factors are to be considered in selecting the type and level of
analysis to be performed. The basic objective of our program is to keep the
analysis as simple as possible and to perform analysis only to the extent
needed to support either the water quality standards decision or to deter-
mine the level of control.
The criteria summarizing the selection of an analytical technique are
shown in Table II. The table lists each criterion mentioned above for the
four levels of complexity and presents the type of problems in water bodies
for which that level of analysis is appropriate, the planning complexity
associated with that level, and the time required for the study.
Type A: Simplified analysis generally has relatively few waste sources,
a lack of multiple stream effects, and the assumption is made that there are
no changes in waste loading and hydrology with time (steady state).
Type B: Steady state linear kinetics has several sources which have
overlapping stream effects, steady-state conditions still hold (i.e., no
change in waste loading and hydrology).
Type C: Transient linear kinetics has multiple sources, which have over-
lapping stream effects. Waste loading and hydrology vary linearly with time
(e.g., storm events, diurnal variations, and highly variable waste discharg-
ers) .
Type D: Time variable and nonlinear kinetics have multiple sources which
have overlapping (two-dimensional) stream, lake, and estuary effects. The
internal mixing process represents a major problem.
There are a number of issues related to the model application which are
important and emphasize the need for judgment and understanding. There are no
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TABLE II. CRITERIA FOR SELECTION OF TECHNIQUES
Model complexity
Water quality Problems
and variable Water body
Simplified
analysis
(Type A)
Steady state
linear kinetics
(Type B)
Transient linear
kinetics analysis
(Type C)
Time variable
non-linear
kinetics analysis
(Type D)
D.O. (carbon &
nitrogen)
a) D.O. (carbon &
nitrogen) tempera-
ture & nonpoint
source;
b) Anticipated or
existing water qual-
ity problems
a) Time varying D.O.
nonpoint source anal-
ysis & temperature.
Simple eutrophication
analysis. Full storm
water overflow analysis;
b) Water quality
problems
a) Detailed eutrophica-
tion analysis et al.;
b) Water quality
problems;
c) High growth of
area projected
One-dimensional
streams &
estuaries (com-
pletely mixed)
One or two-
dimensional
streams, estu-
aries, rivers
lakes
Rivers, lakes,
& estuaries.
One or two-
dimensional
All bodies of
water
Planning
characteristics
Time required
for study
a) Low risk of
capital and/or
environmental
quality degrada-
tion;
b) No alternate
strategies &
control options
available
a) Low to moderate
risk to capital &
environmental qual-
ity degradation;
b) Alternative
strategies & con-
trol options must
be available
a) Moderate to high
risk of capital &
environmental qual-
ity degradation;
b) Alternative strat-
egies & control op-
tions must be
available
a) High risk of
capital and/or en-
vironmental quality
degradation;
b) Varying strategies,
control be available
Days to weeks
2 to 9 months
6 to 24
months
12 to 36
months
-------
simple answers to these issues, most of which stand unresolved. However,
these issues must be addressed either explicitly or implicitly before recom-
mendations on specific control programs can be made.
Definition of the proper design conditions: The model design must ac-
count for extreme or critical stress conditions that will affect the recom-
mendations of the analysis. For example, for stream analysis a low flow/high
temperature period is usually chosen, such as the 7-day duration, 10-year re-
currence low flow. In certain water bodies an analysis based on the 7-day,
10-year time period may not be protective of sensitive biological systems. If
problems are associated with winter ice cover or seasonal discharge activity,
then a different design flow would be appropriate. The wet weather design
conditions are critical for storm water pollution analysis. Inasmuch as the
pollution potential here is not only dependent on the intensity and duration
of rainfall but also on the antecedent conditions, simple rules are elusive.
A better approach to determine the appropriate decision rules for such an area
would consider the treatment costs and the corresponding probabilities of a
stream standards violation causing varying losses of beneficial uses.
Definition of mixing zones: If the discharge is of a different quality
than the receiving water, then at the point of discharge there will be a zone
of differential water quality called a mixing zone. The size of the zone may
vary, but if the zone becomes too large, then the beneficial uses for the
river could be impaired.
Definition of permit conditions: Once an individual effluent limit has
been established, it is transferred to a permit which is legally enforceable.
Treatment plant performance is based on a number of factors, and plant per-
formance can be quite variable. The permit condition may be expressed as a
maximum discharge concentration never to be exceeded, or as a daily or weekly
discharge average.
The validity of the criteria: The goal of our efforts is to improve or
preserve beneficial water uses. Sometimes it may be desirable to step back
and consider the criteria which are assumed to justify use classifications.
An example would be a coliform bacteria limit for swimming. If the limit is
exceeded during rainfall periods, it might be reasonable to ask whether anyone
would swim at this time anyway.
Should there be time variable effluent limits: In a case where effluent
limits are based on water quality, it is reasonable to argue that the allow-
able loading should be based on the existing assimilative capacity of the
water body. For some pollution constituents this assimilative capacity may
vary considerably from season to season.
Definition of the safety factors: Some of the above-mentioned issues
such as design conditions and permit expression can be considered in the con-
text of safety factors. During the course of the analysis, numerous assump-
tions must be made and the question arises if in our concern for safety we
have been too conservative in our estimates. That is, what level of risk are
we willing to accept?
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The above are some of the issues which may be addressed as decisions re-
garding specific enforceable effluent limitations made from model results.
Experience to Date with_Water QualityAnalysis
As we initiated this program, there was a realization that water quality
analysis could not be performed simultaneously throughout the entire country
if the legislative mandates contained in the statutes were to be achieved. As
a first step, the nation's waters were divided into two kinds of segments:
effluent limited segments, where established effluent limitations would
achieve water quality standards; and water quality limited segments, where the
effluent limitations alone would not achieve water quality standards. The
most complex and challenging work is on the water quality limited segments.
With the states providing the lead, some 2,000 water quality limited segments
were identified. Our objective for these segments was to perform intensive
water quality analyses, which were to establish the necessary levels of pollu-
tion control for these segments. The states have completed approximately
1,700 segment analyses since the initiation of this program. The remaining
segments have more complex problems that require further time for analyses.
Although the sophistication and technical expertise of the different states
varies greatly, the last few years have seen an increased acceptance of the
modeling approach for setting effluent controls.
Water Planning Division Case Studies
An analysis was performed using water quality models to set control
levels for five water quality limited segments with severe dissolved oxygen
problems. The purpose of the analysis was to study the effect of the differ-
end methods of establishing the control levels including the effect of differ-
ent waste load allocation procedures. Segments were chosen to represent a
variety of hydrologic and waste-flow conditions. The segments are listed in
Table III, along with the average yearly stream flow in cubic feet per second
and the municipal and industrial contributions in millions of gallons per day.
The basic methodology employed in the analysis was to use a model veri-
fied for each stream segment (which in all cases was a Type B steady state,
linear kinetic model) and perform a water quality analysis of the effect of
daily waste-source inputs into historical stream-flow conditions. The analy-
sis enabled us to identify the impact of various waste treatment plant options
on the water quality levels.
The first step in the procedure was to determine the generalized daily
variation of treatment plant effluents, derived from historical data from
seventeen plants in Michigan and Texas. The treatment costs for these plants
were based upon cost curves for treatment processes as a function of the waste
flow. The costs included amortized construction, operation, and maintenance.
The historical daily flow records for each segment were available from the
Geological Survey for the last 22 years. Daily waste loads were generated
and, with stream flows, information was used as inputs to the verified model
to determine the impact on dissolved oxygen conditions. The next step in the
analysis was to analyze different possible policies in terms of cost and
effectiveness. The effectiveness of policies was evaluated in terms of the
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TABLE III. SELECTED RLVERS FOR CASE STUDIES
Segment
Location of River
Cubic Feet
Per Second
Municipal
(million gals
per day)
Industrial
(million gals
per day)
Chattahoochee
Georgia
3,543
108
Flint
Michigan
344
10
CD
CO
Cache La Poudre
Colorado
130
Reedy
South Carolina
84
15
Upper Mississippi
Minnesota
12,087
250
-------
probability of meeting a stream standard of five milligrams per liter dis-
solved oxygen (DO). The three levels of control of waste sources were
secondary treatment or best practical treatment (B.P.T.) for industry, best
available treatment (B.A.T.) or a comparable level of municipal treatment,
and a level of control that was determined by the necessity to meet the in-
stream criteria of five milligrams per liter of DO. For each of the control
levels, a total cost curve was derived.
From the results of the analyses some conclusions
may be derived. For these segments, the cost of B.A.T. treatment would be
approximately equal to the total cost associated with determining the effluent
control level based on the water quality standard. The total cost of the
waste load analysis for these segments, based on a 7-day duration, 10-year
recurrence critical flow condition, ranged from about 30% to 50% greater than
the total cost of the secondary treatment effluent. At secondary treatment,
the percentage of in-stream violations for these water quality limited seg-
ments range from 1.2% of the time on the Chattahoochee River to 18.8% of the
time on the Flint River, while at the waste load analysis level, the percent-
age of in-stream violations is less than 0.5% in all cases.
A major limitation in the above study is that the analysis only consid-
ered the parameter of dissolved oxygen. It is realized that DO cannot be used
as the sole criteria for determining the biological integrity of a water body.
Aside from the fact that DO may not be the best indicator of the biological
integrity, reliance or? this parameter alone fails to indicate the associated
control level of solids, metals, and other toxics that are derived from more
advanced levels of treatment, and it is these parameters that are likely to
cause many water quality problems.
National Commission on Water Quality Evaluations
The National Commission on Water Quality was established to assess the
implementation of the nation's water bill. It also performed an analysis on
the relationship between alternative control levels and water quality condi-
tions. The Commission looked at 21 rivers covering approximately 4,600 river
miles. At the present time, 380 river miles or about 8% of the total do not
achieve the dissolved oxygen standard of 4 milligrams per liter. This mile-
age is decreased to 36 miles of 0.8% or the river miles studied through the
application of secondary treatment or B.P.T., and 23 miles or less than 0.5%
after the application of B.A.T. These results are demonstrated in Figure 2.
Summary and Conclusions^
The above evaluation indicates the importance of water quality analysis
in determining levels of control. Of the 5,500 water quality segments identi-
fied in the United States, approximately 2,000 were identified as having water
quality problems. Models were applied in the majority of these cases and the
control levels needed to meet water quality standards were established.
The case studies performed by the Water Planning Division and the find-
ings of the National Commission on Water Quality confirm that higher levels
of treatment control are not required in all places. Our estimate is that
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35 to 40% of the nation's waters would require a higher level of control than
provided by secondary treatment of base level industrial treatment require-
ments. The above case studies also indicate that where water quality problems
exist, best available treatment (B.A.T.) will generally solve the problem.
Thus, we find that uniformly higher levels of waste treatment applied
across the nation would not necessarily be the most effective way to proceed
in the near term future. Appropriately applied modeling and water quality
analysis will assist us in determining those areas where higher levels of
treatment will be necessary as well as those areas where such treatment will
not be required.
Concerning the above general conclusions, it must be noted that only
dissolved oxygen was considered in the case studies, and it is generally re-
cognized that reliance on dissolved oxygen alone may not be the best method
for indicating the desirable level of control for a water body.
It should be further noted that the above case studies do not consider
many of the nonpoint source problems, nor do they consider the problem of
toxics and other heavy metals. The refinement of these tools and the expan-
sion of our capability to handle both the transient problems introduced by
storm water and wet weather flow problems, as well as the toxic elements, must
be pursued if modeling techniques and water quality analysis are to be of
further assistance to us in analysis. With explicit recognition of these
additional problems, there is little doubt that we can enhance and improve our
capabilities in this area.
DO
Milligrams Per Liter
101
EOD
Seasonal Low Flow
10% 20% 30% 40°.{
Percent of area with DO less than level shown
J I I L
50%
500 1000 1500 2000 2300
River miles with DO less than level shown
Figure 2. Dissolved oxygen improvement: 21 water bodies, 4600 river miles.
190
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DISCUSSION
THOMANN: Mr. Pisano has just presented the various aspects of the
researcher's responsibility to administrators. I want to use this tribunal
to emphasize points which illustrate an administrator's responsibility to
researchers:
1. Be sincere and listen to the recommendations we give you, even if
you don't like them;
2. Relate to us with understanding. We also don't know everything,
nor do we know how to do everything;
3. Be flexible in the use of your resources when introducing a policy
recommended by the research community;
4. Realize that you are responsible for decisions made, not us;
5. Be magnanimous towards our mistakes.
DELUCIA: I want to add one point to Dr. Thomann's requirements: Give us
enough time to fulfill the tasks before us. I want to remind Mr. Pisano,
the economist, who has reproached researchers for the long continuance of
their research, of the words of Mr. Pisano, the administrator, that a
researcher should not err and his information should be sufficiently complete
and detailed.
I would like to ask Mr. Pisano, the economist and administrator, to
comment on the reasons for the improvement in water quality which has b^en
observed over the last few years in the United States. In particular, I
would like to hear about the role of descriptive, and especially prescriptive
models. In addition, I would like to ask him to comment on the role of the
United States' legislation and other aspects of influence emanating within
the planning framework.
PISANO: I want to answer Dr. DeLucia's question from an administrative point
of view. I already spoke about the complexity of the dilemma of choosing
between the continuance of research and its infallibility. Everything in
this world is relative; there are no absolutely good or bad decisions. All
decisions, expressed when conducting determined reforms, possess a certain
supply of stability regarding their relation to minor unexpected influences.
Administrative personnel cannot understand all the nuances of modeling.
Researchers should and can economize their time and efforts, indicating
details and the basic results of decisions made, and determining the area of
model application where the applied simplifications do not lead to mistakes.
That is, the researcher should indicate to the administrator on what he
should concentrate his attention when analyzing proposed decisions, what
dependencies are significant in the model, and what are secondary.
191
-------
In response to Dr. DeLucia's last request, the more we learn about the
equations we are studying, the more we recognize the complexity of the
problem of preserving the purity of the water environment and how insufficient
are our efforts. Existing U.S. legislation, by special act of Congress,
underwent an evaluation, from the point of view of effectiveness, using
mathematical models. Dr. Thomann and Dr. DeLucia directly participated. I
must note that we still know little about what is useful for such an
evaluation. It is necessary to continue developments in this direction, and
in the direction of changing the law.
Allow me, in conclusion, to express the hope that Soviet-American
cooperation on this question will be useful for our country. I will emphasize
several concrete areas of research which present considerable interest in
this connection: development of optimization models, the fight against
pollutants entering with unconcentrated flow, and evaluation of their
influence.
SUKHORUKOV: I would like to make a few remarks on responsibility during
the decision-making process. Each model is composed of three essential
parts: 1) physical bases, 2) strict administrative limitations, and 3)
criteria for evaluating the quality of the decisions. The responsibility of
the researcher is dependent on the accuracy of his model. The administrator
carries the responsibility for the limitations imposed by him. In addition,
both must strive towards mutual understanding and implementation of the
work.
Unfortunately, Mr. Pisano did not introduce into his presentation his
classification of models by complexity. In the report, he proposed simple
and quick models in small risk situations, but, for more complex and
difficult cases, the preliminary research period can reach one to three years.
It would be possible to develop the model's suitability for a specific case on
the basis of the "information value method" which is well known in cyber-
netics .
BELOGUROV: In Mr. Pisano's report, two approaches to the problem of water
protection were discussed. They both lead to a decrease in the discharge
of polluting substances. In the Soviet Union, one more approach is being
developed. It is connected with the redistribution of a given mass of
pollutants dependent on the change in the assimilating ability of the water
body. We call this water quality regulation. It is possible, for example,
to regulate the discharge from tanks, to change the capacity of aeration
facilities, and to change the graphics of discharge from water resources. It
is even possible to change the level of wastewater treatment during the normal
working processes of the treatment facility.
At VNIIVO, they are estimating the cost of decreasing the maximum
concentrations of pollutants in a water body by decreasing the general mass
of discharged substances and by redistributing the discharges. The results
show that the latter is 4:11 times cheaper. In the Soviet Union, a lot of
attention is given to this approach.
192
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DETERMINING WATER QUALITY CRITERIA FOR WATER FLOW IN
SOLVING THE PROBLEMS OF CONTROLLING WATER POLLUTION
Kh.A. Velner, V.I. Gurariy, A.S. Shayn
INTRODUCTION
In the Soviet Union a great deal of attention is paid to the protection
of the environment from pollution, and in particular, to the problems of sur-
face water quality. The improvement of water quality is considered one of the
most important economic arid political problems at the present stage of social
development. The introduction of methods for the objective assessment of wa-
ter quality could be an effective key factor in solving this problem.
To improve water quality, it is first necessary to define it quantita-
tively. The problems of organizing and introducing a state system of quality
control that includes problems of planning, forecasting, optimization, and
others must be solved by developing objective methods to evaluate water qual-
ity control.
Experience shows that information based only on water quality evaluation
by individual properties does not provide an adequate solution to the problem.
The necessity for developing a complex estimate of water quality is becoming
more and more evident.
Laws, developed and promulgated in the Soviet Union, set maximum permis-
sible concentrations (M.P.C. or water quality criteria) for harmful substances
in impoundments and water courses, and define the water quality in a reservoir
for drinking water, recreation, and fish breeding water uses (Cherkinskiy et.
al., 1947, 1961, 1971, 1975).
However, some deficiencies have been found in these existing criteria for
water quality assessment from the design and maintenance of water installa-
tions .
1. The M.P.C. is not a single determined value that corresponds to some
threshold change in the quality of a substance, for example, the temperature
threshold at which water boils. Therefore, the M.P.C. should be considered
as an organizational artifact rather than a scientific criterion.
2. The M.P.C. has been defined for a large number of parameters, al-
though they cannot be used to compare the qualitative conditions of water.
Two qualitative conditions are identified within the limits of the criterion;
for example, crystal-pure water and water at the limit of the criterion for
acceptable water quality are very distinct, yet they have an identical quality
193
-------
under M.P.C. concepts. However, two waters with similar water quality or with
an index or parameter value outside the M.P.C. value are considered distinctly
different.
M.P.C. criteria allows only opposing estimates: "yes" or "no," accepta-
ble or unacceptable. The qualitative condition of water even slightly exceed-
ing the M.P.C. in a single parameter is not considered. This can lead to un-
necessary fiscal penalties for solving a number of economic problems for water
pollution control.
3. In addition to limits for the value for each parameter, complex
limits must also consider the effect of the combined presence of several sub-
stances. However M.P.C. concepts do not consider potential synergistic ef-
fects. These considerations are the most complex and require long-term com-
plex investigations.
These considerations require the development of some new criteria, in
addition to M.P.C., to guarantee the purity of water bodies. At the initia-
tive of the Central Board of the Hydrometeorological Service of the Council
of Ministers of the USSR (A. Israel), the problem has been formulated, and the
development of the concept of a maximum pollutant discharge (M.P.D.) into
watercourses and water bodies has begun. These M.P.D. (concurrently with
M.P.C.) must be accepted later as state standards. It is evident that M.P.D.
depends not only on environmental conditions, but to a great extent-on tech-
nical economical prerequisites that may have particular problems pertinent to
different pollution sources. The M.P.D. will be developed stage-by-stage on
the basis of available technologies for pollution control in the state. The
M.P.D. will have a complex character, i.e., to provide an integral evaluation
of the quality of discharged waters.
Further, it is necessary to develop the maximum permissible biological
or ecological degradation, the M.P.B., to evaluate the water quality of water
bodies. The problem is complex and requires detailed investigation. At the
discharge area for waste waters with primary treatment, the biological degra-
dation in the water body should not exceed the permissible values [ABi<(ABjJ ]
so that the ecological equilibrium is maintained in the water body.
It must be emphasized that at present, the M.P.C. plays a very important
role as a simple and convenient qualitative indicator of the permissible limit
for contamination of water for sociolegal purposes. However, for a number of
water economy problems (comparison of water quality, complex use and protec-
tion of water, water quality management, mapping of watercourses, and various
scientific studies), it is necessary to develop additional criteria with a
more comprehensive evaluation range.
The concepts of water quality criteria have a long history, but only
recently both Soviet (Drachev and Bylinkina, 1964; Velner and Saava, 1970;
Mitskene, 1974; and Gurariy and Shayn, 1974) and foreign authors have made
attempts to evaluate water quality by a pollution index number to indicate
total quality. The author is familiar with the publications of the American
authors in the field (Brown, 1970, 1972; Harton, 1965; Prati, 1971; Markins,
1974; and others). Initially, Brown (1970) advocated developing a single
194
-------
global index of water quality; but he recently concluded that water quality
indices should be developed for different water uses (1972)-
The essential problem, when developing complex estimates of water quality,
is the choice of the primary indicators that scale the indices and methods of
aggregating estimates.
Depending on the problem formulation and the objectives of the investi-
gators, sets of factors (5 in Drachev, 1964, to 13 in Prati, 1971) and systems
of evaluation will differ. Bylinkina et al. (1964) were among the first to
propose a complex evaluation of water quality in a water body by the physical,
chemical, bacteriological, and hydrobiological pollution indicators. This
evaluation has quantitative indicators developed in detail for each parameter.
The proposed classification underlies the surface water purity criteria ap-
proved by the Council of Chiefs of Water Management Agencies of the CMEA in
1963.
Great attention is paid to the evaluation of water quality in Poland,
Czechoslovakia, the German Democratic Republic, and Hungary. These evalua-
tions are performed by using groups of categorized indicators (by the worst in
a given group, by the average, etc.), and different types of classifications
including from four to six classes.
The existing evaluation criteria represent a set of methods that are al-
most always badly substantiated, although effective in some cases. Their
essential disadvantage is the subjectivism often exercised in the choice of
the main indicators.
Therefore, a theoretical basis should be developed for complex evalua-
tions of water quality to provide a unified approach to the problem and reduce
the element of subjectivism as much as possible.
In general, the main problems of the criteria theory relate to:
1. describing the n-dimensional area &, of acceptable water quality con-
ditions (Xl5...Xn);
2. choosing the class of evaluation K (in the,general case K is the set
of vectors with a dimension less than n);
3. formulating the evaluation itself U(X) depending on the kind of water
usage V, i.e., determining the operator U(X), which mapsfiinto K, Uy(X)
Criteria developed for evaluating water quality can solve the problem of
comparing N qualitative conditions of water by n-indicators. Let us assume
that each of the main factors can be evaluated during successive water quality
examinations; then each qualitative condition of water can be determined as
a point in n-dimensional space.
The problem therefore is to establish the best water quality conditions,
the next best, and so on, to provide an ordering of the objects under inves-
tigation. To solve this problem, it is necessary to obtain a general estimate
of the water quality on the basis of evaluations by separate factors (in other
words, to produce an aggregation of estimates).
195
-------
The solution is determined by using Fishburn's (1968) theory of utility.
This approach assumes the existence of a single scalar value function UOO for
condition evaluations by factors that can produce a complete ordering of ob-
jects. Function UQT) substitutes a scalar estimate for each set of factors
(water quality index, W.Q.I.). The ordering problem is then simply solved.
In this present paper, as in previous studies, it is generally accepted that
the function U(X) exists for any formulation of the decision-making problem.
At the same time, there are a large number of opinions and theories on this
type of function, i.e., the method for aggregating estimates.
The application of separate methods for aggregating estimates is no sub-
stitute for the decision-making exercise, but is used only to acquire addi-
tional information. This fact is determined by the very character of the
problem.
The purpose of aggregating the estimates is to attempt a rational solu-
tion of a problem that is difficult to formalize.
Generally speaking, the decision-making procedure is not purely mathema-
tical in character, but has certain psychological attributes. Recent methods
of aggregating estimates take these aspects into account (Byunayun et al.,
1971). As their principle feature, these promising methods require that the
person defining the problem and using the results (a person responsible for
the decision) is engaged in some way in the process of aggregating the esti-
mates .
Complex estimates of water quality are very complicated to determine, and
their strict and complete formulation is impossible at this time. The imme-
diate problem is to reduce subjectivism to a minimum at all stages in the
development of a complex estimation.
Velner, Kask, and Saava, specialists at the Tallin Polytechnical Insti-
tute, together with members of the Ministry of Soil Reclamation and Water
Management of the Lithuanian SSR (Daubaras and Mitzkene), proposed a system
of pollution classification for water bodies that has been accepted for the
Baltic region. The water quality evaluation is carried out by three groups
of indicators: the first evaluates the general sanitary state of the water
body; the second, the presence of harmful substances; and the third, the level
of pollution by microbiological indicators. The water quality, determined
during the rated low runoff conditions of a river, follows the accepted stand-
ards and the improved classification of river waters of the Baltic region,
polluted mainly by organic substances without a marked toxic effect.
For each group of indicators, the critical parameter which furthest ex-
ceeds the M.P.C. is determined. The integral quality of water of a river (an
index) is determined by the estimated factor of the critical group, taking
into account the category (kind) of water usage (Table I). This "limiting
condition method" gives a strict estimate of water quality. It was tested by
the Ministries of Water Management of the Lithuanian and Estonian SSR in an
evaluation of the Baltic region rivers.
196
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TABLE I. APPROXIMATE CLASSIFICATION OF WATER QUALITY IN WATER BODIES
Pollutant Measurement
group Quality factors units
Organic Dissolved oxygen mg/£
pollutants Ammonia mg/&
(I) BOD at 20°C mg/£
Dichromate oxidation mg/£
Saprobic zones —
Clean
5
>6.0
<0.1
<2.0
<26.0
mostly
oligo-
saprobic
Level of water quality
Slightly
polluted
4
>4.0
<0.2
<5.0
<32.0
mostly
saprobic
Moderately Very
polluted Polluted polluted
3 2
>3.0 >1.0
<0.5 <1.0
<8.0 <20.0
<40.0 <65.0
from mesosaprobic
to ctmesosaprobic
1
<1 0
>1.0
>20.0
>65.0
from amesq
to poly-
saprobic
Toxic Phenols, oil
substances products, cyanides,
(II) mercury, etc.
ratio of actual
to standard max-
imum permissible
concentration
<5.0
<10.0
>10.0
Pollution Coli-titre, number
introduced of colonies, etc.
by micro-
flora
(III)
units/m£
_>1.0 >0.1-1.0 >0.01-0.001 >0.001 £0.001
<103 <10Lf <105 <106 >106
-------
Gurarly and Shayn (1974) in VNIIVO are conducting research on the develop-
ment of methods for constructing water quality indices for separate kinds of
water usage. The W.Q.I, for municipal water usage has been developed. Its
analytical-expert method adequately eliminates the subjectivism in water
quality estimation inherent in other methods and takes into account all avail-
able information on the values of the parameters (not only the worst indicator).
The development of the W.Q.I., as well as other complex quality estima-
tions, is divided into two stages: 1) estimation of the simple properties; and
2) estimation of the complex properties and the quality as a whole.
The algorithm for obtaining the water quality index takes the form:
STAGE I
Determination of water usage conditions
^
1. Construction of a hierarchical diagram of water indices necessary
and sufficient for estimating its quality ____ _ _____________
;zzmi777~r r ..........
; 2. Designation of the measurements units for each index
" "
3. Selection of basic values of indices for comparison
4. Determination of the type of dependence between the index values
and their estimation
5. Computation of estimates of separate indices
STAGE II
6. Selection of the dimension scale for a complex estimate
;_. ....... ___ .......... . ..................... zzzzzzzzzzi
7. Selection of method for determining the weighting of the indices
I
8. Selection of method for aggregating estimates of separate indices; i
to obtain complex quality estimates I I
9. Computation of a complex water quality estimate
10. Analysis of computed quality estimate and decision-making i
198
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Each operation of the algorithm was performed separately. In considering
operations 1, 2, 4, and 7, the expert estimation method was widely used.
The complex estimate of water quality is constructed on the basis of two
indices: a general sanitary index including general sanitary parameters es-
sential for judging the water quality and, partially, the river regime, and
an index of specific polluting substances that takes into account harmful sub-
stances present in the water.
VNIIVO's mathematical modeling laboratory distributed a questionnaire
among experts (the inquiry was carried out in two steps: the second step cor-
rected the results of the first). In the first questionnaire, a group of 50
water pollution experts from the country's leading institutions selected the
most significant parameters, defining river water for municipal water usage,
from a list of 38 parameters (general sanitary indicators and water body pol-
lutants) and evaluated each according to a five-point system (5-max, 1-min).
The questionnaires were processed: The mean statistical value for each
parameter, the frequency of possible maximum estimates, sum of ranks, and also
the probability characteristics, variance coefficient, concordance coefficient,
and the statistical significance of the level of agreement among the experts
were determined. Proceeding from the mean statistical analysis, ten general
sanitary factors (Table II) and oil and phenol—the most widespread pollu-
tants—were chosen as the most significant indices.
TABLE II. WATER QUALITY ESTIMATES
Parameter
Coli-index
Sign, points
BOD5, mg/£
PH
Dissolved
oxygen, mg/£
Color
Suspended
substances ,
mg/£
General min-
eralization,
mg/£
Chlorides,
mg/£
Sulphates
me/£
Weight
H
0.18
0.13
0.12
0.10
0.09
0.09
0.08
0.08
0.07
0.06
5
0-100
0
0-1
6.5-8
<8
<20
<10
<500
<200
<250
4
100-1000
1-2
1-2
6.5-8.5
8-6
20-30
10-20
500-1000
200-350
250-500
Point, W
3
103-105
3
2-4
5-9.5
6-4
30-40
20-50
1000-5000
350-500
500-700
2
105-107
4
4-10
4-10
4-2
40-50
50-100
1500-2000
500-700
700-1000
1
>107
5
>10
<4>10
<2
>50
>100
>2000
>700
>1000
199
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The rank correlation coefficient (concordance coefficient) computed by
the results of the first questionnaire (it is an index of the experts' agree-
ment level) had the value W=0. 725 and was indicative of good agreement (W>0.5).
Confidence in the experts' agreement level was tested by the X2 criterion.
Calculations showed that the estimates of the experts sufficiently coincided
and the results of the inquiry were authentic.
The weight of the chosen parameters was calculated from the formula
si ° -i
1 1SP
p=l
where S. is the sum of ranks set by the experts for the i-th parameter.
S = l * (2)
where -Y .. is the estimate rank set for the i-th parameter by the j-th expert,
IT
m is the number of experts.
The results from the second set of questionnaires confirmed the results
of the first set and allowed the authors to establish the dependence of the
water quality estimates on the pollutant concentration. The water quality
scale is a five-point system: 5—corresponds conditionally to very clean water,
4—clean water, 3—moderately polluted, 2—polluted, 1—contaminated. Each
interval of values in the columns is considered as a semi-interval of the
type (a,b).
For normalized parameters, the superior limits of the interval correspond-
ing to point 4 are the existing standards for municipal water usage.
To account accurately for parameter values exceeding standards (which cor-
responds to point values of Wi=3 and less), we also introduce the so-called
"penalty" function n
F = n n. (W-L,•*.[_).
1=1
The value n^ should decrease with the decrease of W-j_. With equal quality
levels W.p the value ^ should decrease if the relative weight •*-[_ increases.
In addition, the function should be normalized (the values should not exceed
a unit). It is convenient to take function ^ as:
.1 W± > 3 (3)
where 3^. is the relative weight of the coli-index maximum for all -j . .
200
-------
Thus, the formula for the general sanitary index Ta is of the form:
Ta =
n n
1=1
1=1
n-i (W,- , t-i'
1 x 1 ' -L-
n
0 qp<5
qH °r qF = 5 (7)
If the number of ingredients in the pollution index normalized by one
toxic factor is equal to n, then generally,
i = l,...,n (8)
201
-------
Thus, the estimation of water quality leads to the determination of two
indices, a general sanitary index and a specific pollutant index, which are
considered independent characteristics. The water quality index is represen-
ted by a two-dimensional vector. To compare different qualitative water con-
ditions, it is necessary to solve the vector comparison problem. For this,
the range of possible values of each component is divided into a number of
intervals according to Drachev's classification (1964).
Five water quality classes are determined: very clean, clean, moderately
polluted, polluted, and contaminated. Each class has its corresponding range
of pollution and general sanitary indices and associated possibilities for
using water for specific purposes. By using the independence of the indices,
zones of possible values are divided into a number of rectangular areas
(Figure 1); five water quality zones represent the transition from bad to
better quality. The W.Q.I, is denoted by two numbers T = (N,l). The first
number N is the number of the zone in which the end of vector T(TX> Tz) is
found. It represents the possible water use; the second, 1, is equal to the
length of vector T and is necessary only for comparing water quality within
a zone.
From the water quality estimates, it is possible to evaluate the poten-
tial use of water of the given class for other types of water utilization.
Thus far, no W.Q.I.'s have been developed for these types.
An example (Table III) was constructed on the basis of existing standards
and recommendations.
This complex evaluation of water quality can be conveniently used to de-
termine the water preparation required for different types of water consump-
tion, to estimate program efficiencies for water pollution prevention, to
manage water quality, to map rivers, and to compare water quality of different
water bodies.
Water quality management can be defined as the requirement to sustain the
necessary level of the W.Q.I, for a given type of water usage. The water
quality index for municipal water usage was determined by VNIIVO to evaluate
the quality and map the Dnieper, Severskiy Donets, and the Moskva Rivers.
CONCLUSION
1. The development of criteria for water quality evaluation is an urgent
social and national economic requirement.
2. At present, we lack a common system of water quality indexing.
3. The application of analytical-expert methods allows establishment of
a common approach to the problem of developing water quality criteria for
water management systems.
202
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V
1 1.5 2 2.5 3
Figure 1. Water quality index (indicated by Roman numerals). Water quality
indices are derived from a general sanitary index Tx, and a specif-
ic pollutant index Tz.
203
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TABLE III. EFFECT OF POLLUTION OF THE POSSIBILITIES OF USING WATER
Kinds of water consumption
Water
quality
index
Very clean
5
Clean
4
Moderately
polluted
3
o Polluted
2
Contami-
nated
1
Domestic
Tx Tz potable
5 5 suitable
4-<5 4-<5 suitable if
chlorinated
2.5- 3.5- suitable after
<4 <4 standard puri-
fication
1.5- 2- suitable only
<2.0 <3.5 with special
purification
in case of
technico-
economic
expediency
<1.5 <2.0 not suitable
Bathing
sports
fully
suitable
tt
suitable
utiliza-
tion
doubtful
not
suitable
Fisheries Industry
fully suitable
suitable
suitable "
II M
suitable suitable
except
for val-
uable
species
not suitable
suitable for special
purposes
Transport
port instal-
lations Agriculture
fully fully
suitable suitable
II 1!
ii ii
suitable suitable
undesirable limited
utilization
after puri-
fication
-------
BIBLIOGRAPHY
Brown, R.M., et al. 1970. A water quality index—do we dare? Water and Sewage
Works, No. 10, pp. 117.
Brown, R.M., et al. 1972. A water quality index—crashing the psychological
barrier. Adv. Water Pollut. Res. Proc., 6-th Intern. Conf., Jerusalem,
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Byunayun, R., O.I. Larichev, and Zh. de Mongolfye. 1971. Multicriterion
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Cherkinskiy, S.N. 1971. Sanitary conditions of wastewater disposal into water
bodies. Moscow, Stroyizdat, 208 p.
Drachev, S.M. 1964. Pollution control of rivers, lakes, and water storage
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Fishburn, P.C. 1968. Utility theory. Manad. Sci. Vol. 14, No. 5.
Gurariy, V.I., and A.S. Shayn. 1974. Numerical estimates of water quality.
Problemy Okhrany Vod. Issue 5, pg. 136-140.
Harton, R.K. 1965. An index—number system for rating water quality. Water
Pollution Control Federa. Journ., Vol. 37, No. 3.
Markins, R.D. 1974. An objective water quality index. Journal Water Pollut.
Control Federat., Vol. 3.
Mitskene, P. 1974. A method for estimating and calculating the change in
water quality along a flow. Kaunas.
On the common criteria and standards for surface water purification and the
principles of their classification. In: Materialy Po Vodnomu Khozyaystvu,
Part I, Moscow, 1965.
Prati, L., et al. 1971. Assessment of surface water quality by a single
index of pollution. Water Research, Vol. 5, No. 9, pg. 714-751.
Rules for surface waters protection from pollution by waste waters. 1971.
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Velner, Kh.A. 1970. Principles of water resources protection. UNESCO Higher
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205
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DISCUSSION
PISANO: What criteria are used to establish the maximum permissible concen-
trations of substances in a ^ater body and to determine the use of the water
body? How do you determine the value of a certain parameters, or whether one
parameter is equal to or is more important than another?
SHAYN: To establish a water quality index, we use an analytic-expert
approach. Processing the expert's data and literature allows us to establish
a ranking to the chosen indices. In developing criteria for evaluating water
quality, we produce an evaluation which is applicable to an already available
specific water use. The M.P.C., for specific types of water use, already
exists in the USSR.
LOZANSKIY: I want to add the following: Determining the type of water use
for a given water body or separate parts of it is an administrative decision.
Authorized government agencies for water regulation, use, and protection from
the USSR Ministry of Land Reclamation and Water Use are especially con-
cerned with this. Unfortunately, these organizations have still not con-
structed a scientific basis for determining water use and for establishing
limitations on certain water uses. They proceed from general impressions and
the situation at a given water body. They try to satisfy all the require-
ments within a given water basin from the point of view of all the water
users.
THOMANN: Why does your water quality index include BOD? Theoretically, we
should examine the content of dissolved oxygen. Should the index be re-
calculated each time a new substance enters the water body?
BARTSCH: Is there any research in the USSR which is concerned with the
problem of the supersaturation of oxygen in water used by the fisheries?
SHAYN: Most specialists agree that BOD must be included in the water quality
index. Discussions have confirmed that this parameter is not superfluous.
A limited number of parameters are water quality indices (ten general health
and two more which characterize specific pollutants). Therefore, if a new
pollutant appears in the river, the general method for constructing the
index allows us to introduce this parameter into the index.
FUKSMAN: Did you calculate statistical characteristics for evaluating water
quality?
SHAYN: Yes we did. Probability and statistical characteristics, their
average statistical value, dispersion, and the correlation coefficient were
calculated.
206
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LOZANSKIY: The problem of oxygen supersaturation has not been identified in
the USSR. Perhaps we have not sufficiently studied this question. Our
American colleagues can obtain more detailed information from representatives
of the Institute of the Biology of Inland Waters present at the symposium.
RODZILLER: I want to comment on several aspects of the report: First, the
role of the water quality parameters—BOD. I was among those experts who
answered VNIIVO's questions. They unanimously counted BOD among the most
important water quality parameters. The health agencies and fish protection
agencies gave it a high value, since BOD characterizes the potential for
removing oxygen from a water body. In the USSR, when analyzing industrial
waste waters, it is imperative that full BOD be determined, because any
limited BOD measure is not adequate.
207
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MATHEMATICAL MODELING OF EUTROPHICATION PROCESSES IN LAKE ONTARIO
Robert V. Thomann
INTRODUCTION
Purpose of Research
The overall purpose of this research is to structure a mathematical mod-
eling framework of the major features of eutrophication in large lakes. Lake
Ontario, the subject of intensive field work as part of the International
Field Year for the Great Lakes (IFYGL), is used as the problem setting (Fig-
ure 1). The overall objectives of the research include:
1. determination of important interactions in lake eutrophication;
2. analysis of lake water quality and biological responses to natural
and man-made inputs; and
3. formation of a basis for estimating the direction of change to be
expected under remedial environmental control actions.
The problems of impairment of the quality of lake systems are magnified
for "large lakes" such as the Great Lakes. The size of these lakes is such
as to preclude any immediate improvement in quality after control actions are
taken. Further, it is much more difficult to obtain reliable data on water
quality, biological structure, and hydrodynamic circulation, again because of
the difficulty of sampling large lake systems. The general circulation it-
self may not be adequately known in relation to climatological and hydrolo-
gical factors. In the biological area, measures of phytoplankton populations
are temporally and spatially dependent and reflect complex interactions with
nutrients, such as nitrogen and phosphorus, and upper trophic level preda-
tion. This report reviews work accomplished to date on development of a
mathematical model of phytoplankton in Lake Ontario that hopefully will form
a reasonable basis for estimating the effects of nutrient removal programs.
Scope of Research
The scope of this research is therefore lakewide and attention is
directed primarily to the behavior of the phytoplankton for the lake as a
whole. The smallest scale considered in the more advanced model (Lake 3) is
on the order of 10-40 km. Detailed nearshore behavior on a lesser scale is
not a part of this research. Furthermore, this research is concerned with
phytoplankton dynamics as described on a time scale of weeks to months.
Therefore the seasonal progression of the phytoplankton throughout a year
and from year-to-year is the time scale of interest. Short-term fluctuations,
in the order of hours or days, are not described here.
208
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K>
O
INTERNATIONAL FIELD YEAR
FOR THE
GREAT LAKES
Fi[ure 1. General basin map of Lake Ontario.
-------
Finally, the modeling structure developed as part of this research is
aimed /at the phytoplankton biomass as a measure of eutrophication and asso-
ciated water quality. Rooted plants, such as cladophora, are not modeled.
Particular emphasis is placed on the interaction of the passive phytoplankton
biomass (as characterized by the concentration of chlorophyll) with the
nutrients, principally nitrogen and phosphorus, and the grazing by herbi-
vorous zooplankton and higher order trophic levels.
Lake Ontario—Background
With the Niagara River, the outflow of Lake Erie and hence the cumulator
of all the upper Great Lakes outflow, as its principal source of inflow and
the St. Lawrence River its outflow, Lake Ontario is the last in the chain of
Laurentian Great Lakes. Lake Ontario's narrow and deep rock basin was formed
by the action of glacial corrosion.
Morphometry—
Lake Ontario is the smallest of the Great Lakes in terms of surface area,
19,477 km2 with a drainage area of 90,132 km2. The mean depth of Lake On-
tario, 90 meters, is second only to Lake Superior for the Great Lakes and can
be considered one of its most important physical features. Lake Ontario's
volume is 1,669 km3 with a maximum depth of 244 meters. The lake's elevation
above sea level is 74.01 meters which makes its depth of cryptodepression
170 meters. Lake Ontario's length is 307 km with a maximum width of 87 km.
The lake's shoreline length is 1,380 km.
Hydrology—
The Niagara River, the major source of inflow for Lake Ontario, is the
cumulative outflow of the other Great Lakes. The average flow of the
Niagara River is 5,520 kmd/sec., and accounts for 84 percent of the flow dis-
charged via the St. Lawrence River. The average annual precipitation on
Lake Ontario's water surface is 83.28 cm and the average annual evaporation
is 71 cm.
The thermal bar development which precedes the development of the full
thermocline begins in late April or early May in Lake Ontario. The offshore
movement of the thermal bar is such that within 17 to 28 days the nearshore
ring of stratified water covers over half the area of the lake. The average
depth of the thermocline is 17 meters, with its dissipation beginning in
late September. The hydraulic detention time of Lake Ontario (volume divid-
ed by flow) is 8.1 years and is significant in that it gives some indication
of the response time of the lake.
Nutrient Inputs—
Nitrogen and phosphorus are used as nutrients in the present configura-
tion of the model, though silica will most likely be inhorporated in the
future.
210
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Nutrient loads for Lake Ontario are shown in Table I.
Source
Niagara River
Tributaries
Municipal
Industrial
Total
Lake Ontario MO
TABLE I.
Phosphorus
Kg /day
19,100
7,100
7,300
500
34,000
del structure
NUTRIENT LOADS
%
56
21
22
1
100
Nifiugeu '
Kg /day
236,900
86,600
32,900
44,100
400,500
%
59
22
8
11
100
Theoretical Background —
The basic theory for the Lake Ontario model makes use of previous model
structures (DiToro et al. , 1971; Hyd'roscienee, Inc., 1933; Thotnann et al . ,
1974) and essentially represents a mass balance around each ecological or
nutrient compartment and around physical space.
The basic structure consists of three parts:
1. transport and dispersion sub-system,
2. biological subsystem, and
3. chemical subsystem.
The latter two subsystems represent the kinetic behavior or the extent
and complexity of interactions between relevant variables, while the first
system represents the physical processes of water movement and mixing. The
theory of each subsystem is presently developed to a different degree, with
the transport and dispersion theory developed to a more advanced degree than
the biological and chemical systems.
The general equation which results from applying the principle of the
conservation of mass is:
3sk = -3_ (usk) - ^_ (vsk) - ^_ (wSk)+ J_ (Ex 8s
3t 8x 3y 3z 3x 3x
+ <>_ (Ey !\) + !_ (Ez !\)
3y 3y 3z 3z
+ Sk (x,y,z,t,84,sk) ± Wk (x.y.s.t): k-l..m Q)
x/ _L . . in
where s, is the kth dependent variable (biological or chemical), u, v, w are
the velocity components in the x, y, and z directions, respectively, EX, E ,
and Ez are the dispersion coefficients in each respective spatial direction,
Sv represents the kinetic interactions between the k variables and W is the
direct inputs of the substance, sk. It should be noted that Equation 1, as
211
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written, does not include any interaction between the variables s^ and the
transport and dispersion regime. Thus, the physical system is separable, in
the sense that it can be externally supplied and, once determined, it can be
used for all variables. In vector form, Equation 1 is:
Ssk = (V. [E] (Vsk) - V. Usk) + Sk + Wk (2)
3t
where [Ej is a diagonal matrix of dispersion coefficients, (s^) is a column
vector, U is the velocity vector and
V = 3 i+3 1+3 k
3x 3y 9z
In Equation 2, the first term in parenthesis on the right-hand side re-
presents the dispersive and advective field in three dimensions, as given by
the general circulation. In this wo.rk, it is assumed that the circulation is
"known," either through observation or from output from a hydrodynamic model.
The inputted circulation can be validated by utilizing Equation 2 for a trac-
er, such as dye or water temperature, to compare computed results to observed
data. Water temperature has been used in some of this work and the results
of that analysis are given in the Lake 1 model. Primary interest
therefore centers on the reaction kinetics, their functional forms, and nu-
merical values.
A finite difference approximation to Equation 2 is necessary to apply the
equation to an actual water body. Thus, if the lake is divided into a series
of finite, completely mixed volumes, n in number, and each with volume V.,
then it can be shown that:
[V-] d(s)k = [A] (s). + (S). + (W). (3)
dt k tc k
where [vj is an n x n diagonal matrix of volumes, (s), is an n x 1 vector of
the variable s^; [A] is an n x n matrix of advection and dispersion terms;
(s), is an n x 1 vector of kinetic interactions; and (W), is an n x 1 vector
of inputs of variable s^.. For m interacting variables and n physical loca-
tions, Equation J indicates that a total of mxn equations must be solved. In
general, the equations are both linear and non-linear.
When one recognizes that initially up to 10-15 interacting variables can
be specified and that 50-100 initial spatial segments may be required, it is
obvious that the computer program and data preparation necessary to obtain
solutions are of a substantial magnitude. Accordingly, prudent practice
would dictate a modeling strategy that allows one to proceed sequentially
from simple models to the more complex. Such a strategy is utilized in this
work and is illustrated in Figure 2.
As indicated, two parallel paths are being followed. The upper path in-
volves the gathering of data on the lake geomorphology and transport and dis-
persion structure. The model is a single variable model of water temperature
or chlorides both representing tracer variables that can be used to validate
212
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SPATIAL DEFINITION
SYSTEM
GEOMETRY
PREPARATION
Flow
Depth, volume
Dispersion.
Inputs
WATER TEMPERATURE
AND
CHLORIDE MODEL
(67 SEGMENTS)
TEMPORAL AND KINETIC DEFINITION
Solar radiation ..
Temperature
Flow .
Nutrients _
Solar radiation ^
Temperature
Flow
Nutrients
LAKE 1 MODEL
(3 VERTICAL LAYERS)
LAKE 2 MODEL
(7 VERTICAL LAYERS)
Phytoplankton and
zooplankton dynamics
Temperature verification
phytoplankton,
chemistry and sediment
interactions
SYNTHESIS OF KINETIC AND SPATIAL MODELS
LAKE 3 MODEL
(67 SEGMENTS,
10-15 VARIABLES)
Coarse grid model
5000
COMPARTMENT
MODEL
(Future expansion)
Figure 2. Eutrophication modeling strategy.
-------
various sectors of the inputted circulation. A three-dimensional grid of 67
segments is employed.
The emphasis in the lower parallel path is on the subsystem kinetics
with spatial detail held to a minimum. The lake is assumed to be horizontal-
ly completely mixed and is divided into a series of vertical layers. Atten-
tion is specifically directed to the interactions between phytoplankton, zoo-
plankton, and water chemistry. Two models, Lake 1 and Lake 2, are explored
in this path.
The synthesis of the spatial definition of transport and mixing with the
kinetic interactions is accomplished in Lake 3. This model, a coarse-grid
model, is three-dimensional and incorporates the major kinetic features of
the Lake 1 and 2 models. -With 67 segments and 10 variables, a total of 670
equations or "compartments" must be solved. Future expansion calls for an
approximate order of magnitude increase in the number of compartments to 5000.
In this report, primary emphasis is placed on the Lake 1 model. As such,
a detailed review of the kinetics employed in the models is given below for
the biological and chemical subsystems. The systems diagram for the Lake 1
model is shown in Figure 3. The Lake 2 model includes those variables for
Lake 1 and also includes a representation of the carbon cycle.
Phytoplankton Chlorophyll —
The basic kinetic interactions, for phytoplankton chlorophyll (a measure
of biomass), is given for a single volume, j, as:
SP = V. (Gp - Dp) jPj - Vj !ii P. + V± ^ P± (4)
HI H±
where P is the phytoplankton chlorophyll (yg/£) , Gp and D are the growth and
death rate (I/day) respectively and w j ^ and w.^ are the sinking velocities of
the phytoplankton between segments j and i.
Phytoplankton Growth Rate —
The growth rate expression is similar to that developed previously
(DiToro et al., 1971) and SB used in this work is given by:
Gpj = Gl (Gmax'T) ' [l(Ia,Is,ke,P,H)] ' ([N
= (Temperature
effect) • (Light effect) • (Nutrient effect) (5)
where Gmax is the maximum growth rage, T is water temperature, I is light
intensity at Gmax, Ia is incoming solar radiation, ke is the extinction co-
efficient, H is the depth of the segment, kmn is the half-saturation constant
for total inorganic nitrogen, N', N2 is ammonia nitrogen, No is nitrate nitro
gen, 1C is the half-saturation constant for phosphorus, and ?2 is the avail-
able phosphorus. The temperature effect is given by Eppley (1972) as:
= Gmax d,066) (6)
214
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UPPER TROPHIC
LEVEL NO. 2
CARBON
UPPER TROPHIC
LEVEL NO. 1
CARBON
CARNIVOROUS
ZOOPLANKTON
CARBON
HERBIVOROUS
ZOOPLANKTON
CARBON
PHYTOPLANKTON
CHLOROPHYLL
BIOLOGICAL SUB-MODEL
ORGANIC
NITROGEN
NITROGEN CYCLE
AMMONIA
NITROGEN
NITRATE
NITROGEN
PHOSPHORUS CYCLE
ORGANIC
PHOSPHORUS
AVAILABLE
PHOSPHORUS
CHEMICAL-BIOCHEMICAL SUB-MODEL
Figure 3. System diagram, Lake I model.
-------
where Eppley reports Gmax at 0.587 (I/day). This form permits phytoplankton
growth at low temperature and at 20°C results in a growth rate of 2.I/day.
The function form for the light effects is as given in previous work
(DiToro et al. 1971) and incorporates vertical extinction of solar radiation
and self-shading effects. The form is:.
-a, -an , s
I = 2,718f (e * - e °) (7)
TT~H
where ke = ke + °-0088p + 0.05AP0"66
Is
and k^ is the light extinction coefficient at zero chlorophyll and f is the
photoperiod. All pertinent variables are functions of j , the segment loca-
tion. The nutrient effect makes use of product Michaelis-Menten kinetics and
is given by:
The range of values for k and k™, the half-saturation constants for
nitrogen and phosphorus respectively, has been documented previously (DiToro
et al. 1971). More recent other work (Caperon, 1972, and Hendrey, 1973) tends
to support the original range; namely k in the range 5-50 yg N/2, and k^ in
the range 1-10 yg/&. There is an obvious species dependence of this constant
as well as a possible dependence on the eutrophic state of the lake. In the
Lake 1 model, k is used at 25 yg N/&, and k^ of 1 and 10 yg P/& is used as
a range.
At present, silica limitation is not used in this model. Goering et al.
(1973) and Paasche (1973) have estimated the half-saturation value for silica
at about .02-. 08 mg Si/ H for marine species. Under peak growing conditions
in Lake Ontario, surface silica values are about 0.1-0.2 mg Si/£, which tends
to indicate that silica may not have a significant effect on the growth rate.
Work is in progress, however, to include this nutrient since at the lower con-
centration and higher Michaelis levels, it could be important.
The complete growth rate expression, Gpjj is then given by the product of
(6), (7), and (8). Although there is no a priori reason for using a product
formulation (Bloomfield, 1973) , various experiments (DiToro, et al. 1971)
have tended to support the approach (Ditoro, in press, and Hutchinson, 1.967) .
It can also be noted that the gross growth rate of the phytoplankton bio-
mass, G -P. is equivalent to the daily average productivity in the j th seg-
ment. Thus:
216
-------
where v- is the average rate of carbon fixation (mgC/m2-day), acc is the car-
bon to chlorophyll ratio (range 20-100 mgC/mg Chlor).
Phytoplankton Death Rate—
The expression for the death rate of the phytoplankton includes two pri-
mary effects: endogeiieous respiration and predation by herbivorous zooplank-
ton. The death rate is given by:
Dpj = K2(1.08) T~20 + CgZ. (10)
where K2 is the constant (I/day), C is the grazing rate of the herbivorous
zooplankton (liters/day — mg zooplankton carbon) and Z is the carbon concen-
tration of the herbivorous zooplankton. As noted below for the zooplankton
observed in Lake Ontario, a filtering rate of .03-.06 liters/mgC-day-°C ap-
pears appropriate and at 20°C, represents about 2-6 ml, filtered per individ-
ual per day.
Phytoplankton Sinking—
In Equation 4, the sinking of phytoplankton from one segment to another
segment can be a potentially significant sink or source of phytoplankton and
associated nutrients. This phenomena is complex, incorporating vertical tur-
bulence effects, the density structure, and the physiological state of differ-
ent species of phytoplankton. For example, the generation of gelatious
sheaths by phytoplankton has been shown to be of importance in settling (Hut-
chinson, 1967) and apparently the settling velocity of nutrient rich cells is
somewhat less than cells that are nutritionally deficient (Yentsch, 1962).
Published values of the sinking velocity of phytoplankton, mostly in
quiescent laboratory conditions, range from 0.07 - 18 meters/day. In some
instances, the settling velocity is zero or negative.
Burns and Pashley (1974) have investigated the settling velocity profile
of particulate organic carbon in Lake Ontario with an ingenious in situ meas-
uring device. Their results show a general decrease of settling velocity with
depth and a marked seasonal variation of settling velocity. For example, on
19 July 1972, settling velocity in the thermocline (about 10 m) was estimated
at about -0.3 m/day, while at 50 meters depth, the settling velocity was about
+1.3 m/day. But in March 1973, the velocity profile was positive ranging from
0.9 m/day at 20 meters to 0.1 m/day at 140 meters. Overall, the values given
by Burns and Pashley varied between -0.4 and +2.0 m/day. These results and
others tend to indicate that the settling velocity should be related in some
way to the state of the phytoplankton biomass (e.g., actively growing versus
declining phase) or vertical velocity variations. In this work, however, the
vertical settling velocity has been treated as a constant. As such, this
effect is only approximately modeled and, accordingly, the settling velocity
has been treated as a constant over a range from 0.05 m/day to 0.5 m/day.
217
-------
Zooplankton Carbon—
The measure of the herbivorous zooplankton population Z^ ,is taken as
the carbon content of the-biomass and all sources of food for Z1- ' are the
phytoplankton chlorophyll. The general expression for a single volume j is
therefore:
s(l) = v. (G(1) _ D(l)) . Z(D (11)
zj 1 z z J j
where G^ ' and D' ' are the growth and death rates respectively of Z*-1- . The
form of fid' is:
£„ P (12)
where a is the zooplankton carbon to phytoplankton chlorophyll efficiency,
CgZp is the grazing rate of the herbivorous zooplankton and Kg is the half-
saturation concentration of phytoplankton at which the growth of zd) is half
of maximum growth. This latter effect reflects the limiting growth of zoo-
plankton at higher phytoplankton populations.
The mortality of the herbivorous zooplankton is given by two terms:
endogenous respiration and grazing by the next highest trophic level. There-
fore:
D(l} = K<1> (T) + C 21(T) Z(2) (13)
L- L, g'- J-
where K(!) is the endogenous respiration rate as a function of temperature
(l/day-°C), Cg2i is the grazing rate (liters/day-mgC-°C) and z(2) is the next
highest trophic level, designated by superscript (2).
Expressions for all trophic levels above the herbivorous zooplankton
arranged in a linear food chain are similar to Equations 13 and 14. Thus:
S(2) = v. (G<2) - D<2)
^-j J L ^ 1 i
where
r(2) _ „ n 01 7(1) (15)
G z ~ az Lg21 L^ ' ^^'
and
p(2) = K ) (T) + C (T) 2, '
These expressions are an obvious oversimplification of a complex food
web and reflect only a "linear" type of food chain (see Figure 3). Indeed
the effect of the zooplankton on the phytoplankton as formulated above is to
always decrease the population, yet it has been shown (Patalas, 1969) that
zooplankton can also increase phytoplankton population or have no effect.
The mechanism for an increase in the population is apparently the passage of
gelatinous green algae through the zooplankton gut with subsequent breakup of
the colonies and regrowth after excretion. The smaller species are generally
diatoms and Asterionella formosa, which are characteristic of Lake Ontario.
218
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The zooplankton of Lake Ontario are dominated by the crustaceans specif-
ically of Order Copepoda and Cladocera. Most hauls are made over 0-50 meter
depth range. Principal Copepoid species given by Patalas (1969) are Cyclops
bicuspidatus and Tropocyclops prasinus and the cladocerans, Daphnia retrocurva
and Bosmina longirostris. In Patalas' work in 1967, these four species
accounted for 91% of the total zooplankton and averaged about 60/liter from
June - October.
Others (Glooschenko et al., 1972; McNaught et al., 1973; and Watson et
al., 1974) generally show values in a similar range in some cases up to a
maximum of almost 200/liter. For use in biomass computations, conversions
must be made to equivalent carbon. Such conversion depends on age and ac-
companying weight of the animal and can therefore vary over a wide range.
Watson et al (1974) in their conversions used weights of species to arrive at
total biomass and equivalently averaged about 2.8 yg dry weight/individual.
The results, using an average of 40% carbon by dry weight, show peak
values generally in August at lakewide averages of between 0.07 - 0.12 mg
carbon/liter.
At the average dry weight of about 1-5 yg, individual grazing rates of
about 0.8-2 ml/individual-day are appropriate (DiToro et al., 1971 and
Kirby, 1971). Indeed the rate for Bosmina longirostris, a dominant species in
Lake Ontario, has been given as 1-3 ml/individual-day (Hutchinson, 1967).
Grazing rates expressed as liters per mg zooplankton carbon per day have been
assumed to be linearly related to temperature and have been used in the range
of 0.03-0.06 (l/mgC-day-°C), corresponding to the reported grazing ranges at
20°C.
Nitrogen System—
The basic equations for the nitrogen sub-system follow previous work
(Hydroscience, 1933 and Thomann et al., 1974) and include non-living organic
nitrogen, N^, ammonia nitrogen, N2, and nitrate nitrogen, N-^. The equation
for the source-sink term for organic nitrogen is:
%,,. - filDzZ + Mrja-!^cgzpZu>+aiDpP_KlNl
C221Z(2) (16)
ac
where ai is nitrogen to chlorophyll ratio, and ac is the carbon to chlorophyll
ratio, and K-^ is the overall decay of N^.
The first and third terms represent organic nitrogen released through
endogenous respiration by the zooplankton and phytoplankton respectively and
the second term represents the organic nitrogen of the grazed but unassimila-
ted phytoplankton. The last term represents the decomposition, settling, and
other effects that contribute to the decay of organic nitrogen.
219
-------
For ammonia:
= K12(T)N2 - K22(T)N2 -
where Ki2 is the rate of production of ammonia from organic nitrogen, K22 is
the rate of oxidation of ammonia to nitrate, and a is a preference factor
given by:
a= N2
N2 + Kmn
or is alternatively set equal to one-half for an "indifferent" uptake of nit-
rogen. The last term represents the uptake of ammonia for phytoplankton
growth. For nitrate,
c = K~oN, - a, (1-a) GP
N3)_ 23 2 1 p (19)
where K23 = K22, the production of nitrate by ammonia oxidation.
Phosphorus System--
The equations for the phosphorus cycle used in the models are as pre-
viously developed (Hydroscience, 1933 and Thomann et al. 1974) and include
nonliving organic phosphorus and phosphorus available for phytoplankton
growth. For organic phosphorus, the equation for the source and sink for pj
is:
f-. TT f ~] \
S = ^ I DZZ + apPj (1" ZP I Kg Z -KlPl+^Z^) (l-az)
>j ac Kg + e- ac
g (20)
where a is the phosphorus to chlorophyll ratio and Ki represents a general
decay or organic forms.
For the inorganic phosphorus, assumed as phosphorus available to the
phytoplankton, the source-sink equation for p2 is:
S = K]2(T) P] - a G P - K22 (T) P?
P2,j U 1 P P 2 (21)
where K-.^ represents the rate of decomposition of organic phosphorus to forms
available for phytoplankton utilization and ^^2 rePresents an overall loss of
inorganic phosphorus.
220
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LAKE 1 MODEL
Basic Structure
Because of the complexity of the overall modeling structure, a simplified
version of the lake has been developed to explore the kinetic behavior in
greater detail. This is indicated in the modeling strategy of Figure 2. The
simplified model, designated Lake 1, assumes that Lake Ontario is well-mixed
horizontally and gradients are allowed to develop only in the vertical dimen-
sion. Such a simplification obviously does not permit nearshore versus open-
lake comparisons of the effect of the Rochester or Toronto discharges on the
local lake environment. However, an inspection of the data on nutrients and
chlorophyll does not appear to indicate substantial horizontal gradients al-
though variations do exist in certain areas. (Lake Ontario can be contrasted
in this regard to Lake Huron with marked horizontal gradients from Saginaw Bay
or to Lake Erie with important horizontal gradients from the Western Basin to
the Eastern Basin.)
Because of the complexity of the interactive systems and most important-
ly, the computational time involved in obtaining solutions, the horizontally
well-mixed assumption appears to offer a reasonable starting point for under-
standing the dynamic behavior of Lake Ontario. Figure 4 is a schematic of
the Lake 1 model and shows the three vertical segments that comprise the basic
geometry. The principal physical features are 1) horizontal transport, 2)
vertical dispersion between the epilimnion and hypolimnion, and 3) vertical
settling of the phytoplankton. The lake shown in Figure 4 has a defined
epilimnion depth of 17 meters and a hypolimnetic depth of 73 meters. The
Lake 1 model is well-mixed in the winter and spring, stratifies during summer,
and then goes through a fall overturn. The effects are simulated by control
of the vertical mixing.
Results of "Final" Verification Analysis
Data were available for the years 1967-1970 collected by the Canadian
Centre for Inland Waters for this final verification phase. These data were
supplemented by other observations in the literature. Values for the various
coefficients, parameters, and exogenous variables were obtained from published
sources as discussed previously, and all values are used within reported
ranges in the literature. More than 80 runs have been made of Lake 1 to exam-
ine the effects of various phenomena such as settling velocity, vertical
mixing, zooplankton predation, and half-saturation constants for nitrogen and
phosphorus. Out of these runs, a consistent set of parameter values has
emerged to satisfactorily explain the observed data on a variety of different
variables. A plausible explanation for the dynamic behavior of the phyto-
plankton is therefore possible and is discussed below.
221
-------
NUTRIENT INPUTS
NIAGARA RIVER")
TRIBUTARIES MUNICIPAL >
INDUSTRIAL WASTES \
ENVIRONMENTAL INPUTS
fSOLAR RADIATION
= ( WATER TEMPERATURE
(LIGHT EXTINCTION,
(.SYSTEM PARAMETERS
\ 0 EPI LIMN ION
I VERTICAL EXCHANGE
1
1 i
\ '* *
r * * 1
1 y^
— »• TRANSPORT 17 M /?
i
^SETTLING
I
i
\
\
I /
j
\ (2) HYPOLIMNION
/
BENTHOS
73 M
Figure 4. Major physical features included in Lake 1 model.
Figures 5 and 6 show the verification of the Lake 1 model upper layer
(0-17 meters), while Figure 7 shows the verification for the hypolimnion
(data in range, 50-150 meters). Table 2 shows the principal coefficients
used for the runs. As shown, the verification for plankton in the epilimnion
is quite good and satisfactorily duplicates the spring peak, subsequent mid-
summer dieoff, and fall bloom. Five other variables, in addition to the phy-
toplankton chlorophyll are satisfactorily verified in the comparison. It is
generally not possible to obtain the verification shown in Figures 5-7 by
arbitrary specification of the coefficients, so the results do not simply re-
present a "curve fitting" exercise. Rather, the results, based on the theory
discussed earlier, are much more representative of the observed data than ar-
bitrary statistical coefficients. Since the model permits computation of the
components of the dynamic behavior, considerable insight can be gained by
examining the influence of various phenomena on the growth and death rates of
the phytoplankton.
Figure 8 shows the dynamics of the kinetic growth rate of the phytoplank-
ton (Gp in Equation 4). As shown, maximum growth rate reaches a peak in mid-
September in the Lake 1 model at about 1.8/day and then decreases rapidly due
to the fall overturn. The reduction in the maximum growth rate due to light
is substantial and shifts peak growth to early August. A further reduction in
growth rate is due to the nutrient limitation so that the resultant growth
rate, G , is lowest at the end of May and peaks about 0.25/day in August. The
peak value represents an overall reduction of 90% from saturated growth condi-
222
-------
^ o
P Q_
>- O
x:
o
20
15
10
5
0
I °'16
2- °-12
O ~cr>
| E 0.08
5
% 0.04
o
M
0
o
o
ex.
0.6
0.4
0.2
0
Lake 1 Model Verification
(0-17 Meters)
RANGE OF MEAN ± 1 STANDARD DEVIATION
LAKE-WIDE MEAN.
0-17 METERS
I _ I I . I I I
J F M A M J J
0 N D
NOTE: ZOOPLANKTONCARBON
IS FOR 0-50 METER HAUL
• (•ii*iii i i
30 60 90 12Q 150 180 210 240 270 300 330 360
J ' F ' M' A ' M ' J ' J ' A
S 0 N D
Figure 5. Lake 1 model verification, 0-17 meters.
223
-------
Lake 1 Model Verification
(0-17 Meters)
-------
Lake 1 Model Verification
(50-150 Meters)
o
o
a:
0.6
0.4
0.2
0
o
o 0.15
LAKE-WIDE MEAN, 50-150 meters
• 1970
• 1969
o 1967
\ ' F ' M
M1 J ' J ' A ' S ' 0 ' N ' D
<: en
— E
0.10
0.05
0
RANGE OF MEAN ± 1 STANDARD DEVIATION
30 F 60 M90 A120M150J180 J210A2405270°300N330D360
AMJJASOND
JFM
- 1 - 1 - 1 - 1 - 1 - 1
FMAMJ JAS
OND
Fio-ure 7. Lake 1 model verification, 50-150 meters,
225
-------
TABLE II. PRINCIPAL PARAMETER VALUES USED IN LAKE 1 MODEL OUTPUT SHOWN IN FIGURES 5-7
Kmn
Kmp
Cg
Kg
K
2
a
zp
K
z
Kl
K22
K
op
al
"p
a
c
w
Half -saturation constant-nitrogen
Half -saturation constant-phosphorus
Grazing rate for zooplankton
Half -saturation constant-phytoplankton
Endogenous respiration rate-phytoplankton
Zooplankton conversion efficiency
Zooplankton endogenous respiration rate
Decomposition rate of organic nitrogen
Ammonia to nitrate nitrification rate
Decomposition rate of organic phosphorus
Nitrogen-Chlorophyll ratio
Phosphorus-Chlorophyll ratio
Carbon-Chlorophyll ratio
Settling velocity of phytoplankton
25.0
2.0
0.06
10.0
0.1
0.6
0.001
0.00175
0.002
0.007
10.0
1.0
50.0
0.1
UgN/£
ygPM
£/mg C-Day-C°
yg chlor/S,
days"1 (@200C)
_1
days - C
days'1 (@20°C)
1
days -°C
1
days «320°C)
ygN/yg chlor
UgP/yg chlor
ygC/yg chlor
m/day
-------
tlons. Primary productivity values calculated from the growth rate and phyto-
plankton biomass, Equation 9, give peak values of about 600 mg carbon/m2-day
at the end of August and values of about 500 mg C/m2 day at the height of the
spring bloom. The former value is in general agreement with Glooshenko (1974)
while the latter is somewhat lower than the Glooshenko's estimated spring
values of 500-1100 mg C/m2-day.
Dynamics of Phytoplankton Growth Rate
2.0
1.8
J 1.6
£"1.4
<
i1<2
§1.0
o
I 0.8
z
? 0.6
o
0.4
0.2
J 30 F 60 M 90 A120M150J 180 J 210 A 240 S 270 ° 300 N 330 D 360
MAXIMUM GROWTH
RATEATEPILIMNION
WATER TEMPERATURE
REDUCTION DUE TO
LIGHT LIMITATION
RESULTANT PHYTOPLANKTON
GROWTH RATE
REDUCTION DUE TO
NUTRIENT LIMITATION
J J
0 N
2.5
o
2.0
Q_
00
1.5
CD
o
o
i.o
CO
0.5
Figure 8. Dynamics of the phytoplankton growth rate.
Figure 9 shows the dynamics of the kinetic death rate of the phytoplank-
ton, Dp in Equation 4. Three effects are noted: 1) phytoplankton settling,
2) water temperature effect on endogeneous respiration, and 3) effect of zoo-
plankton grazing. Peak resultant death rate is 0.25/day at the beginning of
August and is primarily due to zooplankton predation and quantitatively ex-
plains the mid-summer decline in phytoplankton biomass.
227
-------
Dynamics of Phytoplankton Death Rate
10
-> £ CT c
o_ o n 5
go
0. O
0
J I I L
J L
J 30 F 60 M 90 A120M 150J 180 J 210 A 240 S 270 ° 300 N 330 D 360
" 0.25 -
0.20
g 0.15
o_
O
0.10
0.05
RESULTANT PHYTOPLANKTON DEATH RATE
F M A M
ZOOPLANKTON
GRAZING EFFECT
EFFECT OF WATER
TEMPERATURE
EFFECT OF
PHYTOPLANKTON
SETTLING RATE
(0.1 me\er/day)
J A
0
N D
Figure 9. Dynamics of the phytoplankton death rate.
The dynamics of phytoplankton net production, i.e. chlorophyll/liter-day,
dP/dt, are shown in Figure 10. Net production here includes the kinetic in-
teractions of growth in Figure 8, death and predation in Figure 9, and the
effect of vertical mixing and lake outflow. The results shown in Figure 10
summarize the basic hypotheses of phytoplankton dynamics in Lake Ontario.
The spring growth phase to approximately mid-May is due primarily to in-
creasing light and temperature. There are little zooplankton as yet, so
growth continues until nutrient limitation becomes significant in early June.
(This is principally phosphorus limitation as discussed below.) The spring
growth and spring peak are therefore simply described by a nutrient interac-
tion effect with light and temperature dominating the growth and death terms.
Following the spring peak however, the situation becomes considerably more
complex and indeed a significant effort was devoted to unraveling the complex
interactions which lead to a mid-summer decline and subsequent broad full
peak.
228
-------
Dynamics of Phytoplankton Net Production
J 30 F 60 M90 A120M150J 180 J 210 A 240S270°300N 330D360
ACTIVE
GROWTH
INCREASING LIGHT
AND TEMPERATURE
o
>—
o
ZD
0 2-,
SY
o- ^1
SI
^ 8-
PHYTOPLA!
\iq chlor
fc
^
0.20
0.15
0.10
0.05
0
-0.05
-0.10
-0.15
-0.20
NUTRIENT REGENERATION
NUTRIENT LIMITATION
NUTRIENT LIMITATION
FALL OVERTURN
ZOOPLANKTON GRAZING
DECLINING
GROWTH
Figure 10. Dynamics of the phytoplankton net production
It is hypothesized as computed in Figure 10, that during mid-summer, and
following the nutrient limitation effect, zooplankton grazing becomes in-
creasingly important and accounts for the minimum values of biomass in July.
However, at increased zooplankton grazing, biological recycling of nutrients
becomes more significant. Nutrients are released back to the epilimnion
through excretion as well as phytoplankton endogenous respiration. In late
July therefore, growth exceeds death due to nutrient regeneration and an ac-
tive growth phase begins again. Nutrients, however, are already at low levels
so growth is not pronounced and proceeds slowly (as shown by the net produc-
tion approaching zero in September but not yet entering a negative growth
phase). The fall overturn of the lake then reduces the growth of phytoplank-
ton biomass primarily because of mixing with the colder hypolimnion waters.
229
-------
Figures 11 and 12 explore these dynamic effects at another level of de-
tail. The model permits calculation of nutrient limitation effects as well as
the flux of nutrients due to zooplankton excretion and release of nutrients
due to endogenous respiration by the plankton. Figure 11 shows the nutrient
effects. As seen, the spring bloom is halted essentially by a reduction in
phosphorus that is quite pronounced, and growth is reduced by a factor of
about 0.35 due to the low phosphorus levels. At the same time, nitrogen is
not limiting and is at a level twice that of the phosphorus. Nitrogen, how-
ever, does become limiting during July and again in September at which time
phosphorus also exerts a limiting effect on growth. Figure 11 shows therefore
that the spring peak is primarily phosphorus controlled. The dynamics after
the spring peak are, however, controlled by both nitrogen and phosphorus (as
well as zooplankton predation).
Figure 12 shows the estimated sources of phosphorus recycled to the
epilimnion by both biological effects and vertical mixing effects from the
hypolimnion. As shown, during early spring the hypolimnion contributes almost
twice as much nutrients to the epilimnion as from external inputs, and by
early summer in June-July, biological recycling reaches almost five times the
mass rate of external phosphorus sources due to waste residuals, Niagara
River, and other contributors. Zooplankton excretion recycles almost twice
the input rate and in late summer—early fall, biological recycling reaches
another peak but then drops rapidly due to fall overturn. The flux of
nutrients from the more nutrient rich hypolimnion during October is not suf-
ficient to offset the drop in water temperature in the epilimnion. Biological
recycling therefore drops rapidly in October, and the phytoplankton biomass
declines (Figure 5) . The results shown in Figure 12 indicate quantitatively
the significance of biological recycling and vertical mixing relative to the
level of external nutrient inputs. A reduced input phosphorus load therefore
may not necessarily result in a concomitant reduction in biomass. Further,
the time to reach a new equilibrium biomass level will extend beyond a single
year due, in part, to the recycling effects discussed previously and long
detention times in Lake Ontario.
Summary of Lake Responses to Nutrient Inputs
In addition to increased understanding of the behavior of lake phyto-
plankton, the model also provides a basis for estimating the direction of
future changes due to various nutrient loading scenarios. A range of condi-
tions on the external nutrient inputs has therefore been examined. However,
to place the present loads into an historical perspective, a preliminary anal-
ysis was made of the nutrient inputs that might have existed at some distant
time in the past. These loads were termed the Pastoral Loads. In addition,
a review was made of nutrient reductions suggested by Vollenweider (1968) and
those of the Great Lakes Water Quality Agreement (1973) . A full review of all
simulations is given by Thomann et al. (1975). The simulations were therefore
prepared using a wide range of input nutrient loads with the Pastoral Loads
as a baseline condition.
For phosphorus, the input range was from present conditions (34 metric
tons/day) to Pastoral Loads (9.3 metric tons/day). The Water Quality Agree-
ment and Vollenweider reduction loads were 73% and 63% respectively of
230
-------
Effect of Nutrient Limitation
gd_ 10
3^ ,
Q- o =i 5
o °£
>-
31
Q-
o
0
J
M
M J J A
0 N
1.0
0.8
=; I0-6
o
o
n ..
0.4
0.2
0
o
i—
-------
Biological Recycling
- 5.0
O —
LU cn
O
LU <-)
a: S
1/1 """'
D_
l/l
O
TOTAL EXTERNAL INPUTS
(NIAGARA RIVER.
MUNICIPAL. AND
TRIBUTARIES
EXCRETION BY\
^x/ZGOPLANKTON ^
; V i.o
J I F I M I A
30 60 90 120 150 180 210 240 270 300 330 360
Hypolimnetic Recycling
-
>-
2.0
1-6
1.2
0.8
0.4
0
M ' J
I , I n I
0
N ' D
5.0
4.0
3.0
2.0
1.0
D_
00
O
g
o
LU
O
LU
1/1
C£
o
Q-
1/1
O
CL.
o
Figure 12. Biological and hypolimnetic recycling of phosphorus.
232
-------
present inputs.
Figure 13 shows the results of the simulations under three conditions on
the long-term loss of phosphorus. Using the "reasonable" set of kinetics,
the results indicate that Lake Ontario may not be in equilibrium with the pre-
sent nutrient inputs. As indicated in Figure 13, peak values may reach as
high as 22 yg chlorophyll/£ under the pessimistic assumption and a continua-
tion of present loads. The implementation of the WQA loads would reduce this
"worst" case to only about 20 yg/&. Under the reasonable kinetics that in-
clude organic decay, the change occasioned by the WQA would be less than
1 yg/£. Reductions below present levels of biomass may therefore be difficult
to achieve.
PRELIMINARY LAKE 3 MODEL
As shown in Figure 2, the Lake 3 model represents a synthesis of spatial
geometry and transport structure with the biological and chemical kinetic
interactions of the Lake 1 and 2 models. Lake 3 is therefore a three-dimen-
sional time variable model which does permit some analysis of phenomena in
the nearshore region as compared to the open lake region. The structure of
Lake 3 represents a further step beyond the simple Lake 1 model, the extent
of the step incorporating such factors as computational time for a one year
run and availability of survey data.
The work reported here is of a preliminary nature since considerably more
effort is required to assure the veracity of the computed results both in
terms of the numerical output and the degree to which the model verifies ob-
served data.
The basic segmentation used for the Lake 3 model is shown in Figure 14.
As shown, 67 segments are distributed over five vertical layers. The upper
two layers, 0-4 meters and 4-17 meters, are considered to represent the
epilimnion during periods of vertical stratification. A "ring" of segments
extending 10 km out from the coast is used to represent the nearshore environ-
ment.
All cross-sectional areas for interfaces in three dimensions between the
segments were planimeters and volumes were computed and summed to within 10%
of total lake volume.
The hydrodynamic circulation is considered to be externally supplied
through inferences from observations and/or hydrodynamic model output. The
final summer circulation pattern, as given by inter-segment velocity values
can be seen in Figure 15.
Preliminary Phytoplankton Computations
Several runs have been made of the Lake 3 model using the transport and
dispersion regime discussed previously and the system kinetics given in
Section III.
233
-------
o
QC •=:
x -
°Z
z o
25
20
15
10
I
Q_
LU
Q.
25
20
15
10
5
PRESENT
PESSIMISTIC
REASONABLE
OPTIMISTIC
25
20
15
10
8 12 16 20 24
VOLLENWEIDER
PESSIMISTIC
REASONABLE
OPTIMISTIC
I
25
20
15
10
8 12 16 20 24
YEARS
I I
WQA
PESSIMISTIC
^^B^M
REASONABLE
OPTIMISTIC
I
I
8 12 16 20 24
PASTORAL
PESSIMISTIC
J L
REASONABLE
J L
8 12 16 20 24
YEARS
'Figure 13. Summary of lone t^rm simulations.
-------
Ml
>150 Meters
Figure 14. Lake 3 segmentation.
-------
10 0 10 20 30 40 50
in 0 10 20 30
79*
78*
77<
Figure 15. Assumed summer circulation for Lake 3 model.
-------
Figure 16 shows early very preliminary results of phytoplankton biomass
at three segments across the lake. This run did not include any vertical
settling velocity. Segment #17 represents the Rochester embayment and as
seen reaches peak values of almost 15 yg chlorophyll/5, at the end of May while
the more open lake area (segment #16) lags by about one-half month. Peak
values at the open lake segment are about half of the Rochester segment. At
the end of May therefore, at day 150, a lateral gradient of chlorophyll of
about 10 yg/& is computed. This results primarily from the thermal bar effect
simulated by variations in the horizontal dispersion. Gradients across the
lake in the fall are not pronounced and reflect the general well-mixed char-
acter of the lake at that time.
Output Data Display
The Lake 3 model generates a substantial output totalling thousands of
numbers that represent the time variable response of each variable at each
segment. In addition, output on the growth and death kinetics, nutrient re-
cycle, and nutrient fluxes are also obtained. The size of Lake 3 makes it
difficult to fully comprehend the output so some effort is being devoted to
further analysis and display of the output. In this way, it is hoped that
further understanding and insight can be obtained on the behavior of the
solutions that are generated.
Figure 17 shows the output flow diagram that is presently being utilized.
Output from the model is written on tape in addition to hard copy and digital
overplots. The latter plots are for general screening by the analyst. Plot-
ting software is then employed on the tape output from a model run. Paper
contour plots and three dimensional plotting routines are employed to provide
further perspective on the output. Figure 18 shows a three dimensional plot
of output of phytoplankton biomass in the 0-4 meter layer from a view looking
down the lake from east to west. Plots such as these reveal the structure of
the solution in a meaningful way and also permit scanning of the output to
determine any strange or anomalous results that may indicate a flaw in the
numerical computations.
Returning to Figure 17, the three dimensional plots are also displayed
on a cathode ray tube, microfilmed, and prepared for a motion picture of the
complete output over time. A screening of this film then reveals the dynamic
and spatial behavior of the solution in a very unique way. At this stage, a
film has been prepared of a preliminary run for a full year of the surface
phytoplankton chlorophyll.
237
-------
f-0
U)
CO
en
O
E
CD
30
>= 25
IT
I 20
o
15
10
°- 5
0
5
a.
O
Segment 14
Segment 16
Segment 17
14
30 60 90 120 150 180 210 240 270 300 330 360
DAY OF YEAR
Figure 16. Preliminary results of Lake 3, 0-4 meters.
-------
DATA OUTPUT FLOW DIAGRAM
f-o
LO
Additional Statistical
Analysis
Water Quality
Etrtrophication Model
PRINTED
OUTPUT
LISTING
Plotting Software
(Contour-3D plots)
CATHODE RAY
TUBE DISPLAY
PAPER PLOTS
DIGITALOVERPLOTS
(Observed data vs.
computed values)
\
/
PROJECTION
CAMERA
Figure 17. Data output flow diagram used for verification ar>d display purposes.
-------
LAKE ONTARIO-LAKE 3 MODEL
LAYER 1 (0-4 METERS)
JUNE
NIAGARA RIVER
ROCHESTER
OSWEGO RIVER
ST. LAWRENCE RIVER
PHYTOPLANKTONCHLOROPHYLLa Maximum, 8.2|ig/l
Figure 18. Three dimensional plot of phytoplankton chlorophyll calculated
from Lake 3 model, June, 0-4 meters.
240
-------
REFERENCES
Bloomfield, J.A., et al. 1973. Aquatic modeling in the eastern deciduous
forest biome. U.S. - International Biological Program in Modeling the
Eutrophication Process - Workshop Proc., E.J. Middlebrooks, et al. Ed.,
Utah Water Res. Lab., Logan, Utah, pp. 139-158.
Burns, N.M. and A.E. Pashley. 1974. In situ measurement of the settling
velocity profile of particulate organic carbon in Lake Ontario. J. Fish.
Res. Bd., Canada, Vol. 31, No. 3, pp. 291-297.
Caperon, J. and J. Meyer. 1972. Nitrogen—limited growth of marine phyto-
plankton—II. Uptake kinetics and their role in nutrient limited growth of
phytoplankton. Deep Sea Research. Vol. 19, pp. 619-632.
DiToro, D.M. An evaluation of phytoplankton kinetic behavior in flask experi-
ments. (In preparation)
DiToro, D.M., D.J. O'Connor, and R.V. Thomann. 1971. A dynamic model of the
phytoplankton population in the Sacramento-San Juaquin Delta. Advances in
Chemistry, No. 106. American Chemical Society, pp. 131-180.
Eppley, R.W. 1972. Temperature and phytoplankton growth in the sea. Fishery
Bulletin. Vol. 70, No. 4, pp. 1063-1085.
Glooschenko, W., et al. 1972. The seasonal cycle of pheo-pigments in Lake
Ontario with particular emphasis on the role of zooplankton grazing. Limn.
and Ocean., Vol. 17, No. 4, pp. 597-605.
Glooschenko, W.A., et al. 1974. Primary production in Lakes Ontario and Erie:
A comparative study. Journal of Fisheries Research Board, Canada, Vol. 31,
No. 3, pp. 253-263.
Goering, J.J., et al. 1973. Silicic acid uptake by natural populations of
marine phytoplankton. Deep Sea Research, Vol. 20, No. 9, pp. 777-789.
Great Lakes Water Quality Board. April 1973. Great Lakes Water Quality
Annual Report to the International Joint Commission.
Hendrey, G.R. and E. Welch. June, 1973. The effects of nutrient availability
and light intensity of the growth kinetics of natural phytoplankton com-
munities. Presented at 36th mtg. ASLO, Salt Lake City, Utah.
Hutchinson, G.E. 1967- A Treatise on Limnology, Vol. II, Introduction to Lake
Biology and the Limnoplankton. J. Wiley & Sons, N.Y. pp. 601.
241
-------
Kibby, N.V. 1971. Effects of temperature on the feeding behavior of Daphnia
Rosea. Limn, and Ocean., Vol. 16, No. 3, pp. 580-581.
Limnological Systems Analysis of the Great Lakes. 1933. Prepared by Hydro-
science, Inc., Westwood, N.J. for the Great Lakes Basin Commission, 474 pp.
McNaught, D.C. and M. Buzzard. 1973. Zooplankton production in Lake Ontario
as influenced by environmental perturbations. In: First Annual Reports of
the .EPA, IFYGL Projects, NERC, ORD, EPA, Corvallis, Oregon, EPA-66/3-73-
021, pp. 28-70.
Paasche, E. 1973. Silicon and the ecology of marine plankton diatoms, I.
Thalassiosira pseudomona (Cyclotella Nana) grown in a chemostat with sili-
cate as limiting nutrient. Marine Biology, Vol. 19, pp. 117-126.
Patalas, K. 1969. Composition and horizontal distribution of crustacean
plankton in Lake Ontario. J. Fish. Res. Bd. of Canada, Vol. 26, No. 8,
pp. 2135-2164.
Porter, K.G. 1973. Selective grazing and differential digestion of algae by
zooplankton. Nature, Vol. 244, No. 5412, pp. 179-180.
Thomann, R.V., D.M. DiToro, R.P. Winfield, and D.J. O'Connor. March 1975.
Mathematical modeling of phytoplankton in Lake Ontario, I. Model Develop-
ment and Verification. Environmental Protection Agency, Corvallis, Oregon.
660-3-75-005. 177 pg.
Thomann, R.V., D.M. DiToro, and D.J. O'Connor. 1974. Preliminary model of
Potomac Estuary phytoplankton. Journal of Environmental Engineering Divi-
sion, ASCE. Vol. 100, No. SA3, pp. 699-715.
Vollenweider, R.A. 1968. Scientific fundamentals of the eutrophication of
lakes and flowing waters, with particular reference to nitrogen and phos-
phorus as factors in eutrophication. OECD, Director for Scientific Affairs,
Paris, France, 159 pg., 34 figures.
Watson, N.H/F. and G.F. Carpenter. 1974. Seasonal abundance of crustacean
zooplankton and net plankton biomass of Lakes Huron, Erie and Ontario.
J. Fish. Res. Bd. of Canada, Vol. 31, No. 3, pp. 309-317-
Yentsch, C.S. 1962. Marine Plankton. In: Physiology and Biochemistry of
Algae, Ed. by R.A. Lwein. Academic Press, N.Y. pp. 771-797-
242
-------
DISCUSSION
VASILYEV: ,What is the ratio of the lake's volume to the total annual river
flow to the lake? Did you consider current variability produced by wind
effects? How significant were the diffusion processes and did they make an
essential contribution to the individual compartments into which the lake
was divided?
THOMANN: The ratio of the lake's volume to the annual flow is such that when
the volume is divided by the flow, a hydrologic retention time of eight years
is obtained. Vertical circulation changes under the influence of the wind.
Tests were carried out to take seasonal changes into consideration. We
assume that it is expedient to include the variability of the vertical effect.
The three-dimensional model includes the influence of the diffusion processes.
The influence of horizontal and vertical diffusion are included and their
influence is variable.
YEREMENKO: If Dr. Thomann's three-dimensional model is based only on the
concept of phosphorus as a limiting nutrient, could nitrogen become a
limiting nutrient and if so, what is the influence of interference on
eutrophication? Can the model state when the transition from one factor to
another takes place? What kind of response or reaction takes place in the
lake,in response to a change in the phosphorus content?
THOMANN: The model considered both phosphorus and nitrogen and calculated
complete interference of these factors. We calculated in time steps of one
month over a period of 24 years. There are cases, it would seem, where the
relationship of the.content of limiting elements in the lake were displaced
from.organic phosphorus to organic nitrogen. Actually, transition to
nitrogen limitation takes place in 10 years.
The model does not exclude the influence of various combinations of
phytoplankton or rooted aquatic growths. It would be desirable to perfect a
measure for evaluating lake eutrophication. This has only begun. We are
beginning to attach models to estuaries, bays, and small lakes.
Our method using chlorophyll as the indicator for eutrophication is
somewhat deficient because it is changed by the variability of chlorophyll in
the phytoplankton. The model considers mass transfer from the sediment into
water. The results of other works on lakes conclude that a general phosphorus
regeneration from sediments comprises not more than 10% of all inputs.
Therefore it is not a very important parameter and is considered only in
general terms.
TULUPCHUK: Several species of phytoplankton change their buoyance. This
causes an intense bloom. Have you modeled this?
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THOMANN: The problem is very difficult and completely real in many areas.
We have not modeled it. We have tried to construct the simplest model and
even in this case we have obtained very complex equations. It is our
opinion that when we have information on the actual lake condition, then
everything will be clear and it will be possible to take all these elements
into consideration. The model exists only as a model and the main point in
this is that it is rarely connected with fact, which we actually observe. Our
models have water quality management as their goal. Therefore I would say it
is necessary to be brave enough not to consider all factors.
The process of evaluation of parameters begins with a comprehensive
literature review for each parameter. We tried to distinguish as many para-
meters as possible, and parameters are introduced into the model. After four
model runs, we can decide what parameters must be included. Eighty-two
runs were devoted to the dynamics of the spring bloom and these details are
shown in the graphs. We are now working with a three-dimensional model.
FUKSMAN: What kind of numerical values are assigned to the growth dynamics
parameters and do these change from segment to segment?
THOMANN: In a three-dimensional model, numerical values of parameters are
identical between different segments. In the future, we hope that some
changes from segment to segment will be considered.
FUKSMAN: Were 700 equations actually introduced into the computer?
THOMANN: Yes, each specialist has a parameter which, in his opinion, is the
main parameter. But it is necessary to choose the least possible number of
equations because, given a large number of equations, the results can be
unrealistic.
FUKSMAN: What is the largest size of a calculated system of equations?
THOMANN: From 1000 to 2000.
FUKSMAN: Did you initially evaluate the possibility of combining different
vertical layers? Was a solution obtained?
THOMANN: We are at the stage of solving a three-dimensional model. We mean
that we will be aggregating vertically but perhaps it will be successful to
do it horizontally.
ROZHKOV: Do you take into consideration that there can be models which it is
possible to adapt to the experiment?
THOMANN: Different methods which can describe the phenomenon are being
examined. The basic approach is construction of a model based on the inter-
relationship of substances taking into consideration their internal kinetics.
I include in our model data on flow in the lake which we can somehow control.
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A three-dimensional model requires 40 minutes of machine time. This
costs a lot of money. Each new variable introduced increases the cost of
the computer run.
ZNAMENSKIY: The model described gives us good results. One of them is the
conclusion that phosphorus retention in Lake Ontario continues for eight to
ten years. This corresponds well with the period of complete water exchange.
This confirms the fact that the basic factor is the water exchange factor.
In our research on the "Volga Cascade," we came to the same conclusion.
When we were trying to evaluate the contribution of different factors to the
development of the chemical composition of water, in the most general approxi-
mation, we found that 80% of the control is derived from residence time of
water, 10% from internal water body dynamic process, and all other factors
the remaining 10%. Therefore it must be noted that in Dr. Thomann's model the
most basic hydrological hydrodynamic parameters have been successfully
determined. This allows us to obtain not only scientific conclusions but also
practical recommendations.
TULUPCHUK: A model, not of a general ecosystem but of a specific water body,
has been developed. The interpretation of the dynamics of seasonal changes,
and in particular of algal productivity, is of great interest. This allows
us to make a judgment about the assimilative capacity of a water body, and, on
the basis of the model's reaction to the determined loading by nitrogen and
phosphorus, to make recommendations for establishing loadings to a water
body which will protect water quality.
YEREMENKO: I would like to comment on the role of the suggested models in a
general system for monitoring water quality. The old models included both
BOD and dissolved oxygen; these were preceded by models of algal biomass.
Using the models, it is possible to calculate eutrophication processes. This
is the significant value of Dr. Thomann's paper. Development of these models
is being carried out on a large scale in the United States. We are trying to
work in this direction and we've found difficulty in defining the constants
which are introduced into the equations, and deriving their numerical expres-
sions. We are trying to solve the problem of determining the constants and
have not yet had satisfactory results.
THOMANN: I must fully acknowledge the role of limnologists, biologists, and
other associates who have submitted a wealth of material for processing. I'm
asking that all similar work begin with the simplest models because from them
it is easier to move on to more complex models.
There is very little choice in determining parameters when constructing
models. We chose the coefficients from data published in literature, then
confirmed them in simulated laboratory conditions. This, unconditionally, is
the greatest problem in the creation of models.
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MODELING THE DISPERSION OF POLLUTION AND
THERMAL POLLUTION IN WATER BODIES
O.F. Vasilyev
INTRODUCTION
This survey is compiled from the following eight reports:
1. Barannik, V.A. and Ye.V. Yeremenko (VNIIVO, Kharkov), "Calculations of
Currents in the Nearshore Zone of a Deep Lake."
2. Barannik, V.A., Ye.V. Yeremenko, N.I. Selyuk, and B.S. Sinelshchikov
(VNIIVO, Kharkov), "Calculations of Currents in Shallow Water Bodies with
Estimated Wind Influences."
3. Vasilyev, O.F. and V.I. Kvon (Institute of Hydrodynamics, A. Sci.
USSR, Novesibirsk), "Numerical Modeling of Thermal Pollution in Water Bodies."
4. Yeremenko, Ye.V. and V.I. Karas (VNIIVO, Kharkov), "Formation of
Velocity Fields and Temperature under Wind Influence in a Water Body that
Receives Heated Water."
5. Tolmazin, D.M. (Odessa Division of the Institute of Biology of the
Southern Seas, A.Sci, Ukr, SSR, Odessa), "Calculations of Pollutant Dispersion
in Shallow Basins (with the Kakhovka Water Reservoir as an Example)."
6. Kozhova, O.M., V.I. Gurman, G.I. Gershenhorn, and T.I. Vlasova (BGNII
Irkutsk University Computer Center, Irkutsk), "Study of the Process of Bottom
Pollution Dispersion with a Dynamic Model."
7. Bystrik, P.S. and V.I. Sinelshchikov (VNIIVO, Kharkov), "Transport of
a Pollutant in a Shallow Turbulent Water Body."
8. Labzovskiy, N.A., A.M. Kryuchkov, and N.N. Filatov (Limnological
Institute, A. Sci, USSR, Leningrad), and V.A. Rozhkov (Leningrad Division of
the State Oceanograhic Institute), "The Analysis of Dispersion and Transforma-
tion in Polluted Waters in Large Lakes."
The problem of water pollution caused by discharge of industrial, agri-
cultural, and municipal effluents containing various pollutants is very real.
In some cases pollution can be linked with the discharge of heated water
(thermal pollution).
Evaluation of the degree of influence of the pollution source on the
water environment requires an evaluation of the resultant hydrodynamic and
physical processes, as well as the chemical-biological processes of self-puri-
fication. Mathematical models of water quality have been developed to de-
scribe these processes. They are based on the application of hydrodynamic
equations of turbulent liquid motion and include equations that describe the
mass balance of miscellaneous pollutants as well as heat and oxygen. The
major hydrodynamic question is the definition of diffusion coefficients when
compiling these equations.
246
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These reports studied separate aspects of the common problem of water
pollution. The reports may be conditionally subdivided into three groups.
This survey report is accordingly divided into the following three parts:
1) flow in water bodies; 2) hydrothermal processes in water bodies; and 3)
convective diffusion. Special attention is given to numerical modeling of
these phenomena in shallow water bodies.
The physical processes accompanying these phenomena are described by the
following equations:
Ua
a
Ka
"
9U2
9U
9U2 _
(J,
8P
9X3
9T
9t
9S
= -gp; .EL
i
+ U 9T _
"" 9X
a
, TT 3S =
3Xi
9P
3x7
= 0, P-P0 = -3 (T - T0)p0
9T , F
9t ' "u SX
9X
9
9X
KT
a a
9S
cpPo
(2)
(3)
(4)
(5)
a
9X
a
where
t = time;
X (a=l,2,3) = axes of a rectangular system of coordinates, the Xa axis being
directed vertically upward;
Ua = the components of the velocity vector;
P =- pressure;
p = density;
P = a constant value;
T,S = temperature and concentration of the non-reactive pollutant,
respectively;
T0 = the temperature at the density po;
jKsa = coef f icients, respectively, of the resultant viscosity, ther-
mal conductivity, and diffusion (taking into account both the
turbulent and the molecular exchange processes) ;
g = acceleration of gravity;
1 = the Coriolis parameter; and
B = coefficient of thermal expansion of water. If the pollutant
is not conservative, then the equation of pollutant transport
contains an additional item. In particular, the equation for
BOD is as follows:
K
9L
,
3X
3X
TT
La
8L
(6)
9X
where K is the mineralization coefficient, characterizing the rate of oxygen
consumption due to oxidation of organic pollutants (the deaeration coeffi-
cient) . Here the process of biochemical oxidation of the organic substances
is approximately represented by the linear term KL which corresponds to the
247
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so-called monomolecular reaction (the Streeter-Phelps law). There are some
hypotheses on the application of more complex (including non-linear) depend-
encies for describing the oxidation processes, which correspond, for example,
to bimolecular or polymolecular reactions. If computers are used for cal-
culations, then such complications are trivial.
The equations 1 to 6 describe motion and convective diffusion of heat
and pollutants for a shallow water body. The Boussinesq approximation is im-
plicit in these equations, which assumes that density variations can be neg-
lected except in the terms representing buoyancy. In the case when a pollu-
tant actively modified the dynamics of the current, the term connecting tem-
perature and density is modified to include pollutant concentration,
P = F(T,S) (7)
This can occur when calculating the intrusion of seawater into the mouths of
rivers, or currents in estuaries.
To complete the system of equations 1 to 6, coefficients are defined to
account for turbulent effects. In some of the works summarized here, constant
coefficients are defined, in others, semi-empirical dependencies are used in
the energy equations.
The contemporary models of water quality, in addition to the equations
mentioned above, also contain an equation for dissolved oxygen. In each par-
ticular case, it is possible to add additional transport equations to such a
system of equations. (If we are especially interested in the radioactive con-
tamination of a water body, then an equation for this non-conservative pollu-
tant can be added.) Besides, some components of the pollutant can interact.
Therefore, when compiling equations for each components, these processes must
be taken into consideration.
The specific versions of mathematical models similar to the one described
above, especially the one-dimensional models, are now beginning to be widely
used in engineering hydrology. It should also be noted that recently some
research workers have made an effort to develop models that more completely
describe the process of biochemical oxidation of organic substances (Vavilin,
1974), the process of photosynthesis, and several other processes that take
place in the water ecological system (Umnov, 1971).
Flow in Water Bodies
The indispensible premise used in the study of pollutant dispersion is
the quantitative determination of the velocity profile of flow in the water
body. Therefore, the analysis of flow in the nearshore zone where treated
effluents are usually discharged is of considerable importance. The problem
of the constant density steady-state model has been considered by V.A.
Barannik and Ye. V. Yeremenko (report 1). They used corresponding equations of
the constant density. The terms for inertia and the terms for horizontal tur-
bulent exchange were not calculated. The coefficient of vertical exchange is
constant. The model for steady-state circulation in a deep lake has been
solved using the Birchfield method (1972) for a lake having a bottom with a
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parabolic relief. The authors have generalized the approximation developed by
Birchfield for the specific case of nearshore bottom relief with a rather
well-defined underwater slope. Such is the case, for example, in Lake Baikal.
The Ekman number has been accepted as a minor parameter. The problem actually
was reduced to separate determinations of the pressure fields in the central
and the nearshore parts of the water body; then these determinations were com-
bined .
In accordance with the given wind velocity distribution over the lake,
the pressure field is determined and then, by rather simple relations
(Feltsenbaum, 1960), the velocity field of currents, characterizing the wind-
induced circulation in the lake, is established. The velocity profile of cur-
rents calculated in this way at the free surface of Southern Baikal in its
inner deep water area is shown in Figure la. The wind stress field at the
lake surface is shown in Figure Ib. Such wind stress distribution corresponds
to the southwest wind ("Kultuk") blowing along the lake. The circulation in
the surface layer is such that water masses flow from the western to the
eastern shore.
The dependencies allow us to calculate relatively simply, within the
framework of the model, the steady-state currents in the nearshore part of
the lake zone where water is sufficiently deep.
If the coefficient of the vertical eddy diffusivity is accepted as con-
stant, it is impossible then to obtain the vertical velocity distribution of
the wind- induced current that contains specific features of bottom and surface
turbulent-boundary layers.
Barannik, Yeremenko, Selyuk, and Sinelshchikov, in report 2, attempted to
describe these peculiarities of the current. They determined the vertical
eddy diffusivity coefficient by the depth, as a third power polynomial:
K3 =
XZ
X3Z2
(8)
Its four coefficients, X^(i=l, 2,3,4), were found on the assumption that the
shear stress coefficient in the bottom and surface boundary layers were ex-
pressed in the form of:
K_ = RU.Z
O X
where R is Karman's constant, U, is the dynamic velocity in the bottom (or the
surface) boundary layer, and Z is the distance from the bottom (or from the
water surface) .
The vertical eddy diffusivity coefficient was used for calculating the
stationary wind- induced flow in a shallow water body.
As in the original equations, Barannik et al. used equations 1 to 3 writ-
ten for steady-state flow. In this case, the water density and the coeffi-
cient of horizontal eddy diffusivity are accepted in the equations as con-
stants. A drag coefficient was defined for the bottom and given the condition
of pressure; the free surface, kinematic boundary conditions, pressure, and
249
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wind stress were defined and held constant. The problem was solved with a
numerical algorithm similar to the one proposed by Mikhailova (1968) for cal-
culating oceanic flow currents. Description of the numerical algorithm and
an example of calculating the wind-induced flow in a rectangular water body
of a constant depth were given.
10 cm/second
20 gyne/cm
Figure 1. Diagram of calculated resulting flow velocity.
a. is the velocities activated on the surface layer of Southern
Baikal by a southwest wind.
b. is the values of flow velocity and wind friction stress deter-
mined by vector lengths in the scales presented in the diagram.
Barannik et al. also considered the problem of the wind-induced flow in a
water body with an irregular bottom. In this case, they used the motion equa-
tions which described conditions of dynamic equilibrium between the pressure
force and the viscosity determined by the vertical eddy diffusivity change.
Instead of the condition of adhesion at the bottom, the condition of sliding
is accepted in the form of interconnection between the bottom friction stress
250
-------
and the local velocity. An algorithm of the numerical solution and the re-
sults of these calculations of a flow plan for a certain stretch of the Volga
River are given in Figure 2.
Figure 2. Plan configuration of flows in the channel with a wind direction
opposite to river flow (arrows on the flow line show the direction
of flow, circles designate the boundary flow line, flow function
graphs are constructed on the cross-sections).
In this calculation, a wind direction was selected which produced a nar-
row zone along the left bank of the Volga River in which undesirable effluents
were forced up the river. This picture of flow was formed in accordance with
the integral stream function, which characterizes the flow averaged by depth.
For verifying the above method of assigning the coefficient of vertical
turbulent eddy viscosity, the authors calculated the wind-induced longitudi-
nally uniform flow in a flat trough. From the results of these calculations,
it is possible to conclude that the vertical velocity profile of the wind-
induced flow can be described sufficiently well by the proposed method of pre-
setting the coefficient of vertical turbulent exchange.
Hydrothermal Processes in a Water Body
The hydrophysical problems connected with discharge of thermal water into
water bodies when operating stream and nuclear power stations are becoming
progressively more important. For solving these engineering, hydrological,
and ecological problems, it is necessary to be able to forecast and regulate
the temperature of water in water bodies. Dispersion of heat from a discharge
in a water body is determined not only by the heat exchange processes but to a
considerable degree by the character of specific hydrodynamic phenomenon oc-
curring in the water body (for example, the appearance of a density stratifi-
cation of the water body).
251
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The density of thermal effluent water is temperature-dependent. And,
although the temperature equation actually coincides with the inert pollutant
transfer equation because of the interconnection of temperature and density,
the velocity field and the temperature field are interconnected. As a result,
the velocity field acquired an essentially different character than the flow
of a homogenous liquid with constant density. Such plumes, entering the
cooling ponds of thermal and nuclear electric power stations, are considered
in the report of Vasilyev and Kvon (report 3) . Two specific conditions are
considered: hydrothermal phenomena in a shallow-water body of arbitrary
shape; and a temperature stratified flow in a relatively narrow water body of
oblong shape.
To solve the first (three-dimensional) problem, equations 1 to 4 are
used. In these equations, the terms characterizing horizontal exchange and
heat exchange may be omitted, if we assume that for flow in the cooling ponds,
the coefficients of the turbulent viscosity and thermal conductivity are
scalar quantities or that their values in different directions are of the same
order. However, this assumption is not always correct; for example, the hor-
izontal turbulent exchange is essential in mixing zones, when streams are
spreading on the surface.
In such water bodies as lakes and reservoirs, the flow velocities are low
and, in the equations of impulse flow, it is therefore possible to neglect the
non-linear inertia terms.
In the authors' numerical model of temperature-stratified flow in the
cooling pond, these simplifications of the problem are used; in particular,
the non-linear inertia terms, as well as terms characterizing the horizontal
exchange, are omitted. Then the equations 1 to 4, after simple transforma-
tions, may be written as follows:
7 4. n HY + K 4- 1
- z + r~ Y p dX3 + ^r~ K ~^r~ + 1
at dXi v p o x3 i sx3 3X3
|E« = 0; p-po = - 3p (T-T0)
dxa
_3T , ., 8T
(12)
When developing these equations, it was assumed that the deviation of the
level of free surface Z from its undisturbed level Zo was insignificant (in
comparison with the depth) .
The water body is considered limited by the lateral cylindrical surface
consisting of a hard-shore section and two liquid sections, (one through
which water flows into the water body and the other through which water flows
out) .
252
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The following boundary conditions are defined: the shoreline is imper-
vious. At the water discharge section, normal velocity component and tempera-
tures are given, and in the water intake sections, the normal velocity compo-
nent is given. Kinematic conditions and tangential wind stress on the free
surface are given, as well as the thermal flux. For calculating such impor-
tant meteorological factors as dynamic wind action and heat exchange with the
atmosphere, the connection between the tangential stress and velocity near
the bottom is given as a condition on the bottom. The heat exchange through
the bottom and the lateral walls is considered insignificant in comparison
with the heat exchange through the open free surface, and is therefore not
considered. When solving a non-steady-state problem, it is necessary to
specify the boundary conditions, velocity distribution, temperature, and the
state of the free surface in the beginning of the simulation.
The coefficients of turbulent exchange are determined by applying the
equation of energy balance of turbulence. The problem is solved numerically
by a method of incremental steps with an implicit difference scheme.
Formulation of the two-dimensional problem is similar. When studying
flow and heat exchange processes in relatively narrow water bodies', the prob-
lem, in many cases, may be simplified by transition to a two-dimensional re-
presentation averaging the width of the water body. This method of repre-
senting stratified flow can be useful when examining the flow in water bodies
which are narrow and shallow relative to the longitudinal dimension.
In this model, the channel is relatively straight and has only minor
cross-sectional changes. Since there is density stratification, the dynamic
and kinematic character of flow can be averaged in the longitudinal direction
in each transverse section and thus can eliminate one spatial variable, re-
ducing the problem to a one-dimensional case.
To evaluate the drag caused by friction against the side surfaces of the
bed, the authors used the hydraulic formula for the tangential stress on the
wall. This, added to the differential equations of the conservation of momen-
tum and heat, provided initial and boundary conditions similar to those given
for the three-dimensional problem. This model differed from the three-dimen-
sional problem by setting the conditions at the input and outflow of the cur-
rent in the region under study as well as defining the conditions at the bot-
tom. In this case, the rate of flow was defined at the inflow and outflow
cross-sections of the channel, or at the water surface level and temperature
at those points where water flows into the area under consideration. The
dynamic condition at the bottom was given the condition of adhesion, which may
be considered as the extreme case of the more general condition described
above.
This system was used for several numerical calculations of non-steady-
state situations. The results reproduced the actual conditions observed
during the experiment.
Yeremenko and Karas (report 4) modeled the wind-driven currents in a
shallow water body affected by a thermal discharge with a two-dimensional
model scheme. It was solved by complete thermodynamic equations of turbulent
253
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motion of an incompressible liquid with a scalar coefficient of turbulent vis-
cosity. The coefficient of turbulent heat conductivity (eddy conductivity)
was accepted as equal to the coefficient of turbulent viscosity. The coeffi-
cient of turbulent exchange was determined by using its known empiric depend-
ence on the Richardson number. The differential equations of motion were
written for vorticity and the function of flow. These equations were com-
pleted by the addition of initial and boundary conditions. The initial con-
ditions corresponded to an immobile liquid at a particular temperature. The
boundary conditions balanced the inflow, and outflow of water to the water
body and determined distribution of the velocities and temperature at the
thermal discharge site, uniformity of the tangential stresses at the air-
water interface, constant heat exchange between water and air at the free
surface, and absence of heat exchange at the bottom of the water body.
Using Fromm's method (1967), the authors obtained a numerical solution
to the equations for vorticity and temperature by using the explicit three-
layer method, and solution of the equation of flow functions by using the
explicit iterative method. The results of the calculation of several versions
of horizontal flow are given, from which the authors concluded that wind ac-
tion, thermal water discharge, and heat exchange considerably influence the
character of flow. Depending on interrelationships among these factors,
qualitatively different velocity and temperature profiles are formed.
Convective Diffusion of Non-Reacting Components
Dispersion of non-reactive pollutants in a water body is due to its
advective transfer by flow and by turbulent diffusion. Since the character
of the distribution of the passive substance does not influence the dynamics
of the flow, the motion equations in this case may be solved independently
of the pollutant diffusion equation, and then, using the velocity field, the
equations of pollutant diffusion may be solved.
The problem of pollutant distribution in the Kakhovka water body is de-
scribed by Tolmazin (report 5) when affected by the discharge of mining ef-
fluents. The calculations of water flow were made by the author using the
stationary model of a shallow sea, developed by Feltsenbaum (1960). With this
model, calculations of a flow were reduced to the determination of the flow
integral function from elliptical differential equations and subsequent cal-
culation of the horizontal components of velocity by the analytical dependen-
cies. The integral function of flow ¥ is determined through the components of
elementary flow Qx and Q by the following formulae:
Q = _ 3*. Q = 3*
x ay y 9x
where x and y = the coordinate directions in the horizontal plane. The forces
of inertia and the influence of the salt content of the discharged effluents
on the flow dynamics and processes of horizontal turbulent exchange are not
accounted for by the model. The effects of wind and of a river flowing into
the water body on the currents are considered in the model by means of the
wind stress on the free surface and the river flow at the side boundaries. At
the liquid boundaries of the water body, the condition of free flowing liquid
is accepted. In accordance with this condition, it is supposed that the
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elementary flows- at the liquid boundary are directed perpendicularly to this
boundary. Mathematically this condition is written as:
9HV9n = 0
where n is the direction of the normal to the liquid boundary. By analysis of
the results of calculations of the flows, the author concludes that the steady
state currents, calculated on the basis of this model, are in agreement with
the observed data.
The problem of diffusion of the pollutant was solved in two stages by
Tolmazin. At the first stage, convective diffusion for the pollutant near
the point source was examined. The pollutant was discharged into a flow
moving at a velocity varying only with depth. This problem was solved with
simplified non-time dependent equations for the convective diffusion of the
pollutant in which there are no terms of vertical transfer of the pollutant by
the flow and no terms of turbulent diffusion in the direction of the flow.
Velocity distribution with depth and the coefficients of the vertical and
cross turbulent diffusion were accepted as power dependencies on the vertical
coordinate. Analytical solution of the pollutant dispersion near the point
source was obtained by Tolmazin.
In connection with the solution of the convective diffusion of the pollu-
tant, Tolmazin determined the coefficient of vertical eddy diffusivity by
using the momentum and energy equations. However, the results of this analy-
sis have no application for solving the main problem of pollutant diffusion.
At the second stage, the mean concentration of the pollutant was calcula-
ted in the region of discharge, outside the discharge mixing zone where the
mining effluent was already completely mixed with the water medium.
Kozhova, Gurman, Gershenhorn, and Vlasova (report 6) studied the pro-
cesses of pollutant dispersion in the nearshore zone of a water body. The
problem was reduced to solving the pollutant diffusion equation (6). The
right side of the equation contains a term which represents the presence of a
source of pollutant. The current field was predetermined. There was no ex-
change of pollutants through the water surface or into the bottom, and the
zero values for concentrations at the lateral boundaries of the region of the
water body were accepted as boundary conditions. The problem was solved
numerically by a method of incremental steps. The coefficients of eddy diffu-
sion were constant but differed in direction. The results of a series of
numerical experiments and different physical conditions are described in this
paper.
Bystrik and Sinelshchikov (report 7) considered the problem of selecting
the optimum algorithm for numerical solution of non-reactive pollutant disper-
sion in water bodies. The algorithms constructed on the basis of the explicit
and implicit differential approaches were compared.
The basic idea of the comparison is to evaluate the limitations for a
time step with At for the explicit approach. The authors assumed that the
explicit scheme was optimal if: a) the limitations of the At time step for the
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convective and diffusive transfer are of the same order; and b) the At time
step is of the same order as spacing characteristic for a non-stationary pro-
cess. If not, the authors recommend the implicit method (i.e., the method
based on the principle of separation) or a combination of the methods explicit
for processes of one type and implicit for processes of another type, depend-
ing upon the completion of some characteristic inequalities.
In the work of Labzovskiy, Kryuchkov, Filatov, and Rozhkov (report 8) ,
results of field measurements of the water quality indicators in the Onega and
Ladoga lakes were statistically analyzed. They also studied processes of dis-
persion and dilution of effluents from pulp and paper plants.
Water quality measurement values for lignine, dissolved oxygen, calcium,
pH, electrical conductivity, and turbidity were grouped by geographic proxim-
ity and water depth. This data showed dependent random values correlate high-
ly with the main water pollution indicator, the concentration of lignine in
water.
The influence of the dynamics of hydrological processes on the dispersion
and dilution of effluents was studied. These processes, when statistically
analyzed, are not steady-state and are non-uniform in space. Regularities of
the hydrological processes were established on the basis of frequency with
time, two-frequency auto-and mutual-spectral densities. The values of these
statistical parameters were calculated and analyzed for a set of long term
field measurements (six months) to determine the degree of this equilibrium
and non-uniformity of the thermodynamic state of Lake Ladoga, the degree of
anisotropy, and direction of the maximum flow variability.
Vetrov and Dekin conducted a field determination of effluent dispersion
in deep water lakes with a radioactive tracer. For such a large water body
as Lake Baikal, ordinary calculations (e.g., the equation of the steady tur-
bulent diffusion) are complicated due to time-space changeability of the flow
field and the necessity to consider temperature stratification and complex
bottom relief, etc. In such conditions, the data required for production of
pollutant dispersion in a lake can be obtained with the so-called tracer
method by using a radioactive indicator. This experiment was conceived as a
field study of the dispersion of a conservative pollutant in the lake by dis-
charge of quantities of the isotope in the effluent. Relatively short-lived
isotopes with a large yield of gamma rays are usually used as radioactive
indicators. The concentration of the indicator in the water body is measured
directly with the help of a radiometric probe-transmitter lowered to the re-
quired depth from a ship. In 1972-1975, the Institute of Applied Geophysics
of the Hydrometeorological Service carried out a series of experimental dis-
charges of a radioactive indicator from the Baikalsk pulp and paper works into
Lake Baikal to determine the direction and velocity of the pollutant transfer,
and the degree of dilution of the effluents at different distances from the
point of discharge and during different hydrological seasons.
For some years in the USSR, mixing processes have been studied by fol-
lowing the release of fluorescent dyes into water bodies and monitoring the
dispersion from discharges with towed fluorimeters.
256
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The majority of research on the turbulent diffusion of non-reactive pol-
lutants was connected with determination of the spacial characteristics of the
field of averaged concentration. However, some problems, such as calculating
marine effluent dischargte, also require determination of other probability
characteristics of concentration of fixed points in the water body. The work
of Aitsam, Astok, and Juhat (1973) addressed this problem. On the basis of
analysis of the existing mathematical models of turbulent diffusion, the
authors concluded that these models could evaluate the distribution density of
pollutant concentration probability with satisfactory accuracy only in the
case of high concentrations. At low concentrations, there was a discrepancy
in the results of the experimental observations.
Experimentally determined densities of the dispersion of pollutant con-
centration probabilities are given in Figure 3. The graphs show that the
probability densities are of bimodal character. The authors explained the
bimodal character of the curves by the influence of perturbations of different
scales. They assumed that the probability density at a fixed point of a tur-
bulent flow may be presented as the sum of two probability densities y(s) and
f(s). The first describes the turbulent fluctuations of the pollutant concen-
tration inside the plume, the second reflects the influence of the low-fre-
quency perturbations, deflecting the plume axis. From the results of the ex-
perimental observations, the authors assumed that each of these probability
densities can be approximated by the Gaussian distribution curves.
But, since the pollutant concentration cannot acquire negative values,
the approximation is achieved through the single-sided normal truncated dis-
tribution.
The total probability density of the pollutant concentration is expressed
in the following way:
exP
p(s) = -
FlSzI/|
L- 2DY J
where Sy, Sf, Dy, Df are respective components of the mathematical expectation
and dispersion of the substance concentration, F(x) is Laplace's function;
x^
F = JL rae 2 dx (14)
/2TI 0
where f is determined from the condition of normalization. In this case
f=0.5.
The results of comparing the supposed distribution (equation 13) with the
experimental data are given in Figures 3 and 4.
From the formula 13, it follows that the probability density distribution
of the pollutant concentration is determined by the values of Sr, S , Df and
Dy. Sf is the "background concentration." In practical calculations for sea
conditions, we propose accepting Sf=0. The values of Sy, Df and Dy can be
257
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determined from semiempirical equations describing changes of these charac-
teristics.
P(S)
0.004
0.002
100
200
300
400S
Figure 3. Probability density of the pollutant concentration dispersion in
hydraulic flow at a distance of 1.5 m Erora stationary point
source.
This report also gave some results of field study related to probability
characteristics of the field of velocity in the Gulf of Finland, in particular
dispersions and spectral densities of the turbulent fluctuations of velocity
at the depth of 14 m from the level of the free surface.
258
-------
P(S)
0.02
0.01
25
50
75
100
Figure 4: Probability density of the dispersion of dye concentration in a
nearshore marine site.
259
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BIBLIOGRAPHY
Aitsam, A.M., B.K. Astok, and M.K. Juhat. 1973. Mathematical modeling of the
thermal pollution in a water body. In: Proceedings of the IAHR XVth Con-
gress, Istanbul.
Birchfield, G.F. 1972. Theoretical aspects of wind-driven currents in a sea
or lake of variable depth without horizontal mixing. J. Phys. Oceanogr.
Vol 2, No 4.
Feltsenbaum, A.I. 1960. Theoretical bases and methods of calculating the
steady-state flow. Academy of Sciences, Moscow.
Fromm, G. 1967. Non-steady-state currents of a non-compressible viscous
fluid. "Vychislitelnye Metody V Gidrodinamike," MIR Publishing House,
Moscow.
Karabashev, G.S., and R.V. Ozmidov. 1974. Study of pollutant diffusion in
the sea with the help of luminescent indicators and a towed monitor.
"Issledovaniye Izmenchivosti Gidrofizicheskikh Polyei V Okeane," Moscow,
"Nauka."
Mikhailova, E.P. 1968. Calculation of ocean currents; A non-linear equatori-
al problem. "Problemy Teorii Vetrovykh I. Termokhalinnykh Techinee"
Sevastopol.
Practice of theoretical and experimental study of deep water discharge of
effluents with the example of the Yalta region. 1973. "Naukova Dumka,"
Kiev.
Umnov, A.A. 1971. Use of modeling when studying the role of photosynthetic
aeration in a lake. "Ekologiya," No. 6.
Vasilyev, O.F., V.I. Kvon, and R.T. Chornishova. 1973. Mathematical modeling
of thermal pollution in a water body. In: Proceedings of the IAHR XVth
Congress, Istanbul.
Vavilin, V.A. 1974. Mathematical models of the process of biochemical oxida-
tion of organic substances in flowing water. "Vodnye Resursy," No. 2.
Vetrov, V.A., and S.A. Dekin. Field modeling of effluent dispersion in deep
water lakes with the help of a radioactive indicator (for example, Lake
Baikal). Proceedings of All-Union Conference on the Problems of Water
Quality Management and Self-Purification in Water Streams and Reservoirs.
Tallin, December, 1975.
260
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DISCUSSION
BOYCHUK: What methods did you use for identifying the coefficients of the
mathematical models?
VASILYEV: Common heuristic methods were used in the models. The Hydrodynam-
ics Institute has begun work on applying contemporary identification methods
when determining hydrodynamic system parameters. For the future, a wider use
of identification methods has been proposed for application to the model we
have discussed. Our French colleagues have some interesting examples: They
have used identification methods for determining hydrodynamic and hydro-
physical characteristics of motion in the upper sea layer. They included the
well-known corriolis parameter and obtained excellent results.
THOMANN: Please discuss the model verification.
VASILYEV: Professor Yereraenko can reply to this question for his paper. The
heuristic method is used when applying mathematical models of water quality in
rivers and modeling thermal pollution in water bodies. We are making the
first steps towards transferring models to real situations. For the most
part, we are relying on our experience and intuition. Ecological aspects open
a new area for us; we will be relying on the experience of our American
colleagues.
YEREMENKO: The calculations for currents in Lake Baikal were verified by an
integral evaluation of the water body between middle and southern Baikal. The
results were compared with Dr. Znamenskiy's field data. The characteristic
circulation currents were described for characteristic wind situations on
Baikal. These characteristics coincided with our calculations. The calcula-
tion of turbulent viscosity for depth of flow for given wind currents, allows
us to determine the numerical value of polynomial coefficients used for cal-
culating the velocity profile. A comparison of reference data with laboratory
research gave excellent results.
MORTIMER: Professor Vasilyev has compiled an analysis of a large quantity of
work. This is a great service. Steady states are rarely observed in nature.
In all stratified flows, if the Richardson number is lower than the critical
value, a certain situation takes place. The Richardson number expressed the
balance between the terms which characterize the gravitational forces and
turbulence. The Richardson number is a very important parameter. This number
was included in Karas' and Yeremenko's model. I have three questions for the
speaker in reference to this: (1) Was the Richardson number included in
Professor Vasilyev's model? (2) Does Professor Vasilyev agree with me that
modeling flow without the Richardson number is like playing Hamlet without
the prince himself? and (3) Have non-sloping (non-discharge) streams been
studied in Russia?
261
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LOBZOVSKIY: Over the last few years the Limnological Institute has studied
the dispersion of inflowing polluted waters in lakes. They have searched for
methods and have familiarized themselves with the behavior of the polluted
water. Polluted lake water does not dissipate but behaves in a complex
manner. Heavy particles fall to the bottom forming a bottom flow. The
lighter particles are spread along the surface, and behave as stream currents.
The polluted stream is dispersed along the surface for up to 15-20 kilometers,
assuming the temperature of the environment and having a density somewhat
higher than clean water. The polluted water descends downward and forms
lenses at various levels. We have presented this problem as a random, non-
uniform vector process (Rozhkov method). This is one of many directions which
can be taken to study the problem of polluted flow dispersion in lake water.
LICK: I agree with Dr. Mortimer that steady-state models are not suited to
deep lakes. It is possible to apply them only to shallow lakes where a
steady state is reached in 1 to 2 days. We reviewed the case when the in-
flowing water was warmer or colder than that in the lake. In the case where
it was warmer, the inflowing water rose. And in the case where it was
colder, the same thing took place but at the bottom.
VASILYEV: We have already discussed the role and meaning of steady-state
problems. We have to consider that it is necessary to know how to solve
steady-state problems in order to explain the characteristics of the flows
studied. However, quasi-stationary flow systems can occur in practice. In
regard to Richardson's number, Dr. Mortimer and Shakespeare are correct.
Dr. Yeremenko does take Richardson's number into consideration. Dr. Tolmazin
did not observe any stratification phenomena in his work on the Kakhovka
Reservoir (a shallow, well-mixing reservoir). Therefore, density stratifica-
tion is not considered in his work.
Regarding the Hydrodynamic Institute's model, for the sake of brevity in
the paper, we were not able to fully describe the solution used. The system
of equations in the report is supplemented by the turbulent energy equili-
brium. This equation considers turbulent energy diffusion, turbulent energy
inertia, the Archimedean Force, and turbulent energy dissipation; Richardson's
number is concealed in the equation. However, we can separate it if we
combine two terms—the turbulence energy generation and buoyance force.
Regarding Dr. Labzovskiy's statement, at present Hydrodynamics'
specialists are studying jet flow. I can cite corresponding references.
These specialists have discovered some good approaches to the physical and
mathematical formulation of such problems.
262
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IDENTIFICATION OF ECOSYSTEM MODELS BY FIELD DATA
(INVERSE PROBLEMS OF ECOLOGY)
V.B. Georgiyevskiy
INTRODUCTION
We propose constructing ecosystem models to be identified in this paper
on the basis of conventional differential equations (dynamic models) and in
partial derivatives.
Identification is the determination of numerical values of coefficients
for use in differential equations by firsthand observation of biotic and abio-
tic components of an ecosystem in a certain region.
In ecology, problems of this type are examined by identifying objects of
automatic control (Prober, 1971; Beck, 1974; and Leaky and Skog, 1972). The
usual identification method is to select numerical values for the model para-
meters, which minimize the mean square difference between the observed and
calculated trajectories. That is, an execution of the procedure:
min ) [HJ-HJ] pj [HJ-HJ], (1)
z -'_^
where RJ(R3) represents the j-th realization of the vector of observed (cal-
culated) biotic and abiotic components of an ecosystem, pJ is a positively
defined weighting function, T is a transpose index, and z is a vector of un-
known parameters. These methods of ecosystem identification are based on the
solutions of direct boundary conditions H.
Nevertheless, ecological problems possess some characteristics which
essentially hamper this approach and sometimes render solutions impossible.
The characteristics of inverse problems in ecology, which necessitate identi-
fication without using solutions of the direct boundary conditions, are as
follows:
1. The ecosystem models may be of a very complicated type: the equations
may be non-linear, of partial derivatives, and the number of equations may be
large. ..The. identification in this case may be realized, e.g., by using quasi-
linearization methods, integral transformations of Laplace, Fourier, Mellin,
Kantorowitz, etc. But in all cases, the problem will prove to be multidimen-
sional and quite complicated.
2. In ecology the information to be interpreted is essentially stochas-
tic in character. Two circumstances are important here:
a. The functions measured in a real ecosystem tend to be masked by
variability. This property has an objective nature. It demands application
of calculating procedures which smooth down the random "noise," i.e., mean
263
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square minimization, integration, etc. "The procedure is determined by this
property.
b. The boundary conditions where HP is observed have not been ob-
tained experimentally; in general cases they have a stochastic nature. The
divergence between HJ and HJ are solutions of different boundary characteris-
tic problems. The use of B.I in this case leads to systematic errors caused
by the method of defining parameters and not by objective reasons. The value
of errors may be quite large since the boundary characteristic definition may
prove to be an improperly posed problem. A similar situation is discussed in
the next paragraph.
The circumstance just mentioned is one reason that necessitates the iden-
tification of ecological systems without solving direct boundary characteris-
tics problems.
3. The third essential characteristic is the necessity to define the
type of model more accurately in the course of identification. In a general
case, the structure of ecological models is not initially known. When con-
structing a model, it is necessary to assume a priori every possible trophic
link and the influence of as many parameters as possible, and when treating
experimental results, the hypotheses concerning the type of links should be
verified.
This problem cannot be solved by the standard methods of identification
based on the utilization of theoretical solutions, in particular B.I. More-
over, the designation of model type can prove to be an improperly posed pro-
blem.
If we assume that iKith processes of diffusion and elimination exist in a
steady-state regime,
V2S - US =0 at S =0, (2)
Z=oo
where V2 = A d (zd )
" 0 ^ o z
,2=1
z
we find •# by measurements in two points:
S(z2) K0(^ z2) (3)
But if there is really no elimination and the process is described by the
ion V2S = 0, then we will inevitably
fers as much as it can from the true one.
o
equation V^S = 0, then we will inevitably find by B.3 a parameter which dif
It is evident that similar errors are possible when designating trophic
links and when studying the turnover of organic and biogenic substances.
The example discussed above is quite a simple one, and it would be easy
to show a procedure here which would help correctly solve the hypothesis on
the type of ecological mode. But in a general sense, this cannot be done.
It should be noted that it would be possible to illustrate that the boundary
characteristic scheme, discussed in the previous paragraph, is incorrect.
264
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These characteristics of ecosystem identification lead us to conclude
that it is expedient to find identification methods which would not use the
solution of direct boundary conditions. This method can be based on integral
transformations of differential equations of ecological models (Georgiyevskiy,
1971), aimed at excluding the experimental functions differential operation.
With such an approach, the identification problem changes from a problem
of mathematical physics into an algebraic problem. Some results of the re-
search in this direction will be outlined below. Identification methods are
reduced to constructing an optimum set of integral transformations (in the
sense of the number of arithmetical operations).
It is important to note that the procedure for defining parameters is
deterministic. The presence of only high-frequency masking variability or
"noise" is assumed. More complete information on "noise" ought to be used
when studying the qualitative evaluations of the desired parameters.
Identification based on integral transformations enables us to avoid
these difficulties:
1. Very complex models can be identified, since the difficulties in
solving boundary condition problems are not important in this method;
2. The errors implicit in the designation of boundary characteristic
schemes are excluded; and
3. Hypothesis on the type of models is formally solved, as only members
with coefficients that differ considerably from zero are left in the model.
The identification of differential equations by integral transformations
allows us, with a great degree of objectivity, to approximate real and highly
complicated ecosystems by generalized model parameters. Identification of
this type also allows us to define certain rules for ecological observations.
It should be noted that the inverse problems of ecology are considered here
from the view of interpreting hydrobiological information.
The Type of Differential Equations of Water Ecosystem Models
The identification method is determined by the type of differential
equation terms, subjected to integral transformations. A widely used dynamic
model of ecosystems is:
n
L = f£i - Ci(ai - I ByCj) = 05 i=l,2,...,n (1.1)
dt j
Fairly generalized models, reflecting the dynamics of spatially hetero-
geneous water ecosystems, are described by differential equations in second
order partial derivatives:
L = ii3Ci - div(D grade.,) - div(cv) - C1(a. - J&.C.) = 0,
3t j J J
i = l,...,n (1.2)
In most cases, it is expedient to consider the equations with constant
parameters, linear relative to these parameters, as this essentially simpli-
fies the problem of excluding the experimental function derivatives. Finding
265
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the variable parameters is proposed generally in parametric form, for instance,
by finding the constants for extension to a certain series. Thus, the problem
of defining variable parameters is reduced to the problem of defining con-
stants, which is a natural corollary of the general principle, i.e., rejection
of solutions of boundary conditions.
In paragraph 4.1, a case of variable parameters, identification without
parametrization will be discussed.
Model Parameters as Systemic Characteristic of Water Biocenosis
An important characteristic of inverse-problem solving.is the fact that
the parameters defined by field data are a certain systemic characteristic of
water biocenosis.
Real, complicated hierarchical ecosystems with extremely ramified links
are approximated, when modeling, by generalized groups of producers, consumers,
and degraders. The parameters which identify a complicated ecosystem with a
generalized model in a specific area may differ substantially from practical
coefficients of reproduction, feeding, death rate, migration,, selection, and
other ecosystem components. For this reason, the use of parameters obtained
under laboratory conditions proves to be quite difficult for constructing
models, and also accounts for the fact that model parameters obtained by
field observations for specific ecosystems may differ substantially even
under identical biocenosis.
The systemic characteristization of biocenosis by a certain vector of
generalized model parameters makes it possible to reduce the study of water-
body pathology (toxic effect, pollution, eutrophication) and ecosystem condi-
tion monitoring to operative observation of the model parameters. The last
circumstance defines the importance of inverse problem solution by field ob-
servations which are stochastic in nature.
Method of Intergral Transformations
The method of excluding experimental function derivatives is described
in the following section, i.e., replacement of a differential equation by an
integral algorithm which serves as an algebraic equation relative to the de-
sired parameters.
The exclusion of derivatives may be achieved by a different type of inte-
gral transformation. Transformations differ in terms of the efforts expended
and in limitations necessary to their structure. One method which can make it
possible to solve the problem of identification is the M-method (modulating
functions method) (Loeb 1965; Takava 1968). However, it is not the optimal
method algorithmically.
The method of integral algorithms which is optimal in the sense of ex-
pended efforts is cited below. Its algorithms can be divided into two
classes: (1) minimum smoothing algorithms using minimal required experimen-
tal information; and (2) maximum smoothing algorithms using all the experi-
mental information. The smoothing of information is done with random weight,
and the algorithms under such conditions require minimum effort.
266
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The structure of integral transformations modified for exclusion of
derivatives is:
L = up- - Dy2H - vgradH-? (CR-H) =0, (6)
where H is any of the ecosystem components and Cu is a certain constant.
The structure of integral transformation is defined by the necessity to
exclude the higher derivative from the equation. The integral transformation
of the differential equation L = 0 is done term by term and in succession re-
lative to each independent coordinate. An important feature of the method is
the fact that the integral analog of various expressions (table of images)
may be calculated beforehand and the algorithm in each case can be obtained by
a formal operation.
The method consists of the substitution for a differential equation by an
integral one, and algebraic exclusion of the values not obtained experimental-
ly. Further in the text, we will consider the argument x - any argument for a
second order equation (e.g., x_,y,z), and the argument t - any argument for a
first order equation.
Minimum Complexity Intergral Transformation I for an Equation in Cartesian
Coordinates
A. Image with respect to x. Let us consider the expression
L =~|- f = 0 (7)
3x2
where f represents all members along some coordinate except the second deriva-
tive. Transforming equation 3.2,
x x r>2-u
f f C*H - f) dzdz = 0
x0 XQ 3x'-
we obtain
H(x) - H(x0) - (x-x0) - / / f dxdx = 0.
u
Out of the last equation we compose a system of two algebraic equations
taking x = x^, i = 1,2 and excluding from the system the member conditioned by
the value 3H(xo)/9x, which is not obtained experimentally. The integral ana-
log obtained as a result of the operation is the differential equation L=0
minimum smoothing representation with respect to the argument x. It will be
designed via IX(L),
I*(L) = (x2-Xl) jfl /Q L dzdx - (xrx0) /2 /Q L dzdx. (8)
We put various expressions in 3.3 instead of L and obtain the table of
minimum complexity images for various terms of differential equations by
which algorithms for parameter determination can be formally obtained accord-
ing to the kind of ecosystem model equations.
267
-------
f o
I* (H) = /1Hdx-x[)(x2-x1) + /1xHdx(x1-x2) + 2Hdx- x2
/zxHdx(x -xn);
-X" ID
CH
B. Image with respect to t. Let us now consider the expression:
L = $* - f = 0; (10)
O t
it is evident that
i* CD = Kdt, (ID
t to
3%
where If is the inverse image of differential equation L=0 minimum smoothing
with respect to argument t.
Consequently :
I* (il) = HCtj) - H(t0),
t at
I* (H) = flKdt,
c to
i* (CH) = (trt0) CH (12)
The Intergral Transformation of Complete Smoothing for Equation with
Cartesian Coordinates.
A. Inverse with respect to x. The expression 3.2 should be transformed
°
xxx
/ p / / ( - f) dzdzdz = 0
xO XXQXO 8x
Internal double integration excludes the derivative, external integration
averages H, with the random weight P .
X
Fixing x = x. , i=l,2,3, we exclude from the last expression, the terms
dependent upon the value 3H(xQ)/9x, (this value is not measured) and upon
H(XQ) so that instead of separate values of the functions, only mean values
will be included into the algorithm. The resulting integral analog will bear
the name of the differential equation L=0 inverse with respect to argument x
under complete smoothing and it will be designated via IX(L) . That is:
IX(L) = f1? f / Ldxdxdx-Ix + /2P / / Ldxdxdx-IIx + /3PX/ / Ldxdxdx- IIIx, (13)
x XQ X0X0 X1 X0X0 X
268
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X X X X
where I = /2PYdx/3P..xdx - /2Pxxdx/3P^.dx;
x xi x X2 xi A X2 A
II = /3P dx • /^xdx - /% xdx/1? dx;
x x2 x x X2 x xo x
III = /^dx/2? xdx - /^xdx/2? dx.
x xO x xi x xo x xi x
If we put various expressions in 3.8 instead of L, we will obtain a
table of inverses of different types of ecosystem model differential equation
terms. The table helps obtain the parameter determination of algorithm (see
sections 10 and 11 [Georgiyevskiy 1971]).
B. Image with respect to t coordinate. Expression 3.5 should be trans-
formed into:
t t
/ p / (il - f) dtdt = 0
to ttO 3t
Here internal integration excludes the first derivative, the second integra-
tion averages the experimental information with random weight Pt. Excluding
the term dependent upon H(tg) from the last expression, we obtain the differ-
ential equation L=0 inverse with respect to argument t under complete smooth-
ing.
t. t. t t t., t
I^(L) = /2P1.dt/1P<_/ Ldtdt - flP^dtf'Ptf Ldtdt = 0 (14)
t ti c to tto to n ti Lto
Intergral Transformations of Complete Smoothing for Equations in General
Curvilinear Orthogonal Coordinates
The integral transformations described above are expedient for the cases
where the experimental area is formed by a parallelepiped. When interpreting
information in a complex-form area, it would be inefficient to limit that
information contained in the parallelepiped. Therefore, it is necessary to
construct integral transformations for equations in the general curvilinear
orthogonal coordinates.
Let us consider a differential equation of the second order with respect
to the independent arguments X1,x2,x3. The structure of integral transforma-
tions with respect to X1,x2,x3 is uniform.
Similar to 3.2, we obtain I (L), the inverse equation L=0 with respect to
under complete smoothing:
h h Ldx dx dx
123 1
1
X? A Xf Xnh X! /X', A,
dxidxi/1? dx, + /XP /1f4-—r1 h.h.h.Ldxdxjd:^. /lp dx,/1?
vo Xi 1 J vi XT vnhoh civn 1 z o i 1 ,,o x. -"-^n
h h x2 xl !] xl Xlxoh2hsx0 1 2 3 1 I x2 xi xO xl
2 3 l ' -
269
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X ,
xoh0h
X2 Xlx0h2n3
1 1
x
X2
1
x?
xi \
*! ' * Px dxl\
1 l X0 1 * )
1 l
Ip /llUL—/I hh h Ldxdxdx, /XlP dx,/lpx /i^l
2 Xix0h2h3xo 123 1 1 1 I xO xl *xl x 1^x0^2
1 1 1 ! l 1
x
(15)
where P is a random weighting function for averaging experimental function
values;Xni,h2,h3 are the Lamay coefficients; x°,x},x^,x3 are the vaiues of the
Xi coordinate in points 0,1,2,3.
The transformations I , I . are obtained from I when X1,h1,h2,h3 is
X2' x3
substituted accordingly for X2,h2,h3,hi or for x3 j
,h2 .
With the help of 3.10, we can obtain a table of images of the ecosystem
model equations terms. In particular:
T .
Lxi*
' -, r / \
1 9 / h2hs 3H \
] h1h2h3 1 9x1 I h! 8xJ
xl
/ •"• 1
}- - /^
I V
X?
Hdx,
1
Hdx
1 1
• II
j
Iv »
Xl
x
flV~
xl xl
Hdx,
II,
+
p
-u-
X2 xl
1
l
x
(h h H)
/1hHdxdxi
X0 X1X0 111
v
xl XlxO
1 1
X X
+ f1? /^.Hdx.dx.
x2 xlx0 1 1 1
1 1
III
xr
T [H] = /1P
X
x
X T-~
l xn Xlxnh2-h3
II
X
X
dz.dx.fc, • IIX1 + /IP. ?'-6jL- '
III
1
xl
(16)
For particular cases of equations in cylindrical and spherical coordi-
nates, the table of images is given by Georgiyevskiy (1971) for minimum and
complete smoothing algorithms. The image with respect to argument t remains
in the form 3.9.
I is the Method for Equations of n-th Order
The method described above is equally valid for the equations of the n-th
order (Georgiyevskiy 1971)
P as a Method
When the argument values are equidistant, an image of differential equa-
tion L=0 under minimum and complete smoothing of a certain recurring method
270
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may be obtained (Georgiyevskiy 1971) .
M as a Method (Loed 1963, 1965; Takava 1968)
The M method may be considered as a particular,case of the n method under
the special selection of weighting functions P. The M method is not algorith-
mically optimal, but it is very convenient because it is compact. The table
of images based on the M method is of the following form:
Mv (Mr) = (-D1/1^ dx, i=0,l,2,...,n (17)
x. 3x xQ dx1
where
= 0, 1=0,1,..;n-l.
x=0
x=x
I
Example of Algorithms
The algorithm for- defining the ecological equation parameters is the
equation inverse for several areas defined by number of the desired parameters
if constant, or by the number of expansion terms if variable:
ItIxIyIz(L)i = 0, (18)
1=1,2,... is the number of integral transformation areas.
It is possible to define the algorithm for determining parameters for the
dynamic system (1.1). For this case the derivatives of the first order with
respect to t are present in 4.1, so it is quite sufficient to use 3.9:
t, t0 t0 t, tn t t, t. t t
I,-(L) = /1PC1-dt/2Pdt - /2PC1dt/1Pdt-a-1- (fl7f C,-dTdt/2Pdt - /2P/ C,-dTdt/1Pdt)
L to ti ti to to to ti ti to to
n t t t t t t
+ )3.. (/1P/C,C-dTdt/2Pdt - /2P/ C-iCidTdt/iPdt) = 0 (19)
V ij to to1 J tl tl to J tO
1=1,2,...n.
It should be noted that the parameters a., 3-n are established from the
1-th analog in 4.2,independently of the other analog parameters; i.e., when
establishing parameters the system 1.1 is "disintegrated."
The necessary number of independent algebraic equations according to the
number of desired parameters a., B^ may be obtained in the i-th analog 4.2
by changing the integration intervals t,, t_ or the weighting function P. At
this point the spacially heterogeneous ecosystem model 1.2 identification
algorithm is identified.
For the reasons shown in previous examples, the index i is omitted. For
simplicity, only one space coordinate is considered. Assume that div v = 0,
D = constant, and using the minimum smoothing algorithm 3.4, 3.7, given
x =0, t =0, we obtain:
271
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-ir J£
I*(L) = y{x /2[c(x,t,) - c(x 0)] dx-x x /2[c(x,t ) - c(x,0)] dx + (x -x )
Ixl L i x1 1
^1x[c(x,t1) - c(x,0)] dx}-DJx flc(x.,t) dt - X,/1c(x?,t) dt - (x -x )
0 l 2 0 l L 0 2 1
/1c(0,t)dt}-v{(x9-x1)/1/1c(x,t) • dxdt-x1/1/2c(x,t)dxdt-a{(x1-x9)/1/1
0 2 1 0 0 l Oxl 0 0
t x t x n t x
xcdxdt-x^^/1/ cdxdt + x, J1/2xcdxdt} + Y 3. {(x -x U^xccjdxdt - x,x
1 2 Oxl * Oxl j J 1 2 0 0 l 2
•J- v . -v~
y1/2ccjdxdt + x1/1/2xccjdxdt} = 0 (20)
Oxl 0X1
Determination of Variable Parameters
By parametrization, for example, by expansion into a certain series, the
problem of determining variable coefficients may be reduced to the problem of
determining constant coefficients as described above. The number of expansion
parameters during identification should be sufficiently large, but only para-
meters which differ significantly from zero should remain in the final result.
It is also possible to obtain integral transformations of differential equa-
tions with variable parameters; thus an integral equation relative to the de-
sired parameters will be obtained.
We are limited by a case of one independent variable, because the distri-
bution to a case of several arguments will be quite similar. Assume that
f- {k(x) M } = f(x)
dx L dx J
Multiplying this equation by a system of linearly independent functions
i=l,2 with the conditions Y.(0) = ^-^(O) = 0, and double integration within
the limits (0,x), for determining k(x), we obtain a system of two Volterrian
integral equations:
H(x)¥.(x)k(x) + / H(z) [(x-z)Y.(2)(z)] - 2Y.(l)(z)] k(z)dz + / H(z)[(x-z)
1 0 1 ! 0
• y (7~) — *¥ (?") ^ ]<(^' (•?} d? = T . i—l? f99">
i- \^ / T.\^/JK- \ 6 J U.£ — r-: 1 — -L,/. \Z./.)
To determine k(x) from the equation
k(x) -0 = f (23)
a system of three Volterrian integral equations must be constructed:
-y . -\T
H(x)y±(x)k(x) + /H(z) [(x-z)^ > (z) - 2y(l)(z)] k(z)dz + 2/H(z) [ (x-z)
0 0
272
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1 jT
0
i = 1,2,3 (4.7)
(z) - Y.(z)] k(l)(z)dz + jTH(z) [(x-z)] Y, (z)k(2)(z)dz = F-
1
The accuracy of numerical solutions to integral equations will be deter-
mined by the statistical characteristics of experimental functions when estab
lishing parameters by expansion into a certain series.
Determination of Dependent Parameters for Ecosystem Models
Of all the coefficients in the ecosystem model differential equations,
only one is dependent in an algebraic sense. The rest are independent. By
using the previous methods and only differential equations, it is possible to
determine only independent coefficients, since one of the terms in the inte-
gral analog must be free. However -, a problem may arise in establishing all
coefficients of the equation's terms, for example, y,D,v,?f in 1.2.
It should also be noted that several ecological parameters are included
in the boundary condition values (meteorological factors, substrate effects,
exchange between the epilimnion and the hypolimnion, advective flows) . Con-
sequently, the process of identification in general must provide methods to
determine both independent and dependent parameters of ecological models si-
multaneously, and to use any information on boundary conditions values to
simplify the integral transformation and increase the accuracy of identifica-
tion (however the boundary condition value of the derivative may remain) .
The general method for determining the whole complex of dependent and in
dependent parameters is: the differential equation (for the ecosystem model)
should be replaced by an integral one, and one of the terms, dependent on the
experimental function derivative, should be determined by information on the
boundary condition. In the expression derived, all the parameters will prove
to be independent. It should be noted that after the first integration, a
continuity equation for a finite volume is often guaranteed. Sometimes it is
simpler to construct an equation of this type on the basis of physical con-
siderations .
It should be emphasized that in the process of obtaining parameter deter
mination algorithms, only part of the information on boundary conditions is
used. This information is insufficient for solving a boundary condition p^^b
lem. Therefore, the algorithms obtained make it possible to interpret com-
plicated forms of motion (multidimensional, non-steady-state regime, law of
random movement, etc.). They prove to be much more generalized than the
theoretical solutions in which complete sets of boundary conditions values
are used, but they are not as generalized as algorithms for determining inde-
pendent parameters where boundary conditions values are not used.
This statement is illustrated with a simple example. Suppose that an
ecological situation is described by the equation:
273
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and the value
D^^-=Q(t) (24)
is measured experimentally, then the identification algorithm y,D,R has the
form:
IT IT [/Ldx]. = 0
it J-x 0 1
under 5.1.
Here IT and I are integral transformations, their structure is derived
from 3.9, then the v§riable t is the argument in Ixt, and in !TX it is the
variable x.
The content of 5.2 necessitates integral transformation, with calculation
of the boundary condition. This approach may be easily generalized for a
multidimensional case and for various coordinate systems.
Construction of the Algebraic Equations Systems
In essence, the outlined integral algorithm method reduces the problem of
determining parameters to an algebraic problem, that is, substitution of al-
gebraic equations for differential ones relative to the parameters. The solu-
tion of these equations requires the application of all the technology which
has been developed to the solution of the algebraic problems.
The main requirement is to work out well-specified systems of equations
(the choice of integration intervals and weighting functions, orthogenaliza-
tion of matrixes, etc.), to verify the condition number A= ||A|| ||A~1||,and to
evaluate the error of the solution.
After complete algebraic reduction of the problem, the concept of obser-
vation and identification are applied to the integral algorithms procedure
(Li 1966) .
Calculation of Intervals
The problem is reduced to the use of the Newton-Cotes quadrature and
cubic capacity formulas, Gaussian and Tchebysheff's quadratures with tabulated
coefficients and node, and also to the approximation of certain special exper-
imental functions. The details are given by Georgiyevskiy (1971).
Parameter Determination When Inadequate Information Is Available on Some
Part of the System
When interpreting ecological observations, the information on some coor-
dinates may be insufficient. For instance, the information on process dynam-
ics is often abundant but the number of sampling points is relatively small.
For such cases, the following approach is recommended: the presence of de-
rivatives in the differential equation on coordinates with a "large" amount of
information should not be considered a complication. However, assuming the
exclusion of these derivatives by a certain integral transformation, then the
derivatives on the coordinates with "little" information should be excluded by
274
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using a priori information on the type of movement, by solving boundary con-
dition problems, or by choosing special approximation laws (Georgiyevskiy,
1971).
For example, the algorithm for determining parameters (both dependent
and independent y,D,?0 for the equation describing an ecological situation
which has cylindrical symmetry, is written as:
L = y |£ - DV2c - vc = 0; V2 = I |- (z |^) (26)
given a small quantity of information on z.
In integral form, the minimum smoothing algorithm follows:
%XT & zLdz] = °
•"•t J-z ZZQ
when D • 9C(zQ,t)/9z = Q/2nzQ,
where I& and I* are the integral transformations of 3.7 with respect to t
and z. ifc Tz
Choosing the approximation c = B(t) + D(t) Inz, and finding B and D
according to measurements at two points, we obtain:
y{[c(z1,ti) - c(z150)] + [c(z2,ti) - c(z2,0)]} In ^ - 1)}- D • 4
1
[c(z15t) - c(z2,t)]dt - I/[c(z t) + c(z t).(li£2 _ i)dt
0 i 0 x zl
o Irv^ ti
±( |Itrl) • /"Qdt, i = 1,2,3 (28)
n z o
Statistical Model
It is expedient to construct a statistical model when random values are
defined by the measurement errors. Evaluation of the accuracy of these
ecological parameters is a problem of evaluating the probability properties
of a random value formed as a result of nonlinear operator influence on a
random field. This problem can be solved by existing methods.
Numerical Examples
A direct method of verifying the identification integral algorithms con
sists of para-mete-!: determination from the results of analytical or numerical
solutions of boundary condition problems with preset parameters (test prob-
lems) . Georgiyevskiy (1971) states that such calculations are given for
determination of the parameters (3.1) and for the equation
.( + >-
275
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in Cartesian and cylindrical coordinates by I, M methods; for
3 d2H _ , =
dx7 (30)
for: B1i4-(z4^) - If, n (HV-HV ,) - Tf, (Hv-Hi,+1) =0, k = 1, 2, . . . ,n (31)
kz dz dz ' k-1 k k-l' k k K-I-J.
For these equations, examples of parameter determination from field ob-
servation data are given by Georgiyevskiy (1971).
The variable parameter was determined in the equation:
j TJ
~- (/k(z)dzl?) = 0 O2.)
dx 0 dx
The parameter: k(z) = kQ + kxz + k2z2 + ^3z3 (33)
was derived from data sets which had a great deal of "masking" variability
(for solution of the equation 10.4, the variability was superimposed from the
random numbers table). Up to a certain level of interference, the function
regenerates steadily.
The basic model equation for interpreting ecological information is:
-•- _ -p.^ C ^ C (~\ / o / \
J.L. Fuchsman (Institute of Hydrobiology, Ukrainian SSR Academy of
Sciences) identified this equation's parameters by I* and I-methods, using
an ALGOL program for a BESM-6 digital computer. The program has been verified
by the test data of Georgiyevskiy (1971); the accuracy of the solution of the
hypothesis on the presence of v and a. parameters by I-metbods was verified.
If the solutions of an equation, where a, or a, and v were absent, are used,
then with 10.6 these parameters become equal to zero. This program was used
to interpret the ecological experiment described below.
Division of the Water Body into Segments
To solve the inverse problems using the results by hydrobiological and
limnological observations, and to organize ecological observations on water
bodies, categories which have uniform characteristics must be developed, A
convenient method of grouping for inverse problems of water ecology has been
developed at the Institute of Internal Waters Biology of the USSR Academy of
Sciences by Butorin, Smirnov, Sklyarenko, and Malinin. This method offers:
a) the initial classification of the system of criteria; b) transformation of
the initial grouping into smaller associations using factor analysis and
principle component analysis; c) the classification into approximately similar
parts. This approach was developed by the authors using the data from 22
synchronous surveys during 1960-1964 at observation stations on the Rybinsk
reservoir with a 110-dimensional vector of hydrological and biological indica-
tors .
276
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Method of Organizing Observations
This section reviews the methods of ecological observation by information
interpreted by the inverse problem solution. The general conditions, satis-
factory for natural ecological observations, are formulated by the identifica-
tion and observation criteria (Li, 1966). This method of organizing ecological
experiments established a system of data collection, number of points, number
of measurements, and type of information measured for a simple determination
of differential equation coefficients for ecosystem models.
Several features should be noted:
1. The minimum volume of experimental data is determined to interpolate
to a minimum power by a second order polynomial. For example, to define one
coefficient of the one-dimensional diffusion equation, it is necessary to have
at least three points along the spatial coordinate and two measurements at
each point. The two-dimensional diffusion equation requires six points along
a plan not situated on a second order curve. When the number of parameters
to be determined is larger, the minimum volume of experimental information
must be increased so that a system of independent algebraic equations can be
used.
2. The accuracy of parameter determination, when other conditions are
equal, depends on the accuracy of integral calculation. Therefore, it may be
expedient not to use the common equal-dimensional observation grid; in-
stead, the measurement points and the data-collection features should be
situated in nodes of orthogonal polynoms, if it is assumed that the quadrature
and cubic capacity formulas of Gaussian and Tchebysheff's types are used.
3. To determine the ecological parameters that are independent coeffi-
cients in models, only the information on the equations' functions is neces-
sary. The parameters will be found with the common character of the equations
themselves. Determination of dependent coefficients necessarily requires some
schematization and information on boundary condition values.
Field Experiment for Determining^cologi_cal Equation Parameters
This method of identification serves as a basis for organizing a complex
ecological experiment at the Kremenchug reservoir by the Institute of Hydro-
biology, Ukrainian SSR Academy of Sciences, to determine the ecological model
parameters based on an equation system (1.2). During 1972, five surveys were
performed every three days in May, June, and October to measure hydrochemical,
hydrobiological, and hydrophysical data. The species composition of phyto-
plankton, zooplankton, bacteria, and phytobenthos were measured; volatile
organic substances, ammonia and nitrate nitrogen, iron, dissolved oxygen, pH,
phosphorus, temperature, and volume at 5000x1000x four meters at eight
stations were also measured. The monitoring stations were organized along
two lines at right angles—six stations along one axis and three stations
along the other. Three depths were measured at each station.
D and a,3 parameters are determined (y is accepted to be 1; v is measured
in the course of the experiment), which give a systemic characteristic of the
biocenosis under observation. A dispersion analysis of the experimental data
is made. The program was used in the treatment of the experimental data.
The research will determine the basis for constructing a hydrarchical
series of models: stratification models, dynamic models, models of
277
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dimensionally heterogeneous biocenoses. Aspects of these models (such as the
oxygen stratification model) conform well with field observations.
The inverse problems of ecology have been discussed with reference to
several models of planktonic ecosystems. The models refer to some extremely
complicated water ecosystems. Moreover, they are intended to study and fore-
cast ecosystem behavior under anthropogenic influences. The problem of hydro-
biological field data interpretation, given a description of ecosystems by
differential or stochastic equations, is examined as a problem of identifying
mathematical physics equations by stochastic processes. The identification is
as follows: determination of ecosystem model parameters that contain a priori
systemic characteristic of the ecosystem. A characteristic of the inverse
problem in ecology is the necessity to reveal the structure of models on the
basis of experimental data and to determine the parameters of models for
which solution of direct problems is impossible (non-linear equations,
stochastic boundary characteristic conditions, a priori unknown structure of
models). The method of identification is based on integral transformations
of ecological model equations. The structure or core of integral transforma-
tions of differential equations in general orthogonal curvilinear coordinates
guarantees optimal algorithms in the sense of the amount of arithmetic opera-
tions. Parameters, which are algebraically both independent and dependent,
constant, and variable coefficients of ecological model equations, have been
identified.
278
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BIBLIOGRAPHY
Beck, M.D. 1974. The identification of algal population dynamics. Lund
Institute Techn., Division Auto. C., Sweden.
Georgiyevskiy, V.B. 1971. Unified algorithms for determining filtration
parameters. Kiev, "Naukova Dumka."
Leaky, R.A., K.E. Skog. 1972. Ecology, Vol. 53, No. 5.
Li, R. 1966. Optimal evaluations, determination of characteristics, and
control. Moscow, "Nauka."
Loeb, M., C. Gahen. 1963. Automatisme,- 12.
Loeb, M., C. Gahen. 1965. IEEE J.
Prober, R. et al. 1971. Water. Vol. 68, No. 124.
279
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DISCUSSION
VAVILIN: Will the coefficients which are being determined depend on the type
of interaction among the parameters in complex systems?
GEORGIYEVSKIY: We assume some kind of interaction, which actually does not
exist. In the process of identification, it is necessary to propose a model
by considering many parameters. After identification, it is possible to
discard, according to coefficient value, those bonds which do not exist. The
latter is more complex. If we propose a model which is simpler than reality,
then the coefficient value can be random. The parameter value will not re-
flect a biological and physical meaning, but will be a certain characteristic
more complex than the operator. What short cuts are there in ecology? In
the first method, it is not difficult to propose all the interactions and
find all the coefficients. The second method is considerably more difficult:
When constructing ecological models, we generalize simplified groups.
OSTROMOGILSKIY: Can you give us an example of a real model which illustrates
the basic premise of your report?
GEORGIYEVSKIY: In my report, I talked about real parameters which could not
be determined by other methods. In the model of the Kremenchug Reservoir,
reproduction and consumption coefficients were found which cannot be deter-
mined by any other methods. The reproduction and consumption coefficients
for generalized groups were determined within the framework of the coeffi-
cients which characterize the predominant groups.
THOMANN: I have seen similar work in the U.S. and I consider this to be a
promising approach. I would, however, like to make a few remarks. Inverse
methods, in my opinion, preliminarily assume a certain ignorance of the eco-
system. For example, a large number of experiments show that there are
determined phytoplankton growth limits. Over the last five years, scores of
determinations of the assimilation of nutrients by phytoplankton have been
made. There is also detailed information on zooplankton grazing. Experiments
are extremely valuable when constructing models, A zooplankton would devour
a phytoplankton, not knowing where it was located—in a bottle or in a lake.
The inverse method can lead to a point where we will be able to say nothing
definitive until we obtain more data. Experimental conditions are valuable.
They must integrally be introduced into the model.
GEORGIYEVSKIY: The paper emphasizes that the two methods for studying eco-
system models (direct and inverse) must supplement one another. Let me give
an elementary example. If we have a known predator/prey system, then prior to
the inverse problem solution, we can state that the ratio of the coefficients
of predator reproduction to prey extinction is greater than the ratio of
280
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coefficients of predator extinction to prey reproduction. It is necessary to
expand the model by means of a large number of such relationships.
281
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MATHEMATICAL MODELING OF THE HYDRODYNAMICS AND DISPERSION OF
CONTAMINANTS IN THE NEARSHORE
Wilbert Lick
INTRODUCTION
Because of possible harmful effects of contaminants, it is necessary to
be able to predict the dispersion and fate of contaminants in large lakes.
The nearshore regions of these lakes are especially important since:
1. the nearshore regions are where contaminants are generally introduced
and therefore their concentrations and effects are generally greater than in
the offshore regions; and
2. the nearshore regions are of more particular interest to us for such
uses as recreation, water supplies, and fishing.
Recent studies have demonstrated significant differences between nearshore and
open-lake waters.
After a contaminant is introduced into a lake, it is dispersed by physi-
cal processes and transformed by physical, chemical, and biological processes.
A large amount of work has been and is being done to understand the transfor-
mation processes of importance in lakes. At the same time, it has been recog-
nized that the dispersion process must also be understood. In addition to
field work, considerable theoretical work (much of it numerical) has been
undertaken to predict the circulation in lakes. Until recently, most of this
work has dealt with the overall circulation in lakes, not specifically with
the nearshore. For example, most numerical models of the circulation in
large lakes employ a numerical grid of two to six miles, a grid far too large
to treat the nearshore in adequate detail.
In a part of research effort sponsored by the U.S. Environmental Protec-
tion Agency (EPA), we have developed numerical models which can realistically
describe the currents and the dispersion of contaminants throughout large
lakes and especially in the nearshore hydrodynamic and dispersion models,
which necessarily include procedures for coupling the nearshore and offshore
lake circulation models. Specific applications of these models have been
made to predict the dispersion of contaminants under specific conditions and
to understand the physical consequences of contaminant dispersion.
To illustrate our numerical models and some of their applications, these
specific examples will be presented here. They are:
1. the wind-driven currents, both open-lake and nearshore, in Lake Erie
under steady-state, constant density conditions;
2. the dispersion of a contaminant from a river into Lake Erie by wind-
driven currents under present conditions and as modified by proposed large
man-made islands such as a jetport in the lake; and
282
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3. the dispersion of heat in the discharge from a power plant on Lake
Michigan.
Several different numerical models
and are summarized by Lick (1975). Some
little detail. Others are more complex
latively lengthy computer time. Two of
the examples presented here. They are
and a variable-density, time-dependent
sional and give all three components of
of lake currents have been developed
models are relatively simple and give
and detail the flow, but require re-
our hydrodynamic models are used in
a constant-density, steady-state model
model. Both are fully three-dimen-
velocity at any point in the lake.
The basic equations used in the modeling of lake currents are the usual
hydrodynamic equations for conservation of mass, momentum, and energy plus an
equation of state. In sufficiently general form for most lake modeling, these
equations are:
3x
_9v
3y
3w
3z
+ ™ = o
_3u
3t
3x
3uv
w
3uw
- fv = - -
_3v
3t
9z
3z
+
fu = - I i£
+ £_
IE = _
3z
Pg
3T
3t
3uT
3x
3vT + 3wT =
3y
3z
3x
Mi)
3y
+ -M^r) + S
oZ ,
(1)
(2)
(3)
(4)
3T
(5)
P = P(T)
(6)
where u, v, and w are the fluid velocities in the x, y, and z directions re-
spectively, t is the time, f is the Coriolis parameter which is assumed con-
stant, p is the pressure, p is the density, pr is the ambient or reference
density, A^ is the horizontal eddy viscosity while Av is the vertical eddy
viscosity; K^ is the horizontal eddy conductivity while KV is the vertical
eddy conductivity, g is the acceleration due to gravity, T is the temperature,
and S is the heat source term.
Several approximations are implicit in these equations.
1. The pressure is assumed to vary hydrostatically;
These are:
283
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2. The Boussinesq approximation, which assumes that density variations
are small and can be neglected compared to other terms except in the hydro-
static equation, is valid; and
3. Eddy coefficients are used to account for 'turbulent mixing effects
in both the momentum and energy equations.
The appropriate boundary conditions are dependent on the particular prob-
lem to be solved. At the free surface, z = £ , usual conditions are:
1. the specification of a stress due to the wind,
H . TX, pAv|i + Ty (?)
where T , T are the specified wind stresses in the x and y directions, respec-
tively;
2. a kinematic condition on the free surface,
+ u + v
3t 3x 9y (8)
3. the pressure is continuous across the water-air interface and there-
fore the fluid pressure at the surface equals atmospheric pressure pa,
p(x,y,5,t) = pa; (9)
and 4. a specification of the heat flux at the surface,
r)T
q = - pKv ^ = H
-------
state analysis appropriate and useful.
The problem considered here is a description of the steady-state, wind-
driven currents in Lake Erie under constant density conditions (Gedney and
Lick, 1972; Sheng and Lick, 1975). The numerical model used in the calcula-
tions is an extension of Welander's shallow lake model (Welander, 1957). In
this model, in addition to the constant-density and steady-state assumptions,
it is assumed that the nonlinear convection and horizontal diffusion terms can
be neglected. One can show-that the neglect of these terms is a valid approx-
imation throughout the lake except in narrow regions or boundary layers near
shore, regions which are less than 1 km wide (Sheng and Lick, 1975). These
boundary layers are neglected in the present calculation.
The above assumptions, along with the assumption that the vertical eddy
viscosity is constant or a simple function of depth, allows one to integrate
the equations of motion analytically. This procedure reduces the governing
equationsQto two dimensions. Introduction of integrated velocities defined
by U = J_,udz and V = J_-, vdz, where h = h(x,y) is the local depth, and an
integrated stream function 4> defined by U = 3^/3y and V = -3t^/3x reduces the
equations to the single equation (Gedney and Lick, 1972):
V2^ + YI || + Y2 |4- = Y3 (ID
where YI> Y2> au& YS are functions of the local depth and bottom slopes, and
Y3 is also a function of the applied wind stress. The appropriate boundary
condition is that of no normal flux at the shore, or ijj is constant along
shore. Once this equation has been solved for ty, all three components of the
velocity can be calculated as continuous functions of depth as well as the
surface elevation.
The topography of Lake Erie is shown in Figure 1. Some of its more im-
portant features are as follows. It is approximately 386 km long and 80 km
wide near the mid-point of the long axis. Topographically, it can be sepa-
rated into three basins: 1) a shallow western basin with an average depth
of 7 m, 2) a large, relatively flat central basin with an average depth of
18 m, and 3) a deeper, cone-shaped eastern basin with an average depth of
24 m. The western and central basins are separated by a rocky chain of is-
lands, which creates multiple connections on the surface of Lake Erie.
The value of the stream function on the mainland shore is determined by
the river inflows and outflows. The values of the stream function on the is-
land boundaries are not known a. priori. These values are determined by the
condition that the surface elevation be continuous around each island, i.e.,
<£(3£;/3s)da = 0, where the integration path is around each island. In the
calculations presented here, three islands were incorporated.
The above equation for the stream function was solved by finite differ-
ence methods. At all grid points, a five-point central difference equation
was used. For purposes of the computation, Lake Erie was divided into two
regions. One was a region surrounding the islands approximately 80 km by 64
km in which a 0.805 km (0.5 mile) square-grid was used. The second region was
285
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Ni
00
LAKE ERIE LONGITUDINAL
CROSS SECTION
Figure 1. Lake Erie bottom topography (Monin and Yaglom, 1973)
-------
composed., of the-.remainder, of ,Lake Erie...wh-ere a 3.22 km ,.(2- mile).-squares-grid-
was used. A total 5050 grid-points were used. The 0.805 km grid-size was
found necessary to obtain-consistent and accurate t, line integrals around
each island and also to accurately represent the island boundaries.
A combination of successive over-relaxation by points and lines was used
to solve the system of finite-difference equations. A complete iteration was
first performed in one region. From these values, interim interface values
between the two regions were formed for the second region. A complete inter-
action was then performed in the second region. From these values in the
second region, interface values were in turn formed for the first region.
The process was repeated until the maximum relative error between successive
iterations at any grid point was less than 10~5.
For'-a constant eddy viscosity, numerical solutions for the "stream func-
tion and velocities in Lake Erie were obtained for a variety of;wind direc-
tions and magnitudes. For the calculation presented here, the wind-was as-
sumed to be uniform over the entire lake surface with a magnitude of 10.1
m/sec and direction W50°S. This uniform wind condition was found- to be a
valid approximation for the period for which the calculations and field ob-
servations were compared. A friction depth d ;(d=/2Av/f) of 27.4 m was used
because it provided the best agreement between the numerical results and
current-meter measurements. This value of d,corresponds to an eddy viscosity
Av of 38.0 cm /sec. From the results of many different calculations, it was
found, for best agreement between the calculations and observations, that A.^
must be taken to increase as the wind magnitude increases with either a
linear or quadratic relationship (Simons, 1974, for a similar conclusion).
The results presented here include a Detroit River inflow of 5380 m3/sec and
an equal outflow via the Niagara River.
Plots of the horizontal velocities are shown in Figures 2 and 3. In
these figures, the beginning of the arrow represents the actual location of
the current represented by the arrow. The magnitude of the velocity can be
determined from the velocity scale indicated on the figure. Note that the
velocity scale is different for each figure.
The solution for the velocities represents the sum of the Ekman drift
currents and the gradient current. Figure 2 shows that a top surface mass
flux is being transported toward the eastern boundaries, primarily in the
direction of the wind but deflected to the fight by the Coriolis force. As
shown in Figure 3, a subsurface current driven by the pressure gradient re-
turns the surface mass flux in the opposite direction. In the central and
eastern basins, surface currents are in general smaller in the center of the
lake than near the shore. This effect is essentially due to the relatively
large subsurface return current down the center of the lake which is opposite
in direction to the surface current and subtracts from it.
EPA established a system of automatic current-metering stations in Lake
Erie in May 1964. The EPA data consisted of velocity readings taken'every
20 minutes. To compare the water current measurements with our calculations,
the 20-minute readings were vectorially summed over a 24-hour period to form
a resultant.. cur-rent. The resultant magnitude is equal to the1 vector 'sum
287
-------
0
DISTANCE
20
40 MILES
0 30 60 KILOMETERS
CURRENT Oj| FT/SEC.
MAGITUDE noo CM/SEC.
00
00
Figure 2. Horizontal velocities at a constant 0.4 m (1.5 ft) from surface. Wind direction, W50S; wind
magnitude, 10.1 meters per second (22.7 mph); friction depth, 27.4 meters (90.0 ft); rivers:
Detroit, Niagara.
-------
0
20
DISTANCE
40 MILES
i ' '
0 30 60 KILOMETERS
CURRENT 9—r-j FT/SEC.
11111
MAGNITUDE
r-o
CO
0 20 CM/SEC.
CURRENT METER MEASUREMENTS
•— 5/24/64
0--IO/25/64
Figure 3. Horizontal velocities at a constant 9.9m (32.8 ft) from surface. Wind direction, W50S;
wind magnitude, 10.1 meters per second (22.7 mph); friction depth, 27.4 meters (90.0 ft);
rivers: Detroit, Niagara.
-------
magnitude divided by the number of readings. During the time periods used
for comparison, wind was fairly steady for two or more days so that-any seiche
currents were small. The vector summing of the data over a 24-hour period
should remove part, if not all, of any small seiche current since the largest
seiche period is 14 hr.
Wind used in the numerical calculations for a particular day was deter-
mined by taking the average of both the direction and magnitude of the 24-hour
resultant wind at each U.S. Weather Bureau shore station. This average wind,
with its magnitude increased by 1.48, was then used to determine the shear
stress at the water surface. The 1.48 factor was determined by comparing
shore data with "over-the-lake" wind data taken by the EPA. On May 24, 1964,
the resultant "over-the-lake" wind was determined to be 10.1 m/sec with
direction W50°S. The resultant winds for two days prior were within 20° of
this direction and at somewhat less magnitude.
The current-meter data for May 24, 1964, as measured at 10 m below the
surface, is shown in Figure 3. Note that the positions of the measurements
are different from those of the calculated currents. The agreement is marked-
ly good in both magnitude and direction. The discrepancy between the magni-
tudes of the measurements and calculations at point A at 10 m is believed to
be a measurement error since this measurement became erratic at a later date.
The magnitudes of the measurements in the region of point B at first appear
to be considerably different from the calculated values. However, the agree-
ment is believed to be satisfactory when one considers that the currents are
changing rapidly with distance in the point B area. Also in Figure 3 the
meter measurements are plotted for a W43°S wind at a velocity of 8.6 m/sec
taken on October 25, 1964. Again the agreement is quite good.
Figure 4 shows the vertical velocities at mid-depth in the lake. Arrows
pointing toward the top of the plot represent vertical velocities toward the
lake surface. Large upwellings and downwellings are apparent near shore.
The same basic model described above has also been applied to a nearshore
region of Lake Erie (Sheng and Lick, 1975) . The purpose was to analyze in
more detail the currents in the offshore Cleveland area (Figure 1) in the
absence of, and including, a proposed jetport. Advocates of this airport have
proposed that it be situated on a large man-made island in the lake approxi-
mately eight kilometers offshore in waters approximately 15 m deep.
The steady-state currents were calculated for various jetport configura-
tions and for different wind velocities. In these calculations, a 0.4 km grid
was used nearshore and a 3.2 km grid offshore. Two jetport configurations
were studied in detail: an island approximately 4.8 km by 3.2 km, and this
same island extended with an 0.4 km wide causeway to shore (Figure 5).
An interesting result obtained from the calculations was that the island
configuration did not modify the flow appreciably but the island-with-exten-
sion did. The reason for this follows: calculations determined that in the
absence of the jetport, although currents are strong near the surface at any
horizontal location, opposing currents are generally also strong near the
bottom. The result is that the vertically integrated mass-flux in most loca-
290
-------
VO
20
N
DISTANCE
CURRENT
MAGNITUDE
40 MILES
'
0
0
30 60 KILOMETERS
.004 FT/SEC
i i i i
I MM/SEC.
Figure 4. Vertical velocities at mid-depth. Wind direction, W50S; wind magnitude, 10.1 meters per
second (22.7 mph); friction depth, 27.4 meters (90.0 ft); rivers: Detroit, Niagara.
-------
•X • ••••••X«»» • ••• X • * • • ••• X • • • • •• • X««****« X
X •'
JETPORT
0 I MILE
SCALE
2-MILE GRID POINT
^-MILE GRID POINT
^CUYAHOGA RIVER
Figure 5. Topography of jetport and grid structure in the shore region near Cleveland.
-------
tions near the island and offshore is approximately zero. This is not true
near shore where the vertically integrated mass-flux is moderately large and
directed towards the east. Therefore, the island does not appreciably block
the flow; however, a jetport which extends to shore from the island does
block the flow and extensively modifies the nearshore flow field.
This blockage can be seen in Figures 6 and 7 which show the horizontal
velocities at the surface and at a 10-m depth for an advanced concept of the
jetport. The currents are strongly deflected near the jetport. It can also
be shown that strong upwellings and downwellings occur near the jetport.
In general, the coupling between the nearshore and farshore grids was
treated by an iteration process as described above. A simpler procedure, al-
though approximate, is to first calculate the overall circulation in the lake,
and then calculate the flow quantities in the nearshore region assuming that
conditions at the boundary of the nearshore region remain fixed. No further
iteration is required if the nearshore region is large enough to offset the
effect of the jetport outside the nearshore region. In the above calculations
the nearshore region was taken to be approximately 25.6 km by 22.4 km. The
dimension was sufficiently large so that the island configuration could be
modeled without^ iteration (this was shown by comparison of solutions with and
without iteration). For the island with an extension to shore, however,
iteration was necessary.
Dispersion by Wind-Driven Currents
The problem considered here is the dispersion of contaminants by wind-
driven currents after the contaminants have entered Lake Erie from the
Cuyahoga River (Sheng and Lick, 1975). The effect of a jetport on this dis-
persion is also investigated. The river itself is treated as a point source
of pollution and the effect of the river on the flow in the lake is neglected.
The contaminant is assumed to be discharged from the river at a constant rate
starting at some initial time. It is also assumed that the concentrations of
the contaminant are small enough that the effect of the contaminant on the
flow can be neglected. /
The basic equation describing the dispersion of a substance is
3C
- _ - - - - - - - - -
3t 3x 3y 3z ix\ 3x/ IJyV 3y/ 8z\ 3z; (12)
, 3Cu , 3Cv , 3Cw 3 /DU3C\ , 3 /DU8C\ , 3 /D,T3C\ ,
J_ - T - T - = - 1 ft - | -]- - 1 tl - I ~t~ - 1 V - I ~T
3x 3y 3z ix\ 3x/ IJyV 3y/ 8z\ 3z;
where C is the concentration, u and v are particle (or fluid) velocities in
the (horizontal) x and y directions respectively, w is the particle velocity
(the sum of the fluid velocity and the settling velocity ws of the contaminant
relative to the fluid) in the z direction, D^ is the horizontal eddy diffu-
sivity, Dv is the vertical eddy diffusivity, and S is a source term.
A boundary condition sufficiently general to describe practically all
situations encountered in the dispersion of a physical substance can be writ-
ten as (Monin and Yaglom, 1971) :
293
-------
WIND
0
FT/SEC. 0
I MILE
Figure 6. Steady-state nearshore currents in the presence of a jetport island
with extension to the shore with (TX, xy) = (1.55 dyne/cm2, 1.1
dyne/cm2). Horizontal velocities at the surface.
294
-------
WIND
ft/SEC
MILE
s/ss///////
////////
///////
1 ////////
7 / / / / // /
Figure 7. Horizontal velocities at a constant depth of 30 ft. from the
surface.
295
-------
wsC - Dv If = -BC + E
= -3(C - Ceq) (13)
On the left-hand-side of this equation, the first term represents the flux to
the boundary due to gravitational settling, while the second term represents
the flux to the boundary due to vertical turbulent diffusion. On the right-
hand-side, the first term depends on the "porosity" or "stickiness"(of the
wall. The case g=0 corresponds to perfect reflection of the substance from
the boundary while the case Q-^° corresponds to perfect absorption at the
boundary. For 0<8<0°, partial reflection and absorption occurs. ' The second
term on the right-hand-side is due to entrainment. In the second form of the
equation, Ceq represents an equilibrium concentration which in general depends
on the shear stress and sediment composition. When the concentration is
greater than C , net deposition results while when the concentration is less
than Ceq, net erosion results.
In the example considered here, it is assumed that the boundary condition
at both the upper and lower surfaces is that of zero flux. This corresponds
to setting the right-hand-side of Equation 13 to zero, i.e., no entrainment
and a perfectly reflecting boundary. Other conditions assumed for the cal-
culation are the following:
1. a steady-state velocity field is caused by a W32°S wind with a velo-
city of 5.2 m/sec and a wind stress of 0.9 dynes/cm2;
2. variable density effects are negligible;
3. the mass flux of the Cuyahoga River is 27 m/sec;
4. the concentration of the contaminant in the river is 10 units; and
5. the vertical eddy viscosity and diffusivities are 17 cm?/sec while
horizontal mixing is neglected.
For these conditions, the velocities u, v, and w were first calculated
using the hydrodynamic model described in the previous section. The surface
velocities are shown in Figure 8 while the near-bottom (1/6 of the depth from
the bottom) velocities are shown in Figure 9. Strong currents in the along-
shore direction are apparent at the surface and it can be shown that they ex-
tend almost to the bottom. However, the near-bottom currents, especially in
the vicinity of Cleveland, are directed away from shore. From the calcula-
tions, it can be shown that downwelling is present throughout the water column
over the entire nearshore area shown in the figures. Due to the sharper
bottom-slopes near shore, the vertical velocities in regions near shore are
larger than those in regions far from the shore.
Once the velocities are known, the dispersion of a contaminant can be
calculated by means of Equation 12. In all cases, a strictly conservative
numerical scheme was used. In addition, a stretching of the vertical coordi-
nate proportional to the local depth was used. With this transformation, the
same number of vertical grid points are present in shallow as in deeper parts
of the basin. Thus, in the shallow areas, there is no loss of accuracy in
the computations due to lack of vertical resolution, a significant factor in
nearshore calculations.
296
-------
WIND
x\xxxxx\\\\x\\\xx\xx
xxxxxx\xxxx xxxxxxxxx
xxxxxxxxx\\xxxxx x x x x
CUYAHOGA RIVER
2 MILES
CURRENT
MAGNITUDE
OF I FT/SEC.
o
Figure 8. Horizontal velocities at surface in the nearshore with i^ =? 0.9 dyne/cm .
-------
MD
OO
WIND
^v X
V V V.
V X X
X X
X
xxxNXX \\
VVXXVXXXvXX\X\
XVXVVVN\\\\\\\
\\N\\\N\\\X\\\
\ \ \- \ \
t f ///////
I 1 h
2 MILES
CURRENT
MAGNITUDE
OF I FT/SEC.
9,
Horizontal velocities at 5/6 - depth in the near-shore with ty_ = 0..9 dyne/cm2.,
-------
In the present calculation, the dispersion of a conservative, dissolved
substance was considered. Relative concentrations at the lake surface at
128 hours are shown in Figure 10. It can be seen that the contaminant is
transported along shore towards the east but with some transport in the off-
shore direction. This offshore transport is mainly due to the bottom currents
in the offshore direction followed by vertical mixing. A horizontal diffu-
sivity of 2xlOs cm2/sec made little difference in the results.
To show qualitatively the effects of an offshore jetport on the currents
and the dispersion of a contaminant, two additional cases were calculated
under the same conditions as above but with two different proposed configura-
tions of a jetport. These configurations were:
1. an island 6.4 km square approximately 9.6 km offshore of Cleveland,
and 2. a landfill extension of this island to shore.
The results of the calculations for the surface concentrations at 128 hours
are shown in Figures 11 and 12. The island configuration, as seen in Figure
11,.does not affect the contaminant transport appreciably, except in the
immediate vicinity of the island, This is due to the fact that the flow over
most of the nearshore region is not appreciably modified by the presence of
the island. In the case of the island with an extension-to-shore, the con-
taminant is transported much further offshore than in the previous two cases.
The extension blocks the flow in the long-shore direction. Very little con-
taminant is found at the downwind side of the jetport. It can be shown for
this case that there are strong sub-surface, offshore currents in regions
near the river mouth and upwind of the jetport.
Calculations were also made for the same conditions as above except that
it was assumed that the contaminant had an average settling velocity of
-0.5 mm/sec relative to the surrounding fluid. Without the presence of a
jetport, the concentration distribution after 128 hours at the surface and
near the bottom is shown in Figures 13 and 14. By comparing the surface con-
centration distribution with that of the neutrally buoyant contaminant (Figure
10), it can be seen that the along-shore surface transport is reduced appre-
ciably. Near the bottom, the contaminant was transported much farther in the
offshore direction than in the previous case. This was due to the settling
and the offshore currents present near the bottom under these wind conditions.
Thermal Plumes from a Power Plant
A time-dependent, variable-density model (Paul and Lick, 1973, 1974;
Lick, 1975) has been used to calculate the flow-field and temperature distri-
bution in the discharge from the Point Beach power plant on Lake Michigan.
The calculations include realistic geometry, buoyancy effects, wind stresses,
and crossflows in the lake. The basic numerical procedure of solution and
details of the calculations are presented in the references mentioned above
and so, except for a brief statement, of the assumptions used in the model,
the model will not be discussed in detail here. However, as examples of the
dispersion of heat, the results of two calculations and their comparison
with field observations will be presented.
In addition to the usual assumptions stated in the introduction, the
present hydrodynamic model assumes that the rigid-lid approximation is valid,
299
-------
WIND
MILES
OJ
o
o
RELATIVE
CONCENTRATION
A -~-«
Figure 10. Surface concentration contours in the nearshore at 128 hours.
-------
WIND
MILES
RELATIVE
CONCENTRATION
U)
o
Figure 11. Surface concentration contours at 128 hours, island jetport.
-------
WIND
MILES
RELATIVE
CONCENTRATION
A
B
C
D
E
F
IO'5
ID'4
10"3
5xlO'3
10-2
3*IO'2
Figure 12. Surface concentration contours at 128 hours, island with extension to shore.
-------
WIND
0
4 MILES
O
LO
RELATIVE
CONCENTRATION
IO"5
JO'*
IO"3
5 xlO'3
10-2
Figure 13. Surface concentration contours in the nearshore at 128 hours. Settling velocity = -0.5 mm/
sec.
-------
WIND
MILES
(-0
O
.p-
RELATIVE
CONCENTRATION
10-5
IO'4
IO'3
5x I0'3
10-2
3x I0'2
ID'1
Figure 14. Near-bottom concentration contours in the nearshore at 128 hours. Settling velocity = -0.5
mm/sec.
-------
i.e., w(z=0) = 0. This approximation is used to eliminate surface gravity
waves and the small time scales associated with them, greatly increasing the
maximum time-step possible in the numerical computations. In this approxima-
tion, only the high frequency surface variations associated with gravity waves
are neglected while the steady-state results,, with which we are solely con-
cerned here, are calculated correctly and are the same as for the free-surface
case. As in the previous calculation of contaminant dispersion, we used
strictly conservative numerical schemes and stretched the vertical coordinate
proportional to the local depth.
Calculations have been made for two cases of the discharge from the Point
Beach power, plant. These discharges flow into a quiescient lake and into a
lake with a crossflow and a surface wind. The relevant parameters, based on
field measurements taken at the site of the discharge (Frigo et al., 1974),
are listed in Table I. The bottom topography is shown in Figure 15. The out-
fall extends into the lake and the discharge forms a 60° angle with the shore.
TABLE I. PARAMETERS _OF_DISC_HARGE_FROM POINT BEACH PLANT.
Flow rate
Outfall width
Outfall depth
Ambient lake temperature
Vertical eddy coefficient
Horizontal eddy coefficient
Surface heat transfer coefficient
Temperature variation
Maximum velocity
Densimetric Froude number
Equation of state
24.7 m /sec
10.8 m
4.2 m
9.5°C
(50 - 200 ^1) cm2/sec
1000 cm2/seg
30 watt/m2 - °C
8.5°C
0.9 m/sec
4.2
Ap = -1.25 x 10~3AT
The boundary conditions are as follows: the inlet velocity profile v.Tas
specified as a smoothed average of that measured at the outfall. The inlet
temperature was taken as the constant value measured. A surface heat transfer
proportional to the difference in temperature between the surface water and
the air was assumed with the heat transfer coefficient determined from the
work of Edinger and Geyer (1965). The stress acting on the water surface due
to the wind (measured) is calculated by the formulae developed by Wilson
(1960). The outer x and y boundaries for the numerical calculations must be
taken at some finite distance. Roache (1972) discusses outer boundary condi-
tions and concludes that the actual boundary conditions are not that important
as long as the boundary conditions do not severely restrict the flow. The
boundary conditions used here are that the normal derivatives of the veloci-
ties and the temperatures are zero.
The vertical eddy coefficient is taken as dependent on the local vertical
temperature gradient and is assumed to be given by:
AV = a~^ (14)
where a and 3 are constants depending on local conditions. The constant a is
equal to the vertical eddy diffusivity under vertically stable conditions.
305
-------
4.5
METERS
1 I
25 50
1.5 METERS
SHORE
Figure 15. Bottom topography for the Point Beach
power plant,
-------
For the above boundary conditions and parameters, time-dependent calcula-
tions started with an initial guess of the flow and proceeded until a steady-
state had been reached. The first case was that of the discharge into a
quiescent lake. No winds or crossflows in the basin were present. From the
calculations, one can show that as the flow is discharged from the outfall,
it is forced towards the surface by the decreasing depth (see Figure 15) as
well as by buoyancy. 'The large densimetric Froude number of 4.2 indicates
that buoyancy effects are not initially dominant. After about 75 m from the
outfall, the depth begins to increase. After this point, the discharge tends
to remain very near the surface. The resulting surface temperature field is
shown in Figure 16.
A second calculation was made for the case when a crossflow (current of
9.1 cm/sec at an angle of 125°) and a wind (approximately 5 m/sec at an angle
of 270°) were present. In this case, the discharge is physically swept in the
direction of the crossflow. The temperatures (Figure 17) are displaced in the
direction of the crossflow and towards the shore.
For this second .case, a comparison of the calculated results with field
observations (Frigo et al., 1974) is shown in Figures 18 and 19. Figure 18
shows the temperature decay along the centerline while Figure 19 shows the
isotherm areas for various temperatures. It can be seen that there is good
agreement between the predicted results and field observations. Similar
agreement was obtained between predicted results and field observations for
the first case above. The calculations are presently being extended to in-
clude more of the flow field.
Additional field data are available (Frigo et al., 1974) from which one
can determine more details of the flow field. However, the flow, due to its
turbulent nature (Csanady, 1973 for a general description of the turbulent
diffusion and nature of plumes), is highly variable both in space and time.
Continuous field measurements must be made over a sufficiently long period of
time to average out these variations before more general comparisons between
our calculations and observations can be made. This has not yet been done.
However, the above calculated results, although limited, seem more than ade-
quate at this point and establish confidence in the numerical model.
307
-------
CO
o
CO
Figure 16. Surface temperature distribution, no wind, no cross-flow.
-------
6°C ABOVE AMBIENT
WIND
NORTH
CURRENT
Figure 17. Surface temperature distribution with wind and cross-flow.
-------
LO
I—"
O
1.0 i
.8 -
.6 H
i .4 -
.2-
0
• COMPUTED
D FIELD OBSERVATION
100
200
300
CENTERLINE DISTANCE (METERS)
400
Figure 18. Surface temperature decay along centerline with wind and cross-flow.
-------
1.0 -i
.8 -
.6-
AT
AT:
.4-
,2-
0
10'
D
n
A COMPUTED
D FIELD OBSERVATION
10
D
D
ISOTHERM AREA (m2)
10
Figure 19. Surface isotherm areas with wind and cross-flow.
-------
REFERENCES
Csanady, G.T. 1973. Turbulent Diffusion in the Environment. D. Reidel Pub-
lishing Co., Boston, Maryland.
Edinger, J.E. and J.C. Geyer. 1965. Heat exchange in the environment. Publi-
cation No. 65-902, Edison Electric Institute, New York, N.Y.
Frigo, Frye, and Tokar. 1974. Field investigations of heated discharges from
nuclear power plants on Lake Michigan. Argonne National Laboratories ANL/
ES-32.
Gedney, R. and W. Lick. 1972. Wind-driven currents in Lake Erie. J. Geo-
physical Research, 77.
Lick, W. 1975. Numerical models of lake currents. U.S. Environmental Protec-
tion Agency Report.
Monin, A.S. and A.M. Yaglom. 1973. Statistical Fluid Mechanics, Vol. 1. The
MIT Press, Cambridge, Maryland.
Paul, J.F. and W. Lick. 1973. A numerical model for a three-dimensional,
variable-density jet. Technical report, Case Western Reserve University.
Paul, J.F. and W. Lick. 1974. A numerical model for thermal plumes and river
discharges. Proceedings 17th Conference on Great Lakes Research.
Roache, P.J. 1972. Computational Fluid Dynamics. Hermosa Publishers. Albu-
querque, New Mexico.
Sheng, Y.P. and W. Lick. 1975. The wind-driven currents and contaminant
dispersion in the nearshore. Case Western Reserve University Report.
Simons, T.J. 1974. Verification of numerical models of Lake Ontario, Part I,
Circulation in spring and early summer. Journal of Physical Oceanography,
Vol. 4, No. 4.
Welander, P. 1957. Wind action on a shallow sea. Tellus 9, No. 1.
Wilson, B.W. 1960. Note on surface wind stress over water at low and high
speeds. J. Geophysical Research, 65.
312
-------
DISCUSSION
BARANNIK: My question concerns modeling thermal streams. Several models
have accepted that the eddy viscosity coefficient depends on the Richardson's
number. Dr. Lick uses a linear dependency between the eddy viscosity coeffi-
cient and the temperature coefficient. I would like you to comment on this
in more detail.
LICK: We had only two field observations, and we determined the parameters
on the basis of field observations. Therefore, we did not use very compli-
cated formulas.
LOZANSKIY: Why does the range of parameters within which field research work
was carried out not correspond to the range of parameters within which cal-
culations were carried out?
LICK: This is due to the insufficient quantity of field data. The calcula-
tions gave those fluctuations which are not far from the field observations.
DAVIES: The majority of hydrodynamic specialists criticize ecosystem models
because of inadequate verification. However several circulation models pre-
dict flows which, in actuality, are not observed. How do you reconcile these
inconsistencies?
LICK: We assume that we have working numericals models. Further efforts
must be directed to verifying these models.
VASILYEV: In your discussions, there is a discrepancy between thermal vis-
cosity and conditions where there is no bottom friction. How do you use
information on wind? How many monitoring stations do you have to obtain in-
formation on the lake's surface?
LICK: We applied non-bottom friction conditions because Lake Erie is shallow.
In this case, it is better to apply conditions of non-bottom friction than
conditions of bottom friction. If we use a boundary condition, then for
stationary flow the general results differ. If the pressure is proportional
to the velocity, then it is necessary to determine two parameters: bottom
pressure and viscosity. And we think that it is not reasonable with a
limited amount of field data to use a lot of parameters.
The wind was measured at different points over the lake, and a steady-
state situation was assumed. The average wind velocity value for two days
was taken. The wind was measured at six reference points on the shore, and
the wind velocity value at any point on the lake was interpolated between
these points. We accept the wind-driven stress as proportional to the
velocity square, multiplied by the coefficient, which was determined by
313
-------
field measurements. We introduced the correction coefficient in order to
increase the wind velocity over the lake. It is interesting to note that the
latter is always larger over the lake than over dry land. Experiments prove
that it is 1.5 times larger.
MORTIMER: Dr. Lick described one of the principle difficulties in.,a model of
lake hydrodynamics, that of weak use of meteorological data, not only the
wind velocity value over dry land and over the lake, but also its direction.
It is also interesting to note, that wind velocity and direction are changed
when there is a transition from a smooth to rough surface. It is necessary
for us to study wind-driven motion. Flows are measured using instruments.
We need "LaGrangian measurements with floating instruments, transmitted
through earth satellites. Reception stations can be located in the mountains.
For this, Baikal is an ideal model.
BARANNIK: The model compares vertical turbulent exchange coefficient values
which .were selected using a comparison of field observation data, with cal-
culation results. In similar models for modeling flows in large water bodies,
the dependency between the vertical turbulent exchange coefficient and the
wind velocity was based on field data. This evaluation showed that the
vertical turbulent exchange coefficient which Dr. Lick used is close to our
dependencies. It is possible to conclude that Dr. Lick's model is close to
those models which are used in the USSR for calculating flow in large lakes.
YEREMENKO: Dr. Lick's method is a fully acceptable method for solving com-
plicated problems. We have developed a similar full flow method. I would
like to briefly discuss the research which we conducted using the Azov Sea
as an example. The problem was solved where there were conditions of bottom
adhesion and steady viscosity. No essential deviation in circulation was
observed. When there were adhesion and steady viscosity conditions, we
essentially overestimated the bottom resistance. Divergence with the Chezy
conditions took place 5 to 7 times. In this case, I think that Chezy's law
for wind-driven flow is also an assumption. Researching the turbulent
viscosity distribution, given wind-driven flow, we were successful in solving
a problem with bottom friction, but with turbulent viscosity data obtained
for wind-driven flows. It turned out that in this case, the divergence
between the adhesion conditions when there is steady viscosity was decreased
in comparison with the case when Chezy's law was used.
ZNAMENSKIY: It is possible to distinguish steady state-processes from field
data. In this case, the larger the averaging level, the better it is.
On Lake Baikal, the basic air currents are stable. The division line for
two multi-directional air flows over a period of several years varies insig-
nificantly. We used this principle when constructing a hydraulic model of
Lake Baikal. Using the established steady state systems, we modeled drift
and compensation flows. We obtained results which are close to the field
observations. Dr. Lick's model allows us to formulate a series of delicate
questions on flow structure in water bodies. Using this model, we obtained
an evaluation of the rise and fall of the water masses in the nearshore
regions. I think that the models presented are sufficiently well evaluated,
not only from a practical aspect, but also from an aspect of knowledge of
natural processes.
314
-------
ROZHKOV: If you examine flow in large lakes, then it is necessary to turn
your attention to their probability structure. I will describe the flow
spectrum in a stationary approximation using data from field observations in
Lake Ladoga. The observations were taken over several months. The diagrams
clearly show the maxima of synoptic changes with the cycle of operations
lasting three days, the inertia fluctuations over a period of 14 hours, and
the inertia fluctuation overtone. If you examine a non-stationary probability
model, then it is possible to note that the intensity of all these fluctua-
tions is changed in time. The frequency spectral density will be the most
complete characteristic of the two. The synoptic fluctuations are changed
with a half-month cycle of operations. These are the basic characteristics
of the flow structure in large lakes as a non-stationary non-uniform vector
process.
LICK: For deeper lakes, we used a three-dimensional model and developed
several models for steady-state conditions. We computed several examples
using these models. Additional difficulties take place when modeling in
nearshore zones, when the size of the grid disguises different flows, and
when we apply valid limited conditions—coefficients of fluctuation and en-
trainment. Little is known about the entrainment and adhesion coefficients.
Still, the numerical models are useful for reconstructing real processes for
understanding situations which arise when water bodies are stratified.
315
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MODELING AND VERIFICATION; THE TWO INDISPENSABLE LEGS FOR PROGRESS
TOWARD UNDERSTANDING AND MANAGEMENT OF AQUATIC SYSTEMS AS
DEMONSTRATED BY A STUDY OF COASTAL CURRENTS IN LARGE LAKES
C.H. Mortimer
INTRODUCTION
Today, computerized mathematical modeling of aquatic systems, fed by
some successes, partial successes, and more promises, has become fashionable
with environmental managers in governmental agencies and industry. This
development, although welcome in many ways, contains two serious risks: 1)
that limited funds for research will not be spent in an optimum, interactive
balance between model development and field experiment; and 2) that, when it
eventually becomes apparent that the hoped-for or promised predictive capa-
bility is all too often a mirage, adverse reaction will starve the development
of better models. This paper is addressed to the avoidance of the first risk.
If this risk can be avoided, as I believe it can, the second risk becomes
smaller.
Considering lake examples we must not, on one hand, create a state of
affairs in which limnological modelers and managers prefer to put questions
to their models rather than to lakes. Nor must we, on the other hand, waste
effort in unplanned or random collection of data. Our best hope of progress
toward deeper knowledge and eventual predictive capability lies in a proper
balance between questions directed both to models and to lakes, either
simultaneously or, more often, in an iterative two-legged sequence in which
models assist the planning of optimized schemes of data acquisition. Models
can then be tested and improved. "Classical limnology" has not lacked for
models, concepts, and hypotheses to guide (and misguide) it; but the prime
lesson to be learned from a century of lake research—from Forel to modern
times—is that soundly conceived and adequately based "hard" sets of data,
from experiments and lake surveys, are the only grindstones on which models
can be sharpened to the point at which their full power can be brought to
bear.
As you see, I am discussing models used as scientific cools to define
the most revealing questions we can ask of lakes. While eventual predictive
capability is the ultimate goal for environmental management, even incomplete
and invalid models can generate new ideas and better define the questions.
But, above all, I would urge careful and thorough listening to the answc "s
that will put us more quickly on the right road to knowledge than modeling
alone. We need two legs to walk that road—modeling and verification;
theory ami experiment. I could give many examples—from hydrobiology, hydro-
chemistry, and hydrophysics—of how listening to signals transmitted by the
lakes themselves can shortcut a great deal of unproductive work and put us
316
-------
much further forward on the road to productive modeling and understanding.
But for brevity, and because the results may turn out to be very relevant to
Lake Baikal, I have chosen an example from the physics of large lakes, namely
the responses of the stratified basin to wind impulses.
Models of Thermocline Dynamics
Summer wind storms, the driving force on large stratified lakes and
reservoirs, produce displacements of the thermocline level which, in turn,
generate internal waves. In the largest basins, the Coriolis force of the
earth's rotation combines with the wind to force the thermocline to the sur-
face, causing extensive upwelling in coastal waters in a nearshore band some
20 km wide along the shoreline to the left of the wind direction. A Lake
Michigan example, produced by strong wind from the north, is shown in Figure
1. Similar upwelling from strong wind was noted on Lake Baikal by Skabit-
schevsky (1929); and many other examples have been described, for example,
the detailed monograph of Rossolimo (1957).
With upwelling as the "hammer blow" and the thermocline as the "gong,"
let us listen for the vibrations which follow. This we can do, at little
cost, by plotting (Figure 2) the temperature readings at city water intakes
on opposite sides of the basin. When intake pipes are near thermocline
depth, temperature records from water intakes on opposite sides of the lake
basin—three pairs in Figure 2—show a flip-flop motion of the thermocline
(down one side, up the other) after strong wind. If a period of weak winds
follows, a slow-moving internal wave is generated and progresses counter-
clockwise around the shore until destroyed by the next storm.
The simplest analytical model for this wave response is a combination of
two internal Kelvin waves (Figure 3) in a channel of uniform depth. I wish
to stress here that it is not always necessary, at the outset, to construct
an elaborate numerical model for each particular lake. Much physical insight
can initially be gained from simple generalized analytical models.
The model in Figure 3 tells us that the Kelvin wave amplitude is greatest
at the shoreline and decreases exponentially away from the shore, at a rate
which depends on latitude and on the density difference across the thermo-
cline. The Kelvin wave currents are everywhere shore-parallel; but beyond
a certain distance from shore, the Kelvin wave currents become negligibly
small. In Lake Michigan, with a typical internal Kelvin wave speed of about
1.6 km/hr, the associated current amplitude decreases with increasing dis-
tance from shore, falling to 5% at about 14 km from shore. Because of the
higher latitude and smaller density-difference across the Lake Baikal thermo-
cline, the internal Kelvin wave speed is about 1 km/hr in late summer*, and
the offshore distance of the 5% amplitude level is about 7 km. Internal
Kelvin waves are therefore trapped near the shore. They are set in motion by
winds strong enough to bring about upwellings on one side of the basin and
downwelling on the other. In the northern hemisphere such internal Kelvin
waves travel counter-clockwise around the basin, but they complete the circuit
Assuming a thermocline at 20 m, and upper and lower layer temperature at
11° and 4°C, respectively.
317
-------
SURFACE TEMPERATURE
•c
SYNOPTIC 21
9 AUGUST I9S5
km
400
-300
-200
-100
u 0
Figure 1. Lake Michigan, 9 August 1955. Distribution of surface temperature,
°C, derived from seven "synoptic" temperature surveys by Ayers et
al. (1958). Vessel tracks are shown as broken lines.
318
-------
20 25 ' 30
5 10 15 20 25 30,
TWO RIVERS 11 m
5 ' I0 15 20' 25 30
LUDINGTON
14 m
.._ A/ -/->
10—\-J- —'— -
USKEGON
I5m
BENTON HARBOR
13 m
U'M /-'
• DATE OF FIG.l
CRUISES
RACINE
' 9m
MICHIGAN CITY
12m
SEPTEMBER
AUGUST I955
5 10 15 20 25
Figure- 2. Daily or twice-daily .temperature readings at the water intakes of
cities around Lake Michigan, 1955 (Mortimer, 1971). The intake
pipes extend about 1000 m into the Lake; the depths are shown on
the diagram. The temperature records for the intakes on the west-
ern shore (unbroken line) are paired with the corresponding record
(broken line) for the opposite intake on the eastern shore (see
•Figure 1). The vertical arrows at .9 August denote the time of the
Figure 1 cruises. The thick, sloping dotted line follows the pro-
gress northward along the eastern shore of a presumed internal
Kelvin wave (see text) generated by the Figure 1 upwelling.
319
-------
Figure 3. (a) Two identical Kelvin waves traveling in opposite directions.
These combine to form the amphidromic standing wave and associated
currents, successive (1/8 cycle) phases of which are illustrated
in (b) assembled from Mortimer (1975) .
320
-------
only if not disturbed by another storm. Completion of. the circuit in Lake
Michigan would require a month, and the Lake is unlikely to remain undis-
turbed for so long. In Lake Baikal, I calculate the period for one circuit
to be two months. Therefore the clue provided by Schostakovitsch (1926) in
Figure 4 is of interest. During the months of July to September 1914, the
4°Cisotherm depths at opposite ends of the Baikal basin rose and fell in
opposition and with apparent periodicity of about 40 days. The data are not
complete enough to decide whether an internal wave or an upwelling interpreta-
tion is appropriate (see Rossolimo, 1957, p. 548).
If we increase the frequency-resolution of our lake-signal receiver, as
in the Figures 5 and 6 recordings of waterworks intake temperatures, snatches
of higher frequency signals come through, when the thermocline stays for
several days near intake depth. The thermocline "gong" is heard to be vi-
brating at a set of frequencies (free modes) which are closely tuned to the
local inertial frequency, but which never exceed it. The equivalent periods
are 14 to 17 hours for Lake Michigan, which has an inertial period of 17.4
hours, i.e. 12 hours divided by the sine of the latitude. For Lake Baikal
(inertial period 15 hours) the preferred frequencies are expected to lie
between 12 and 15 hours. The simple model, proposed in 1963 for this near-
inertial wave response and later verified (Mortimer, 1963, 1971, 1974), is
that of an internal Poincare wave in a channel of uniform depth (Figure 7).
JULY 1914 AUG. SEPT. OCT
5 IO 15 20 25 30 5 IO 15 2O 25 3O 5 IO 15 2O 25 3O 5
O M.-
5OM.
IOOM.
I5OM.
Figure 4. Lake Baikal 1914. Variation in depth of the 4°C isotherm, July to
October, at Marituj (M) and Douschkatschan (D), i.e. at opposite
ends of the lake, redrawn from Schostakovitsch (1926), with a 38-
day time interval added.
In contrast to the slow-moving internal Kelvin waves confined to a near-
shore strip about 15 km wide and with wave currents always shore-parallel,
the Poincare wave is a cross-channel internal seiche which vibrates the whole
thermocline in a cellular pattern with clockwise-rotating currents. As with
seiches in small lakes, several modes can be forced into oscillation by a
storm, but in very large lakes (Baikal and Michigan, for example) all observed
modes possess periods close to (but never greater than) the local inertial
period.
321
-------
20°
Cr'
10 —
SKEGON 8 m
,'V '"''
.'Vv
-^
.^
.j- — -\ , — -.-
/./ '•^^
^./
-^0
, MILWAUKEE 17m
\ /
sx
15*
10°
Figure 5. Hourly temperatures at Lake Michigan water intakes, 1963 (Mortimer,
1971).
70f
16 HR. INTERVALS
17 HR. INTERVALS
14 ' 15 ' 16 ' 17 i 18 ' 19 ' 20 I 21 I 22 n23
Figure 6. Hourly mean temperatures, during various episodes, at the Milwaukee
Wisconsin water intake (Mortimer, 1971).
322
-------
Figure 7. A standing Poincare wave in a wide, rotating channel of uniform
depth. Separated- by 1/4 cycle, two phases of the oscillation are
shown for the cross-channel trinodal (3rd mode) case, with a ratio
of a long-channel to cross-channel wavelength of 2/1. The clock-
wise rotation and cellular distribution of the current vectors are
illustrated on the horizontal planes below the channel (Mortimer,
1974).
323
-------
We must therefore infer from the lake's responses and from our simple
models of those responses, that summer storms over large lakes will generate
three types of current responses: 1) shore-parallel patterns within 15 km
of the shore, produced by upwelling/downwelling events (geostrophic currents)
generating internal Kelvin waves; 2) rotary (Poincare) currents at offshore
distances greater than 15 km; and 3) a mixture of these responses at inter-
mediate distances. The various forms which this mixture can take are modeled
by simple combinations of rotary and unidirectional components, as in Figure
8. With increasing proportions of unidirectional current added to the rotary
(Poincare) current, we obtain meandering, cusping, and looping current tracks,
illustrated in later examples.
4
0 -*
10
circling , v = 0
looping, v= r/2
cusping, v= r
meandering, v= 2r
Figure 8. Combinations of a rotating current vector of .constant speed _r
(shown at 2 hr intervals and corresponding to a Poincare wave of
16 hr period) with increasing proportions of a unidirectional
current vector of speed v_. Current "roses" and progressive vector
diagrams are shown for the following models: v_ = 0 (motion in the
inertial circle) ; v_ = 1/2 r_ (looping current track) ; v_ = r
(cusping current track) ; and ^v = 2r_ (meandering current track) .
Signals from Large-Basin Coastal Currents
With the above models to guide us, my student G. K. Sato and I were in a
position to "listen" to the motions in the coastal zone. Although complex,
these motions are of great practical importance. For this is the zone into
which most man-made wastes are discharged and dispersed, from which water is
extracted for industrial purposes, and in which most recreational uses
develop.
Our listening post for signals from coastal currents occupied a near-
shore area of Lake Michigan about 15 km square (Figure 9), with ten current
meters moored in depths ranging from 10 to 26 m (Sato and Mortimer, 1975).
324
-------
CURRENT METER STATION LOCATIONS
LOCATION OF
STUDY AREA
CUDAHY
'////A
SOUTH
MILWAUKEE
y////,
• CURRENT METER STATION'*
UXATION
DEPTH CONTOUR INTERVAL !0 METER
Figure 9. Location of instruments to measure current speed, current direction,
and temperature off Oak Creek, Wisconsin, western Lake Michigan
(Sato and Mortimer, 1975).
325
-------
The following diagrams present extracts of continuous records of water flow
(speed and direction) past the fixed instruments moored at various depths at
the station positions shown in Figure 9. The water .current speeds and direc-
tions are compared, in these diagrams, with wind speed and direction at a
nearby land station. When the coastal waters were unstratified, the currents
followed the local winds relatively closely and reproduced the episodic char-
acter of the wind force (Figure 10). This- was also true in early spring
(June 1972, Figure 11), when, although the shallow nearshore waters were
beginning to stratify and form the thermal bar first described in detail by
Tikhomirov (1963) in Lake Ladoga, the main lake basin was still isothermal
without a thermocline. The episodic character of the wind-responsive currents
is well displayed in Figures 10 and 11. When the wind direction changed and
picked up speed, the current also changed direction, after a few hours lag,
and also picked up speed, often very rapidly. Changes from northgoing to
southgoing currents were accompanied by sharp rises in temperature (down-
welling motions) while northgoing winds brought lower temperatures. This
suggests that the development and fate of the thermal bar will be determined,
not only by the sequence of heat input during the spring months and by the
convergence model of Tikhomirov (1963), but also by episodic wind influences.
After -whole-basin stratification became well-established with a distinct
"permanent" thermocline by mid- or late-June, the current responses in the
nearshore/offshore intermediate region (5 to 10 km from shore) became modified
by internal Poincare wave dynamics (Figure 12, 18 June to 17 July 1972,
Station 6, 23 m, 10 km offshore), i.e. they were less directly dependent on
local wind forcing (see 24 to 27 June). Depending on the relative proportions
of the unidirectional (wind-driven or geostrophic) component and the rotary
Poincare component of the current, the resultant current was either "mean-
dering," "cusping," or "looping" (see Figure 8). Examples of looping currents
are shown for Lake Michigan (Figure 13, 12 m, August to September 11, km off-
shore) with a current track showing loops (produced by a rotary Poincare wave
component) every 16 to 17 hours. The only continuous current records availa-
ble to me from Lake Baikal (Figure 14, September 1960, two stations, 6 and 21
km offshore) clearly show Poincare wave motion, with about four rotations in
2.5 days, corresponding to an approximate period of 15 hours, close to the
inertial period at 54° North latitude. I predict that this type of rotary
current will be conspicuous in Lake Baikal, particularly as the width of the
coastal zone of Kelvin wave influence is relatively narrow, about 7 km.
Much nearer shore in Lake Michigan in August (Figure 15, 3 km), the
rotary current component disappears and the currents are more nearly shore-
parallel, related not so much to the wind as to flows associated with up-
welling and downwelling motions (see temperature record in that figure).
A summary of what we have learned about coastal currents is contained
in spectra of the kinetic energy associated with the currents at two stations,
respectively 6 km and 11 km from shore in Lake Ontario (Figure 16, Boyce,
1974, unpublished observations by J.O. Blanton). There is a distinct gap in
both spectra between low-frequency motions with periods of 3 days or more and
Poincare-wave motions with periods close to the inertial period,
17.5 hr. At the 11 km station, the near-inertial energy component is greater
and the low-frequency energy component is less than at the 6 km station.
326
-------
1 to 25 November 1972: wind (at Mitchell Field) compared with current at Station l.^ 10 m depth.
Wind direction towards:
N
W
u>
N3
. 1 , I , IIM
• 0 1-0 2.0 S-0 4-0 6.0 6-0 7.) 1-0 8.0 10.0 II. 0 12.0 II.0 14-0 16-0 11-0 17.0 16-0 19-0 ZO-0 21.0 22-0 28- D 24-0 26.
TIME IN DflYS
N Current direction
W
S
E
N
towards:
IjVtaKt
s Wivj
ogi
ifij
.jte'Effgi'''!
<£¥MS
feY&ir
iV
!UJ
.0 UO 2-0 S.O 4.0 6.0 6-0 7.0 1-0 »-0 10.0 11.0 12.0 11-0 14.0 16-0 11.0 17-0 1«-0 It-O 20.0 21-0 22-0 2B.O 24-0 26
Wind speed squared: . TIME IN DflYS
1.0 3.0 1.0 4.0 C.O C.O 7.0 1.0 B.O 10.0 11.0 iS.O iMJ lV.0 1E.O 11.0 17.0 li-0 1».0 20.0 21.0 22.0 2IJ
Current speed M™E IN OflYS
30 r -l
cm s
O 2*.0 2E
15 10 Time in Days 15 20 " 25
Figure 10. Direction and speed of wind and current compared to Station ^2 at 10 m depth, 1 to 25 Nov.,
1972. The directions shown are those toward which both the current and wind are moving.
-------
1 to 26 June 1972: Wind (at Mitchell Field) compared with current and temper-
ature at Station 3, 16 m depth.
Wind direction towards
N ,--,-. -.. , • • • , •-, -, ,..-,--,— -,.-...-.-,-.,--, , -T-, —,--._-,- , ._-_,._,....,. ,.-, __--,-- ,_-r. — r_
W
I "•.''.',-,
.,,_,-
/" A
I
E u^,l_* i
N :.^ •'•
B-G 9 0 ',0-0 II D \?-Q 13.0 H-0 16-0 IE C 17 0 16-0 19.0 20- 0 Zl-u 22-0 23-0 *« 0 Zb 0 ?6.0
Current ^ /•" ,i/v HV /*
direction ; / ^ - I1 ' " , '
; towards- \ | . • ' ;
! :,;',,, ;. ' '',!,
'1 /^ !
f r
S
E
i . ._,
•W .c i.c : 3 3.0 4.0 & : e.s i.o a.a 90 ic o no 12 o 13 o K.O 16 o ie o i" o ie.0 is.o 20.0 21.0 22.0 n:o 2* o 26 o 26.o
_1 2 " Tly'E I;" DRYS
150 -Wind speed (m s )
j
100
50 : ., . , ((\ t / " "\
0
30 rCurrent speed
20 L cms"1
101 .;
^ "C Temperature
i 10
2.0 12 0 H.O IB 0 !E.O 17. y 16.0 19
T i^E IN OnYS
0 7E, 0 26
S.
Time in days
1 5
- i .. i .j _. j. i i
15
20
25
j_J._._J__i J_i_l. i 1. i i .. ± t_J .....1
Figure 11. Direction and speed of wind and current compared (plus tempera-
ture), at Station 3 (see Figure 9), 16 m depth, 1 to 26 June 1972
(Sato and Mortimer, 1975).
328
-------
21 June to 16 July 1972: Wind (at Mitchell Field) compared with current and
temperature at Station 6, 23 m depth.
Wind direction towards:
W
VfW " ' '
v
N ^.
21.v 22. 0 29.0 24.0 26.0 29.0 21.0 20. a 2B.O 90.10 1.0 1.0 9.0 4.0 6.0 1.0 1-0 C-0 8.0 10-0 11-0 12.0 19-0 II.0 16.0 1
N -,,,,, Current direction CURRENT DIRECTION T^ROS
A , Kin t I ' I 'I ' ' 1 ' I ' I rTI '
22-0 29.0 24.0 26.0 28-0 27-0 28.0 29-0 30-.D 1.0 t.O 9.0 4.0 G-0 B.O 7.0 6.0 9.0 10.0 11.0 12-0 11.0 14.0 *t-0 1
150 Wind speed squared WINO SPEEO SQURRED IN m/sEc>..2
, -1,2
(m s )
0 22.0 13.0 24.0 2E.O 2*3.0 27.0 28.0 2B.O 30..0 1.0 2.0 1.0 4.0 t.O 1.0 1.0 1.0 1.0 It. 0 11.0 11.0 II.0 14.0 J6.0 I
CURRENT SPEED IN CH/SEC
Current speed
premature
release V
30 5
Time in days
10
°C Temperature
'20 'jj0 jjo 24.0 26.0 Zt.O J1.0 J«.D «-0 »-<0 1.0 2-B t-O 4.0 i.O |.« T .0 1.0 ».0 l»-0 11-0 U.O 11-0 K-0
Figure 12. Direction and speed of wind and current compared (plus tempera-
ture) at Station 6 (see Figure 9~) 23 m depth, 21 June to 16 July
1972 (Sato and Mortimer, 1975).
329
-------
UJ
UJ
o
I ' i A1 rt'i ' ' !i ' ' ' "
[WlflWind
direction
towards:
^•"nyvi
i.
1C-D 17-0 JS.Q 18-0 20. D 21-0 22-0 23-0 24-0 25-0 2G-0 27-0 2S-0 28-0 &0.0 31-0
-0 17.0 i«.0 H-0 20.0 21.0 22.0 23.0 24.0 26-0 29-0 27.0 28.0 29.0 80.0 31-0
18-0 17.0 !«.0 18.0 20.0 21.0 32.0 23.0 2«.0 26.0 38.0 27.0 2«.S 29.0 SO .0 31.G
30
20
Current speed
-1
15
Figure
25
30
13.
20
Time in Days
Direction and speed of wind and current compared (scales are the
same as in Figures 10 to 12) at Station 5 (see Figure 9) 12 m
depth. 15 to 31 August 1973. A progressive vector diagram is
also for Station 5, 12 m depth, 17 August to 13 September 1973
(Sato and Mortimer 1975) .
0
km
10
-------
Trekhsutochnaya Station, 6 km southwest of Bol.
Uschkaru'y Island, 8-11 September 1960.
II. IX.60
24 MRS
Scale
0 40 cm/sec.
8.1.60
12 MRS
Peninsula anchor station, 21 km north of Choboi Point
(Olchon Island), 16-18 September 1960.
N
Scale
0 25 cm/sec
17. IX.60
12 MRS
Figure 14. Progressive vector diagrams of currents at 10 m depth near the
"Academic Ridge," Lake Baikal (from "Atlas of Baikal," Limnologi-
cal Laboratory, Acad. Sci. USSR, Siberian Section, 1969).
331
-------
1 to 23 August 1972: Wind (at Mitchell Field) compared with current and tem-
perature at Station T-j_, 11 m depth.
Wind direction towards:
.u i.u 2-0 3-D 4-0 6-0 6-0 7.0 8.0 8-0 10.0 11-0 12-0 13-0 14.0 16.0 16.0 17.0 H.O 1B-0
N CURSENT DIRECTION TOHBROS TIME IN UHYS
~|—r-|—i—[—'—T
w'
S
E
N
' / I r*"M
Current direction towards:
.0 1.0 2-0 3-0 1-0 6-0 B-0 7.0 «.0 B.O 10-0 M.0 12-0 19-0 14-0 16-0 16-0 17.0 1«-0 19-0 20-0 21-0 22-0 73-C
TIME IN OHYS
[ Wind speed squared
(ms"1)2
50-
.0 6.0 S.O 10.0 11.0 13.0 II.0 14.0 IS.O IS.O 17.0 1«-0 11.0 20.0 21.0 320 J3.c
30 Current speed cm s
-1
TIME IN OflYS
•0 i.u I.0 3.0 4.0 S.O B.Q 7.0 B.D 9.0 10.0 II.0 12-0 13.0 14.0 IS.O 11.0 17.0 1«,0 LI.O 20.0 21.0 72.0 73.
15 r Temperature °C
10
Time in days
0 7-0 e-0 8.0 10.0 11.0 12.
-I 1 . I . I
0 13-0 14.0 16.0 18.0 17.0 1B.O 19.
0 2f.O 21.0 J2.0 !3.0
Figure 15. Direction and speed of wind and current compared (plus tempera-
ture), at Station T^ (see Figure 9) 11 m depth, 1-23 August 1972
(Sato and Mortimer, 1975).
332
-------
104
10
CO
~z.
LU
Q
O
cc
LU
6km OFFSHORE
DEPTH : 8m
TOTAL VARIANCE
107cm2 -s-2
11 km OFFSHORE
DEPTH : 7m
TOTAL VARIANCE : 129cm2-s'2
10Z
PERIODS (h)
10'
Figure 16. Frequency spectra of kinetic energy in the currents at two sta-
tions in Lake Ontario, 3-17 July 1970 (Boyce, 1974).
Lessons Learned from Listening to Lakes
Listening to the lake itself has put us a useful step forward on the
road toward physical understanding, upon which realistic three-dimensional
models of motions in particular basins can be soundly based. For our next
state of modeling—i.e. for the next step in our two-legged progress—the
lake's signals have told us that:
1. During seasons of no stratification, nearshore currents are episodic,
shore-parallel, and directly responsive to wind, with characteristic steep
rises and slower falls in speed (Figure 11). The rate of speed decrease pro-
vides information on coastal friction.
333
-------
2. During stratification, nearshore currents are still partially shore-
parallel, episodic and wind-responsive, but also respond (as shore-parallel
geostrophic currents) to upwelling/downwelling events and to the low-frequency
internal Kelvin waves generated by the upwelling/downwelling motions.
3. Beyond 15 km from shore, Poincare wave dynamics, with nearshore
inertial clockwise rotation of currents, are dominant. In the range 3 km to
20 km offshore, combinations of shore-parallel and rotating current patterns
are encountered.
4. The simple, linear Kelvin and Poincare wave models do not fully
explain the observed nonlinear features of the internal waves arising because
the (linear) assumptions, a) of uniform depth and b) af. small wave amplitude
relative to total depth, do not hold. Nonlinear models must therefore be
developed.
5. The final lesson is that we must continue to maintain our listening
posts—and develop new ones—to receive signals from the system we are trying
to understand. We now perceive some features in outline—and we can recognize
some of the impediments to clearer vision—but we are reminded by lake mes-
sages from new listening posts (namely satellites, Figure 17) that there is
still a great deal to be learned about large-scale motions, particularly
mechanisms and scales of horizontal dispersal.
It is clear that, for a long time to come, we shall need the two legs of
theory and experiment, i.e. modeling verified by signals from nature.
Some General Remarks on Lake EcosystemModeling
While the examples given here of interactive iteration between listening
and modeling have been physical, it will be profitable to adopt a similar
two-legged approach to modeling of chemical and biological events and proc-
esses. Models applied to aquatic ecosystems fall into three general classes:
1. hypotheses or mathematical models, not necessarily valid, which
suggest new ideas or paths to explore;
2. tunable mathematical models, which can be adjusted to the conditions
in a particular lake or river, but are limited in application elsewhere; and
3. models based on more fundamental knowledge of the controlled mecha-
nisms in a wide range of real systems. Such models will be eventually capable
of providing reliable predictions in a wide range of examples.
Of class 1. little need be said; it is part of the general method of
advancement of scientific knowledge. The "curve-fitting" models of class
2. often are useful in environmental management, but independent verification
of them is rarely possible; and, in using them, it should not be forgotten
that the environmental manager's most pressing need is to be able to predict
the effects of large perturbations of the system. In lakes at least, large
perturbations are characterized by sweeping changes in species composition.
The stage is swept clear, to be invaded by a different set of hungry actors,
to which the earlier tunable model can no longer be made to fit.
Class 3. models of lakes, capable of universal, independent verification,
are still in embryonic or component stages, with hydrodynamic modeling the
most advanced component. It is clear that the most rapid advance will come,
not through separate elites of "classical limnologists" and modern systems
analysts, but through an intimate, interactive partnership between the two.
334
-------
.
Figure 17. .LANDSAT '(ERTS' satellite,)
images* of; (above) lower Lake Huron
and Lake St. Clair, 27 March 1973; and
(left) Lake Michigan, 21,August 1973.
The "white water" in Lake Michigan has
•been attributed, to calcium carbonate
pre.cipitation. The turbidity in Lake
Huron was produced by wave erosion of
the shoreline.
*from'negatives supplied by the U.S.
National Aeronautic and -Space Administra-
tion.
335
-------
The question is not which partner is superior or more up-to-date, but which
partner is lagging behind the other. Models can be powerful, interpretative
tools, but the discoveries and the verifications are still to be sought and
listened for in the lakes themselves. The availability of modern tools does
not diminish the usefulness of signposts to limnological progress erected by
the pioneers.
"Un lac se rapproche par ses proportions d'un laboratoire
ou le naturaliste peut repeter a volunte ses recherches, instituer
des experiences, interroger la nature au lieu de se borner a en
ecouter les lecons." Forel (1892, Vol. I, p. vii).
which, freely translated, reads:
"In its dimensions a lake may be compared to a laboratory,
in which the naturalist can repeat his researches at will, set up
his experiments, and, interrogate nature, rather than confine himself
to listening to lectures."
This seems to be a fitting note on which to end this lecture!
336
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REFERENCES
Ayers, J.C., D.C. Chandler, G.H. Lauff, C.F. Powers, and E.B. Henson. 1958.
Currents and water masses of Lake Michigan. Univ. Michigan, Great Lakes
Res. Div., Publ. No. 3, pp. 169.
Boyce, F.M. 1974. Some aspects of Great Lakes physics of importance to
biological and chemical processes. J. Fish. Res. Bd. Canada, 31:689-730.
Forel, F.A. 1892-1904. Le Leman; monographic limnologique. F. Rouge, Lau-
sanne. Vol I (1892), pp. 539 and map; Vol II (1895), pp. 651; Vol III
(1904), pp. 715.
Limnological Institute, Acad. Sci., USSR, Siberian Section, 1969. Atlas of
Baikal. Govt. Dept. Geodesy and Cartography, Irkutsk and Moscow, pp. 30.
Mortimer, C.H. 1963. Frontiers in physical limnology with particular refer-
ence to long waves in rotating basins. Proc. 5th Conf. Great Lakes Res.,
Univ. Michigan, Great Lakes Res. Div., Publ. No. 9, 9-42.
. 1971. Large-scale oscillatory motions and seasonal temperature
changes in Lake Michigan and Lake Ontario. Spec. Rept. No. 12, Ctr. for
Great Lakes Studies, Univ. Wisconsin, Milwaukee. Part I, Text, 111 pp.,
Part II Illustrations, 106 pp., with collaboration, in Part III on internal
wave theory, of M.A. Johnson.
. 1974. Lake hydrodynamics. Mitt. Internat. Verein. Limnol.
20, 124-198.
. 1975. Substantive corrections to SIL Communications (IVL
Mitteilungen) Nos. 6 and 20. Mitt. Internat. Verein. Limnol. 20, 60-72.
Rossolimo, L.L. 1957- Temperature regime of Lake Baikal. Papers of the
Baikal Limnological Station, 16, 551 pp., Acad. Sci. USSR, E. Siberian
Section.
Sato, G.K. and C.H. Mortimer. 1975. Lake currents and temperatures near the
western shore of Lake Michigan. Spec. Rept. No. 22, Center for Great Lakes
Studies, Univ. Wisconsin, Milwaukee, pp. 314.
Schostokowitsch, W.B. 1926. Thermische Verhaltnisse des Baikalsees (Russian
with German summary), Verh. Magnet, met. Observat. Irkutsk, 1:1-30, 1 pi.
Skabitschewsky, A.O. 1929. Uber die Biologie von Melosira baikalensis
(K. Meyer) Wisl. (Russian with German summary), Russ. Hydrobiol. Z.,
Saratov,'8, 93-114.
337
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Tikhomirov. A.I. 1963. The thermal bar in Lake Ladoga. Bull. (Izvestiya),
All Union Geogr. Soc. 95:134-142 (Amer. Geophys. Un. Trans., Sov. Hydrol.,
Collected Papers No. 2).
338
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DISCUSSION
ROZHKOV: First, if the reaction in the lake is determined and the influence
force is random, will a model based on stochastic hydrodynamics levels be
suitable? Second, in the report, flow spectra obtained by field data are
presented. Do you have any internal fluctuation spectra? You did not
illustrate them in the report. How do you relate them?
MORTIMER: Due to insufficient time, I did not speak about quasi-stationary
conditions and geostrophic flows near the shore. The impulse of any natural
system can be two-fold: quasi-impulse and fluctuating. I spoke about one
determined group of lake frequencies; the second group of frequencies is
surface fluctuations.
The second question relates to spectral data on flow. The model used is
simple and based on influx theories. We had this type of model 50 years
ago. If you know the basin dimensions and the density differences in the
thermocline, it is possible to calculate the frequency of the discharge motion
fluctuation. When the basin is large, then the fluctuation frequencies are
close to inertia frequencies. Therefore, you see such a large peak on the
spectrum. In small water bodies, there is a better division of the model ac-
cording to frequency. The spectrum shows us a series of peaks.
GAVRILOV: I would like to mention the results obtained from research on the
Mozhayskiy water reservoir. Unfortunately, I cannot show you any graphic
materials, but analysis shows that automatic measurement of parameters as pH,
temperature, dissolved oxygen, light penetration, and so forth, is subordi-
nated to the same spectrum and carries impressions of the same phenomenon
which has been examined from the hydrodynamics point of view. One main
question is what sampling method should be used for hydrodynamic analysis?
Data obtained from 1974 to 1975 show that during the spring algal bloom,
fluctuations in the parameters examined above have quasi-sinusoidal character.
ROZHKOV: Stochastic hydrodynamic equations should be the basis for models of
natural processes (flows, internal waves). Such models already exist. The
principles of internal fluctuations, about which Dr. Mortimer spoke, are
valid not only for lakes, but also for oceans.
MORTIMER: Large lakes have their own characteristics, which differ from
those of the ocean, but they are often presented as models for oceans. I
do not think this is valid.
ZNAMENSKIY: I would like to emphasize that it is important that mathematical
modeling and field research go band in hand. When everyone understands the
danger of pollution, questions of environmental protection will be extremely
important.
339
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CONSTRUCTION OF A MODEL OF LAKE BAIKAL
ON PRINCIPLES OF SELF-ORGANIZATION
A.G. Ivakhnenko
INTRODUCTION
Maintenance of water purity in Lake Baikal, which contains 80% of the
freshwater reserve in the USSR and 20% of the world source, is a task of
vital urgency. The Limnological Institute of the Siberian branch of the USSR
Academy of Sciences, was organized to monitor and study the processes in Lake
Baikal. During the years of its activity it has undertaken a considerable
amount of work which is published in many scientific publications (see Bibliog-
raphy) . Galkin has published methods of modeling the lake self-purification
processes using differential equations and utilizing Monte-Carlo statistical
methods to determine average characteristics of the Lake Baikal pollution
zone. Galazi (1960-1964) also published the results of lake pollution studies
on the basis of the research conducted in Baikal. Shimarayev (1971) showed
that the forecasting of the hydrobiological processes in the lake should be
divided into two parts: the forecast of a small number of low-frequency
factors and the forecast of a large number of high-frequency factors.
The Biological Faculty of Irkutsk University, the Institute of Cybernet-
ics of Ukr. SSR Academy of Sciences, and the Limnological Institute have join-
ed in the effort to study the application of the self-organization method to
forecast the Melosira biomass yield in Lake Baikal (Kozhova et al. 1973).
Irkutsk University and the Institute of Cybernetics of Ukr. SSR Academy
of Sciences have also carried out joint research aimed at a better understand-
ing of correlation links between phytoplankton and zooplankton in Lake Baikal.
The results of this research are shown in Appendix I.
The main part of this report reflects the results of a joint investiga-
tion carried out by the Novocherkassk Hydrochemical Institute and the Insti-
tute of Cybernetics of Ukr. SSR Academy of Science (1973-1975). It is con-
cerned with the construction of three mathematical models of Lake Baikal mix-
ing zones on the basis of self-organization principles.
Construction of Models of Lake Baikal's Self-Purification Process by
Self-Organization Algorithms"
For a number of years, the Novocherkassk Hydrochemical Institute has been
conducting observations and investigations of Lake Baikal mixing zones in that
*The authors are: Zenin, A.A., A.A. Matveyev, V.L. Pavelko, A.G. Ivak-
hnenko, and P.I. Koval'chuk.
340
-------
region of the lake which receives the effluents of the Baikal pulp plant (BPP)
and the paper factory (on the Selenga River)—the main "polluters" of the
lake (Limnological Institute Report, 1968).
The results of these observations have been used to construct mathemati-
cal models of Lake Baikal's mixing zones by the following three hydrochemical
parameters: (1) dissolved solids; (2) suspended solids; (3) chemical oxygen
demand (COD).
It was assumed that these parameters do not interact with one another.
(To refine the models these interactions will be taken into consideration.)
The models obtained are intended to:
1. develop methods to forecast changes in the concentration of hydro-
chemical parameters;
2. develop methods to identify the mixing zones and to extrapolate them
beyond the region of available measured data;
3. determine the maximum permissible concentrations that do not induce
lake eutrophication ("critical quantities problem").
These results were derived from the latest measured data, least subject
to the influence of wind-induced currents. In addition, the effect of only
one point source of pollution was considered. The problems involving several
sources of pollution, when there is an inflow of freshwater, are now in their
definition stage (problems of Lake Balkash, Selenga River, etc.) and will he
solved with methods similar to those described below.
Because of the absence of a deterministic theory on self-purification
processes, the relatively short time series of observations, and a small num-
ber of the measured points, the most acceptable method of mathematical model-
ing is the principle of computerized self-organization (Ivakhnenko 1971,
1975). This principle implies that all the information on the value of co-
efficients of cause and effect linkages and on the model structure is already
available in the field data, and deep insight into the mechanism of self-
purification is not required.
The principle of self-organization maintains that with gradual sophisti-
cation of the mathematical model structure, the heuristic criteria of selec-
ting the best models ("external supplements") reach their minimum, and
further sophistication of the model is useless. Thus, a model of optimum
complexity is selected. Most often, a so-called "regularity criterion" is
used as the criterion of the selection. This criterion is the value of mean
square error determined on a separate test series of points. Some other
criteria are also applied.
Complete scanning of models is not always possible when models are non-
linear and when the number of input arguments is large. In these cases, it is
more rational to use algorithms of gradual model sophistication which are
called algorithms of the Group Method of Data Handling (GMDH). They are based
on the principle of multi-series selection of models according to the chosen
heuristic criterion. Two types of the GMDH algorithms will be used below:
(1) with sequential separation of trends, and (2) a generalized algorithm with
341
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selection of projectors....More detailed information ,on the principle of self-
organization,, and GMDH algorithms based upon it with their application, can
be found in a series of papers by Ivakhnenko (1971, 1972, 1973, 1974, 1975).
Synthesis of a Two-Dimensional Mathematical Model of Dissolved Solids-—
Models are written in the form .of a self-^regressing function (Ivakhnenko
et al., 1973).
Uj,t + At ~ f(Uj,t - KAt'"-'Uj + iAx,t)s (1)
K = !,...,!;
i = 1,...,m.
It should be noted that the dynamic ratio is not attributed to any
definite physical sense. What this implies is a self-evident concept that
any value of the field is a function.of the proceeding values in time and
space. The number of arguments can be unlimited since the self-organization
arguments have regularizing properties (Tikhonov, 1963). They are resistant
to multicollinearity of variables and are consequently of poor determinancy.
When synthesizing the model of dissolved solids concentration, the ex-
pression Uj: (-+1' normalized with respect to the average value of concentration
at the j-tn point at the next (t+1) moment of time, was adopted as the output
value. The following variables "close" in space and time (Figure 1) are used
as input arguments:
U
Uj,t - At'Uj
Ax,t' and Uj - Ax,t' with At =
The initial data of the dissolved solids parameter V are shown in Table I.
t+A t- -
t-- Vj+Ax, t I-
t-At--
vj ,t+At
V(j)t
-I Vj-A.x. t
Vj,t-At
Figure 1: Diagram of the selection of points to synthesize dynamics equations
of the (dissolved solids) mineralization conductors and suspended
solids.
342
-------
TABLE I. INITIAL DATA ON DISSOLVED SOLIDS
Poiiit
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Vi,t-At
93.3
90.1
88.7
97.4
99.8
•98.9
89.1
97.6
98.1
95.0
97.8
100.2
93.1
95.7
96.0
104.5
107.9
103.2
105.8
101.2
104.9
Vi+Ax,t
99.8
98.9
97.8
97.6
98.1
95.7
97.0
96.2
98.7
95.7
96.0
96.4
97.9
94.7
102.2
89.9
88.1
86.4
93.5
87.7
91.6
Vi,t
97.4
99.8
98.9
89.1
97.6
98.1
94.9
97.0
96.2
93.1
95.7
96.0
95.2
97.9
94.7
94.1
98.9
88.1
93.0
93.5
87.6
Vi-Ax,t
98.5
97.4
99.8
96.1
89.1
97.6
100.2
94.9
97.0
99.8
93.1
85.7
97.1
95.2
97.9
96.0
94.1
89.9
93.0
93.0
93.5
Vi,t+At
89.1
97.6
98.1
94.9
97.0
96.2
98.0
77.0
94.0
95.2
97.9
94.7
95.0
93.0
98.0
92.4
90.2
94.6
94.0
90.9
92.5
343
-------
The synthesis resulted in models of optimum sophistication (as far as the
regularity criterion is concerned) in the form of systems of nonlinear differ-
ence equations with retarded arguments (Ivakhnenko et al., 1973, 1974):
(a) according to the generalized GMDH algorithm with selection of- pro-
jectors
U-j^t + ]_ = 0.0083 - 0.266-U1jt _ L - 0.037 -Uj+ljt - 0.0069 - 0.063-U1jt
+ 6.395-U2j)t (j = l...n); (2)
(b) according to the GMDH algorithm with sequential separation of trends
U- «-+! = 0.0082 - 0.282-Ui r_i - 0.0069 - 0.0609-U. t + 6.439-U2, t
J,t+l ],t i J,L J,L
(j = 1...n) (3)
where U. , *. are the values of the variable V-*f normalized with respect to
j > t ] i L
the average
n. = V-i r ~ vi*i- r i •
],t J ' J 'c . V-*t are average vaiues of the concentration in
V-!*<- question;
J 5 L
l*L^™:
•].t W
V- t is the concentration at t-moment of time at the j-th point of the
field.
The residual variances obtained for the normalized data on the test set
are equal to 62 = 0.03540; S?; = 0.03505, respectively, which characterizes
the accuracy of the models obtained.
Synthesis of Mathematical Model of a Two-Dimensional Field of Suspended
Solids—
Equations for suspended solids (Ivakhnenko et al., 1974) have been de-
rived in a similar way (Table II).
U-! f+1 = -0.022 - 0.053-U, r_i - 0.0215'U^_i f + 0.272 + 0.155-11, i r
i » L^l J,LJ. JJ.,L JJ.,L
- 0.224-U2- _i r + 0.155 - 0.791-U-.1 ^ - 0.799'U2, ,1 ,.
J 1 , L JT1 , L J Tl , U
+ 0.52-U3j+1>|. (j = l,...,n; (4)
This model is more sophisticated than the model of dissolved solids,
since it contains more nonlinear terms.
Synthesis of the Mathematical Model of a Three-Dimensional Field of Chemical
Oxygen Demand (COD)—
The COD model is synthesized for the case of a three-dimensional field;
where, as input variables, besides "close" values of this parameter, cartesian
coordinates of the measured points (x,y,z) EBEE(3) have also been selected.
It has enabled more detailed investigation of dynamic properties of the
mixing zone in different areas; further, it has been possible to obtain
344
-------
TARLF TT. INITIAL DATA ON SUSPENDED SOLIDS
Point
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Vi,t-At
4.
6.
1.
1.
0.
0.
1.
3.
1.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
3
8
4
2
0
0
3
3
4
0
6
7
4
0
4
0
0
1
0
6
3
2
7
2
Vi+Ax,t
0.
0.
0.
6.
1.
5.
2.
3.
2.
16.
6.
2.
4.
2.
2.
4.
2.
2.
4.
2.
2.
3.
2.
2.
0
0
6
8
4
2
12
23
0
4
9
0
0
8
0
0
61
0
4
0
0
6
4
2
Vi
1
0
0
4
6
1
2
2
3
4
16
6
4
4
2
2
4
2
2
4
2
2
3
2
,t
.0
.0
.0
.3
.8
.4
.02
.42
.23
.0
.4
.9
.0
.0
.8
.8
.0
.61
.0
.4
.0
.4
.6
.4
Vi-Ax,t
0.
1.
0.
2.
4.
6.
4.
2.
2.
4.
4.
16.
1.
4.
4.
4.
2.
'4.
2.
2.
4.
4.
2.
3.
0
0
0
0
3
8
82
02
42
8
0
4
2
0
0
8
8
0
8
0
4
4
4
6
Vi,t+At
0.
1.
0.
1.
0.
0.
0.
1.
1.
0.
1.
1.
0.
1.
1.
0.
1.
1.
0.
0.
1.
2.
5.
3.
0
0
8
0
0
0
8
5
7
5
3
6
3
2
5
3
0
4
3
9
2
0
2
1
345
-------
quantitative characteristics of the process ;under investigation in terms of
short- and long-term forecasting.
One hundred observations have been used as input information: 50 points
were treated as a training set, the remainder as a test set.
The optimum equation (according to "global" minimum of the regularity'
criterion or the mean sequare error on the test set) was obtained after the
third selection level of the main network with subsequent projection of the
remainder according to an algorithm with sequential trend separation. The
equation is as follows:
D
l,J,k,t+l
l
where y^ = 0.0013 + 0.29-Z3 + 0.80-Z10 is the equation of the third selection
level.
The best variables of the second selection level are as follows:
Z3 = 0.002 + 0.402 -y + 0.776-yg;
Z10 = 0.006 + 0.822-y9 + 0.626-yy
The best variables of the first selection level are as follows:
= -0.01 + 0.0358-Ui
2 -. .-i j + 1 k t
yy = -0.015 + 0.0272.U± .+1 k t
yg = -0.016 + 0.062-Z +'0.056'x
k t - 0.059^ _
0.070-x;
During sequential trend separation from the remainder y - y., , the fol-
lowing equations have been obtained:
fl = °-010 + 0.071.Uitj+1>k>t - 0.084.U
f2(Ui-l,j,k,t) =-0-016 - O^S-Ui-ij^.t + 0.458 -
_1 .
J- ) J
Here
X = [XQ + (i-1) AX - 4.32] /4.32;
Z = [ZQ + (k-1) AZ - 3.12] /3.12;
where XQ , ZQ are initial values of the region
along the spatial coordinates X and Z.
AX,AZ are the step length
It should be noted, that unlike the preceding models (2), (3), and (4),
the function of the pollution source capacity here is not constant but depends
on spatial coordinates.
346
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Study of,(Stability and Other Dynamic Properties of the Processes Concerning
the Change in Hydrochemical Parameters on the Basis of the Models Obtained—-
Natural waters can be considered as multicomponent equilibrium systems
(Vel'ner et al., 1968). For each water body, the equilibrium is determined by
the combination of physico-geographical, hydrological, and meteorological con-
ditions and is established over a long period of time. When pollutants enter
the water body, the natural equilibrium is shifted .and the water body "mobi-
lizes its forces" to combat the distortion of the natural conditions and
strives to return the whole system to its stable state.
If the amount of pollutants exceeds "the self-purifying1' capacity of a
water body, it results in an irreversible distortion of the biological equi-
librium and, as a consequence, the degradation of water quality properties
(Messineva, 1973).
The effect on water quality can be described by a nonlinear operator A,
which enables us to transform the equations (2), (3) or (5) as follows:
U(t+l) = A[U(t),...,U(t-l)J, (6)
where U(t) is a vector characterizing the state of the field at t moment of
time.
To understand the character of pollutant transformation, it is necessary
to study the dynamic properties of the operator equation (6). First, it is
necessary to determine the set of equilibrium points U-^ EU where
U- = A[U_p . . . ,U.j_] and to understand the character of each of them. Then it is
necessary to determine the stability regions Q^ with respect to the initial
conditions
U(t)|t=1 = UQ; U(t_1)|t=2l =U1
Of some interest also is the dependence of the stability region critical para-
meters on the change of the A operator structure, e.g., due to change of "the
pollution-source capacity function."
Research on the stability of the operator equation (6) in the vicinity of
a stationary point U^ carried out by using Lyapunov functions for the linear
systems, U(t+n = AU(t) + BU(t-l) were selected (Ivakhnenko, 1974).
Z(Zk) = max { |Uk| ],
}
(7)
Thus, the research on stability is reduced to verification of the condi-
tion
max
y>max
(8)
which is valid only in case | A
B
<1
Investigation of the Dynamics of Dissolved Solids Field Model —
If boundary conditions, for example, in the form Uo = U^; Un = Un+i ,
are united with the systems (2) or (3) , the following matrix equation is
obtained:
347
-------
Un+1 = AUn
(9)
where
1
A =
=
0
0
0
•
0
0
0
c
c
0
0
0
0
z
z
0
•
0
0
0
0
0
c
0
0
0
q
q
z
•
0
0
0
0
0
0
0
0
0
00 ...
00 ...
qO ...
00 ...
00 ...
00 ...
* * • • •
000
000
000
...
zqO
Ozq
Ozq
000
000
000
cOO
OcO
OcO
B =
p 0 0 ...
p 0 0 ...
0 p 0 . . .
000 ...
000 ...
0 0 0 ...
000
000
000
Poo
OpO
OpO
b =
Here z = -0.063; q = -0.037; p = -0.266; c = 6.395, b = 0.0014.
Application of the first Lyapunov method has shown that if
2 max { , A + 2CUp
= max;
B } <1 where the matrix norm is defined as
then the system (9) is stable in a small increment area,
i.e. in a linear approximation.
Determination of the ultimate stationary level has shown that all the
trajectories of the free motions used by initial deviations, at t ->- °° approach
the value
+ VJ_F = 94.75 mg/£
Qj =
The free term in the right part of the equation (9) corresponds to the
capacity of a pollution source. Analysis of the equation (9) solution with
pollution source capacity b = + O.OOL4 has shown that the process trajectories
converge to a stationary level Vj m = 94.75 mg/£. With initial deviation
V < 114 mg/2. , the processes have a damping character, that ist they are stable
(Figure 2), with a Larger value of the initial deviation, they are unstable.
With further change of the pollution source capacity b, the stationary level
gradually increases or drifts up to the value V-j ^ < 103 rug/?. (Figure 3-2', 2")
while the upper limit of the process (of the admissible pollution threshold)
is gradually lowered to the same level (Figure 3-l'.l" curves).
Coincidence of the dynamic characteristics of the equations (2) and (3)
(Figure 3), obtained with the help of different GMDH algorithms, is direct
evidence of the validity of the dissolved solids model,
The stability region Q of the system of equations (2) or of its matrix
analysis (9) was determined in two-dimensional space (AV*-D,
with respect to "disturbances" of the initial point (Figure 4a)
i.e.
V,(t) = 94
Vj(t-l) = 94 +
(j = 1 ..... n).
348
-------
Vj(t)m2/l
Figure 2. The process of change in dissolved solids' concentration given
different initial deviations.
V
114-
110-
106-
102-
98-
94
112-
106-
102-
98-
94
r
2'
1"
0.02
0.04
0.06
Figure 3. Dependence of the upper limit (l', 1") and stationary level of the
dissolved solids process (2!, 2") on the pollution source capacity
a) l',2' according to the generalized GMDH algorithm; b) 1",2" ac-
rording to the GMDH algorithm with sequential separation of trends
349
-------
Figure 4. Change in the stability region of the dissolved solids parameter
with increase in the pollution source capacity.
The investigations of the stability region limits in the coordinates
and AV' ) with changing pollution source capacity are of interest. As
the value of b increases, the stability region decreases: at values b>_0.07,
it was already impossible,to determine (Figure 4). As is clear from the
analysis of equation 9, if one of the coordinates exceeds the threshold value,
then a process of continuous concentration begins to increase. This process
results in the loss of the self-purifying property in a particular locality
of the lake. The permissible level of pollution should be somewhat lower than
the threshold value, since a locally developed point of eutrophication can
gradually expand and cover a much larger zone.
Though it is not possible to directly identify the terms responsible in
the equation (9) for self-purification or the terms that reflect multiannual
fluctuations of dissolved solids concentration in Lake Baikal, the research
shows that the self-purification mechanism does work at low concentrations
for dissolved solids.
Study of the Dynamics of Models of Suspended Solids and Chemical Oxygen Demand
(COD) Fields-
Numerical experimentation similar to that described above has been car-
ried out for the system of equations (4), which characterize the field of
suspended solids. In this case, the stable equilibrium point is equal to
1.5 mg/fc. Figure 5 shows how the suspended solids concentration changes in
time at a certain initial deviation.
350
-------
Figure 5. Process of change in the suspended solids concentration.
As far as COD is concerned, the upper limit of the stability region is
equal to 7.5 mg/£. The ultimate concentrations of the field in a vertical
section (Table III) have been used as a basis for a long-term forecast of
CODs. Using the equation 5, my experimental calculations showed that, within
the. interpolatdon zone, the process of the COD change is stable.
Conclusion
This paper reviews a method of synthesizing equations that describe
dynamics of the main hydrochemical parameter- fields. The- equations are
synthesized on the basis of the self-organization principle for the ca-se of
only one pollution source. The first simple dynamic models of the dissolved
solids, suspended solids, and COD fields have been obtained. The models are
based on the regularity criterion; Dynamic properties of the models have
been investigated; a long-term, forecast concerning COD was obtained within the
interpolation zone of the field.
The regularity criterion, on which the selection of the above mentioned
models is based, does not permit distant extrapolation of the mixing zone.
To do this, other selection criteria or other regularization methods are re-
quired.. Further refinement of the mathematical models of mixing zones will
be connected with the application of the unbiased criterion, as only unbiased
equations can be assumed to be a law describing the zone.
Forecasting the Yield of Melosira Biomass in Lake Baikal by Self-Organization
Algorithms*—
Methods .of self-organization of forecasting equations ensure the1 most
exact and objective forecast, since the model uses only actual values of the
*The authors of the study are: Kozhova, O.M., A.G. Ivakhnenko, A.G.
Yeremenko, N.N. Todua, M.N. Shimarayev.
351
-------
TABLE III. CHANCE IN THE STABILIZED LEVEL OF COD AT DEPTH (Z) AND AT (X)
DISTANCE FROM SHORE
X
1
2
3
4
5
3.68
3.70
3.71
3.73
3.74
3.74
3.75
3.77
3.78
3.79
3.77
3.79
3.80
3.82
3.83
3.82
3.83
3.85
3.86
3.87
3.86
3.87
3.89
3.90
3.92
3.90
3.91
3.93
3.94
3.96
3.94
3.96
3.97
3.99
4.00
3.98
4.00
4.01
4.03
4.04
4.03
4.04
4.05
4.07
4.08
4.
4.
4.
4.
4.
07
08
10
11
12
variables characterizing the process. It is not necessary a priori to set up
the equations of the system elements as in the dynamic modeling method.
The following self-organization algorithms will be compared in a specific
example of a problem concerning medium-term forecast of Melosira biomass yield
in Lake Baikal."
1. forecast according to the harmonic trend (time function) of optimum
complexity;
2. multiple (that is, repeating year after year) systematic differential
forecast according to "differential equations of regression" synthesized for
the "residual," that is, for deviation of the parameter from the harmonic
trend mentioned above.
The validity of the forecast is tested on two data sets: the first and
second (verification) sets.
The first test set is selected according to the "seventh" (harmonic
trend) and to the "third" (differential equations of regression) method of
regularization. For the second verification set, data gathered during 1946-
1949 are used.
As will be shown below, with the prediction interval Ty=5 years, if the
"residual" (a multivariable function) is calculated, the precision of the
forecast would increase by approximately 50%, as compared with the forecast
according to the trend (a time function).
To verify the validity of our forecasting model, we shall use a forecast
with time counting "in reverse" (Ivakhnenko, 1971).
Melosira and Cyclotella play a leading role in the biomass of the primary
producers of the open water in Lake Baikal. But the plankton biomass succes-
sion is not the same in different years. High-yield years are alternated with
two or three low-yield ones (Kozhov 1972). This phenomenon can be attributed
to a number of factors: temperature conditions, illumination, availability
of biogenic elements, and other factors considered as governing abiotic fac-
tors. Nevertheless, no singular explanation has been found for the sharp
fluctuations of Melosira yield in different years. The first suggestion that
the Melosira yield may be governed by environmental factors was first sugges-
ted by Kozhov (1965). The idea was further developed by Shimarayev (1971)
in a qualitative analysis of the relationship between the changes of the
Baikal plankton yield and the hydrometeorological characteristics of the lake.
352
-------
The paper maintains that the Melosira yield is a random process consisting of
two components: a low-frequency and a high-frequency.
The low-frequency component should be treated as a function of certain
arguments (their number being small; one or two), and the high-frequency com-
ponent as a function of other arguments (their number being much larger).
The optimum separation of the process into components with different
arguments enables us to define self-organization algorithms, such as the al-
gorithms of the Group Method of Data Handling (GMDH) (Ivakhnenko 1971).
Solving the Problem of the Medium-Range Forecast of the Melosira Biomass—
This paper solves the problem of medium-range forecast of the Melosira
average annual biomass (F) by self-organization algorithms. The Melosira
biomass is assumed to be a random process that can be represented as a sum of
two components: the trend, a time function (FT(t)), and the remainder—a
function of the variables of the system V1, V2,..., Vm
F(t) = FT(t) + f(Vi, V2,..., Vm).
Thus, the low-frequency component _of the process will represent a time
function—the trend; a high-frequency component will represent a function of
the variables of the system.
The selection of the high-frequency components arguments is governed both
by the existing general concepts and the available observation data.
The arguments have been composed of the following parameters (data from
1950-1971):
Vi - average annual precipitation on the lake surface;
V"2 - average yield of Cyclotella minuta biomass during 2nd-4th months;
V3 - average yield of Gymnodinium biomass during 2nd-<4th months;
VLJ - average annual tributary inflow into Lake Baikal;
¥5 - monthly average temperature in April at the depth 0 m;
Vg - monthly average temperature in April at the depth 50 m;
V7 - yearly average evaporation from the lake surface;
VB - average temperature in the 8th - 9th months at the depth 50 m;
Vg - amount of heat at the end of the year (determined according to the
technique described by Shimarayev (1971) ;
VIQ - average annual water discharge from the lake.
The initial data were taken from-observations made by the Baikal biologi-
cal station of the Biological Geographical Institute affiliated with Irkutsk
State University (variables F, ¥2, Vs, ¥5, Vg, VB), and by the Limnological
Institute of the Siberian branch of the USSR Academy of Science (variables
Vl, V4, Vg, V7, Vg, VIQ).
The solution of the problem of the Melosira average annual yield medium-
range forecast is carried out in the following two stages:
1. separating the process low-frequency component (i.e., the harmonic
trend of optimum complexity according to GMDH) and forecasting this component;
2. synthesizing "differential equations of regression" (Ivakhnenko et
al.» 1973) for "the remainder" (i.e., the deviations of the variable from its
353
-------
harmonic trend) and forecasting this remainder.
Forecast on the Basis of the Harmonic Trend of Optimum Complexity (in time
function)—
In the problems of forecasting random processes that contain latent
periodic components, harmonic trends with non-multiple quantization steps,
should be separated (Kimura 1913; Serebryannikov and Pervozvanskiy, 1965).
Self-organization algorithms enable us to optimize the problem concerning
the selection of the process harmonic components (Koppa and Todua, 1972).
In this particular case the random process is defined by the discrete
time series F. = f(jAt), where At is a quantization interval (equal to one
year); j = l,2,3,...n (n is the number of interpolation points). The value
F. in this expression will be approximated by the harmonic series
F (t) = 7 + $ f '^
where fi(t) = BQ^ + a^ sin(— t) + a.^ cos(_t);
T
i
n
is the random mean (¥ = i_ Y F.);
n J
i is the number of harmonic components of the . approximating expression; T is
the period; t is time.
The harmonic trends have been separated according to the program (devel-
oped by Koppa and Todua, 1972) in the following order. The time series of the
data obtained during 1950-1971 (Figure 6) have been split into two succes-
sions: a training one and a test one. The points, following the method
described by Ivakhnenko et al. (L973) have been separated according to the
"seventh" regularization method, i.e., the test succession was comprised of
the apexes of the first harmonic o,f random process (1950, 1955, 1961, 1971).
These points formed a training succession. '
The criterion indicating cessation of the selection levels of build-up,
i.e., the criterion of the complexity of the approximating expression F" (t)
is the magnitude of the relative mean square error A(l) on the first test
succession P A
AT<1) -I (F. - 1)2
P
I,
.. ,
where p is the number of points of the first test succession; F... ,i$. the actual
yalue of the parameter of the test succession points;
F^ represents the evaluations of these points.
354
-------
Lo
Ui
100.00- - \ \ //
t 200 years
Figure 6. Change in average annual Melosira yield (training and first test succession; forecast
T =5 years: 1 - actual values; 2-- harmonic trend and forecast on the basis of the . -
trend (A(O = 0.194); 3 - forecast on the basis'of trend and remainder (A^U1) = 0.039-).
-------
If the error on the S level of selection is bigger than on the S-l level,
i.e.
A(l)
A(l)oi or if the error A(l) decreases negligibly with the in-
crease in the selection levels (t.he value of the difference A(l)g - A(l)g_|
is an indication of this), then it is necessary to stop at the (S-l)th
selection level (Figure 7).
Figure 7.
Dependence of the relative mean square error A(l) (on the first
test succession) on the number of selection levels S.
After the seventh level, the error AT(!) = 0.194 and its further decrease
is insignificant. Thus, it is evidently expedient to stop at the seventh
selection level. The harmonic trend of optimum complexity F ( t) is separated
in the figure.
Multiple Systemic Forecast by Differential Equations of Regression- —
Differential equations of regression are a specific variety of difference
equations that at At ->• dt° are reduced to common differential equations with
continuous argument t in Cauchy's normal form.
The differential equations of regression are synthesized for "the re-
mainder" (Ivakhnenko et al . , 1973) i.e., for the deviation of the output
parameter of the system from its harmonic trend F°(t) = F(t) - F (t).
In its general form, the differential equation of regression is written
in the following way:
where parenthesized indices determine the value of the retardation of the
arguments .
356
-------
In agreement with the algorithm of the multiple differential forecast
(Ivanhnenko et al., 1973) as a partial description in the case of medium-
range forecast, we recommend selection of "partial" descriptions of the type
. + a^X-^X • , where i,j = l,m.
y = a + a-iX- +
The points were split into a training succession (R=14 points) and a test
succession (Rl=4 points) using the "third method of regularization" (Ivak-
hnenko, 1971).
The optimum complexity of the regression differential equation of the
remainder is selected on the basis of the mean square error A2(l) on the first
test succession: P
A2(l) = I
1=1
(AF°. - AF°.
I (AF°.)2
1=1 1
where p is the number of points in the first test succession,
AF°. is the actual value of the first difference of the remainder,
AF°. is the determined value of the first difference of the remainder.
The criterion for the optimum complexity of the equation 6 is the minimum
of the error A2(l), and in our case it is achieved on the third level of
selection at A2(l) = 0.039 (Figure 8).
0.20 --
0.10--
0.039
Figure 8.
1234 5 &
Dependence of the relative mean square error A2(l) on the first
test succession on the number of levels S.
The following equation of the remainder has been obtained by using the
program described by Todua and Gulyan (1973).
AF
At
= 0.47892 + 0.63344 Y! + 0.82953 Y2 + 0.02152 YiY2,
(1)
where YI = 0.64411 + 0.61788 Xj + 0.75660 X2 - 0.01058 XiX2,
Y2 = 0.04844 + 0.43567 X3 + 0.89378 X4 + 0.02866 X3X,,,
357
-------
where KI = 1.47473 + 19.42608 V7/.)- 1-07258 V2(_2) - 25,48776 V7(_2)V2 (_2) ,
X2 = -0.615839 + 18.09545 V?(_2) - 0.14528 V3(-1y
- 19.67778 V7(_2)V3(_1)5
X3 = 1.23069 + 29.11694 V7(_2) + 0.36028 V3(_2) + 12. 94899 V?(_2)V3 (_2) ,
X^ = 0.52410 + 33.66962 V8 3) + 17.16956 V8 + 69.51791 V8(_3)V8' 2
The differential equation of regression (2.1) is written in normalized
deviations from the average, that is: R+Rl
v. V^OK.J) - Vi - _ _i _ I v.
Vi(-j) - =- - - ; Vi ~ R + Rl 1=1
vi
where i is the variable number; j is the time shift; a is the absolute value
of the parameter; R and Rl are the numbers of points in the training and test
successions.
It is possible to pass from the trend to absolute values of the deviation
in the following way:
= " (F° + !>• *° = F° ± AF°>
AF° is that formation of the difference increment;
"p° is the arithmetic mean of the remainder.
The meaning of the indices is as follows:
(H) is the normalized value of the parameter; t,(t+l) designates the sequence
of the parameter values. Calculations based on such complicated equations
are possible, only by use of a computer.
In its general form, the structure of the forecast equation (trend + re-
mainder) can be written as follows:
= FT(t) + f V3(_2), Vy(_2), V8(_2), V8(_3))
The algorithm of self-organization has automatically selected the fol-
lowing arguments (except for time) :
V2/_2) = average yield of Cyclotella minuta biomass during the 2nd to
4th months two years ago;
V3/_1x = average yield of Gymnodinium biomass during the 2nd to 4th
months a year ago;
^3 (-2) = same two years ago;
V? (_2) = evaporation from the lake surface two years ago;
v8(-2) = average temperature during the 8th to 9th months at a depth of
50 m two years ago;
Vs(_3) = same three years ago.
The first three arguments V2(_2), V3/_1) , V3(_2) characterize the role of
phytoplankton in the yield of Melosira. The arguments Vy,^, Vo , V /_ \
358
-------
are temperature conditions. Change in the Melosira biomass may depend on
temperature conditions, connected in turn with the vertical water exchange
(especially at greater depths H>50 m) ; it is important to note that their
influence is manifested every two or three years (which is in agreement with
the actual frequency of Melosira biomass yield) .
Thus, it follows from the forecast that, as far as Melosira is concerned,
1972 and 1973 should be low-yield years, followed by twin high-yield 1974 and
1975 and low-yield year 1976. Observations in 1972 supported the forecast.
Forecast with Time Counting "In Reverse" to Check the Algorithm Validity —
The accuracy of the forecast was checked on the second test (verifica-
tion) succession which was composed of the data collected during 1946-1949
(Figure 6) . The validity of the forecast algorithms was checked by means of
time counting, "in reverse" (Ivakhnenko 1975)'.
The accuracy of the forecast by the harmonic trend with time counting
"in reverse" is characterized by the relative mean square:
1949
I (F - Ffc)2
= t=1946 fc =5.5,
1949
I
where Ft is the actuaLvalue of the parameter;
"F* is the evaluation according to the trend.
The equation of the remainder forecast with time counting in reverse is
of the following form:
AF°*
At
= -0.49860 + 0.39861 Y1 + 0.88788 Y2 + 0.03276
where Yj = 0.28378 + 0.22782 X: + 1.06805 X2 - 0.15408 XiX2,
Y2 = -0.84958 + 0.45462 X3 + 0.57152 X^ + 0.01856 X3X4
where Xx = 0.07592 - 1.58986 FO*^ - 4.04214 V*(_2) + 0.02588 F°*)V* 2)
X2 = 0.02237 - 1.34101 V*(0) + 1.36748 V2(_3) + 1.30351 V2(0)V2(_3),
X3 = -0.98033 - 18.408 Vj 2 - 0.28931 V*(_^ + 200.93788 V^^V^
X4 = -0.04640 - 1.68006 F°*} _ 9.47947 Vg(_3) - 0.32228 F°J)Vg(_3)
The superscript * means that the arguments are taken with time counting
in reverse.
The accuracy of the forecast on the basis of the trend, taking the re-
mainder into account (i.e. the trend + the remainder), is characterized by
the mean square error:
359
-------
A(2) = t = 1946 = 2.73
1949
I (Ft
t = 1946
A 2
- V
1949
I Ft
t = 1946
A
where F. is the actual value of the parameter; F are the estimates on the
basis of the trend, with a calculation of the remainder.
Thus, the mean square error on the second test succession has been re-
duced approximately by 50% if the remainder is taken into account. According
to the forecast, low-yield years were 1947 and 1948 and high yields were 1946
and 1949, i.e., there was discrepancy with the actual data only for 1947.
Hence, it can be concluded that the forecast algorithm, described above,
is characterized by the trend separation with forecasting the remainder as a
function of the system parameters and yield results that are accurate enough
for practical requirements.
360
-------
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Energii V Vodokhranilishchakh. Listvenichnoye Na Baikale.
Popovskaya, G.I. 1973. Seasonal and yearly fluctuations of phytoplankton in
the Chivyrkuysk Bay of Baikal. Gidrobiologicheskiy Zhurnal, Vol. IX, No. 5.
S'erebryannikov, M.G., A.L. Pervozvanskiy. 1965. Distinguishing latent perio-
dicities. Nauka, Moscow, USSR.
Shimarayev, M.N. 1971. Hydrological factors and fluctuations in counts of the
Baikal plankton mass. Reports of the Limnological Institute, Vol. 12, No.
32.
Some elements of the hydrometeorological regime of Lake Baikal. 1964. Reports of
'Limnological Institute of the Siberian Branch of the Academy of Science,
Vol. XXV, Nauka, Moscow-Leningrad.
Tarasova, Ye.N. 1972. Relationship between the organic and mineral components
of suspended solids in the waters of the Southern Baikal. Gidrobiologiche-
skiy Zhurnal, Ukrainian Academy of Science, Naukova Dumka, Kiev, Vol 8:5.
Tarasova, Ye.N. 1972. The relation between organic carbon and different kinds
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Okhrane Poverkhnostnyk Vod Ot Zagryazneniya, Tallinn.
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363
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APPENDIX I
CORRELATION LINKS OF PHYTOPLANKTON AND ZOOPLANKTON IN LAKE BAIKAL*
The following problem is discussed below. Determine the possible links
between the number of zooplankton and phytoplankton on the basis of the meas-
urements carried out at the permanent monitoring station in Bolshiye Koty
(Southern Baikal) in 1964-1966.
The number of zooplankton depends upon the number of phytoplankton cur-
rently available, but it also depends upon the number of phytoplankton which
was available in the past, i.e., there is a certain delay in the algae-zoo-
plankton system. So, it is necessary to find out not only what types of algae
have the greatest effect on separate species of zooplankton, but also the
"lag" values corresponding to them. These questions can be answered-with the
help of correlation functions.
N-T
/N-T _ 2
- X.) I (X.
1 h=l T'u J
where X^,/^_|_TN is the time succession of initial data of the i-th species of
zooplankton;
X- ^ is the time succession of initial data of the j-th species of phy-
toplankton;
T is delay time (lag time);
N is the number of measurement intervals.
After calculating the correlation functions, i,e., the values of the cor-
relation coefficient at different values of delay, we can select those values
of delay, at which the absolute value of the correlation coefficient is great-
er than a certain preset value 0 (usually 0>0.6).
The following species of zooplankton and algae have been selected to
construct the correlation functions:
KI - Epischura baicalensis Sars
X2 - Epischura baicalensis nauplii
X3 - Cyclops kolensis Lill
Xit - Cyclops kolensis Lill nauplii
"Authors of the study are Ivakhnenko, A.G., Yu,V. Koppa, and T.V.
Makukhiva.
364
-------
Xs - Bosmina longirostris (O.F. Mull)
XG - Keratella quadrata Mull
Xy - Kellicottia longispina Kell
XB - Filinia terminalis Plate
Xg - N. grandis Vor
IQ - Synchaeta pachypoda Jaschn
n - Melosira baicalensis
M. islandica
- spores of M. islandica
Synedra acus.
Xis - Synedra ulna.
XIG - Stephanodiscus
X17 - Cyclotella baicalensis
Cyclotella minuta
'Gymnodinium
X20 ~ Anhistrodesmus
For this example i =1,2...,10; j = 11,12,...20. The number of measure-
ment intervals in the available data N = 180 (measurements- were carried out
every 10 days). When constructing the obtained correlation functions, it was
assumed that T = 0,1,2...,73. Examples of the obtained correlation functions
are shown in Figures 9 and 10.
The results of the computations for all the species of zooplankton and
phytoplankton are given in Table IV. The positive correlation coefficient
shows that zooplankton biomass increases with the increasing number of species;
a negative coefficient means that the zooplankton biomass decreases with
algae increase. Table IV enables us to draw the following conclusion: that
the Epishura baicalensis Sars (X]_) biomass is most severely affected by the
quantity of the following species:
Xjy - 380 days ago;
X18 - 310 days ago;
X19 - 390 and 350 days ago.
Similar conclusions can also be drawn for other variables.
365
-------
K
0.8 -
0.6-
0.4 -
0.2 -
0
-0.2 -
-0.4 -
-0.6 -
-0.8 -
V
Figure 9: Correlation coefficient of Bosmina longirastris (O.F. Mull)
with Stephanadiscus
K
Figure 10: Correlation coefficient of Bosmina longistrostris (O.F. Mull)
with Synedraulna
366
-------
TABLE IV. CORRELATION COEFFICIENTS BETWEEN THE NUMBER OF ALGAE AND DIFFERENT
SPECIES OF"JZOCTPLANKTQN*
Zooplankton
Algae K! X2 X3 X4 Xfe ; X;6 Xy Xg Xg X10
Xn 11 11 4_ 53 53 1_ 0_
O..J659 0'.709 0.638 0.691 0.624 0.263 0.815
12 12 1:4 4 1
0.860 . 0:784 0.394 0.756 0.625
13 53 27 11 38
,0.608 ,.0,785 0..779 0.973 0.665
53 5'4 29 39 39
0.752 0.799 0.621 0.302 0.782
54 70
0.948
11
0.
0.
687
73
379
15
53
X12 45 11 11 15 53 62 53 2 0
0.661 0.687 0.850 0.657 0.794 0.321 0.770 0.285 0.800
12 24 3_ 37
0.705 0.676 0.713 0.682
13 53
0.665 0.680
20
0.613
53
0.783
54
0.707
35 0 10 12 53 0
-0.628 0.680:0.832 0.621 0.696 0.663i
1_ 11 13
0.652 0.658 0.331
10 14
0.897 0.797
(continued)
*The numerator is the lag in measurement interval (10 days). The denominator
is the correlation coefficient which exceeds 0.6.
367
-------
TABLE IV (continued)
Algae X: X2
Xl3
0.
0.
0.
0.
0.
Xm 95
0.601
0.
0.
0.
0.
X16
0.
0.
0.
0.
x3
11 .
792
14
773
18
632
19
861
33
690
13
783
14
809
55
909
66
745
13
628
16
746
54
761
55
677
58
X,
25
0.771
67
0.970
13
0.802
14
0.672
25
0.702
55
0.830
16
0.643
25
0.690
54
0.770
58
0.644
Zooplankton
X-5 Xg X7
13 18 52
0.618 0.732 0.739
53 53
0.725 0.716
16 55 55
0.705 0.666 0.645
18 1 54
0.652 0.695 0.303
54 62
0.669 0.768
55
0.811
58
0.641
62
Xg Xg X1Q
18 38
0.757 0.718
53
0.661
55 4 0
0.770 0.905 0.681
5 1
0.832 0.790
11 2
0.801 0.627
38
0.670
56 3 C
0.255 0.612 0.673
4 3
0.810 0.726
8
0.605
13
0.878
0.601 0.737
(continued)
368
-------
TABLE IV (continued)
Zcoplankton
Algae Xj X2 X3 X4 X5 X6
XT 7 38 53 18
0.722 -0.608 0.650
X1B 31 52 70
0.618 0.628 0.725
64 73
0.683 0.633
X]q 39 9 21
0.887 0.857 0.617
53 21 22
0.642 0.614 0.805
25
0.661
X9n 63 13 32 0
0.625 0.600 0.657 0.782
1
0.600
36
0.600
-36
0.710
38
0.894
0
0.731
34
0.760
35
0.867
36
0.710
37
0.618
70
0.821
71
0.822
25
0.675
51
0.658
53
0.735
66
0.606
X7
1
0.814
0
0.684
35
0.784
36
0.659
54
0.740
70
0.905
71
0.759
21
0.787
25
0.608
36
0.659
54
0.660
51
0.623
xe x9 xro
35
0.933
36
0.892
32
0.614
34
0.603
66
0.336
70
0.869
11
0.776
2
0.718
1 %0
0.695 0.629
369
-------
DISCUSSION
ROZHKOV: How are algorithms forecasted when there are few data? Have you
tried to establish an equation of the hydrodynamic type as a source equation
system to inject purely hydrodynamic research into your work?
IVAKHNENKO: The self-organization method allows us to organize very large
surpluses. It is necessary to introduce all known data into the computer.
These methods are combined self-organization methods. For example, to con-
struct an economics model of England, we take a determined model with 21
variables. The computer reviews all possible equation variances. The equa-
tion form is selected as a partial derivatives equation. There are no
unusual terms in these equations.
MORTIMER: Did you allow for absence of motion under the ice?
IVAKHNENKO: The presence of ice does not mean that we fully avoided flow,
turbulence, and so forth. In March the ice is still thick on Lake Baikal, and
it is possible to freely walk on it. To some degree, this decreases the
difficulties which each of us knows.
LEBEDEV: Does your approach eliminate inaccuracies in the regression models?
I am referring to the following: 1) the necessity for long series of obser-
vations; 2) the impossibility of transferring models from one basin to
another; 3) the regression dependency that is destroyed with the appearance of
extremely high values for specific pollution indices. Why was a three-dimen-
sional model used for calculating the chemical oxygen demand and a one-dimen-
sional model used for calculating the distribution of suspended substances?
IVAKHNENKO: Our approach eliminates the specified inaccuracies in classic
regression models. Long series obtained by using a multiseries model are not
necessary. For example, five observation points are sufficient to partially
describe a determined member from four coefficients. A complete regression
equation can have a million members, constructed along five points. The
authenticity of the model is proportional to the square root of the number of
points. Twenty points are better than five. But five points are that
minimum that allows us to describe the dependency.
It is possible to construct models with 1,000 incoming variables. But
man cannot devise equations with a large number of delayed elements. The
computer selects all the partial polynomials and finds the optimal density
equation. I think it is possible to discover principles according to the non-
displacement criterion and to transfer the data from one water body to
another.
370
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THOMANN: The statistical analysis described in the report answers many
questions such as how to obtain the maximum quantity of information from
field data.
MATVEYEV: Allow me to make several comments on Academician Ivakhnenko's
report, since I am a representative of an interested group of specialists
working on the Baikal problem. The Hydrochemical Institute, which I represent,
is an organization which has been monitoring the condition of the lake for
eight years. I agree with Dr. Labzovskiy, it is possible to name around 15
components that could fail in this model. But three components, characteris-
tic for discharges from the pulp and paper industry, were chosen. At the
time we were required to forecast the possible influence of man's activities
on the water ecosystem of Lake Baikal, this question was an entirely new
concern for us both from the standpoint of complexity and components
examined. We faced the problem of choosing a model that could forecast with a
comparatively small series of observations. The first difficulty which we
encountered was to overcome the specialists' opposition to models which
required submission of data on frequent observations both .in time and space.
The second difficulty was the correct understanding and choice of coefficients
for equations for the models selected. The continuous dialogue which we
carried out with specialists who used different models turned out to be very
useful.
The Cybernetics Institute's model allowed us, on the basis of field data
gathered for three to five years, to establish a forecast equation for three
parameters (components of the environment). A verification by one of the
parameters of the forecast showed a similarity to the forecast for one time
step; that is, for one year, the divergence was about 9%. From contacts with
specialists from the Cybernetics Institute, I have formed an opinion that
the interpretation of several coefficients (i.e., the identification of the
free term in equations with the source function) is still possible. If so,
the importance of this work will grow immeasurably.
My impressions about the symposium follow: the main impression is that
the problems of the Soviet and American scientists working on the ecosystem
model are practically the same.
371
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MANAGEMENT DECISIONS FOR LAKE SYSTEMS BASED ON A SURVEY OF
TROPHIC STATUS, LIMITING NUTRIENTS, AND NUTRIENT LOADINGS
A.F. Bartsch and Jack H. Gakstatter
INTRODUCTION
The National Eutrophication Survey was started by the U.S. Environmental
Protection Agency (EPA) in early 1972. Its broad objectives are to identify
lakes and reservoirs which receive effluents from municipal sewage treatment
facilities and to determine the significance of these point-source nutrient
inputs through field investigation. As the program progresses, these objec-
tives were modified by new legislation to include assessment of relationships
between drainage area characteristics and nutrient runoff and to develop
water quality criteria for nutrients. A number of lakes, not impacted by
municipal wastes, were also included in the survey.
The 812 water bodies selected for the continuing study are distributed
across the contiguous 48 states as shown in Figure 1.
Information sought for each lake was needed to make management decisions
for control of point and non-point sources of phosphorus and other nutrients.
An annual nutrient budget was first estimated for each water body with dif-
ferentiating inputs originating from point and non-point sources. Next, the
existing trophic condition was evaluated by sampling, and, finally, an algal
assay was performed to determine whether phosphorus, nitrogen, or other ele-
ments limited the productivity of the system.
Because a large geographic area was covered in the short period of four
years, UH-1H Bell helicopters were used in the sampling program. They were
equipped with pontoons so they could land on lakes. They also carried both
automated and manually operated water sampling gear to make the appropriate
water quality measurements. The sampling teams of limnologists and techni-
cians were supplemented by a mobile, wet analytical laboratory, chemists,
electronic specialists, and other staff involved with helicopter maintenance
or program coordination. The total staff in the field usually ranged from
12-14 people.
A central headquarters at an airport served as a base for sampling all
lakes within a 100-mile radius. When all of the water bodies within the area
had been sampled, the support staff moved to a new central location and began
sampling on a different set of lakes. In this manner, 150-250 lakes were
sampled three times each year, and the survey was completed for all of the
812 lakes in four years.
372
-------
U>
^J
00
NATIONAL EUTROPHICATION SURVEY
NUMBER OF LAKES a YEAR SAMPLED
1975-152
1973-250
GRAND TOTAL- 812
Figure 1. Number of lakes and reservoirs sampled in each state and year of sampling by the National
Eutrophication Survey.
-------
The quality characteristics measured in each lake are listed in Table I.
They are the usual.ones-needed to assess trophic condition and to provide
additional information needed to characterize each lake. Coincidently, sam-
ples were collected monthly from the significant tributaries and outlet of
each lake. About 4,200 stream sites were sampled nationwide. The tributary
sampling was done in each state by National Guard volunteers who received
on-site training by EPA to assure acceptable procedures. Immediately after
collection, the stream samples were preserved with mercuric chloride and were
shipped to the Corvallis Environmental Research Laboratory to be analyzed for
nitrogen and phosphorus.
The U.S. Geological Survey provided flow estimates for each of the sam-
pled streams. These data were used in conjunction with concentration values
to derive nutrient-loading .estimates.
The respective state water pollution control agencies arranged for the
monthly collection of effluent samples from municipal sewage treatment plants
to determine the nutrients they contribute to the surveyed lakes. This sam-
pling of .about 1,000 municipal sewage treatment plants usually was done by
the plant operators.
The field aspect of the survey is completed and data analysis is in
progress. In the remainder of the paper, I will share some of our findings
and those of other investigators as they relate to lake management for
eutrophication control.
TABLE I. WATER QUALITY CHARACTERISTICS MEASURED BY THE NATIONAL EUTROPHICA-
TION SURVEY IN LAKES AND RESERVOIRS
Physical-Chemical
Biological
Alkalinity
Conductivity*
Dissolved oxygen*
pH*
Phosphorus:
Ortho
Total
Nitrogen:
Ammonia
Kjeldahl
Nitrate
Secchi depth
Temperature*
Algal assay
Algal count and
identification
Chlorophyll a.
* Determined on-site by electronic probes.
Trophic Conditions—Phosphorus Concentration Relationships
Most of the lakes and reservoirs studied during the first year of the
survey were eutrophic; only about 20% of them were either oligotrophic or
mesotrophic. This was to be expected because the majority selected for study
are enriched by effluents from waste treatment facilities. The trophic con-
ditions, therefore, are not typical of all lakes and reservoirs in the United
States.
374
-------
Determining the existing quality of each lake is a matter of both prac-
tical and scientific interest. From a control aspect, however, it is pro-
bably more significant and productive to slow the eutrophication process in
a high quality, but threatened lake, than to restore one in advanced dete-
rioration. As with many environmental endeavors, the practice of prevention
is far preferable to remedial efforts. Unfortunately, in this regard, strong
public and political pressures for corrective action are usually not imposed
until the symptoms of eutrophication become very obvious.
Data from lakes and reservoirs in the northeastern United States were
used to examine relationships between total phosphorus concentrations and
other aspects of water quality which are indicative of the trophic state.
Such relationships, once derived, are useful in establishing water quality
criteria for phosphorus as well as in divulging more precisely the character-
istics of different trophic conditions. Figure 2, for example, shows the
relationship between chlorophyll a. and total phosphorus concentrations for
a number of phosphorus-limited lakes. The symbols differentiate oligotrophic,
mesotrophic, and eutrophic lakes on the basis of nutrient levels, chlorophyll
concentrations, Secchi disc depths, and the degree of dissolved oxygen de-
pletion during periods of thermal stratification. Admittedly, these three
trophic terms are subjective and difficult to define in absolute terms.
Nevertheless, they are widely used by the limnological community and are
generally understood to designate three approximations of water quality.
Several observations can be made concerning the relationships in this
figure. Lakes classified as oligotrophic did not exceed a median total phos-
phorus concentration of 10 yg/£ or a mean chlorophyll ji concentration of
7 yg/£. Median total phosphorus concentrations in the mesotrophic lakes did
not exceed 18 yg/£ and chlorophyll ji did not exceed 12 yg/£. Most eutrophic
lakes had median total phosphorus concentrations greater than 18 yg/£.
These lake data also give a reasonable fit with chlorophyll a_ ranges
suggested by others for various trophic levels (Sakamoto, 1966; National
Academy of Sciences, 1972; Dobson et al., 1974):
Chlorophyll a (yg/£)
Lake
Trophic Level Sakamoto Academy Dobson Survey
Oligotrophic 0.3-2.5 0-4 0-4.3 0-7
Mesotrophic 1-15 4-10 4.3-8.8 7-12
Eutrophic 5-140 >10 . >8.8 >12
If a working guideline for phosphorus concentration control was estab-
lished empirically on the basis of the data presented in Figure 2, the fol-
lowing would be concluded:
Trophic Level Chlorophyll a (yg/£) Total Phosphorus (yg/£)
Oligotrophic 0-7 0-10,
Mesotrophic 7-12 10-20
Eutrophic >12 >20
375
-------
100.0
cc
LJ
o
o
o
_J ^
> ^
X
CL
O
o:
o
X
o
LJ
|0.0
1.0
1 1—1—I I I I I I 1 1—I—I—I I I I
Log|0 ChlA =0.807 Log|QTP - 0.194
1 ix I I I
Oligotrophic
Mesotrophic
Eutrophic
i I i i i i i i i
1.0 10.0 100.0 1000.0
MEDIAN TOTAL PHOSPHORUS CONCENTRATION (//g/l)
Figure 2. The relationship between chlorophyll A and total phosphorus concentrations in northeastern
U.S. lakes and reservoirs included in the survey.
-------
The data presented in Figure 2 also agree with phosphorus guidelines
suggested by other investigators. Sawyer (1947) suggested nearly 30 years
ago that inorganic phosphorus concentrations at spring overturn should not
exceed 10 yg/£ if algal blooms were to be avoided. Translated into terms of
total phosphorus, Sawyer's values would probably be 15 to 20 yg/£. Bachmann
and Jones (1974) recently concluded that total phosphorus concentrations
would have to be reduced below 20 yg/& before significant increase in water
transparency could be achieved as a result of reduced standing crops of algae.
They also presented a phosphorus-chlorophyll relationship which demonstrated
that 20 yg/£ total phosphorus corresponded to a summer chlorophyll a. con-
centration at about 7 yg/£.
Vollenweider (1968) and Dillon (1975) also concluded that 20 yg/£ total
phosphorus at spring overturn approximates the transition between mesotrophic
and eutrophic waters.
In summary, it becomes obvious that there is substantial evidence to
indicate that a concentration of 20 to 30 yg/£ total phosphorus in lakes
should not be exceeded at any season if eutrophic conditions and attendant
nuisance algal blooms are to be avoided. The key to success, however, is
capability to determine the degree of phosphorus control necessary within the
drainage area to avoid exceeding critical nutrient levels in the lake.
Nutrient Loading—Lake Response__Re_laj:_ionships
The relationship between the magnitude of nutrient input to a lake or
reservoir and its trophic condition has been known for many years. Specific
aspects of this relationship have been probed by several investigators during
the past seven years. Although results do not answer all questions, excellent
progress has been made in providing improved understanding of nutrient dynam-
ics in the eutrophication process.
Vollenweider (1968) appears to have been the first to examine the rela-
tionship between loading and lake response using data from a number of Central
and Northern European and North American lakes. Initially, he established an
empirical relationship between total phosphorus loading (g/m2/yr), trophic
condition expressed as oligotrophic, mesotrophic, or eutrophic, and lake mean
depth. Later (1974), he modified the original relationship to account for
the fact that lakes with equal mean depths might have different flushing times
and, therefore, different nutrient assimilation capacities. His revised model
plotted the annual total phosphorus load against the quotient of mean depth
in meters divided by the mean hydraulic retention time in years. Vollen-
weider 's work in this area was extremely important, not only because he was
the first to investigate the loading-response relationship, but because others
became interested in his approach.
Stimulated by Vollenweider's earlier work and by the fact that hydraulic
retention time was not adequately accounted for in the model, Dillon (1975)
recently published his version of a phosphorus-loading-concentration rela-
tionship. He related a factor which included total phosphorus loading, the
phosphorus retention coefficient, and the hydraulic flushing time to lake
mean depth.
377
-------
Independently in 1974, Larsen and Mercier of our laboratory also began
an analysis of the Vollenweider model and some of its inherent limitations.
Their model (in press) relates the theoretical average incoming total phos-
phorus concentrations to the phosphorus retention coefficients of lakes.
The average incoming total phosphorus concentration is defined as the total
annual phosphorus load divided,by the total hydraulic inflow which is also
equivalent to the factor:
L
p.-z- .
where, L = annual total pho.sphorus are?! loading (g/m2/yr)
p = hydraulic flushing time (exchanges/yr)
~z - mean depth (m)
Therefore the Larsen and Mercier model incorporates the same parameters
as the Dillon model. Both of these models, in effect, predict the total
phosphorus concentration of a lake based on its annual phosphorus loading,
mean depth, hydraulic retention time, and phosphorus retention coefficient.
Figures 3, 4, and 5 compare the application of Vollenweider's, Dillon's
and the Larsen-Mercier models to actual data from 23 lakes studied by the
National Eutrophication Survey in the eastern United States. The lakes re-
present a range of trophic conditions (6 oligotrophic, 9 mesotrophic, 8
eutrophic), mean depths (2.3 to 30.8 meters), mean hydraulic retention times
(0.01 to 15.6 years), and total phosphorus loadings (0.03 to 16.94 g/m2/yr).
Figure 3 shows these water bodies plotted against the Vollenweider
model. The y-axis represents the annual total phosphorus loading rate
(g/m2/yr) and the x-axis represents the mean depth in meters divided by the
mean hydraulic retention time in years. All points above the upper curved
line are in the eutrophic category, and all points below the lower curved
line are in the oligotrophic loading category. Between the two lines is the
mesotrophic loading range.
For this set of lakes, the Vollenweider model does not fit very well
with observed loadings and trophic conditions. Half of the eutrophic lakes
lie within the eutrophic zone of the relationship, while three are in the
mesotrophic area and one is in the oligotrophic area.
Only two of the mesotrophic lakes are within the mesotrophic category
while the remaining seven are in the oligotrophic category according to the
model. All of the oligotrophic water bodies fall into the oligotrophic
category as they should.
At least for this group of lakes, the "permissible loading" which corre-
sponds to the transition from oligotrophic to mesotrophic conditions would
seem to be a better cutoff point on the Vollenweider relationship than the
"dangerous loading" which represents the transition from mesotrophic to
eutrophic conditions.
The same lakes are plotted against the Dillon model in Figure 4 in
which the eutrophic zone corresponds to the area above the upper line, the
mesotrophic zone to the area between the two diagonal lines and the oligo-
378
-------
10.0-
-------
1.0
CO
CO
o
E
\
o»
O.I
0.01
I I/II I I
'EUTROPHIC
0 27A8
D 2750
D 2618
D3639
D43I5
D27CI
D45I2/ A23I3
O 2695
02309
03303
A 2306
AI3I8
02311
OLIGOTROPHIC'
O Oligotrophic Lakes
A Mesotrophic Lakes
D Eutrophic Lakes
j I
10.0
MEAN DEPTH ( METERS)
100.0
Figure 4. The Dillon model applied to a number of eastern U.S. lakes and reservoirs sampled by the
survey.
-------
o
LJ
O
g 100.0
O
I
O en
10.0
O
o
o
<
LJ
.0
D 2750
"EUTROPHIC"
D 27A8
*27B4A|3|6
"MESOTROPHIC"
.A ""
2309
O 2314
D26I8
D3639
D 4515
°45I2 3617
D27CI A 2696
O 2695
A J3I8
2306^02311 ''OLIGOTROPHIC"
O Oligotrophic Lakes
A Mesotrophic Lakes
D Eutrophic Lakes
0.0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PHOSPHORUS RETENTION COEFFICIENT ( REXp)
1.0
Figure 5. The Larsen and Mercier model applied to a number of eastern U.S. lakes and reservoirs sampled
by the survey.
-------
trophic zone to the area below the lower diagonal line. The upper and lower
lines correspond to in-lake total phosphorus concentrations of 20 yg/£ and
10 yg/£ which roughly correspond to the transition between mesotrophic-
eutrophic and oligotrophic-mesotrophic conditions. In general, the model
well describes the conditions observed by the survey in the 23 lakes. All
of the eutrophic lakes fall within the eutrophic area of the graph, while
only two of the mesotrophic lakes are out of place. One oligotrophic lake
plots within the mesotrophic category, while the rest are where they should
be.
The Larsen-Mercier relationship is illustrated by Figure 5. Although
their approach is slightly different from Dillon, the net result is nearly
identical. With the exception of one oligotrophic lake and two eutrophic
lakes, the conditions observed by the survey fit the model well. The upper
and lower curved lines in Figure 5 correspond to theoretical in-lake total
phosphorus concentrations of 20 yg/£ and 10 yg/£, respectively, as in the
Dillon relationship.
Since both the latter two models predict in-lake concentrations of total
phosphorus, the vertical distance from an observed point, representing a lake,
to one of the transitional lines is at least a semi-quantitative measure of
the degree of oligotrophy or eutrophy.
The vertical distance from a given point to a transitional line in the
Vollenweider relationship has less meaning in terms of the degree of oligo-
trophy or eutrophy because the model does not directly relate total phos-
phorus loading to in-lake phosphorus concentrations.
In summary, the models developed by Dillon and Larsen-Mercier, which
relate total phosphorus loads to lake phosphorus concentrations, should prove
to be very useful lake management tools. The Vollenweider model, at this
time, is less precise because it considers only total phosphorus loading
without regard to in-lake processes that can reduce the effective phosphorus
concentration.
Nutrient Sources
It is obvious from the foregoing that among the first questions of con-
cern when dealing with eutrophication management are magnitude and sources of
the annual nutrient input. In general, sources can be either point or non-
point in character. Point sources include discrete municipal and industrial
waste discharges. Non-point sources include a number of categories, but
usually represent runoff within the drainage area which either enters the lake
through tributaries or as diffuse runoff along the lake shore. Another non-
point input is the nutrient load from precipitation or dry fall-out directly
on the lake surface. In areas where cultural influences in the drainage area
are minimal, precipitation and dust fall, while of limited actual amount,
may represent a significant percentage of the annual nutrient budget (Vollen-
weider and Dillon, 1974).
Municipal waste treatment plant discharges represent a concentrated
point source of phosphorus. It has been estimated that about 1.6 kilograms
382
-------
pf phosphorus per .person per year are discharged in raw wastewater (Bartsch,
1972). Even where raw wastes, are treated before discharge, only from 20 -
30% of the phosphorus usually is removed by the .treatment process. As a re-
sult, unless a treatment plant was specifically designed to remove phosphorus,
most of it moves through the treatment process and is discharged in the
effluent.
Effluent data, collected by the survey for 505 municipal waste treatment
facilities, gave 1.0 kilogram per year as the median per capita contribution
of total phosphorus remaining in treated wastes. The median total phosphorus
concentration in.., the.;same effluents was 5.9 mg/Jl. Corresponding figures for
total nitrogen were 2.7 kilograms per.capita and for median concentration,
15.0 mg/£ (EPA, 1974). A wide range of per capita nutrient loads between
plants was found to be common, which suggests that actual data, rather than
estimates, should be used when important control decisions are involved. The
fluctuations are believed to be related primarily to industrial contributions
of phosphorus to the wastewater. However, localized domestic practices which
either increase or decrease phosphorus contributions to wastewater should not
be discounted.
Another significant .point to be considered in dealing with municipal
wastewater is that effluents contain an abundance of phosphorus in relation
to nitrogen. In general, algae require a ratio of inorganic nitrogen to
orthophosphorus-P, by weight, in the range of 11:1 to 15:1 for growth (Miller
et al., 1973; Porcella et al., 1974). In natural waters, unimpacted by man,
the N:P ratio is usually higher than 15:1 which suggests a relative abundance
of nitrogen and algal growth limited by the scarcity of phosphorus. In
treated municipal waste, however, the average ratio of N:P is about 2.5:1.
This then offers an abundance of phosphorus supplied in an available form
since more than half of the P_ is present ^as dissolved orthophosphate-P (EPA,
1974). In municipal waste effluents, detergents :account for about 50% of the
total phosphorus in areas where phosphate-based detergents .are still used.
Although detergent phosphorus certainly is not the sole.cause of accelerated
eutrophication, substantial reductions in phosphorus loads to some lakes
could be achieved by the removal of phosphorus from detergents.
Figure 6 depicts the significance of phosphorus from municipal waste
sources to the total annual phosphorus load for 132 surveyed lakes and reser-
voirs in the eastern United States. The frequency histogram shows that a
number ,of water bodies would have their annual phosphorus loads reduced by
80-100% if the input originating from municipal wastewater were eliminated.
The reduction in lake loadings would be half as much if only the detergent
fraction of the phosphorus were eliminated. For .all-of the lakes, the mean
phosphorus contribution from domestic wastes was 40% of the total load.
Too often, studies designed to delineate lake problems as a step toward
management ignore drainage area characteristics and land use. Such charac-
teristics, including size, geology, climate, land use, and population density,
largely determine the quantity of nutrients that enter a lake annually.
It would be extremely enlightening if one could go back,to the early
17th Century in the United States and make a lake survey similar to the
383
-------
LO
00
-P-
CO
UJ
_l
1 1
L^B
o
a:
UJ
m
ID
2
45
40
35
30
25
20
15
10
5
0-20 20-40 40-60 60-80 80-100 ^ Existing Condition
0-10 10-20 20-30 30-40 40-50 ^ With Detergent P Removed
PERCENT OF TOTAL PHOSPHORUS LOAD FROM MUNICIPAL WASTES IN
NORTHEASTERN U.S. LAKES AND RESERVOIRS
Figure 6. A frequency histogram representing the percent of total annual phosphorus load attributable
to municipal wastes for a number of northeastern lakes and reservoirs included in the survey.
-------
present one. Many questions concerning man's impact on lakes could then be
easily and accurately answered. It is likely the 17th Century lakes pre-
sented a diversity of trophic conditions simply because then, as now, a di-
versity of geological areas in the United States provided a range of base-
line nutrient levels. On such base levels are superimposed the secondary
impact that comes from land use patterns within the area. Mean export levels
for total phosphorus have been determined for southern Ontario watersheds of
different land use types and geological classifications (Dillon and Kirchner,
1974). As shown by Table II, in forested drainage areas of basically igneous
origin, only 4.7 kg/km of total phosphorus are exported per year. However,
this value increases to 10.2 kg/km2 when forest and pasture uses are combined.
In sedimentary watersheds, total phosphorus exports range from 11.7 kg/km2
for forest areas to 46 kg/km2 for agricultural areas, while urban land use
contributed an average of 1,050 kg/km/year.
In these cases, geological differences, such as igneous vs. sedimentary
origin, accounted for a 2.5-fold difference in phosphorus export while land
use patterns within one geological type (sedimentary) accounted for a 3.9-
fold difference (between forested and agricultural land use types).
TABLE II. MEAN VALUES FOR EXPORT OF TOTAL PHOSPHORUS FROM 43 WATERSHEDS IN
SOUTHERN ONTARIO (DILLON AND KIRCHNER,_JL974)^
Results in kg/km2/year
Geological classification
Land use Igneous Sedimentary
Forest 4.7 11.7
Forest and pasture 10.2 23.3
Agriculture 46.0
Urban 1050.0
As part of the survey, a study is underway to relate drainage area char-
acteristics to nutrient concentrations and loads in streams as a step toward
estimating annual lake and reservoir inputs. At this point, the survey
evaluations have mainly examined land use rather than geological aspects,
although the latter are being considered in future evaluations. Preliminary
data for 473 drainage areas on the eastern United States support the view
that forested areas contribute the lowest amounts of nutrients to surface
waters and agricultural lands contribute the highest. However, some other
interesting facts, as well, are indicated in Figure 7 which relates the pre-
dominating land use types in the watershed to mean total phosphorus and total
nitrogen concentrations in the streams draining the area. The mean total
phosphorus concentrations range from 0.014 mg/£ in forested drainages to
0.135 mg/£ in agricultural areas, nearly a 10-fold difference.
Similar examination shows that mean total nitrogen concentrations range
from 0.850 mg/£ in forested drainages to 4.17 mg/£ in agricultural areas, a
5-fold difference in this case.
385
-------
LO
00
NUMBER
OF SUBS
MEAN TOTAL PHOSPHORUS CONCENTRATIONS
vs
LAND USE
NUMBER
OF SUBS
S3
170
52
11
96
91
FOREST
olhfi >,.••. .••'.•." y . ••••;' J] i.sij
' • " **••*•?• ?' -i ^- "" ^ 'i,'!C, ' rf ,'. --•'.*'\^'- ^ > V '.v ' "-, " "••'' ""' . \J 4-170
1 I 1 1 1 1 1 <
1.0
2.0
Mil LIGRAMS PER LITER
3.0
4.0
Figure 1. Mean total phosphorus and total nitrogen concentrations in streams draining different land
use categories.
-------
If stream flow factors are applied to the above values to calculate
annual nutrient loads carried by streams draining the different land use
types, the land use impact, while still important, is not as great as con-
centrations suggest. Figure 8 indicates that 3.7 times as much phosphorus
is exported from agricultural watersheds as from forested watersheds while
the difference in total nitrogen export is, 2.2-fold.
It is evident from the above discussion that both geology and land use
of lake drainage areas can play a significant role in determining the quanti-
ty of nutrients which are input to a lake. While obviously the basic geology
cannot be easily manipulated for management purposes, land use planning
efforts must consider the nutrient aspect.
Limited Nutrients
The quest for effective procedures to curb eutrophication has been
given increased impetus in many countries. Much has been written during the
past few years about the roles of phosphorus, nitrogen, carbon, and trace
elements in enhancing or limiting the productivity of aquatic systems. It is
apparent that under the right set of circumstances, any of the essential
elements might be growth limiting. However, the preponderance of evidence
suggest that, in the majority of cases, phosphoruis is the element with which
lake managers should generally be concerned.
The role of phosphorus in eutrophication and its significance to lake
management has been discussed recently by Bartsch (1972). Experimental work
by the Fisheries Research Board of Canada. (Schindler et al., 1973; Schindler
and Fee, 1974), clearly supports previous theories that phosphorus control is
of singular importance in slowing lake eutrophication-. The U.S. Environmental
Protection Agency's current project at Shagawa Lake (Malueg et al., 1973) in
Ely, Minnesota, is successfully demonstrating on full scale the capability to
reserve the eutrophication process through phosphorus removal from municipal
sewage.
Algal assays were performed for each of the surveyed lakes and reservoirs
to determine if productivity was limited by phosphorus or nitrogen exhaustion
at the time of sampling. The assays were done according to a standard pro-
cedure (EPA, 1971) using a Chlorophyte, Selenastrum capricornutum Printz, as
the test organism. Assays have now been completed for 623 lakes! and the
results are summarized in Table III.
That assay results such as these should be evaluated with care has been
discussed by Schindler and Lean (1974) . Where human impact on both point
and non-point sources of nutrients in the watershed is inconsequential,
primary production in surface waters generally is controlled by the supply of
phosphorus, that is—phosphorus then is the limiting nutrient. Man's pertur-
bations in the watershed, such as establishing waste treatment plants or
pursuing agricultural activities, frequently accelerate the influx of phos-
phorus relative to other nutrients. Thus a lake, once phosphorus-limited in
its undisturbed state, may become eutrophic and nitrogen-limited because
abundant phosphorus supplied by man's activities far exceed the algal needs.
Because the algal ass'ays generally are performed after such changes have
387
-------
NUMBER
OF SUBS
53 FOREST
other types negligtbte
170 MOSTLY FOREST
52 MIXED
11 MOSTLY URBAN
other types present
96 MOSTLY AGRIC.
other types presanl
91 AGRICULTURE
other ryp«i negligible
8.3
TOTAL PHOSPHORUS EXPORT
vs
LAND USE
DATA ON 473 SUBDRAINAGE AREAS IN
EASTERN UNITED STATES
17.4
"1 18.4
22.7
J 3
-------
occurred, they show that a number of lakes have not reached a nitrogen-
limited state. It is important to note that this does not impair the possi-
bility to shift back, through appropriate phosphorus control mechanisms, to a
mode in which phosphorus again assumes a limiting nutrient role.
The algal assays revealed that phosphorus was the limiting nutrient in
about 67% of the 623 lakes examined, as shown in Table III. This is remark-
able when one considers that the majority of the lakes were enriched by munic-
ipal effluents. Included in the phosphorus-limited category were lakes in
which both phosphorus and nitrogen were present in the optimal ratio for
algal growth. There were 28 lakes of this type. These lakes would benefit
immediately from phosphorus reductions.
TABLE III. SUMMARY OF ALGAL ASSAY RESULTS FOR LAKES AND RESERVOIRS IN THE 37
STATES EAST OF THE ROCKY MOUNTAINS
Limiting nutrient Number of lakes % of all lakes
Phosphorus
Nitrogen
Other
417
186
20
67
30
3
Total 623 100%
About 30% of the water bodies were found to be nitrogen-limited. Many
of them _could be_expected tq_become phosphorus-limited following sufficient
reduction of phosphorus loading. A small percentage (3%) of surveyed water
bodies were limited by an element other than phosphorus or nitrogen, such as
cobalt, iron, molybdenum, or manganese.
Morphometric Considerations
Data on lake morphometry also are essential to sound management deci-
sions. A deep lake with steep bottom contours is less likely to develop
problems with aquatic weeds than a shallow lake where light penetrates through
a large percentage of the water volume. For various reasons deep lakes can
also absorb greater nutrient loads without serious symptoms. In shallow
lakes, there is greater opportunity for nutrient-rich sediments to recycle
into the lake biomass through wind action whereas in deeper lakes the nutri-
ents may be removed from the cycle through the sedimentation process. Given
two lakes with equal drainage areas and surface areas but different mean
depths, the deeper one will have the longer hydraulic flushing time which
will permit a greater percentage of the nutrient load to be removed by sedi-
mentation. It also will have a greater volume and therefore a greater
diluting effect on the incoming nutrient load.
The ratio of drainage area to lake surface area should also be consid-
ered. A high ratio is generally accompanied by a proportionately higher
nutrient and hydraulic loading and a proportionately lower hydraulic reten-
tion time. For lakes lacking good morphometric data, drainage areas to sur-
face area ratios from better known lakes in the area can be useful in approx-
imating values. For example, Figure 9 relates drainage area-surface area
389
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10,000
to
^o
o
CO
9 1,000
LJ
L±J
I-
LU
o:
o
a
>-
x
<
LJ
100
10
Loglo HRT=-I.I77 Log,0 -^ + 4.077
I i i i I I n
i i i i i I 11
I 10 100 1,000 10,000
LAKE DRAINAGE AREA / LAKE SURFACE AREA
Figure 9. The relationship between mean hydraulic retention time and the drainage to lake surface area.
ratio for a number of northern U.S. lakes and reservoirs.
-------
ratio to mean hydraulic retention time for several northern U.S. lakes and
reservoirs and could be used to approximate the hydraulic retention time of
a. water body which had not been bathymetrically mapped.
: The types of lake restoration techniques which might be applied to a
Take with serious eutrophicatioh problems would depend upon lake morphometry
to a great extent. For example, one would not choose to employ aeration
techniques to a lake which is shallow and well-mixed; whereas in a deep ther-
mally stratified lake, aeration might be considered a potentially useful tool
in a restoration program.
Another consideration is the irregularity of the lake shoreline in the
form of embayments or peninsulas. Some lakes almost resemble a series of
smaller connected lakes rather than a single large one. A good example is>
Lake-Minnetonka, Minnesota, U.S.A., which is so irregular in shape that for
study purposes it was divided into 15 nearly separate basins (Megard, 1972).
Because most models for predicting lake response to nutrients assume that the
system is relatively uniform and completely mixed, shoreline irregularity
becomes a factor worthy of serious reflection. Unfortunately, such reflec-
tion is often impeded because adequate morphometric and bathymetric informa-
tion is lacking.
Summary
Several factors have been discussed as important considerations in
managing a lake to a desired trophic level. ' They are summarized below.
1. Total phosphorus concentrations in a lake ranging from 20-30 yg/£
or greater are usually associated with eutrophic conditions. These phos-
phorus concentrations correspond to chlorophyll _a concentrations of 9-12 yg/£
or more.
2. To determine the extent to which phosphorus must be controlled to
obtain the desired trophic condition, one must be able to relate total in-
coming phosphorus to the lake response. Three general models from the liter-
ature were presented which relate total phosphorus loading to lake response.
3. Ratios of inorganic nitrogen to dissolved orthophosphorus-P in sur-
face waters are usually greater than 15 to 1 unless impacted by point-source
discharges, particularly municipal wastes. The average N to P ratio of munic-
ipal waste is 2.5 to 1, thus an abundance of phosphorus relative to nitrogen
is supplied.
4. The National Eutrophication Survey results demonstrate that a number
of lakes and reservoirs in the United States receive a substantial part of
their total phosphorus load from municipal wastes.
5. Phosphorus is the element with which lake managers will usually be
concerned in controlling eutrophication. A survey of 623 lakes in the United
States indicated that phosphorus was the .limiting nutrient for 67% of these
and nitrogen for 30%. Many of the nitrogen-limited lakes could be shifted to
a phosphorus-limited mode through appropriate phosphorus control measures.
6. Drainage area geology and land use are important factors that in-
fluence the total phosphorus load to a lake. Phosphorus concentrations in
tributaries draining agricultural land in the northeastern United States
averaged ten times greater than in those draining forested land, while nitro-
gen concentrations were five times gi'eater.
391
-------
7. Before lake restoration is attempted, the morphometry of the lake
and its drainage area characteristics should be examined in detail because
these factors significantly influence lake trophic condition.
392
-------
REFERENCES
Bachman, R.W. and J.R. Jones. 1974. Phosphorus inputs and algal blooms in
lakes. Iowa State J. of Res. 49:155-160.
Bartsch, A.F. 1972. Role of phosphorus in eutrophication. U.S. Env. Prot.
Agency, Ecological Res. Series No. EPA-R3-72-001. pp. 45.
Dillon, P.J. 1975. The phosphorus budget of Cameron Lake, Ontario: the
importance of flushing rate to the degree of eutrophy of lakes. Limnol.
Oceanog. 20:28-39.
Dillon, P.J. and W.B. Kirchner. 1974. The effects of geology and land use
on the export of phosphorus from watersheds. Water Res. 9:135-148.
Dobson, H.F., H.M. Gilbertson, and P.G. Sly. 1974. A summary and comparison
of nutrients and related water quality in Lakes Erie, Ontario, Huron, and
Superior. J. Fish. Res. Bd. Can. 31:731-738.
Larsen, D.P. and H.T. Mercier. 1975. Lake phosphorus loading graphs: an
alternative. National Eutrophication Survey Working Paper No. 174 (In
press).
Malueg, K.W., R.M. Brice, D.W. Schultz, and D.P- Larsen. 1973. The Shagawa
Lake project: lake restoration by nutrient removal from wastewater
effluent. U.S. Env. Prot. Agency, Ecological Research Series No. EPA-R3-
73-026. pp. 49.
Megard, R.O. 1972. Photosynthesis and phosphorus in Lake Minnetonka. Limnol.
Oceanogr. 17:68-87-
Miller, W.E., J.C. Greene, T. Shiroyama, and E. Merwin. 1975. The use of
algal assays to determine effects of waste discharges in the Spokane
River system. Proc. Biostim. Workshop, Corvallis, Oregon, October 16-18,
1973.
National Academy of Science and National Academy of Engineering. 1972. Water
quality criteria: a report of the committee on water quality criteria.
Washington, D.C.
Procella, D.B., et al. 1974. Comprehensive management of phosphorus water
pollution. U.S. Env. Prot. Agency, No. EPA-600/5-74-010. pp. 411.
Sakamoto, M. 1966. Primary production by phytoplankton community in some
Japanese lakes and its dependence on lake depth. Arch. Hydrobiol. 62:
1-28.
393
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Sawyer, C.N. 1947- Fertilization of lakes by agriculture and urban drainage.
N. England Water Works Assoc. 61:109-127.
Schindler, D.W. and D.R.S. Lean. 1974. Biological and chemical mechanisms
in eutrophication of freshwater lakes. Ann. N.Y. Acad. Sci. 250:129-135.
Sdrintile'rT U.W. and E.J. Fee. 1974. Experimental lakes area: whole-lakes
experiments in eutrophication. J. Fish.' Res. Bd. Can. 30:937-953.
Schindler, D.W., H. Kling, R.V. Schmidt, J. Prokopowich, V.E. Frost, R.A.
Reid, and M. Capel. 1973. Eutrophication of Lake 227 by addition of phos-
phorus and nitrate: the second, third, and fourth years of enrichment,
1970, 1971, and 1972. J. Fish. Res. Bd. Can. 30:1415-1440.
U.S. Environmental Protection Agency. 1974. Nitrogen and phosphorus in-
wastewater effluents. National Eutrophication Survey Working Paper No. 22
pp. 19.
U.S. Environmental Protection Agency. 1971. Algal assay procedure bottle
test. National Eutrophication Research Program, Corvallis, Oregon.
pp. 82.
Vollenweider, R.A. and P.J. Dillon. 1974. The application of the phosphorus
loading concept to eutrophication research. National Research Council
Canada No. 13690. pp. 42.
Vollenweider, R.A. 1968. The scientific basis of lake and stream eutrophica-
tion with particular reference to phosphorus and nitrogen as factors in
eutrophication. OECD, DAS/CSI/68-27. pp. 159.
394
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DISCUSSION
LEBEDEV: It was stated in the report that the element which limited algal
development was determined. I am interested in knowing how these determina-
tions were carried out and the percentage of cases in which insufficient
phosphorus was available for plankton growth?
BARTSCH: We can use many methods to determine the limiting factor. The
method we used was jointly developed by EPA, three universities, and private
industry. An algae serves as the control organism; water samples are taken
and treated by different methods. Of the 812 lakes examined by this method,
67% were limited by phosphorus, 30% by nitrogen, 3% by microorganisms. There
is a supplementary method for seawater using diatoms.
BOYCHUK: What do you mean by the "introduction of phosphorus?" Is it a
quantity at a given moment in time or the distribution of this quantity in
time? When studying the influences of phosphorus, do you consider the time
factor?
BARTSCH: We monitor eutrophication by introducing phosphorus. A large
phytoplankton bloom is a symptom of eutrophication. For growth, phytoplankton
requires a series of elements. If we take away any of these elements, growth
stops. One of the problems is slowing down or stopping the bloom by phos-
phorus retention; we do not introduce phosphorus on purpose.
LEBEDEV: What element limiting the plankton development in water bodies
could help determine the direction for modeling processes which take place in
water ecosystems? The question on the limiting action of one or another
element is not new, and, unfortunately, is complicated. Determination of a
deficiency in an element depends on the selection of organisms tested, the
length of sample incubation, and type of incubation container. It is
possible to say that water bodies under a strong eutrophication influence
suffer primarily a deficiency in nitrogen. At this time, a definitive
deficiency in phosphorus is observed in lakes where there has been no
anthropogenic influence. Our conclusions that nitrogen is the limiting
factor where there is strong anthropogenic influence support the results
obtained from the mathematical model for forecasting water quality in
reservoirs. In the Mozhayshiy reservoir, we obtained good results verifying
the limiting action of nitrogen. However, I do not insist on my conception of
the limiting action of nitrogen, and I do not negate Dr. Bartsch's conception
of the limiting action of phosphorus. The question is too uncertain to
construct a model now.
BARTSCH: Let me answer the question about the limiting element. I am not
surprised to hear that some lakes are limited by nitrogen, others by phos-
phorus, and a third group by carbon. From a practical point cf view, there
395
-------
is no difference as to what element limits a lake. If the lake is limited by
phosphorus, then it is our goal to retard lake eutrophication by means of
controlling phosphorus content.
TSYTSARIN: My question is related to the methods for evaluating non-point
sources of phosphorus. In Dr. Bartsch's paper, I did not discern a system
for collecting samples. During the last 5 to 6 years, our group in the
Moscow area labored over this question. We were successful in measuring the
flow of phosphorus for determined periods. When evaluating the phosphorus
loads, the greatest errors are possible in the spring when 70-80% of the
annual flow occurs and in the fall when 10-15% of the flow occurs. During
the low-flow period, it is more difficult to evaluate the phosphorus loads
because phosphorus transport takes place through the atmosphere. A very
high concentration of phosphorus (0.2-0.3 mg/£) is observed in all atmospheric
precipitations. The reason for this phenomenon is the terpene-type products
that enter the atmosphere during the degradation of vegetation.
BARTSCH: These data are extremely interesting. Sampling water in rivers to
determine the phosphorus loss from the drainage basin does not give us ac-
curate results. These values would be characteristic if the sampling day
coincides with the flow maximum. Other basins in the United States are
studied with a higher accuracy from the flow and chemical points of view.
Regarding the phosphorus in the atmosphere, we admit its existence,
but research has not been specifically carried out in this field. Other
reports have noted high concentrations of phosphorus in atmospheric
precipitations in the U.S. In Oregon large quantities of phosphorus are
found far from the ocean. The concept of phosphorus loss from surface
films is accepted.
396
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MATHEMATICAL MODELING STRATEGIES
APPLIED TO SAGINAW BAY, LAKE HURON
Victor Bierman, William Richardson,
and Tudor T. Davies
INTRODUCTION
The objective of this modeling research is to develop quantitative tools
which can be used to supplement intuition and scientific judgment in policy
decisions concerning water quality. This research is directed initially to-
ward water quality problems concerning the international waters of the North
American Great Lakes. These waters have a very broad range of uses, some of
which induce water quality problems and environment degradation. Use of these
waters is economically and socially significant to the area, and the conse-
quences of decisions aimed at preserving or even improving water quality af-
fect the well-being of the people in the drainage basin and beyond. Manage-
ment decisions on major changes in treatment technology and enforcement ac-
tions should be based on the results of simulations of the effects of these
changes on water quality and the ecosystem.
To apply modeling techniques to the problems concerning the Great Lakes,
it is essential to categorize the problem into a series of time and spr.ce
scales. This series ranges from the short-term transport (on the scale of
hours or days) of some hazardous material within a very localized area, to the
long-term effects (ranging from years to decades) of control or limitation
of nutrients or toxic materials within the drainage basin on the ecosystem of
individual lakes or of the whole system of lakes. Two papers in this sympo-
sium by Dr. Lick and Dr. Thomann represent both ends of the spectrum, the
short-time and small-space scales and the long-term responses of the whole
lake.
This paper will attempt to integrate these different approaches to prob-
lems of a small area of the Great Lakes which has suffered specific water
quality degradation. This study is part of an intensive effort by the U.S.
and Canada to study and report on the so-called upper International Great
Lakes which are considered to be largely unaffected by cultural degradation
but remain in an oligotrophic state in contrast to the lower Great Lakes. Al-
though the main lakes are considered to be largely unaffected, some of the
coastal areas and embayments do have significant problems. The Saginaw Bay
system is perhaps the most highly degraded.
This study attempts to apply transport models and algal growth modeling
concepts to Saginaw Bay to describe prevailing conditions in the bay. The
model developed is based on available data on the bay, and the modeling
397
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concepts will be verified by using an extensive set of field data collected
from a sampling design aimed specifically at this modeling program.
This modeling system will then be used to evaluate the response of the
system to a range of management alternatives for enhancing water quality.
SAGINAW BAY—BACKGROUND
Morphology and Geographical Setting
Saginaw Bay is a shallow SW extension of Lake Huron, which is a deep
oligotrophic lake. The bay, which has a surface area of 2,960 km2 and is 21-
42 km wide and approximately 82 km long, receives drainage from an area of
21,000 km2. The bay has two distinct parts, an inner or southern zone which
is very shallow with a mean depth of 4.6 m (maximum depth 14.0 m) and an
outer bay with a mean depth of 14.6 m and a maximust depth--of- 40.5 m. The
inner bay constitutes 30% of the total baywater volume and is dominated by
extensive shallows.
Hydrology
The Saginaw River which has an average flow of 3850 CFS is the major
tributary draining 80% of the drainage basin of the bay. The mean annual pre-
cipitation of 189.5 cms is evenly distributed throughout the year. Average
water levels are controlled by Lake Huron water levels and a predominantly
wind-driven circulation system normally influenced by SWS and SW prevailing
winds.
Deepwater temperatures remain at about 4°C, while the shallow and surface
waters undergo change from season to season. In spring, the total water mass
is homothermous at 1-4°C. Shallow waters are warmer, but only in the extreme
outer bay does a well-formed thermocline develop in the summer and fall. Con-
sequently, the bay may be considered as vertically well-mixed. Complete ice
cover and limited stratification is typical of the bay in winter.
The prevailing circulation is considered counter-clockwise with Lake
Huron waters entering from the NE and flowing along the northern bay shore.
Saginaw Bay water flows out to Lake Huron along the south shore of the bay.
However, this typical circulation pattern is swiftly upset by changing wind
conditions.
Land Use
Land use in the drainage basin has a considerable influence on water
quality in Saginaw Bay: 50% of the drainage basin is considered to be agri-
cultural, mainly cropland; 35% is forested; only 3.5% is urbanized. This
area has significant industrial, chemical, mining, and food processing indus-
tries .
398
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Water Uses and Water Quality
The bay and small reaches of the Saginaw River have extensive diverse
uses, including mining, commercial and sports fishing, and general recreation.
In addition, these waters are used for cooling electrical power plants, dis-
posing of municipal wastes, and producing industrial goods. Some of these
uses degrade water quality seriously enough to interfere with''municipal water
supply, recreation, and the fisheries.
Water quality in Saginaw Bay shows a significant deterioration from Lake
Huron water in both physical, chemical, and biological characteristics. High
concentrations of dissolved substances that occur in the inner bay are pro-
gressively diluted by high quality'Lake Huron water in the outer bay. Exces-
sive nutrient concentrations support prolific growths of algae, and other
biologic criteria indicate degraded conditions. Although water quality has
improved slightly in the past few years, commercial fishing has been curtailed
because of a major shift to low value species. This shift occurred because
of changes in the food web, changes in habitat, predation, competition from
introduced and invading species, and commercial over-fishing.
SAGINAW BAY MODEL STRUCTURE
Objectives
The ultimate goal of this work is to develop a mathematical model that
can be used to describe the physical, chemical, and biological processes in
Saginaw Bay and to predict the effects of reduced waste loadings. Specifi-
cally, the modeling effort will focus on phosphorus, nitrogen, and silicon
loadings to the bay and the resultant production of phytoplankton biomass.
Model development is proceeding along a series of pathways which are pro-
ducing and evaluating alternative submodel designs for the two major submod-
els: (1) physical circulation (transport and dispersion) and (2) chemical
biological processes. To model the temporal and spatial transport of material
and nutrient-phytoplankton dynamics, both submodels are essential.
Alternative Transport Schemes
The inadequacy of field measurements in Saginaw Bay requires a mathemati-
cal determination of circulation which will be verified with all available
field information.
Two basic approaches have been identified:
1. Fine-grid hydrodynamic approach.
This involves a complex mathematical formulation of the physical forces
that influence circulation. This approach is outlined by Lick (in this vol^-
ume). The approach is being pursued for Saginaw Bay as a future resource,
particularly to address fine space and time resolutions, but is inapplicable
to the present relatively long-time and the broad geographical scale of this
study.
2. Coarse-grid, mass balance approach.
A more economical approach to quantifying circulation has been used by
399
-------
Thomann (1972). This approach utilizes one of the key hydrodynamic princi-
ples, mass balance, to trace a material through a system which has been seg-
mented on a relatively coarse-grid. In the case of Saginaw Bay, the most ap-
propriate substance is the chloride inflow from the Saginaw River. Chloride
provides a continuous tracer over the entire year, is non-reactive, and can be
conveniently measured at its sources and in the bay.
The coarse-grid approach provides sufficient detail to define problems
within large time and space scales. This approach is appropriate for this
study's consideration of the water quality problems that require definition
over time periods of weeks to months and a space scale of 5 to 50 kilometers.
While the chloride mass load and the resulting chloride concentrations
are known in the bay, the circulation parameters are unknowns. These para-
meters are determined through a simple iterative calibration and verification
process using a computer solution to the mass balance formulation which re-
quires less than a minute of computer time per run.
Basic Mass Balance Formulation
The basic mass balance equation is formulated as follows:
£ = ° = IT ^ - IT (vc) + IT ^ IT + ^ ~
at ax 9y 9x 9x 3y (1)
where:
u = velocity in x direction (L)
v = velocity in y direction (L)
EX = dispersion coefficients in x direction (L2/T)
E = dispersion coefficients in y direction (L2/T)
c = concentration of substance (M/L3)
K = decay rate of substance c (1/T)
The approach used to solve equation (1) is described by O'Connor and
Thomann (1971) and summarized as follows:
The water body is divided into completely mixed segments, and a mass
balance is defined for each by converting equation (1) to the form:
|f = 0 ~j [Qkj(«kjck + 3kC)
'
kjck + ^kjcj) + E kj (Cj - ck)]~ VkKkck + Wk
where:
V = volume of segment K (L3)
E'kj = bulk dispersion coefficient - Ek.A,./L (L3/T)
r = average length of adjacent sections"1 (L)
Akj = cross sectional area between segments k and j (L2)
akj = dimensionaless mixing coefficient; 0, .
Q^j = advective transport parameter (L3/T) ^
Wk = source of c in mass units per time (M/T)
Units: L = length
T = time
400
-------
•M = mass
The parameters a and 3 are weighing factors which correct cases of adja-
cent segments of unequal length.
L. where positive solutions are maintained by the
kj — stability criteria:
Lj + Lk akj >l-E'kj
Qkj (4)
Alternative solutions:
The system of simultaneous equations (one equation for each model seg-
ment) can be solved in two ways, steady-state (dc = 0) or time variable
(dc 1 0). dt
dt
The steady-state approach is used to investigate average conditions
throughout a defined time interval. All model inputs must be defined in terms
of averages for this period. Model output, i.e., the concentrations in each
model segment, also is defined in terms of a time average.
Model Application to Saginaw Bay
Whether the steady-state or time variable solution is used, the final
circulation scheme is determined by using the same iterative calibration and
final verification process. As shown in Figure 1, the model formulation
requires:
(1) segmentation of the water body,
(2) determination of segment physical characteristics,
(3) measurement of chloride loads,
(4) measurement of chloride concentrations in most segments,
(5) specification of the circulation parameters.
Segmentation—
Specification of a particular segmentation scheme depends on several
factors including:
(1) the desired spatial resolution,
(2) water quality gradients,
(3) available research time,
(4) available computer time.
Based on these considerations, Saginaw Bay was divided into 16 segments
as shown in Figure 2.
Chloride Loads—
Chloride loads are obtained by field measurements of chloride concentra-
tion and of flows at all major sources. For Saginaw Bay, the primary source
is the Saginaw River; however, loads from all tributaries and major point
sources have been accounted for and are included as input into the appropriate
model segment. The average chloride load in 1965 was 2.8 million kg/day
401
-------
MODEL CALIBRATION PROCESS
INFLUENCE FROM
LAKE HURON
SEGMENTATION
Depth D
Area A
Volume V
Length L
W(
BAY CIRCULATION
Advection Q
Dispersion E
BAY CHLORIDE CONCEN-
TRATIONS CM
CHLORIDE LOADINGS Wk
INPUT
A
V
D
STOP
COMPUTE CHLORIDE
CONCENTRATION
RESET>
Figure 1. Model calibration process.
402
-------
0.00000 Runoff, m3/km2-see
(0.000) Runoff, ft3/mi2-sec
00 Model segment number
Figure 2. Saginaw Bay Drainage Basin indicating average 1965 runoff for model
segment sub-basins.
403
-------
(Richardson, 1974). By 1974, the average annual load had been reduced to 1.2
million kg/day, as shown in Figure 3.
vd
O
X
-o
£j
Q
O
O
LLJ
O
12
Saginaw Bay
Chloride Load from Saginaw River
SECOND HALF AVERAGE
FIRST HALF AVERAGE
ANNUAL AVERAGE
0
SO N D
vO
O
5
x
CD
4 1,
O
o <
-> O
LU
O
1 <
1
0
Figure 3. Chloride load in 1974 for Saginaw Bay.
Segmental chloride concentrations—
Results of historical chloride data have been reviewed by Freedman
(1974). The specific data sources for this investigation include the 1965
FWPCA chloride data measured at about 20 sampling stations (USDI 1969) and
the 1974 Cranbrook Institute of Science data measured at the 59 stations
shown in Figure 4 (Smith, 1975). In addition to monthly-to-bimonthly cruise
stations in 1974, four stations were sampled at water intakes on a one to
three-day frequency. The cruise data are averaged on a segment basis for
comparison to calculated concentrations.
The calibration process proceeds with initial estimates of the transport
parameters E^. • and Q^-, and an initial calculation of segment concentrations
is made. The results are compared to the measured segment concentrations.
Adjustments of E^j and Q^- are made, and the process is repeated until an
acceptable comparison is obtained. Verification of the previously calibrated
404
-------
WATER QUALITY MODELING PROCESS
NUTRIENTS
TEMPERATURE
WIND
DEPTH
CORIOLIS FORCE
CHEMICAL-BIOLOGICAL
PROPF^Q MOnFI
(Batch)
HYDRODYNAMIC
PROCESS MODEL
(Fine grid)
REACTION RATES
rniiRSFHRin
TRANSPORT MODEL
t
VELOCITIES
I
_^. WATER QUALITY
MANAGEMENT MODEL
1
1
SIMULATIONS
pRpnirTioNjt;
o
Ui
Figure 4. Saginaw Bay 1974 sampling network.
-------
circulation scheme is attained when a new set of load and concentration data
is used, as above, and an acceptable comparison is obtained without adjust-
ments .
S_teady-State Calibration
This process was first used to test the hypothesis that the circulation
in Saginaw Bay could be described by one average annual pattern (Richardson,
1974). The rationale was that circulation is highly random and closely
related to meteorological events. These events, however, should average out
over a period of time, within the time and space scales of interest to this
study (i.e., weeks-to-months and 5-to-50 kilometers). The highly fluctuating
and random events would be included in the dispersive transport parameters,
EJH . Any prevailing pattern (i.e., net circulation) would be included in the
advective parameter Q^-j .
The calibration process was performed over a range of values for the
transport parameters. The initial advective pattern was that suggested by
Ayers (1956). The final scheme (Figure 5) was determined by matching com-
puted and measured concentrations, as shown in Figure 6. For this final
calibrated scheme, inflow from Lake Huron was 6400 m /sec and dispersion,
5 km2/day, set at all interfaces, except for those in the outer bay which
were set at 15.6 km /day.
Steady-State Verification
The verification process simply involves one calculation using the final
advective and dispersive scheme from the 1965 calibration and the 1974 chlo-
ride load average of 1.2 million kg/day. The chloride concentrations com-
puted in each segment were then compared to those measured in 1975 and the
results plotted in Figure 7-
These results indicate that the model predicts the general decrease in
chloride concentration one would intuitively expect with the reduced load;
however, the spatial comparison significantly differs, most distinctly in the
inner bay.
Further analyses conducted by Richardson (1974) verified the steady-
state circulation pattern for the summer-fall period when most 1965 data were
collected. The spring period, however, appears to be too dynamic for a
steady-state analysis (Figures 8a and 8b).
Time Variable Analysis
Using a computer program to solve equation 1 with ^L / 0, a time-variable
circulation pattern was calibrated (Richardson, 1974) . dt
This was accomplished using time-variable chloride loads shown in Figure
9. A single circulation pattern used for the entire year with time variable
loads resulted in the computed versus measured chloride time series (Figures
12 and 13). The initial (uncalibrated run) reasonably compares to the chlo-
ride levels in the summer-fall but was much too low during the spring. This
406
-------
Legend
000
*• Net Advective Flow (mVsec)
00) Model Segment Number
LAKE HURON
jlOKM
—110 Ml
Figure 5. Calibrated average advective transport scheme for Saginaw Bay
407
-------
CHLORIDE, mg/l
00
COMPUTED CHLORIDE, mg/l
Figure 6. Saginaw Bay calibrated versus measured 1965 average chloride
isopleths.
408
-------
-P-
o
—00-—Computed chloride, mg/l
Measured chloride, mg/l
Figure 7. Computed versus measured chloride isopleths for Saginaw Bay.
-------
LEGEND
-00— Computed chloride, mg/l
~00--- Measured chloride, mg/l
(b)
KM
10 Ml
Figure 8. Computed versus measured chloride isopleths
(a) March through June 1974
(b) July through November 1974.
410
-------
,50
40
<
£30
o
g20
c_>
LU
2 10
(a)
MEASURED AT WATER INTAKE
T MEAN i 0.5 STANDARD DEVIATION
J F M A M J J A S 0 N D
1974
^50
E
| 40
$
£30
o20
o
LU
-10
(b)
J F M A M J J A S ' 0 ' N ' D
1974
o
<
UJ
O
25
20
10
- 5
a:
g
5 o
j ' F'M'A'M' j ' J'A'S'O'N' D
1974
Figure 9. Chloride concentrations measured 1974 in Saginaw Bay model segments
(a) segment 2; (b) segment 4; (c) segment 13.
411
-------
substantiates the conclusions drawn from the steady-state analysis.
Since all chloride loads were accounted for, some additional force must be
present during this period to effect a different circulation scheme.
For the next computation, the dispersion transport parameter E^ was re-
duced at all interfaces by 50% for the period March 1 to June 1. The ration-
ale for this change was that ice cover and floes would tend to dampen the dis-
persion transport.
This solution shows little change from the initial result for the outer
bay segments. However, chloride concentration was substantially reduced in
segments near the Saginaw River, while concentration in segments along the
southeast shore (segments 3 and 9) substantially increased. In the segments
along the northwest shore (segments 4 and 13), concentration decreased prima-
rily in spring, when the dispersion coefficient had been reduced but when
loads were high. This effect was contrary to initial expectations. It was
apparent that a change was necessary in the advective scheme. However, some
rationale or underlying physical reason was sought for this recalibration.
After consideration of all circulation influences, it was decided that
only temperature would regularly influence circulation on a seasonal basis.
It was suspected that a thermal bar formed at the mouth of the bay during the
spring warming period. This hypothesis was reinforced by plotting some of the
thermal structure during this period (Figure 10).
Final Calibration
Through the previously described iterative process, a newly calibrated
time-variable circulation scheme was determined. The final time-variable
model results are shown (Figure 11) with the general indication of the time
periods when different circulation regimes are used.
The final spring circulation pattern was used between March 1 and May 20
and is characterized by reductions of advective flow from the outer to inner
bay (segment 13 to 10) of approximately 75% of the advective transport, used
for the remainder of the year, and of the dispersive transport, by a factor of
50%. From May 20 to June 1, the advective transport was returned to normal
and the dispersion increased by a factor of 2. In an attempt to more accu-
rately match chloride levels in the fall, dispersion again was reduced by 50%
from October 1 to November 15.
Conclusions
A simple coarse-grid, mass-balance model is an economical and invaluable
tool for deducing quantified circulation characteristics of Saginaw Bay, as
well as a unique methodology for interpreting other limnological data (i.e.,
thermal bar effect). The average circulation scheme calibrated with 1965
data (Richardson, 1974) was verified as valid for most of the year, except for
spring. Using a time-variable solution to the model equations, it was deter-
mined that a 75% reduction of advective transport from outer to inner bay must
be included, as well as a 50% reduction in dispersion, to reasonably
412
-------
DISTANCE FROM SAGINAW RIVER, km
A 25 35 45 55 A
1
Q.
LU
Q
10
5°C
4°C
4-29-74
DISTANCE FROM SAGINAW RIVER, km
A 25 35 45 55 A
Q_
LJJ
Q
1
5
10
5-13-74
DISTANCE FROM SAGINAW RIVER, km
A 25 35 45 55 A
DISTANCE FROM SAG INAW RIVER, km
A 25 35 45 55 A
1
E 5
31
I—
Q_
LU
Q 10
I I I
20° C
i i r
7-9-74
Figure 10. Saginaw Bay temperature profiles alone transect A-A.
413
-------
herm Baro|NO| Normal 0
Q |NO| Normal 0
EF s EF 2 EF 1
—CompoleO
Choride neasured * *j(er inlake
J Mean chlonfl* f^eawreO at al
J'F'M'A'M'J VA'S'O'N'D
JFMAMJJASONO
Therm Bar Q INQ( Normal 0
EF-5 Ef-2 EF-1
I I
J'F'M'A'M'J'J'A'S'O'N'D
J FMAMJ J ASONO
NorinQ TrwrmbrOkd NorrnXQ
EF-1 Ef-5 EF-J EF-1
Chloride Loao for
Saqinaw Rivtr
J ' F ' M T A M ' J ' J ' A ' S ' 0 ' N ' 0
JFMA M JJ A S O.ND
j'F'MAMjj 'A'S'O'N'o
Figure 11. Final computed chloride concentrations vs measured concentrations for selected model segments
with indicated circulation schemes (Therm Bar Q - period when advective scheme influenced by thermal bar;
normal Q or NQ - period of normal advective scheme; EF-X adjustment factor to dispersion transport).
-------
approximate chloride levels measured in spring. This reduction in transport
is;; indicated by the development of a thermal bar structure which can form a
vertical barrier to mass exchange between the nearshore and open lake. A
reasonable agreement of computed yers'us measured chloride concentrations was
ob-tained for most segments throughout the year. The circulation scheme is
considered to be calibrated and the necessary verification will be the subject
"of•' future research. Further research, will also be conducted using, a fine-grid
hydrodynamic model which should provide more insight into the finer physical
processes involved. However, it is concluded that the presently calibrated
circulation scheme is adequate to combine with the biological process model
to form the first spatially segmented version of a nutrient-phytoplankton
water quality management model.
CHEMICAL-BIOLOGICAL PROCESSES
Two basic approaches are being used in this study of Saginaw Bay. The
first basically imitates the modeling approaches used for Lake Ontario by
•Thomann (this volume) who describes phytoplankton biomass as a measure of
eutrophication and associated water quality. Further discussion is unneces-
sary here except to note that the phytoplankton biomass model has spatial
definition.
The second approach does not include spatial detail but is a research-
oriented process-model with biological and chemical detail. This multispecies
phytoplankton model investigates the relative importance of various processes
in providing competitive advantages for the development of nuisance algal
growth.
MODEL CONCEPTS
The present version of this model evolved from earlier work which dealt
initially with microbial substrate uptake kinetics (Verhoff and Sundaresan,
1972; Verhoff et al. , 1973) and was expanded to include phytoplankton growth
kinetics modeling (Bierman et al., 1973; Bierman, 1974). The compartments in
the model are four phytoplankton, two zooplankton, and three nutrients (Figure
12). The four functional groups of phytoplankton were chosen primarily for
water quality considerations and for the fact that phosphorus, nitrogen, and
silicon are usually considered to be the major nutrients limiting algal
growth. Carbon is not considered important in limiting either the growth
rates or the maximum sizes of the algal crops in the simulations. Consider-
able support for this assumption can be found in the recent in situ work by
Schindler et al. (1973a, 1973b) in the Canadian Shield and in the recent
laboratory work by Goldman et al. (1974). Each nutrient is assumed to exist
in two different forms: an available form which the phytoplankton can di-
rectly absorb and an unavailable form not directly assimilable. .,No distinc-
tion is made between the dissolved and particulate fractions of the unavail-
able forms. Two of the phytoplankton groups, diatoms and green algae, are
assumed to be grazed by two zooplankton types, differentiated on the basis of
their maximum ingestion rates. It is recognized that zooplankton grazing is
a complex phenomenon and involves, among other processes, phytoplankton size-
selectivity (McNaught, 1975). However, the present version of the model is
415
-------
only intended to investigate gross changes in grazing among the phytoplankton
groups.
HIGHER PREDATORS
ZOOPLANKTER
1
-
ZOOPLANKTER
2
, Jv-/? <
X
DIATOMS
/ \
1
r
AVAILABLE
SILICON
*
/•
S.
NON AVAILABLE
SILICON
GREEN ALGAE
I
AVAILABLE
PHOSPHORUS
I
-
,
NON-AVAILABLE
PHOSPHORUS
A i
i
OTHERS
BLUE-GREENS
INON H-flXIHCI
;
BLUE-GREENS _~.
fN FIXING}
i
I
AVAILABLE
NITROGEN
1
ATMOSPHERIC
NITROGEN
NON-AVAILABLE
NITROGEN
,
,
Figure 12. Principal compartments of the Saginaw Bay eutrophication model.
A unique feature of the model is that cell growth is considered to be a
two-step process involving separate nutrient uptake and cell synthesis mecha-
nisms. The motivation for this "variable stoichiometry" approach is that an
increasingly large body of experimental evidence indicates that the mechanisms
of nutrient uptake and cell growth are actually quite distinct (Dugdale, 1967;
Fuhs, 1969, 1971; Droop, 1973; Azad and Borchardt, 1970; Caperon and Meyer,
1972a, 1972b; Eppley and Thomas, 1969; and Halmann and Stiller, 1974). The
model includes carrier-mediated transport of phosphorus and nitrogen using a
reaction-diffusion mechanism and possible intermediate storage in excess of a
cell's immediate metabolic needs. Specific cell growth rates are assumed to
be directly dependent on the intracellular levels of these nutrients, in con-
trast to the classical Michaelis-Menten approach which relates these rates
directly to extracellular dissolved phosphorus.
Net specific phosphorus uptake rate is a response to the balance between
extracellular- and intracellular-dissolved phosphorus:
416
-------
'ALGAL 'CELL
PCM (1)
External ^
Dissolved * "
Phosphate
PCA
Internal
Dissolved ^
Phosphate
(2)
fr Internal
Phosphorus
Storage
•
(1) Net specific uptake rate =
Maximum uptake rate • f(T),f(L)|
(2) PCA = (PCAMIN)
1
1
+ (PK1)(PCA) 1 + (PK1)(PCM)
where PK1 = affinity constant between phosphate and an assumed membrane
carrier
P = actual total phosphorus per cell
PO = minimum stoichiometric level of total phosphorus per cell
PCAMIN = minimum possible value of PCA
f(T),f(L) = temperature and light reduction factors, respectively.
An identical approach is used for nitrogen uptake kinetics (Bierman, 1974).
The quality PK1 has actual physical significance because it is the equilibrium
constant for the reaction between phosphate and an assumed membrane carrier
molecule in the cell. Such a molecule has been isolated in the bacterium
Escherichia coli, and its binding constant with phosphate has been measured
(Medveczky and Rosenberg, 1970, 1971). Since insufficient knowledge of com-
plex biochemical processes exists at this time, attempts to develop mechanis-
tic theories of these processes usually lead to empirical assumptions at some
point. Equation 2 relates actual intracellular phosphorus concentration, PCA,
to a minimum value, PCAMIN, which is assumed to be a small number. This is
not a rate equation, but is an empirical device to instantaneously calculate
PCA as a function of the cell's total phosphorus storage. Intracellular phos-
phorus concentration must be determined in some manner so that feedback can
be obtained.
A unique value for net specific uptake rate does not exist for a given
concentration of phosphorus in solution. Instead, there is a family of val-
ues, each corresponding to a different level of intracellular phosphorus
(Figure 13). Negative values for net specific uptake rate correspond to the
phosphorus leakage back to solution that can occur under certain conditions.
If the cells are assumed to be starved, that is, if P = PO, then the net
specific uptake rate becomes half-maximum at 30 yg-P/Liter in the example in
Figure 13. Since the traditional Michaelis-Menten approach to nutrient uptake
kinetics does not include a feedback mechanism, Michaelis-Menten kinetics is
actually a special case of this theory which assumes the cell's nutritional
state is constant. As PCA approaches PCAMIN in Equation 1, the first term in
brackets approaches unity and the equation for net specific uptake rate re-
duces the Michaelis-Menten equation to the familiar hyperbolic form.
417
-------
+ 1.PQ
a:
UJ
o
UJ
Q_
I/O
2 +o-50
i/o
I -0.50
x
<
-1.00
0 100 200 300 400
CONCENTRATION OF DISSOLVED PHOSPHATE PHOSPHORUS,
Figure 13. Degree of saturation of the phosphorus uptake system for the blue-
green algae in the model as a function of both extracellular and
intracellular phosphorus.
The two-step approach presented here is far from a rigorously correct
treatment of phytoplankton kinetics. However, it is more realistic and con-
sistent with the experimental evidence than the traditional single-step
approach. Further, in contrast to empirical attempts to develop two-step ap-
proaches (e.g., Huff et al., 1973), the carrier-mediated mechanism used here
is only one application of a more general substrate uptake theory (Verhoff et
al. , 1973). The latter provides a systematic methodology for investigating
alternative and more sophisticated uptake mechanisms which can serve as a
framework for future research.
Model Implementation
A major problem in implementing a complex chemical-biological process
model is that such models usually contain coefficients for which direct
measurements do not exist. It is possible that more than one set of model
coefficients could produce an acceptable fit between the model output and a
given data set. In the transition from single-class to multi-class models,
this problem becomes particularly acute because it is no longer sufficient to
ascertain a range of literature values for a given coefficient. Multi-class
models necessitate the definition of class distinctions within this range.
418
-------
Given such circumstances, many coefficients in multi-class models must be
estimated.
The primary operational differences among the phytoplankton types in the
model are summarized in Table I. Class differences in phosphorus uptake af-
finities are the least well established. Bush and Welch (1972) and Hammer
(1964) have recorded strong circumstantial evidence that indicates blue-green
algae have higher phosphorus uptake affinities than other classes of algae.
Shapiro (1973) has reported that the phosphorus uptake mechanisms of blue-
green algae reach half-saturation at significantly lower extracellular phos-
phorus concentrations than do the phosphorus uptake mechanisms of green algae
over a wide range of environmental conditions.
TABLE I. OPERATIONAL DIFFERENCES AMONG PHYTOPLANKTON TYPES
Phytoplankton type
Characteristic Blue-Greens Blue-Greens
property Diatoms Greens (non n-fixing) (n-fixing)
Nutrient requirements
Phosphorus
nitrogen
silicon
Phosphorus
nitrogen
Phosphorus
nitrogen
Phosphorus
Relative growth rates
under optimum condi- Moderately
tions at 25°C High High Low Low
Phosphorus uptake
affinity Low Low High High
Sinking rate High High Low Low
Grazing pressure High High None None
The first step in the development of a comprehensive water quality model
is to apply a spatially simplified version of the model to the system in
question. The successful results of this application can then be incorporated
in a spatially segmented version. This approach is used for the inner portion
of Saginaw Bay which is assumed to be a completely mixed reactor.
Preliminary simulations are presented using only chlorophyll a_. The
model output describes the chlorophyll a_ pattern reasonably well in the spring
and late summer-fall periods, but there is a significant discrepancy in June
(Figure 14). Richardson (1975) has shown two distinct seasonal flow regimes
in Saginaw Bay with a turbulent transition period in June. This suggests a
discrepancy due in large .part to the assumption of steady hydraulic flow
between the inner and outer bay.
419
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50
40
I"30
2 o
20
&
o
o
10
MEAN± ]/2 STANDARD DEVIATION
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
TIME
Figure 14. Seasonal chlorophyll a. distribution for 1974 in Saginaw Bay, inner
portion, as compared to model output (solid line).
Sensitivity Analyses
Approach—
The class composition of the Saginaw Bay model indicates that the early
phytoplankton crops are dominated by diatoms and green algae and the broad
summer-fall peak is dominated by blue-green algae (Figure 15). This succes-
sional pattern has been observed in the inner portion of the bay by Vollen-
weider et al. (1974). Chartrand (1973) has reported significant late summer
crops of Aphanizomenon, a filamentous, blue-green algae, in the outer portion
of the bay.
The development of multi-class phytoplankton models creates serious re-
search problems due to their comprehensive data requirements. The approach
adopted here is to perform sensitivity analyses of blue-green algae to estab-
lish the relative importance of several assumed competitive advantages. The
coefficients used in the model give the blue-green algae higher phosphorus up-
take affinities, lower sinking rates, and freedom from grazing, as compared to
the diatoms and green algae. These advantages will be removed individually
from the blue-green, and the resulting simulations will be compared to the as-
sumed baseline case (Figure 15). Although the use of recycled phosphorus is
420
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not restricted exclusively to blue-green algae, the importance of this;process
will also be investigated.
50
40 -
.30 -
oo 20
S
10
DIATOMS
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
TIME
Figure 15. Phytoplankton class composition of model output.
for sensitivity analyses.
Baseline case
Effects due to class differences in nitrogen dynamics will not be inves-
tigated. In developing the model to this point, it is assumed that class dif-
ferences in nitrogen dynamics are not as important as differences in phospho-
rus dynamics and that the primary mechanism for nitrogen competition is the
ability of some blue-greens to fix atmospheric nitrogen (DePinto et al.,
1975).
Results—
Phosphorus recycle occurs in the model as a result of phytoplankton death
and zooplankton excretion (Figure 12). Since the phytoplankton cells are
assumed to have a variable phosphorus stoichiometry, the recycle mechanism has
two components: recycle of the cell's minimum stoichiometric phosphorus level
to the unavailable pool and recycle of all cellular phosphorus above this
minimum level directly to the available phosphorus pool. Such a two-component
nutrient release upon cell death has been reported by Foree et al. (1970)- If
both phosphorus recycle components are "turned off," the early crops of dia-
toms and green algae are little affected, but the subsequent blue-green crop
shows a significant decrease, compared to the baseline case (Figure 16).
421
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50
40
30
il20
£
O
O
10
DIATOMS
BASELINE CASE
-—NO PHOSPHORUS RECYCLE
JAN FEB MAR APR MAY JUN JUL AUG
TIME
SEP OCT NOV DEC
Figure 16. Comparison between baseline case and case for which no phosphorus
is recycled upon phytoplankton death or zooplankton excretion.
The choice of coefficients used in the model for the phosphorus uptake
mechanisms is based on the assumption that competitive differences primarily
result from differences in phosphorus affinities, rather than in maximum phos-
phorus uptake rates. The actual situation is probably some combination of
these factors. Maximum phosphorus uptake rates have simply been chosen in
proportion to the minimum stoichiometric requirements of each of the four
classes of algae. If the cells are phosphorus-starved, the uptake mechanisms
of the diatoms and green algae reach half-saturation at approximately 60 yg-P/
liter and the uptake mechanisms of the blue-greens reach half-saturation at
approximately 30 pg-P/liter. The phosphorus affinity of the nonnitrogen-
fixing blue-greens is set slightly higher than for the nitrogen-fixing blue-
greens (Fitzgerald, 1969). This range of values is consistent with the data
of Fuhs (1971), Healey (1973), and Rhee (1973).
When the phosphorus affinities of both blue-green algae are reduced so
that their phosphorus uptake mechanisms half-saturate at 60 yg-P/liter, it
causes a significant reduction in the late summer-fall total crop (Figure 17a)
and also changes the class successional pattern. The reduction in the blue-
green component of the total crop (Figure 17b) is more significant than appar-
ent from Figure 17a because the diatom-green component develops an extra peak
in August (Figure 17c).
422
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50
40
30
20
10
_ 0
40
Q.
O
30
S
o
oc.
8
20
10
40
30
20
10
TOTAL CROPS
; BASELINE CASE
,' ---REDUCED PHOSPHORUS
AFFINITY IN BLUE-
GREENS
i i -r r
i i
BLUE-GREEN CROPS
BASELINE CASE
---REDUCED PHOSPHORUS
AFFINITY IN BLUE-
GREENS
DIATOMS AND GREENS
(C)
BASELINE CASE
---REDUCED PHOSPHORUS
AFFINITY IN BLUE-
GREENS
JAN ' FEB ' MAR ' APR1 MAY ' JUN ' JUL ' AUG ' SEP ' OCT ' NOV ' DEC
TIME
Figure 17. (a) Comparison of total crops between baseline case and case for
which phosphorus uptake affinity has been reduced in the blue-
green algae; (b) comparison of the blue-green components of the
total crops for cases in (a); (c) comparison of the non-blue-green
components of the total crops for cases in (a).
423
-------
The sinking rate for the diatoms and green algae is 0.40 meters/day.
This value is consistent with the laboratory data of Smayda (1974) and with
the range of sinking rates used by Thomann et al. (1975) for simulations of
Lake Ontario phytoplankton. Blue-greens are arbitrarily sunk at 0.15 meters/
day. If both of the blue-greens are also sunk at 0.40 meters/day, their con-
tribution to the total crop becomes completely negligible and diatoms and
green algae dominate the simulations throughout the growing season (Figure
18). Similarly, if all four algal types are grazed in the same manner as the
diatoms and green algae, the blue-green component of the total crop again be-
comes negligible (Figure 19) .
o'
uu :
£!
O
O
50
40
.30
20
10
BASELINE CASE
INCREASED BLUE-GREEN
SINK ING RATES
DIATOMS
GREENS
PREDOMINATELY
DI ATOMS -
GREENS
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
TIME
Figure 18. Comparison between baseline case and case for which blue-green
sinking rates have been increased.
Discussion—
A general picture emerges: Blue-green algae, although their growth rates
are slower than other groups of algae and tend to occur at the end of the
typical successional pattern in eutrophic environments, are nonetheless armed
with a variety of competitive advantages which can lead to their dominance
under conditions of nutrient enrichment. Freedom from grazing and lower sink-
ing rates were the most important of the competitive advantages considered
here, although more subtle effects involving phosphorus dynamics were also
significant in certain circumstances. The conclusions presented, however,
must be qualified by noting that they only apply to the ranges of coefficients
and to the particular structure of the model.
424
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50
40
^30
oo20
8 10
0
DIATOMS/ GREENS
— BASELINE CASE
-- BLUE-GREENS
GRAZED
BLUE-X
GREENS
PREDOMINATELY
DI ATOMS -
GREENS
1 \
N— _ --*
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
TIME
Figure 19. Comparison between baseline case and case for which blue-green
algae are subject to the same grazing pressure as the diatoms and
green algae.
Phosphorus recycle was shown to be an important factor in the development
of late-growing blue-green algae, even in the Saginaw Bay system where there
are continuous sources of external nutrient supply. Since little quantitative
information is available on the mechanisms of phytoplankton decay, zooplankton
excretion, and the role of bacteria in nutrient recycle, this area deserves
more research attention. Other studies have also identified the importance
of nutrient recycle (Thomann et al., 1975; DePinto et al., 1975). Gunnison
and Alexander (1975) have suggested that blue-green algae, as a group, are
more susceptible to bacterial decomposition than other algal types. A com-
plete treatment of phytoplankton growth and nutrient recycle may even necessi-
tate the inclusion of heterotrophic activity as a separate state variable in
systems models.
The simulations which compared the effects of changes in phosphorus af-
finities between algal types, even though based on incomplete information,
showed that these differences could be important for both total crops and
class successional patterns.
425
-------
Relatively small differences in sinking rates between blue-green algae
and non-blue-green algae showed significant effects in the simulations.
Modelers frequently think in terms of order-of-magnitude differences, but
when blue-green sinking rates were reduced by less than a factor of three in
the simulations, previously large blue-green crops were reduced to almost
negligible sizes. Unfortunately sinking rates are difficult to measure accu-
rately in the natural environment, but laboratory studies with a variety of
algal types under different sets of conditions should at least provide useful
comparative data.
426
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Bierman, V.J., Jr., F.H. Verhoff, T.L. Poulson, and M.W. Tenney. 1973. Multi-
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Bierman, V.J., Jr. 1974. Dynamic mathematical model of algal growth in
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Bush, R.M. and E.F. Welch. 1972. Plankton associations and related factors
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Canale, R.P-, S. Nachiappan, D.J. Hineman, and H.E. Allen. 1973. A dynamic
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Caperon, J. and J. Meyer. 1972b. Nitrogen-limited growth of marine phyto-
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Chartrand, T.A. 1973. A report on the taste and odor in relation to the
Saginaw-Midland supply at Whitestone Point in Lake Huron. Saginaw Water
Treatment Plan, Saginaw, Michigan.
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Chen, C.W. and G.T. Orlob. 1972. Ecologic simulation for aquatic environ-
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DePinto, J.V., V.J. Bierman, and F.H. Verhoff. 1975. Seasonal phytoplankton
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sented at the 169th Meeting of the American Chemical Society, April 6-11,
Philadelphia, Pennsylvania.
Droop, M.R. 1973. Some thoughts on nutrient limitation in algae. Journal of
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Dugdale, R.C. 1967. Nutrient limitation in the sea: dynamics, identifica-
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Eppley, R.W. and W.H. Thomas. 1969. Comparison of half-saturation constants
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Fitzgerald, G.P. 1969. Some factors in the competition or antagonism among
bacteria, algae, and aquatic weeds. Journal of Phycology 5:351-359.
Foree, E.G., W.J. Jewell, and P.L. McCarty. 1970. The extent of nitrogen and
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Freedman, P.L. 1974. Saginaw Bay: An evaluation of existing and historical
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Fuhs, G.W. 1969. Phosphorus content and rate of growth in the diatoms
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312-321.
Fuhs, G.W., S.D. Demmerle, and M. Chen. 1971. Characterization of phosphorus-
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Goldman, J.C., W.J. Oswald, and D. Jenkins. 1974. The kinetics of inorganic
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tion 46:554-574.
Gunnison, D. and M. Alexander. 1975. Resistance and susceptibility of algae
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Halmann, M. and M. Stiller. 1974. Turnover and uptake of dissolved phosphate
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774-783.
428
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Hammer, V.T. 1964. The succession of "bloom" species of blue-green algae and
some causal factors. International Association of Theoretical and Applied
Limnology 15:829-836.
Healey, P.P. 1973. Characteristics of phosphorus deficiency in Anabaena.
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Huff, D.D., J.F. Koonce, N.R. Ivarson, P.R. Weiler, E.H. Dettman, and R.F.
Harris. 1973. Simulation of urban runoff, nutrient loading, and biotic
response of a shallow eutrophic lake. In: Modeling the Eutrophication
Process. Proceedings of a Workshop, September 5-7, 1973. Utah Water Re-
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McNaught, B.C. 1975. A hypothesis to explain the succession from Calanoids
to Cladocerans during eutrophication. Verb.. Internat. Verein. Limnol. 19.
Medveczky, N. and H. Rosenberg. 1970. The phosphate-binding protein of
Escherichia coli. Biochimica et Biophysica Acta 211:158-168.
Medveczky, N. and H. Rosenberg. 1971. Phosphate transport in Escherichia coli.
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Middlebrooks, E.J., D.H. Falkenburg, and T.E. Maloney (ed.) 1973. Modeling
the Eutrophication Process. Proceedings of a Workshop, September 5-7, 1973.
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O'Connor, Donald J. and Robert V. Thomann. 1971. Water quality models:
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An Assessment (Ed. George H. Ward), Chapter 3. U.S. EPA Research Series,
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O'Connor, D.J., R.V. Thomann, and D.M. DiToro. 1973. Dynamic water quality
forecasting and management. Environmental Protection Agency Ecological
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Rhee, G. 1973. A continuous culture study of phosphate uptake, growth rate
and polyphosphate in Scenedesmus Sp. Journal of Phycology 9:495-506.
Richardson, William L. August 1974. Modeling chloride distribution in Saginaw
Bay. Proceedings Seventeenth Conference on Great Lakes Research, Interna-
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Richardson, W.L. 1975. An evaluation of the transport characteristics of
Saginaw Bay using a mathematical model of chloride. Paper presented at the
169th Meeting of the American Chemical Society, April 6-11, Philadelphia,
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Schindler, D.W., H. Kling, R.V. Schmidt, J. Prokopovich, V.E. Frost, R.A.
Reid, and M. Capel. 1973a. Eutrophication of Lake 227 by addition of phos-
phate and nitrate: the second, third, and fourth years of enrichment, 1970,
1971, and 1972. Journal of the Fisheries Research Board of Canada 30:
1415-1440.
Schindler, D.W. and E.J. Fee. 1973b. Diurnal variation of dissolved inorganic
carbon and its use in estimating primary production and CC>2 invasion in
Lake 227. Journal of the Fisheries Research Board of Canada 30:1501-1510
Shapiro, J. 1973. Blue-green algae: Why they become dominant. Science 179:
382-384.
Smayda, T.J. 1974. Some experiments on the sinking rates of two freshwater
diatoms. Limnology and Oceanography 19:628-635.
Smith, V. Elliott. April 1975. Annual Report—Upper Lakes reference study:
a survey of chemical and biological factors in Saginaw Bay (Lake" Huron).
Cranbrook Institute of Science, Bloomfield Hills, Michigan, for the U.S.
Environmental Protection Agency, Grosse lie Laboratory.
Thomann, R.V. 1972. Systems analysis and water quality management. Environ-
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Thomann, R.V., D.M. DiToro, R.P. Winfield, and D.J. O'Connor. 1975. Mathe1'
matical modeling of phytoplankton in Lake Ontario. I. Model development and
verification. Environmental Protection Agency, Ecological Research Series
EPA-660/3-75-005.
U.S. Department of the Interior, Fish and Wildlife Service. 1956. Surface
Current Studies of Saginaw Bay and Lake Huron.
U.S. Department of the Interior,,F.W.P.C.A. 1969. Lake Huron—Michigan,
Water quality data; 1965 data. Lake Huron Basin Office.
U.S. Department of the Interior. April 1969. Water quality control plan,
Lake Huron. Federal Water Pollution Control Administration. Unpublished
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Verhoff, F.H. and K.R. Sundaresan. 1972. A theory of coupled transport in
cells. Biochimica et Biophysica Acta 255:425-441.
Verhoff, F.H., J.B. Carberry, V.J. Bierman, Jr., and M.W. Tenn'ey. 1973. Mass
transport of metabolites, especially phosphorus in cells. American insti-
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Vollenweider, R.A., M. Munawar, and P. Stadelrnann. 1974. A comparative review
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430
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DISCUSSION
DeLUCIA: I would like to have Dr. Davies return to his first statements
describing the undesirable phenomena and relate them to biological and chemi-
cal characteristics.
DAVIES: The model I am describing is a research tool at this stage of
development. It is not yet ready to be used for management purposes; it is
not sufficiently understood. In Saginaw Bay, the most apparent problem is
the algal succession, which leads to problems of taste and odor in the water
supply. Saginaw Bay also has a series of other problems connected with toxic
substances, but these are only under study.
MORTIMER: My question relates to the circulation model. What measures were
taken to make observations in the open part of the water body?
DAVIES: In 1974, EPA expended considerable resources establishing current
measurement stations in a series of transects in the bay. Many measurements
were taken throughout the year at the mouth of the bay and in Lake Huron.
In conjunction with our studies at Saginaw Bay, we conducted a series of
research studies and water quality monitoring studies in the southern part of
the lake, and the Canadians studied waters in the northern part.
VASILYEV: In the report, there is a mass balance equation which includes dis-
persion terms. There is no clarification in the relation of details con-
nected with calculating the dispersion coefficients. Since in this case flows
are caused by wind influence, the velocity profiles must be rather compli-
cated. I would like to know how you evaluated the dispersion coefficients in
such conditions.
THOMANN: We tried to distinguish averaged transport conditions. The essence
of the approach is that when possible, we calculate active circulation to
determine the dispersion coefficient.
LOZANSKIY: What practical work has already been done on the basis of models;
what can be done in the future to manage water quality; that is, what kind of
practical actions will be realized to reach the goal set for water protection?
What effect, and I also mean economic effect, can result when the model has
been verified?
DAVIES: At present, this model is being examined as a research model and
does not yet have a full application. However, the present research will be
included in a report to the International Joint Commission, but will not be
used until verified to project the bay's response to changes in nutrient
loadings.
431
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ZNAMENSKIY: I would like to note two interesting aspects in the model pre-
sented by Dr. Davies. First, in the complexity of the model, dynamic chemical
and biological parts are considered and the fact that it is a research model
does not decrease its value. Second, a good method used in the model, namely
use of a conservative substance, makes it possible to calibrate the model.
YEREMENKO: Questions of forecasting flow and questions of forecasting phyto-
plankton are based practically on those developments reported by Dr. Thomann.
Several findings are characteristic of the work. First, the hydrodynamics of
flow are not considered in the usual way. Balance ratios are taken, where
concentration, dispersion, and convective transport are considered. Since
the equations are written factually for a finite sector, we do not have any
derivatives here, particularly if this equation is for a stationary flow; we
simply have an algebraic equation. Having collected experimental data on the
concentration, we can write a necessary system of closed equations and thus
determine the coefficients, not yet obtained, which interest us. Hydro-
dynamics usually give us this data.
In reference to this paper, chlorides are used as an indicator, and they
are suitable for determining marked parameters. The fact that not one but
four types of phytoplankton are examined here is also a positive note.
Likewise, the two-step model is interesting. We understand it now, after a
corresponding explanation. The report presented indicates a good future for
developing ideas that have been promulgated in the U.S., and can be useful in
that area of science in which we are working.
MORTIMER: It is necessary to acknowledge the necessary development of three
circulation models for three periods of the year—characterized by stratifica-
tion in the lake, no stratification in the lake, and a thermocline. Bierman's
model imitates the realistic introduction of phosphorus and the fact that all
the algae are not examined in one lump, but as subclasses.
THOMANN: I would like to emphasize that these models must be verified. Only
when their reliability is established will it be possible to apply the models
to a practical situation. We would like to see more diagrams comparing the
data obtained with practical results. This is especially important for the
transfer to a large number of phytoplankton species. A good data base is
necessary in order to test the multitype model.
If there is information on the flow velocity, then we always use it
instead of building a hydrodynamic model. The shortage of adequate models
for verifying hydrodynamics is disquieting. The basic model structure,
when accepted as valid, provides a basis for administrative decisions.
DAVIES: I would like to reinforce Dr. Thomann's statement that verification
is the key to acceptance of the program. One of our scientific duties and
responsibilities is to decide when to recommend using specific models as aids
in decision-making.
432
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DESCRIPTIVE SIMULATION MODEL FOR FORECASTING
THE CONDITION OF A WATER ECOSYSTEM
A.B. Gorstko
Highly sophisticated computers are required to forecast the consequences
of human activity on aqueous ecological systems. After developing a mathe-
matical model, scientifically valid recommendations can be developed to help
prevent these adverse consequences.
The development of a mathematical model depends upon information avail-
able on processes occurring in the ecosystem under study. Model selection is
based essentially on the volume of accumulated information. For unique water
bodies where the ecosystem has been studied for dozens of years (the Azov Sea
and Lake Baikal, etc.), the most appropriate model is the so-called descrip-
tive simulation system. These model systems fulfill certain general condi-
tions that must be satisfied by any acceptable model of the ecosystem used for
forecasting.
The model must fulfill the following requirements:
1. be derived from a set of submodels individually verified before in-
clusion in the ecosystem model;
2. include and accommodate the results of field and laboratory experi-
mental work;
3. contain a large number of state variables, their functions (often
non-linear) and their limitations;
4. include consideration of the possible extreme values of many para-
meters;
5. be developed as a complex set of computer programs.
One of the disadvantages of the descriptive model is that it must be
reconstructed for each unique objective. Fortunately few unique objectives
require the construction of special mathematical models.
The simulation model of the Azov Sea, developed at the Scientific Re-
search Institute of Mechanics and Applied Mathematics, is described as an
example.
The Azov Sea is subdivided into seven segments (Figure 1). The eco-
system in each segment is described by a state vector, the components of
which are concentrations of different substances in water and the biomass of
specific animals and plants. It is assumed that each segment is completely
uniform with respect to these parameters. Conditions in the river estuaries
and in the Kerchenskiy Strait zone are modeled separately, and are exogenous
factors for the Azov Sea. The dimension of the state vector is equal to 120,
433
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the distribution of the components is listed below:
1-20 concentration of the chemical elements;
21-22 bacteria biomass;
23-34 biomass of the basic types of phytoplankton;
35-46 biomass of the basic types of zooplankton;
47-61 biomass of the basic types of benthos;
62-113 biomass of the basic types of fish, subdivided into age groups; and
114-120 reserved components.
Figure 1. Diagram of the division of the Azov Sea into regions.
434
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The components change with time (the time* is assumed to be discrete with
a time step of five days) under the influence of various-physical, chemical,
and biological processes. It is expedient to separate the Azov Sea model into
partial models to sufficiently describe similar processes and phenomena.
These submodels will be referred to as: 1).: mixing, 2) .biogenous>. elements,
3) polluting substances, 4) phytoplankton, 5) zooplankton, 6) benthos, and
7)"fish. " The fish submodel in fact consists of eight models, one for each
main species. - ,
In the mixing model, the volumes of water exchanged during respective
five-day intervals between the open-sea segments are calculated according to
given external conditions, wind velocity and direction, water exchange with
the Black Sea, and the flow of the Don and the Cuban rivers. This overflow
determines the exchange of the parameters 1-46.
The biogenous parameter model describes the nitrogen, phosphorus, and
silicon cycles, and the processes of decomposition, transfer, removal, and
consumption. "Temperature and'" insolation are the very significant parameters
affecting this submodel.
The polluting substances submodel determines concentration changes during
the five-day time step. In situ decomposition processes and transport into
the segment of pollutants are included in the submodel; the controlling fac-
tors are similar to those in the preceding submodel.
The phytoplankton, zooplankton, benthos, and fish submodels include
feeding and predation, respiration, mortality, reproduction, and migration
from the segment. The fish submodels also include the impact of fisheries
on the populations.
The processes modeled in each submodel act .at different rates under the
influence of external factors. Those considered"'in the Sea of Azov are:
1) flow from the Don River. 2) flow from the Cuban River, 3) water exchange
with the Black Sea, 4) insolation, 5) wind velocity and direction, 6) water
temperature.
These factors are external to the Azov Sea (not influenced by the sea)
and are modeled separately.
In the submodels for mixing, biogenous elements, and polluting substances,
the normal mass balance relationships use finite or differential forms. In
contrast, the submodels for phytoplankton, zooplankton, benthos, and fish are
modeling algorithms based on a complex set of observational data. The data
set determines the dynamics of the specific trophic level relative to external
and internal variables in the system. These algorithms are developed as com-
puter programs.
The Sea of Azov model operates in the following manner: if the state
vector of the ecosystem X0 is known at time To, the state vector KI can be
calculated at To+5 days after one time step. It can take one of two possible
courses: a consecutive course of the process, or a consecutive and parallel
course.
435
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SYSTEM A
1. The external factors for the time interval (TQ,T1) where 'T!=To+5 days
were modeled. For those presumed random factors, this is accomplished by
extrapolating respective time series. The following procedure assumes a se-
ries of observations of a certain random value [S(t-j_)] at the intervals
t ,t ,.,»., tjj. These values can be approximated by means of the polynomial:
n . m
X(t) = y a-rt1 + T b-i cos (2Kfjt +n-;),
t_i-L i->J I I
1=0 j=0
where X(t) is the value of the polynomial at the moment t;
a± are the coefficients of the approximating polynomial;
n is the power of the polynomial; and
b. is the amplitude of the cosine harmonics with an f^ frequency and
an l^j phase.
The polynomial coefficients X(t) are selected from the condition of
functional minimum
N
F = I [S(t±) - X(t±)]2
1=1
Knowing the polynomial X(t), it is possible to approximate the desired
value S(T]_) = S(TN+1) with the value X(tN+1) .
Non-random external factors, (e.g., water exchange with the Black Sea)
are either calculated, or their forecasted values for the period of (TO,T^)
are introduced into the model.
The external factors are introduced into all the submodels, where re-
quired.
2. Calculations in the mixing submodel require modification to the first
46 state vector components in each of the seven compartments.
3. Then the state vector Xo in each compartment is transformed by the
remaining submodels in the following sequence: biogenous elements < polluting
substances < phytoplankton < zooplankton < benthos < fish.
The state vector transformations are carried out in sequence as required
by each submodel for the period of (To,Tj_). The resulting vector X^ is the
system state vector at the time T]_. The order of information exchange be-
tween the submodels is important for understanding the multiple functioning
of the simulation system (SS), as described in Figure 2.
One of the disadvantages of this system is that it does not replicate
actual mechanisms because certain processes in all the submodels take place
simultaneously during the (TO,T-L) time interval. To overcome this difficulty,
ordering of the submodels is introduced. Therefore, the Azov Sea contains
interacting ingredients pertaining to different time periods. If the time
436
-------
interval is sufficiently small (e.g., about five days), any inaccuracy is
insignificant and will not exceed the field measurement error for the para-
meters. A large interval is not acceptable.
N: = Number of Five-Day Periods
External Factors (N)
| MIXING
j Changed concentrations of biogenous elements, salinity,
I polluting substances, phytoplankton, and zooplankton by
I segments.
BIOGENOUS ELEMENTS
~ Modified concentrations of biogenous elements by para-
. meter and segment.
<- • !
«- i
POLLUTING SUBSTANCES
; i Modified concentrations of the polluting substances by
! types and by segments.
PHYTOPLANKTON
Biomass of species and segment quantity of biogenous
elements consumed by phytoplankton, mass of dead phyto
plankton.
ZOOPLANKTON
Biomasses of live organisms by species and segment;
biomass of dead organisms; biomass of sedimented mol-
luscan larvae.
BENTHOS
Biomass of live organisms by species and segment mass
of dead organisms, biomass of emerged molluscan larvae.
I"
FISH
Biomass and population size estimate; dead biomass by
species and segment: state of the forage reserve.
Iterative order
N: = N + 1
Figure 2: Diagram of information exchange,
437
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SYSTEM B
The external factors and the "mixing submodel" are calculated as de-
scribed for System A. However, each process submodel is placed in one of two
classes, a simultaneous class or a sequential class. The state vector X0 is
sequentially modified by the output of each submodel calculation for the
(To,Ti) time step of five days. The state vector is transformed from Xo to
X;L by the Z]_,...ZZ processes which act simultaneously. In each process, some
parameters increase and others decrease, the exact quantity of change is not
known, but the possible dimensional limits are known, depending on a number
of factors.
XIT° /zu . . . . z±j . . . . zlz \
[am(FTo),amz(FTo)]
]/
where F^Q is the vector of factors at time To, and "aij and ajj are the upper
and the lower boundaries, respectively. The major difficulty in the modeling
is to determine the intensity P]^...PZ of the processes. This difficulty can
be overcome by the following algorithm:
1. The FTO vector is calculated;
2. The values of ajj (FTo) > anc^ aij (^To) are determined;
3. If a random value is assumed for aij (Fxo) with a distribution
LsLij (FTo) >aij (FTo) 1 it can then be calculated by a Monte-Carlo method;
4. Boundaries of possible changes X>j , Xj , and xi are determined;
5. The respective components of the vector X^ are selected as the ini-
tial values of the desired intensities for Pi,P2, . . . ,PZ;
6. In the (a^j) matrix, the positive and negative elements are defined,
then the values A-^a^jX^ + X^1 - Sa.^.X^J for the obtained values
7. The vectors for each parameter are formulated from real observation
, Ui2, 01,... 0?) where
UU = U±2 =
X
IT"
where 0^ is the experimental value for the K-th component of the FT
vector for the i-th parameter;
8. The parameters are ordered from actual observations to determine the
preferential development of related forms, as well as tendencies for
the increase or decrease of each parameter;
9. The increment steps for P. are selected;
10. The values for P-j^ are refined by the logic steps outlined above in 4
through 9.
438
-------
11. The process intensity values outside the limits of expected values
actually observed are defined by a repetition of the logic steps 5
through 10 outlined above;
12. The values for P-^ are selected either as the respective mean values
or probability values within the range of acceptable rates; and
13. The Xj1 vector is determined.
A scan of the matrix elements will derive (XT-) as sets of probable
states at time T^. With the modeling approach, a family of projections is
necessary to compare with the real system. Statistical analysis of this
family of model projections to evaluate the impact of various management
strategies on the system is the most important part of the modeling program
for the Azov Sea.
Author's Note: The algorithm presented in this paper requires some clar-
ification. There is no method to rigorously validate the model. It is
actually an imitation of the real process of feeding and mortality. However,
since the processes are treated at the population level and not as individual
organisms, it is necessary to apply the iterative procedure described in
steps 4 through 10 to determine process rate values which in fact do not con-
tradict accepted biologic concepts.
The most acceptable verification and model refinement can be achieved by
testing the algorithm with real data. The basic concepts of this paper will
be used on a simulation model for Lake Baikal.
439
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DISCUSSION
ROZHKOV: Is it possible to examine the model as a Marcovian bond? 'If so",
then what is its finite length? Have the divergence effects been studied?
GORTSKO: Yes, it is possible to examine the model as a Marcovian bond. The
question about the divergence of forecast lines has not been examined.
THOMANN: Is there any information over a long period of time on the 120
variables which you include in the model? If not, then in my opinion, it
does not pay to include them.
GORTSKO: We chose that number of variables as a result cf a very lively
discussion among specialists—mathematicians, biologists, and others. All
variables are guaranteed by information.
IVAKHNENKO: Were the units realized on a digital computer or as analog units?
GORTSKO: On a digital computer.
LOZANSKIY: My question is concerned with the principle of equal stability of
the areas. I would like you to specify what criteria were used to realize
this principle.
GORTSKO: The principle of equal stability in the ecological system can be
evaluated. Biologists know a lot about the water bodies; the Azov Sea, in
particular, has been studied by scientists for several years.
IVAKHNENKO: The author's confidence in introducing all 120 variables into the
model frightens me; this can complicate the work. The number of variables in
the model can be less, since they are all interrelated. Any five out of 120
variables can represent the given system. Belief that all the variables must
be introduced into thf. computer is erroneous.
DeLUCIA: I support the majority of principles presented in the report. My
colleague is discovering for himself principles that were discovered earlier
in the United States. I think that it is necessary to differentiate the
goals of science from the goals of administration. Authenticity in a model
is required. We need complex models to describe with accuracy the phenomena
studied.
ROZHKOV: The Marcovian bond will be a good approximation in one step. The
Marcovian bond requires a transitional probability matrix that is multi-dimen-
sional. We do not comprehend these probabilities very well.
440
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LOUCKS: I think that all the participants in the symposium are interested in
developing a model for practical applications. I am confident that the
models presented here are useful and will be used in our countries. But the
paths used by scientists in various countries to develop models differ.
GORTSKO: In conclusion, I would like to make a few observations. First,
regarding the doubts expressed by Professor Ivakhnenko, it is incorrect to
assume that we included all the variables. If we had, the variables would
not total 120, but 1000. In the future, we plan to use statistical analysis
for the results of a model's run.
441
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MATHEMATICAL ECOSYSTEM MODELS AND DESCRIPTION
OF THE WATER QUALITY IN WATER BODIES
R.A. Poluektov
INTRODUCTION
Intensified production in the past decades has been characterized by a
significant increase in the rate of man's reconstruction of natural communi-
ties. The destruction of existing natural linkages and interactions evolved
through time, and their recreation, allow the growing population of our planet
to provide itself with the quantity of resources necessary for compatible
development. But these complex processes of interaction between man and na-
ture have another side. Because we are ignorant of the detailed picture of
interaction between numerous animate and inanimate components of a natural
community and cannot adequately evaluate the new equilibrium that follows our
intervention, we encounter unexpected effects. Consequently, the development
of a theory of dynamic behavior, both of the biosphere as a whole, and the
biogeocenoses, (i.e. those elementary natural communities that constitute the
earth's biosphere), has been pushed into the foreground. The essential prob-
lem is to construct quantitative models to reflect, with a certain degree of
accuracy, the behavior of existing natural systems and to forecast the impact
of man's controlling actions on these systems.
The theory of mathematical modeling of ecosystems has a 50-year history
but is only now emerging as an independent branch of science. At present we
can distinguish only a cycle of work of a general theoretical character pro-
ceeding in two basic directions. V. Volterra (1931) and A.N. Kolmogorov
(1972), and a small number of investigators, (Poletayev, 1966; Menshutkin,
1971), were concerned with modeling specific ecosystems. In theory, the prob-
lem of the specific processes of interactions among different biological spe-
cies and environmental factors has not even been considered. In this work,
the possible outputs from specially constructed systems of differential equa-
tions were studied on a comparatively abstract level. The biological inter-
pretation of results reflects subjective views of the investigator rather than
the existing situation. Research on any specific problem should be based on
an explicit parametrical notation of the equations of the system being mod-
eled. Subsequent determination of the unknown parameters is based on the
available experimental data. Since the experimental points fit sufficiently
smooth curves, the type of approximating functions may be chosen arbitrarily
by the investigator. Consequently, the number of different models is unneces-
sarily increased, since the comparison of results related to various models is
frequently impossible. Such a situation necessitates a common point of view,
or more precisely, a common approach for describing ecosystems. The general
outline of such an approach, applicable to the theory of water ecosystems, is
summarized below.
442
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The Principles of Ecosystem Modeling
When modeling ecosystems, as any other dynamic system (Kalman et al.,
1971), the major problem is choosing the variables to form the state vector.
The state vector, understood to be a combination of values, reflects the in-
fluence of the system's prehistory on its future development. Since we assume
that the ecosystem state vector is finite, the systems being modeled are con-
sidered a finite number of compartments, each uniquely characterized by the
concentration of its substances or by the quantity of its energy.
The important considerations in research on the dynamics of biological
associations are: How does the general number and composition of these as-
sociations change with time? What is the equilibrium state of the association
after a long period of time? What are the reaction rates and equilibrium con-
centrations of mineral components in the system? If we tried to describe the
fate of each member of the community, the resulting model would evidently be
immense. Therefore, we must divide the live organisms into groups, so that
members of each group possess an identical or similar set of features that
determine their fate in the given community. The above-mentioned basic as-
sumption means that the number of groups in a community is finite. If the
feature associated with the survival of the individual is distributed contin-
uously, then the given combination of the individual should be divided into a
finite number of classes*.
The assumption that a finite number of compartments are in the system's
inorganic and detrital organic parts has been reached in a similar manner.
For example, when considering vertical stratification of the biogenic part of
the system, it is necessary to divide the water body into a sequence of lay-
ers, each with its corresponding average level of concentration. Hence, we
have as state variables:
x(t) = n-dimensional vector, the components of which are the numbers
(concentrations) or biomasses or a group of members which constitute the
biological part of the system in the different compartments, and
S(t) = m-dimensional vector, the components of which are the concentra-
tions of environmental extensive parameters in the same compartments.
Since -we are investigating a thermodynamically open system, it is neces-
sary to take into account the input [xi(t), Sj_(t)] and withdrawal [xg(t) ,
Sn(t)] of both the biological and mineral components, respectively. In addi-
tion, it is necessary to consider the dependence of processes taking place in
the system on the value of intensive abiotic factors which we denote by c(t).
In view of the concepts introduced,, the problem of modeling physical-chemical
conversions and biological transformations of a substance and energy in the
ecosystem is reduced to the description of their movement from one compartment
to another. The level of the environmental intensive factors determines the
rate reactions occur in the system.
*We note that the assumption of finiteness of the number of classes is
not unnecessarily limiting because it corresponds to the experimental data
available to the biologist.
443
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The general system of equations of an ecosystem is:
x = Y[x,S,c(t),x1(t),xQ(t)
S = ^[x,S,c(t),S1(t),S0(t)
(1)
where Y(...) and ij>(...) are, correspondingly, n-dimensional and m-dimensional
functions of the arguments in parentheses. A schematic diagram that repre-
sents the processes taking place in the system is shown in Figure 1. The
solid-lined arrows in the diagram signify the trophic interactions of the
system's elements.
c(t)
X0(t)
Figure 1. Interaction of biotic and abiotic elements in the ecosystem.
From the material presented, the following stages of constructing a model
scheme of the ecosystem are apparent:
1. determination of the combination state variable (functionally homo-
geneous groups of biological objects, biogeneous, and other factors);
2. determination of chronological (trophic) links and construction of a
system diagram; and
3. determination of the dependence of the exchange rates among compart-
ments on the values of extensive factors and the level of effect of the in-
tensive factors.
To clarify the nature of the problems and the difficulties involved, we
shall first examine the simplest example: A population of heterotrophic
microorganisms of one species develops in a flowing system on a certain sub-
strate. The concentration of the microorganisms is x; the concentration of
the substrate is S (here x and S are sealer quantities). The equations of
this system (Malek, 1968) are of the form:
x =.y(S)x - dx,
S = (S1-S)d - -4i(S)x,
y
(2)
444
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where d is the flow rate, y(S) is the specific rate of the microorganism's
growth, and y is the utilization coefficient. These equations already re-
flect two main points which also arise in a general case.
1. The concept of specific growth rate. As is evident from equation 2,
in the absence of flow (d=0) the relative growth rate of the microbial popula-
tion is determined by the concentration level of substrate S:
ix = y(S)
x
Usually this dependence is evident (as in Figure 2) where different
curves correspond to different levels of intensive factors (i.e., temperature).
It is essential to note here that numerically the specific growth rate is the
value used _ for constructing the model for a general case (Poluektov 1974).,
Thus, if "S"1 is the combination of biogenic factors of the i-th group of the
community but Ic1 and x1 are the combination of organisms adjacent to the i-th
group on the lower and higher trophic levels, correspondingly, then,
1 Y - ,, rei vi xi -x 1 ' CT)
— X-! ~U-J_\3»A,Ji,X£y \->S
Here, as a rule, y^ is a monotonic increasing function of the variables
TP- and x"i and a monotonic decreasing function of the variables x^ and its
"own" variable x-[. A more detailed investigation of the dependence of the
specific growth rate on the concentration of biogenic elements and organics
is described by Poluektov et al. (1974).
(S)
max
Figure 2. Dependence of specific growth rate on the substrate concentration
(S) given different levels of the intensive factors (c).
445
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2. Utilization coefficient- The introduction of this coefficient allows
us to "calculate" the consumption of substrate by biomass growth. In the
given case, this coefficient has the 'sense of the partial derivative:
y = _3j£ ~ AX
8S AS
and is equal numerically to the value of increased biomass per unit of con-
sumed substrate. It is evident that the concept of the utilization coeffi-
cient (yield coefficient) also has a profound biological sense. Its use
allows us to connect the adjacent trophic levels of the ecosystem into a
single chain. As experiments demonstrate, the utilization coefficient in a
number of cases depends slightly on the values of the intensive and extensive
factors.
3. The problem in generalizing the model (2), as in the case of complex
trophic interactions, is clarifying the dependence of the specific growth rate
of each population modeled on the state parameters of adjacent , trophic levels
and on the value of the environmental intensive factors. Indeed, the know-
ledge of the dependence (3) and the utilization coefficient is in principle
sufficient to close the system*. In V. Volterra's simplest model (1931), the
specific growth rate of prey is a linear decreasing function of the number of
predators, whereas • the specific growth rate of predators is a linear inc'riSas-
ing function of the number of prey. Obviously different schemes of consump-
tion are possible in the presence of a number of substrates consumed by one
species (Figure 3a) or a number of species of organisms consumed by the preda-
tors of the next trophic level (Figure 3b) . One scheme — the completing dia-
gram — is found when, for organisms to live, all components of an allowance
must be available. The growth rate is limited by the substrate (or species of
"prey") which is in the relative minimum, so that:
and
where functions f._-,(x._-,) and f . (x . ) are of a form similar to curves in
Figure 2. Another extreme case Is sequential consumption, when at first one
kind of substrate (species of "prey") is consumed until it vanishes from the
system, then a second, and so on. A mixed case is possible with various
a priori probabilities of consumption of one or another substrate (different
"affinity" for the substrate) , whereas the real consumption is determined by
the available composition of substrates.
Finally, it is necessary to examine some problems of describing energy
processes — photosynthesis and chemosynthesis of autotrophs and respiration.
Modeling photosynthesis processes are based on the use of the light and carbon
dioxide photosynthesis curves, and the calculation of the absorption of photo-
synthesis radiation by the turbidity of the absorbing water column. It is
somewhat more difficult to model respiration processes. Recent studies have
*It is assumed that the principles of physical and physical-chemical
conversions of energy and matter in the system are known.
446
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established that the respiration of sessile organisms can be divided into
two components. One, "existence breathing," is proportional to the accumula-
ted biomass, and the other, "growth breathing," is proportional to the inten-
sity of biosynthesis. In other words it is necessary to write
(4)
Figure 3. Diagram of the parallel consumption of abiotic (x^) and biotic
(x.) elements in an ecosystem.
The breath of actively mobile organisms possesses a third component con-
nected with the energy expended on movement. This component is evidently con
nected with the necessity of pursuing prey or hiding from the predator and
should be on the same structure as the specific growth rate y^Csi, x^, xA» x-jj
Apparently this breathing component can be connected in the first approxima-
tion with the second component of the formula (equation 4).
Modeling Methods
In discussing the modeling of such complex phenomena as the dynamics of
water ecosystem development, we, of course, cannot hope to develop a model
which adequately reflects the entire assemblage of complex interrelations of
the system's elements. A model is a certain view of a phenomenon from a
definite standpoint. For research objectives, the most important processes
must be described in sufficient detail while less significant processes may
be described in general terms or even disregarded. Therefore, before dis-
cussing a model of one or another phenomena, modeling objectives must be
established.
447
-------
If problems concerning environmental protection and increased biological
productivity of water ecosystems are considered, the models should "conform"
to one or several economic management decisions. Therefore, before catego-
rizing the types of models, it would be advisable to classify such decisions.
All the economic management decisions adopted in environmental protection
and natural resource management can be divided into three categories, depend-
ing on the length of time in which the consequences of these decisions are
observed. These are:
1. feasibility and project decisions, the consequences of which will not
be revealed for several years. Examples of such decisions are water reser-
voir-lake construction projects, and the construction of the dam between the
Black and Azov Seas now being discussed;
2. planning decisions, the consequences of which will be revealed within
one year. An example of such a decision is the establishment of annual catch
quotas for fish and other aquatic animals; and
3. operative decisions, consequences of which are revealed during a part
of the annual cycle. Basically such decisions are connected with the tempo-
rary realization of planning measures covered in item 2.
Further, the models under development can also be divided into three
categories:
1. models of interseasonal dynamics of water ecosystems. Basically
these models are mass-balance relationships that describe the accumulation or
loss of basic material components in the system, the change in the structure
of water bodies and their biological productivity, eutrophication, etc. The
time step of such models fluctuates from one to two months, to one year;
2. models of seasonal dynamics of water ecosystems used to describe the
periodicity of water body bioproductivity within one year. The time step of
the model fluctuates from one to several days; and
3. models of diurnal dynamics describing the daily fluctuations of the
photosynthetic activity of phytoplankton and other water plants, and the daily
cycles of water animals connected with these fluctuations. The time step of
the model varies from one to several hours.
Models of the three above-mentioned types can generally be examined as a
hierarchical system of models. The coefficients of the interseasonal dynamic
models are functions of the outputs of models of diurnal dynamics. An analogy
can be drawn between the hierarchy of models and the hierarchy of the degree
of turbulence in the atmosphere. The modeling diagram is shown in Figure 4.
The basic subsystems of models from each of the three levels can be seen on
the left.
448
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Basic Subsystems
Integrated Models
Heat and Mass Transfer
Balances of Biogenic Elements
Light Regime
Gas Regime
Phytoplankton and Other
Bioproducers
Zooplankton
Consumants (predators of
higher orders)
Biodegradators
Models of Interseasonal
Dynamics of Water Ecosystems
Model of Seasonal Dynamics of
Water Ecosystems
Model of Diurnal Activity
Figure 4. Hierarchical diagram of dynamic models of water biogenesis.
449
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BIBLIOGRAPHY
Kalman, R., P. Talb, M. Arbib. 1971. Excerpts on the mathematical theory of
systems. MIR, Moscow, pp. 398.
Kolmogorov, A.N. 1972. Qualitative study of mathematical models of the
dynamics of population. Problemy Kibernetiki, No. 25. NAUKA, Moscow,
pp. 100-106.
Malek, I. 1968. Continuous cultivation of microorganisms. Pishchevaya
Promyshlennost, Moscow, pp. 168.
Menshutkin, V.V. 1971. Mathematic modeling of populations and communities of
water animals. NAUKA, Leningrad, pp. 196.
Poletayev, I.A. 1966. On mathematical models of elementary processes in bio-
genesis. Problemy Kibernetiki, No. 16. NAUKA, Moscow, pp. 171-191.
Poluektov, R.A., Editor. 1974. Dynamic theory of biological populations.
NAUKA, Moscow, pp. 456.
Poluektov, R.A., I.A. Shvytov. 1974. On mathematical modeling of microbial
ecosystems. From the All Union Seminar "Mathematical Models of Marine
Ecological Systems," NAUKOVA DUMKA, Kiev, pp. 16-17.
Volterra, V. 1931. Lecons sur la theorie mathematique de la lutte pour la
sie. Gauthier-Villav, Paris, pp. 214.
450
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FINAL REPORT
OF THE AMERICAN-SOVIET SYMPOSIUM ON MATHEMATICAL
MODELS TO OPTIMIZE WATER QUALITY MANAGEMENT
In accordance with the intergovernmental agreement between the USA and
USSR on Cooperation in the Field of Environmental Protection, signed in Moscow
on May 23, 1972, the Memorandum of Implementation o'f the said agreement,
signed on September 21, 1972, and the Minutes of the Second Session of the
USA-USSR Joint Working Group, September 27, 1974, the American-Soviet Sympo-
sium on Mathematical Models to Optimize Water Quality Management was held in
Kharkov and Rostov-on-Don, USSR, from December 9 through 16, 1975.
Discussions of the application of mathematical models to optimize water
quality of river basins took place in Kharkov December 9-11. The mathematical
modeling of water quality in lakes and impoundments was discussed in Rostov-
on-Don December 12-16, 1975. Nine summary reports from the USSR, based on 24
reports prepared by Soviet scientists and nine reports of American scientists,
dealing with various aspects of the theme of cooperation, were presented and
discussed.
USSR specialists from 35 scientific research institutions and representa-
tives of the USSR Ministry for Reclamation and Water Management, the USSR
Council of Ministers, Main Office of the Hydrometeorological Service, the
USSR and Ukrainian SSR Academies of Sciences,'. and representatives of higher
educational establishments took part in the work of the symposium. American
specialists from two research laboratories of the U.S. Environmental Protec-
tion Agency and five universities also participated in the symposium.
Scientific problems worked out under the terms of American-Soviet cooper-
ation on projects II-l, "Studies and Modeling of River Basin Pollution," and
II-2, "Protection and Management of the Water Quality in Lakes and Estuaries,"
were considered, along with reports reflecting a general state of development
and application of mathematical models that predict and optimize water quality
management in the USA and the USSR.
Participants noted with satisfaction signs of progress in the statements
of problems and in the development of mathematical models, and observed that
the bilateral projects and exchange of scientific information within the
framework of the USA-USSR cooperation have facilitated this progress.
In water quality modeling, American specialists have developed and
applied deterministic models based on the conservation of mass of various
substances significant in the description of physical and biochemical pro-
cesses. These models were designed for a large number of water bodies and are
widely applied to problems.
451.
-------
In the USSR, the above-mentioned approach has been practically applied
to plan water protective measures for some water bodies, concurrent with non-
deterministic and non-formal methods to afford the advantage of flexibility.
The necessity for this approach has been dictated by extreme involution of
the processes in aqueous ecosystems caused by human activities. It is assumed
that this approach will aid understanding of the processes in water bodies
and development of efficient methods of optimal management of large scale
ecological and economic systems. It is extremely important to compare appli-
cability of the two approaches to provide practical systems for management of
water protective complexes.
Information provided on problems of mathematical modeling is inadequate
at present. Therefore, better methods must be developed to mathematically de-
fine data collection plans, to acquire field data promptly and inexpensively,
and to evaluate water quality. Mathematical models that are based on the
principles of conservation of mass, and define physical and biochemical pro-
cesses involved, have been successfully applied to optimize control of water
quality. Water quality control can be improved by the use of more complex
models which describe the condition of ecosystems.
To advance development, improvement, and application of mathematical
models for water quality control optimization, and also to enhance mutual
benefits of American-Soviet cooperation, the participants of the symposium
propose the following objectives within the framework of joint cooperation:
1. organize a joint comparative evaluation of mathematical modeling
methods;
2. continue to develop improved techniques of water protective programs
based on systems analysis of ecological, economic, and social phenomena;
3. initiate projects to develop methods for estimating the stability and
resilience of natural aquatic ecosystems;
4. promote joint work on data management and modeling;
5. intensify development of models and algorithms for the economic opti-
mization of water protective measures for river basins; and
6. arrange mutual, long-term exchange of scientists for the development
of water quality models in both countries.
Participants in the symposium consider it beneficial to acquaint scien-
tists of both countries with materials of the symposium and request that the
USSR Council of Ministers, Main Office of Hydrometeorological Service, and the
U.S. Environmental Protection Agency, publish proceedings of the symposium.
452
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The symposium has demonstrated that mutual discussions of questions of
mathematical modeling relevant to the problem of water protection are useful
and fruitful. Participants propose a symposium to convene on the theme,
"Methodology and Practice of Water Protection Activity Planning," in 1979,
in the USA.
Further, the participants of the symposium wish to express their grati-
tude for the high level of organization and for the gracious hospitality
shown them during their visit.
This final report was signed in Rostov-on-Don on December 16, 1975, in
two, copies, Russian and English, both copies being equally valid.
Co-Chairmen of the Symposium
rn -n .« \T TT7> o
-L . iA-iv I ' iO
f/^^^
V.LOZANS
A. ZENIN
453
* U.S. GOVERNMENT PRINTING OFFICE: 1978—740-263/1562 Region No. 4
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TECHNICAL REPORT DATA
(Please read Instructions <>n the rtivm1 before i amp/dinf!
1. REPORT NO.
EPA-600/9-78-024
4. TITLE AND SUBTITLE
AMERICAN-SOVIET SYMPOSIUM ON USE OF MATHEMATICAL
ODELS TO OPTIMIZE WATER QUALITY MANAGEMENT
|3, RECIPIENT'S ACCESSION NO.
i
IB. REPORT DATF
6. PERFORMING ORGANIZATION CODE
. ALH HOR(S)
T.T. Davies and V.R. Lozanskiy, Symposium Co-chairmen
8. PERFORMING ORGANIZATION REPi'MiT NO.
. PERFORMING ORGANIZATION NAME AND ADDRESS
U.S. Environmental Protection Agency
Environmental Research Laboratory
Office of Research and Development
Gulf Breeze, Florida 32561
10. PHHGRAM ELEMENT NO.
1BA608
7T~ CO"NT~R"ACT/GRANT NG\
Joint U.S.-U.S.S.R. Project
II-l
12. SPONSORING AGENCY NAME AND ADDRESS
U.S. Environmental Protection Agency
Environmental Research Laboratory
Office of Research and Development
Gulf Breeze, Florida 32561
i
3 TYPE OF REPORT AND PERIOD COVFRED
Final, December 9-16, 1975 _
14 SPONSORING AGENCY CODE
EPA/600/04
15. SUPPLEMENTARY NOTES
Project Officer: T. T. Davies, 8-686-9011, Gulf Breeze, FL
16. ABSTRACT - -
The American-Soviet Symposium on Use of Mathematical Models to Optimize Water
Quality Management examines methodological questions related to simulation and optimi-
zation modeling of processes that determine water quality of river basins. Discu-
sants describe the general state of development and application of mathematical models
designed to predict and optimize water quality management in the USA and USSR.
Subjects of papers presented by American and Soviet specialists include: planning
comprehensive water quality protection systems; modeling water quality in river
basins; geographic-economic aspects of pollution control systems; identification of
ecosystem models by field data; management decisions for lake systems on a survey of
trophic status, limiting nutrients, and nutrient loadings; a descriptive simulation
model for forecasting the condition of a water system; mathematical ecosystem models
and a description of the water quality in water bodies; mathematical modeling
strategies applied to Saginaw Bay, Lake Huron, to eutrophical processes in Lake
Ontario, to hydrodynamics and dispersion of contaminants in the nearshore, and to
coastal currents in large lakes; and the construction of a model of Lake Baikal on
the principles of self-organization. Publication of the proceedings held December
9-16, 1975, in Karkov and Rostov-on-Don, USSR, is in compliance with the Memorandum
from the Fourth Session of the Joint American-Soviet Committee on Cooperation in the
Field of Environmental Research.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
mathematical models
lakes
rivers
water quality
water pollution
water flow
b.lDFNTIFIERS/OPEN ENDED TERMS
US-USSR Agreement in the
Field of Environmental
Protection
Eutrophication
Ecosystems
. COSATl Held/Croup
06/D
08/H
12/A
13/B
8. DISTRIBUTION STATEMENT
Release to public
19. SECURITY CLASS (Tins Report)
unclassified
21. NO. OF PACES
453
L
20. SECURITY CLASb /Tills page ;
unclassified
EPA Form 2220-1 (Rev. 4-77)
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