y
FINAL REPORT
A PROBABILISTIC METHODOLOGY
FOR
ANALYZING WATER QUALITY EFFECTS
OF
URBAN RUNOFF
ON
RIVERS AND STREAMS
JULY, 1989
OFFICE OF WATER
U S ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
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DISCLAIMER
This report has been reviewed by the U.S. Environmental Protection Agency and
approved for release. Approval does not signify that the contents necessarily reflect any
policies or decisions of the U.S. Environmental Protection Agency or ariy of its offices,
grantees, contractors, or subcontractors.
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ABSTRACT
The Probability Dilution Model provides a screening level methodology for
determining the effect of intermittent pollutant discharges on the water quality of streams
and rivers. The processes analyzed in the model (urban runoff, streamflow, and water
quality) are probabilistic by nature and are treated as such by the model. The method uses a
determirdsitic mass balance approach and accounts for the natural variabilities in the model
inputs to estimate a cumulative probability distribution of in-stream concentration due to
runoff into the stream. This model is not meant to replace more sophisticated deterministic
or probabilistic models, but is an extremely useful screening level tool for initial planning
purposes to help determine problem areas and to screen possible control alternatives.
Three methods, the Analytically Derived Distribution Approach, a numerical Gaussian
Quadrature method and the Monte Carlo simulation approach, are presented.
The derived distribution method is based on analytically estimating the mean and
variance of the downstream concentration assuming that the upstream flow, runoff, and
downstream flow as well as pollutant concentrations are Ibgnormally distributed. The
method is conducive to hand calculations.
The Gaussian Quadrature Method assumes that the upstream flow and concentration
and the runoff flow and concentration are lognormally distributed but makes no
assumptions about the downstream flow and concentration. This method is more accurate
than the previous method, but requires more computational effort.
The Monte Carlo Simulation Approach allows flexibility in specifying the input
distributions. This method does not make any assumptions about the downgradient
distributions, is easy to understand and general in its approach. However, it requires the
repeated application (500 to 1000 times) of the deterministic mass balance approach and
hence requires more computational effort.
The Probability Dilution Model is easy to implement and use. A computer code has
been written in FORTRAN 77 and tested on IBM-compatible computers. Comparison of
results obtained using the three approaches discussed above using identical input
parameters is also presented. A copy of the source code and an executable code compiled
using the Ryan-McFarland compiler is included along with sample input and output files.
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TABLE OF CONTENTS
CHAPTER
Disclaimer i
Abstract • ii
Table of Contents i iii
List of Tables ; v
List of Figures vi
Acknowledgements viii
1.0 INTRODUCTION 1-1
1.1 Background 1-1
1.2 Nonpoint Source Technical Assistance 1-2
1.3 Organization of this Report 1-2
2.0 TECHNICAL OVERVIEW 2-1
2.1 Introduction 2-1
2.2 The System Being Simulated 2-1
2.3 Relevant Management Decision 2-1
2.4 Classification of Models 2-2
2.4.1 Approaches to Runoff Modeling.^ 2-3
2.4.2 Regression Models 2-3
2.4.3 Simulation Models 2-4
2.4.4 Probabilistic Methods 2-4
2.5 Characterization of Rainfall and Runoff Events : 2-5
2.5.1 The Rainfall/Runoff Event... 2-5
2.5.2 Selection of Interevent Duration i 2-5
2.5.3 Estimation of Runoff Statistics from Rainfall Statistics....! 2-6
2.5.3.1 Quantity Statistics 2-6
2.5.3.2 Quality Statistics 2-6
2.6 Summary... ; 2-7
3.0 THEORETICAL BASIS OF MODEL • 3-1
3.1 Introduction 3-1
3.2 Model Framework and Statistical Concepts 3-1
3.3 The Analytically Derived Distribution Approach 3-3
3.4 Key Assumptions for the Analytically Derived
Distribution Approach 3-7
3.5 The Direct Numerical Solution , 3-7
3.6 Key Assumptions in the Direct Numerical Solution 3-11
3.7 The Monte Carlo Simulation Approach 3-11
3.8 Key Assumptions of the Monte Carlo Simulation Approach 3-12
3.9 Recurrence Interval Calculation 3-12
TTT
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TABLE OF CONTENTS (concluded)
CHAPTER
PAGE
4.0
APPLICATION OF METHODOLOGY 4-1
4.1 Introduction 4-1
4.2 Data Needs and Sources 4-1
4.3 Computation Procedure Using the Analytically Derived
Distribution Approach 4-3
4.4 Illustrative Example of Analysis 4-7
4.5 Comparison of the Three Probabilistic Methodologies 4-8
4.6 Interpretation and Evaluation of Results 4-13
5.0 EVALUATION OF METHODOLOGY 5-1
5.1 Introduction 5-1
5.2 Verification of Computed Concentrations 5-1
5.3 Verification of Implied Effect on Beneficial Use... 5-5
5.4 Sensitivity Analyses 5-7
5.4.1 Analytically Derived Distribution Approach Vs. Direct Numberical
Approach 5-7
5.4.2 Correlation between Streamflow and Runoff Flow 5-12
5.4.3 Effect of Upstream Concentrations 5-12
5.5 Summary and Discussion 5-15
6.0 USER'S GUIDE 6-1
6.1 Model Acquisition and Installation Procedures 6-1
6.1.1 Model Acquisition 6-1
6.1.2 Model Installation 6-1
6.1.3 Testing the Model 6-2
6.2 Model Structure 6-2
6.3 Common Blocks and Parameter Statements 6-2
6.4 Model Input 6-7
6.4.1 Analytic Model Data Requirements 6-7
6.4.2 Monte Carlo Data Requirements 6-11
6.5 Example Application 6-18
REFERENCES R-l
NOTATIONS N-l
APPENDIX A : Mathematical Details of the Numerical Procedure.. A-1
APPENDIX B: Example Application of the Method of Moments
Method B-l
APPENDIX C: List of Subroutines Included in the PDM Model... C-1
APPENDK D: List of Variables Included in the PDM Model D-l
IV
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LIST OF TABLES
TABLE PAQE
4-1 INPUT DATA REQUIREMENTS AND RELATIONSHIPS FOR
LOGNORMAL DISTRIBUTIONS j 4-2
4-2 INPUTS USED IN SENSITIVITY ANALYSIS OF PDM MODEL 4-14
4-3 COMPARISON OF RESULTS BETWEEN MONTE CARLO
METHOD, ANALYTICALLY DERIVED DISTRIBUTION
APPROACH, AND DIRECT NUMERICAL APPROACH 4-15
5-1 RAPID CITY VERIFICATION TEST j. 5-3
6-1 INPUT FORMATS FOR ANALYTICAL PDM MODEL., 6-8
6-2 SAMPLE INPUT FILE PDM1.IN FOR ANALYTIC DAM
MODELS 6-10
6-3 SAMPLE OUTPUT FILE PDM1.OUT i 6-12
6-4 FORMAT FOR READING SIMULATION CONTROL
PARAMETERS (DATA GROUP 1) 6-13
6-5 FORMAT FOR READING DISTRIBUTION PARAMETERS
FOR INPUT VARIABLES (DATA GROUP 2) j 6-14
6-6 FORMAT FOR READING EMPIRICAL DISTRIBUTION!
DATA (DATA GROUP 3) 1 6-15
6-7 FORMAT FOR READING OUTPUT OPTIONS (DATA
GROUP 4) 6-16
6-8 FORMAT FOR READING CORRELATED VARIABLES
INPUT (DATA GROUP 5) 6-17
6-9 SAMPLE OUTPUT FILE PDM1.OUT 6-19
6-10 SAMPLE OUTPUT FILE PDM1MC.OUT 6-22
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LIST OF FIGURES
FIGURE
PAGE
3-1 SCHEMATIC REPRESENTATION OF WATER QUALITY
EFFECTS OF URBAN RUNOFF ON RIVERS AND
STREAMS 3-2
3-2 REGIONS OF INTEGRATION FOR THE JOINT PROBABILITY
DENSITY FUNCTION 3-9
4-1 SCHEMATIC OUTLINE OF PROBABILISTIC ANALYSIS
METHOD 4-4
4-2 CONSTRUCTION OF PROBABILITY DISTRIBUTION PLOT... 4-6
4-3 PROBABILITY DISTRIBUTION OF POLLUTANT
CONCENTRATIONS DURING STORM RUNOFF EVENTS 4-9
4-4 RECURRENCE INTERVALS FOR STREAM
CONCENTRATIONS 4-10
4-5a COMPARISON OF CDFS FOR THREE PROBABILISTIC
METHODOLOGIES 4-11
4-5b COMPARISON OF CDFS FOR THREE PROBABILISTIC
METHODOLOGIES (HIGHEST 10% OF VALUES) 4-12
4-6 USE OF THE METHODOLOGY TO EVALUATE THE WATER
QUALITY IMPACT OF URBAN RUNOFF CONTROL
PROGRAM (I) 4-16
4-7 USE OF THE METHODOLOGY TO EVALUATE THE WATER
QUALITY IMPACT OF URBAN RUNOFF CONTROL
PROGRAM (II) 4-17
5-1 MAP OF RAPID CREEK AND MAJOR DRAINAGE BASINS
WITHIN RAPID CITY PROPER: SAMPLING SITE
LOCATIONS 5-2
5-2 PROBABILITY DISTRIBUTION OF STREAM AND RUNOFF
FLOWS 5-3
5-3 COMPARABLE INFORMATION ON CONCENTRATIONS IN
RUNOFF AND UPSTREAM LOCATIONS AND THE
PROBABILITY DISTRIBUTION 5-4
5-4 RESULTS FOR TOTAL P 5-4
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LIST OF FIGURES (concluded)
FIGURE PAGE
5-5 RESULTS FOR COD 5-6
5-6 RESULTS FOR LEAD '. 5-6
5-7 COMPARISION OF RESULTS FROM THE DERIVED ,
DISTRIBUTION AND NUMERICAL SOLUTION APPROACH -
PROBABILITY DISTRIBUTION FOR SELECTED INPUT
CONDITIONS 5-8
5-8 COMPARISION OF RESULTS FROM THE DERIVED
DISTRIBUTION AND DIRECT NUMERICAL SOLUTION
APPROACHES - RECURRENCE INTERVALS FOR SELECTED
INPUT CONDITIONS 5-9
5-9 DIFFERENCE IN RESULTS BETWEEN DERIVED DISTRI-
BUTION AND NUMERICAL SOLUTION APPROACH OVER
A WIDE RANGE OF INPUTS ; 5-11
5-10 EFFECT OF STREAM/RUNOFF FLOW CORRELATIONS 5-13
5-11 EFFECT OF UPSTREAM CONCENTRATION ON MEAN
RECURRENCE INTERVAL 5-14
5-12 EFFECT OF CONTROL WITH NO BACKGROUND
CONTRIBUTION 5-16
5-13 EFFECT OF CONTROL WITH HIGH BACKGROUND
CONTRIBUTION 5-17
6-1 FLOWCHART OF THE PDM MODEL 6-3
6-2 FLOW CHART OF THE ANALYTICAL PDM MODEL 6-4
B-1 EXAMPLE APPLICATION OF THE METHOD OF MOMENTS
METHOD :... B-l
B-2(a) PROBABILITY DISTRIBUTION OF POLLUTANT !
CONCENTRATIONS DURING STORM RUNOFF EVENTS.... B-7
B-2(b) RECURRENCE INTERVALS FOR STREAM
CONCENTRATIONS ; B-7
YTI
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ACKNOWLEDGEMENTS
1
This report was prepared by Eugene D. DriscoU (Woodward-Clyde Consultants),
Philip E. Shelley (EG&G Washington Analytical Services Center, Inc.), David R.
Gaboury, and Atul Salhotra (Woodward-Clyde Consultants). The PC-based computer
program was developed by Phil Mineart and Atul Salhotra also of Woodward-Clyde
Consultants. Dominic M. DiToro (HydroQual Inc.) conceived the approach, developed
the methodology, and provided comment and advice on preparation of this manual. Much
of Chapter 3 describing the theoretical basis of the model has been extracted from his
original technical paper on the method. John L. Mancini (John Mancini Consultants)
made significant contributions to the application of the methodology as presented herein.
Special thanks are due the members of the S AB's Probabilistic Methodology
Subcommittee—-B. C. Dysart, IE, Chairman, R. A. Conway, R. C. Loehr, D. J.
O'Connor, and C. R. O'Melia—and their consultants (B. Adams and M. Small) for their
careful review of the draft report. Acknowledgements are also due to Mr. Tom Barnwell
and Rudy Parish of the U.S. EPA Environmental Research Laboratory at Athens, Georgia
and Tom Davenport of U.S. EPA Region VI for reviewing the draft report and providing
many useful comments.
The efforts of Dennis N. Athayade of the Office of Water's Nonpoint Sources
Branch are especially appreciated, both for providing the vision which led to the
development of the methodology as well as serving as Project Officer.
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Chapter 1.0
INTRODUCTION
1.1 BACKGROUND
This report presents a methodology for assessing the effects of intermitttent, highly
variable discharges, such as urban runoff, on the water quality of rivers and streams. The
methodology was developed under the sponsorship of the U.S. Environmental Protection
Agency (U.S. EPA) over the past 15 years, through a number of contracts, and traces its
roots to the statistical methods effort for urban stormwater initiated by the (then) Water
Planning Division in 1974. The method was also discussed in the Areawide.Assessment
Procedures Manual published by the Office of Research and Development, Municipal
Environmental Research Laboratory, in 1976. Much of the additional work that brought
the methodology to its present form was supported by the Nonpoint Sources Branch of the
Office of Water through its Nationwide Urban Runoff Program (NURP), which was
concluded in early 1984, and subsequent work assignments under a series of nonpoint
source technical assistance contracts. i
The Nationwide Urban Runoff Program (NURP) was conducted by the EPA and
many cooperating federal, state, regional, and local agencies, distributed widely across the
United States. The program yielded a great deal of information, much of which will be
useful for many years, for a broad spectrum of planning activities. Among the many
tangible products of the program are the Final Report, the reports of various grantees
(available from National Technical Information Service, NTIS), and several technical
reports that focused on specialized aspects of the program, its techniques, and its
findings.
One of the final NURP products was the first draft of a report entitled, "A
Probabilistic Methodology for Analyzing Water Quality Effects on Rivers and Streams,"
dated February 15,1984. In keeping with established NURP practice, the document was
issued as a draft report to allow maximum opportunity for peer review and comment.
Because of the perceived importance of the methodology and its potential for widespread
application in conducting screening level analysis for nonpoint source impacts on rivers and
streams, the Office of Water also requested the Environmental Engineering Committee of
the Agency's Science Advisory Board (SAB) to provide a formal review of the draft report.
This was done by an ad hoc Probabilistic Methodology Subcommittee, which held hearings
on the subject and retained consultants for specialized advice. The collective comments
resulting from this process were consolidated in a report that was sent to the Administrator
on July 31,1985. The letter forwarding the report to the Administrator stated that "The
Committee believes strongly that statistically based approaches to water quality
management are an important tool for the decision-maker, and commends the Agency for
supporting the effort under review."
The present document represents the final revision of that earlier report and
incorporates the SAB comments as well as those received from other reviewers. As a
result of the extensive review process, the Agency considers the method to be technically
sound and recommends its use as a screening level tool in dealing with urban runoff and
similar nonpoint source discharges. Practitioners are urged to read this document
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carefully, since the Agency does not believe that the method should be used by individuals
who do not fully understand the approach and the assumptions inherent therein, nor '
should it be applied in situations for which it is not technically suitable.
1.2 NONPOINT SOURCE TECHNICAL ASSISTANCE
The possible deleterious water quality effects of nonpoint sources in general, and
urban runoff in particular, were recognized by the Water Pollution Control Act
Amendments of 1972. Because of uncertainties about the true significance of urban runoff
as a contributor to receiving water quality problems, Congress made treatment of separate
stormwater discharges ineligible for federal funding when it enacted the Clean Water Act in
1977. With the enactment of the Water Quality Act of 1987, Congress expressed its
deepening concern about the water quality effects of nonpoint sources, stating that "it is the
national policy that programs for the control of nonpoint sources of pollution be developed
and implemented in an expeditious manner so as to enable the goals of this Act to be met
through the control of both point and nonpoint sources of pollution." As part of the
implementation of this policy, Congress added the new Section 319 and 402 to the law.
One of the major features of Section 319 is its call for states to prepare, and submit to the
Agency, Nonpoint Source Assessment Reports and Management Programs. A key
requirement is for the states to identify those receiving waters that cannot meet the goals of
the Act due to the effects of nonpoint source discharges.
Technical Assistance is an integral component of the Agency's National Nonpoint
Source Program's Agenda for Action, prepared by the Nonpoint Source Task Force. This
report is one of a number of specific technical assistance products being prepared by the
Agency to assist state and local government officials with the planning and implementation
of NPS programs. Under the new mandates of the Water Quality Act of 1987 states are
given the primary responsibility for identifying receiving waters that are being impacted by
urban runoff and other nonpoint sources, and for developing management programs to
control these nonpoint sources so that the goals of the Act can be met.
This report is addressed to those engaged in water quality management planning
and implementation, and is intended to enhance their ability to deal with such issues as
establishing receiving water body priorities, determining the significance of urban runoff
discharges, performing point versus nonpoint source comparisons, and determining the
effect of control measures on water quality. To facilitate the utilization of the methodology
presented in this report, an interactive, user-friendly computer program that performs the
analysis is included as part of this report.
Users are strongly encouraged to read this document carefully. A valid technique
with deceptively attractive simplicity has a greater than normal potential for misapplication
and use beyond its intended capabilities. Despite its ease of use, the Agency does not
believe that this method should be used by individuals who do not fully understand the
Screening level approach and the assumptions inherent therein, nor should it be applied to
situations for which it is not technically suitable.
1.3 ORGANIZATION OF THIS REPORT
Chapter 2 of this report provides a technical overview, compares different
approaches to urban runoff modeling, and outlines the rationale behind the development of
the methodology and the benefits of a probabilistic approach. The theoretical basis of the
methodology, including the development of equations and the assumptions made to
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facilitate its use, is presented in detail in Chapter 3. Chapter 4 addresses the application of
the methodology including data needs and sorces of data; computational techniques; and the
interpretation of the results. Field verification of the methodology and conclusions of the
study are discussed in Chapter 5. Chapter 6 presents guidance for the use of the computer
program and includes example input and output files. Additional mathematical details and
derivations are presented in Appendix A while Appendix B presents step-by-step
calculations based on the Derived Distribution Approach. Appendix C and D include the
list of variable and subroutines used by the computer program.
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Chapter 2.0
TECHNICAL OVERVIEW
2.1 INTRODUCTION
This chapter presents an overview of different approaches used for urban runoff
modeling and some considerations that lead to the selection of the methodology presented in
this report, for estimating the impacts of stormwater discharges from urban areas on rivers
and streams. Before discussing the different approaches to modeling urban runoff,
however, there are some contextual issues that must be addressed. These are to define both
the system being simulated and the management decision that will be based on the outputs of
the chosen methodology.
•
2.2 THE SYSTEM BEING SIMULATED !
Prior to the selection or the development of a model, it is essential that the various
components of the system being simulated be identified. The model presented in this report
addresses the urban runoff system. This is characterized by various components: (i) inputs
to the system, (ii) the land/conveyance system, (iii) outputs from the system, (iv) the
receiving water body, and (v) impact of the outputs on the receiver quality and the use.
Identification of these components of the system is an essential first step in the model
selection and development process.
In the present case, the inputs to the system are the meteorological conditions and in
particular the precipitation rates. The land/conveyance system is comprised of the
catchment, characterized by the slope and soil characteristics, type of land use and associated
activities, and natural and engineered components including lakes and rivers, detention
basins, highways, etc. The urban runoff system includes a number of point and nonpoint
source discharges; that can be quantified as time series of water and pollutant masses. These
outputs from the catchment are discharged into a receiving water body (lake, stream) for
which there is some water quality concern. The water quality of the receiving water body
reflects the combined characteristics of the receiver and the outputs, and impacts the
beneficial uses of the receiver. A literature review indicates that each of these elements has
been modeled individually as well as a coupled system.
2.3 RELEVANT MANAGEMENT DECISION !
i
A somewhat simplistic formulation of the management decision to be addressed by
the methodology presented in this report is (i) whether urban runoff is a. significant
contributor of pollutants in a particular river or stream, and (ii), what level of control
(reduction in runoff) will be required in order to meet the specified water quality goals.
In situations where the stormwater discharges from an urban area cause an obvious
negative impact on the realization of a designated beneficial use (e.g., use impairment or
denial), the answer to the first decision is obvious. However, in most cases, such a
determination is less obvious. In such cases, it is necessary to use a quantitative
methodology, such as the one presented here. Typically, water quality criteria are used as
surrogate measures of the degree to which the level of use of the receiving water body is
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achieved. Thus, decisions on the impact of urban stormwater discharges upon beneficial
uses can be made based upon in-stream water quality that has been estimated using a
quantitative model.
It is important that the information provided by a model be appropriate to the
decision to be made. Favorable consideration should be given to models that provide
information for a number of different alternatives as opposed to those that are restrictive in
their capabilities. For example, a model that estimates only end-of-pipe discharge
information for a "design storm" is much less desirable than one that can estimate the
continuum of discharges. Similarly, a model that can also estimate the effect on receiving
water conditions is even more desirable.
Other important considerations are the time and spatial scales associated with the
decision problem being addressed. Time scales for model outputs range from the duration
of an individual event to long-term steady-state models with time scales of eternity, i.e., the
entire continuum. For the decision being addressed here, models that simulate long-term
conditions are particularly relevant Similarly, spatial scales over which models operate
range from micro-catchment, through catchment and watershed, to political jurisdiction
(e.g., city, county, state). The methodology presented herein is appropriate for scales
ranging from the size of the watershed to jurisdictional spatial scales.
2.4 CLASSIFICATION OF MODELS
There are a number of ways in which models can be classified. One is to classify
them according to their formulation and use as either (i) physical/mechanistic models or (ii)
decision models for either screening level anlaysis or detailed analysis. Physical models
focus on relating cause and effect in terms of the physical variables of the phenomena
involved and the processes being simulated. Decision models focus on relating cause and
effect in terms of the decision variables of the alternatives used to control the processes
being modeled.
Generally, a decision model is less appropriate than a physical model for explaining
the cause and effect relationship. However, it is more appropriate for evaluating the cost-
effectiveness of management options.
It must also be recognized that physical models and decision models are implemented
under different conditions. Most physical models simulate details of the process, including
specific water quality phenomena, under very specific prescribed conditions (e.g., a
"design" storm, 7Q10). Thus, for such models the assumption is that the system behavior is
known in sufficient detail though possibly for very limited conditions. On the other hand,
screening level decision models simulate an aggregate of processes for a range of conditions
(e.g., die entire probability distribution of flow conditions in the receiver) so that the system
behavior is known in less physical detail, but for a wider range of conditions.
Thus, physical models are generally limited to the analysis of specific events, while
decision models analyze a continuum of events. The trade-off in model selection is between
more detailed analyses of behavior for less general conditions and less detailed analyses of
behavior for more general conditions.
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In order to address the decision problem, it is necessary that the; level of
information is most appropriate for the decision at hand. Is it preferable to have detailed
information on what happens at the end-of-the-pipe (which may then be used as an
acceptable surrogate for in-stream water quality) or information on resulting water quality,
or even information on the level of beneficial use? As we move from input to use, we gain in
detail but lose in scope. There is always a trade-off between detail and scope.
I
2.4.1 Approaches to Runoff Modeling j
Even a cursory review of the literature indicates that a number of different modeling
approaches are available (EPA 1985a and b). These modeling approaches can be grouped
into the following categories: (i) Regression methods, (ii) Physically-biased deterministic
models, and (iii) Probabilistic techniques.
These different methods are complimentary to each other and represent varying
levels of complexity and detail. The following sections present brief discussions of well-
documented and tested examples of each of these three types of models and evaluate their
potential use for predicting pollutant discharges from urban areas. A detailed description and
evaluation of each type of model can be found in Miracle (1986).
2.4.2 Regression Models
Regression models provide a quick and simple means for estimating urban runoff
quantity and quality. They are based on the assumption that explanatory variables, such as
drainage area, basin imperviousness, and mean annual rainfall, can explain variation in
stormwater discharge loads on a specified time scale such as long-term, annual or seasonal.
The typical procedure is to perform a regression on natural or transformed data set(s) and to
determine the percent of the variability (usually expressed as R2) that can be explained by
the variables used in the regression. Although causal hypotheses may be advanced to
justify variable inclusion, actual physical mechanisms are not involved, and actual
cause/effect relationships cannot be determined. Caution should be used when applying
regression models outside of the particular study for which they were developed.
Two additional general comments should be made. First, the use of transforms
can alter significantly the form of the regression equation and can introduce bias that must
be properly corrected for. For example, the use of the log transform converts an additive
regression model to a multiplicative one and, although the correct median is preserved, it
introduces a bias in the predicted mean. The second point has to do with spurious
correlations. If one regresses loads against rainfall there will be a spurious self-correlation
because the load is the product of discharge quantity and concentration,, and discharge
quantity can be expressed as the product of rainfall and a runoff coefficient. Thus one can
say, without even performing a regression, that rainfall quantity will be the single most
important variable in predicting runoff loads. The recent United States Geological Society
(USGS) study cited below came to this conclusion. The correct way is to perform the
regression on concentrations, not loads, and thus avoid the spurious effect. Note that a
correlation coefficient based upon transformed data does not accurately reflect the
correlations between untransformed data.
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2.4.3 Simulation Models
Numerous models have been developed to simulate stormwater runoff quantity and
quality from urban and nonurban areas. However, there are relatively few models that are
truly operational; that is, models that have documentation, user support, support by a
government agency, and are widely used by more than just the model developers. Models
meeting these requirements are SWMM (EPA's Stormwater Management Model), STORM
(U.S. Corps of Engineers, Storage, Treatment, Overflow, Runoff Model, HEC (1977)),
and HSPF (EPA's Hydrologic Simulation Program—Fortran, Johanson, et al. 1980).
SWMM and STORM include routines to compute the effect of certain controls.
Another example is the USGS model developed for urban quality predictions by
Alley and Smith (1982). However this model has not been applied as widely as SWMM,
STORM, or HSPF, and its quality routines are derived basically from those in SWMM and
HSPF (Huber 1985). Similarly, the Illinois State Water Survey's model ILLUDAS
(Illinois Urban Drainage Simulator), the Southeast Michigan Council of Government's
model RUNQUAL (Runoff and Quality Model), the Massachusetts Institute of
Technology's model, MITCAT (M.I.T. Catchment Model), EPA's model HVM-QQS
(HVM-Quantity-Quality Simulator) have not been used widely.
Generally, these models simulate the buildup of pollutants during dry periods,
followed by washoff during storms. This procedure was first implemented in the original
EPA Storm Water Management Model, SWMM (Metcalf and Eddy, et al. 1971). The
formulation of the buildup and washoff equations varies from model to model, ranging from
highly flexible in the SWMM to rigid in STORM. Another approach, besides modeling
buildup and washoff, is the use of a rating curve methodology. This is an option in
It is expected that complex simulation models should represent an improvement
over simple regression equations for prediction of runoff quality, however, simulation
models are very data intensive. These models may be viewed as "very complex regression
equations" that must be calibrated in the same way that least squares is used to fit ordinary
regression relationships to data. In addition to the advantage that simulation models
provide a better estimate of storm event loads, these models provide a more physically
realistic predictive mechanism which, when calibrated, may be altered more easily to
examine the effects of changes and abatement practices. Also, simulation models can be
used to estimate temporal and spatial distribution generally not possible using regression
techniques. For example, continuous simulation models may be used to generate a time
history of pollutant loads from which a frequency analysis may be conducted. This permits
an analysis on the basis of pollutant characteristics rather than on rainfall or runoff
characteristics. (The probabilistic method, described below, approaches this same end by
means of a derived distribution, rather than simulation.)
2.4.4 Probabilistic Methods
Probabilistic methods are based on the realization that nonpoint sources in general,
and urban runoff in particular, provide highly variable inputs (concentrations and loads) to
receiving water bodies. They are generated intermittently at highly variable intervals, and
when they do occur, are highly variable in both time and space. Furthermore, the impacts of
nonpoint sources are strongly influenced by significant differences in important
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characteristics of individual receiving water bodies and by the inherent variability of natural
water systems.
For these reasons we have chosen to develop a probabilistic methodology,
described in detail in subsequent chapters, for analyzing the water quality effects of urban
runoff on rivers and streams. Our methodology starts with classical deterministic water
quality models of pollutant fate and uses probability distributions of the relevant input
parameters to estimate the probability distribution of instream concentrations. The critical
situations (e.g., low flows, high loads) are all considered, but they are given their proper
weight as dictated by their observed frequencies of occurrence and intercorrelations. The
approach, therefore, does not depend upon the prespecification of critical conditions for the
analysis; a significant advantage over the other approaches.
Probabilistic methods have the advantage of being able to consider either very long
or short periods as appropriate. Sufficient data for input parameters must be available to
provide reasonable confidence in the distributions developed. This is especially true if
extreme events are of concern, and a high degree of confidence in their values is desired.
Note that monitoring constraints are much less restrictive than those required by simulation
models. Further, probabilistic data (e.g., streamflow, effluent characteristics) are much
more transferable than specific sequences and sites. For example, the assembly of data to
permit reliable estimates of discharge characteristics of urban runoff, CSOs (Combined
Sewer Overflows), and POTW (Publicly Owned Treatment Works) discharges utilizing a
specific process design will allow a rational, compatible examination of point versus
nonpoint source trade-offs.
2.5 CHARACTERIZATION OF RAINFALL AND RUNOFF EVENTS
I
2.5.1 The Rainfall/Runoff Event
The rainfall or runoff may be viewed as a series of independent, randomly
occurring events. Each event can be schematized as a uniform, rectangular hydrograph,
characterized by its duration, volume, average flow rate or intensity, and the time elapsed
since the last event (interevent time). The interevent time is measured between the
midpoints of event. Rainfall parameters for a given site may be estimated from historical
records obtained from National Oceanic and Atmospheric Administration (NQAA), e.g., by
using the SYNOP program (WCC 1988).
I
I
2.5.2 Selection of Interevent Duration
The usual approach used for convenience in processing hourly rainfall data is to
choose a minimum interevent time, such that rainfall values separated by less than this
duration are considered part of the same storm. Values separated by times greater than or
equal to this duration are considered to be independent. Several methods for choosing this
duration exist (Heaney, et al. 1977), but the most common is to assume that the interevent
times are exponentially distributed (Hydroscience 1979; Restrepo-Posada and Eagleson
1980). An exponential distribution is a special case of the gamma distribution when the
coefficient of variation is unity. Note that the gamma distribution is typical of rainfall
parameters. The SYNOP program iteratively estimates the interevent period until the
coefficient of variation of time between event midpoints equals 1.0 Resulting values are
usually in the range of 3 to 24 hours (Hydroscience 1979).
2-5
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2.5.3 Estimation of Runoff Statistics from Rainfall Statistics
2.5.3.1 Quantity Statistics. As a first approximation the runoff coefficient method is
used to calculate the mean runoff event volume and flow from the corresponding rainfall
statistics.
VR = Rv*VP (2.1)
where:
VR = mean runoff event volume [L3]
Rv = average runoff coefficient [dimensionless]
VP = mean rainfall event volume [L3]
The runoff coefficient, Rv, represents the ratio of average runoff to rainfall and
neglects depression storage. This ratio varies from storm to storm, but for preliminary
analyses a representative constant value is usually adequate (Nix 1982). The value of Rv can
be estimated by an analysis of local rainfall/runoff data or estimated from one of several
simple techniques (Hydroscience 1979).
The mean runoff flow rate is calculated as:
QR = Rv*QP*(DP/DR) • (2.2)
where:
QR = mean runoff event flow rate [L3/T]
QP = mean rainfall event intensity [L3/T]
DP = mean rainfall event duration [T]
DR = mean runoff event duration [T]
The ratio DP/DR is included to account for the fact that runoff continues after the rainfall
event has subsided. A method for estimating DR using unit hydrograph analysis is presented
by Hydroscience (1979).
The mean interevent time for runoff events, TR, is assumed to equal the rainfall
duration, TP. The coefficients of variation for runoff event flows, CVQR, and volumes,
CVVR, are also assumed to equal to their rainfall counterparts.
2.5.3.2 Quality Statistics. The mean pollutant load for all runoff events is
determined by the mean pollutant concentration and the mean runoff event volume:
MR = CR*VR (2.3)
where
MR = mean runoff event pollutant load [M]
CR = mean runoff event concentration [M/L3]
This equation assumes that the pollutant event mean concentrations are independent of
runoff volumes. If this is not true, an adjustment is available as discussed below.
Similarly, the mean pollutant load rate can be estimated by:
2-6
-------
WR = CR*QR (2.4)
where:
WR = mean runoff event pollutant load rate [M/TJ
Again, the assumption of independence between pollutant concentrations and runoff flow is
made. If the independence assumptions are inadequate, the following corrections can be
made (Hydroscience 1979).
MR = CR * VR * (1 + CVCR * CVVR * RCV) (2.5)
WR = CR * QR * (1 + CVQR * CVQR * Rcq) (2.6)
where:
CVCR = coefficient of variation for runoff event mean pollutant
concentration [dimensionless]
RCV = linear correlation coefficient between pollutant concentrations
and runoff volumes [dimensionless ranging from -1 to +1]
Rcq = linear correlation coefficient between pollutant concentrations
and runoff flow rates [dimensionless]
A positive value of RCV or Rcq indicates that higher flows or volumes produce higher
concentrations. A negative value indicates that the dilution effect of large runoff events is
dominant.
Finally, it is assumed that all the variation in loads is due to variation in runoff
volumes. Thus,
CVMR = CWR ("2.7)
and i
CVWR = CVQR (2.8)
where ;
CVMR = coefficient of variation of runoff pollutant loads [dimensionless]
CVWR = coefficient of variation of runoff pollutant load rates [dimensionless]
CVVR = coefficient of variation of runoff volume [dimensionless]
CVQR = coefficient of variation of runoff flow [dimensionless]
The probability distribution of loads and concentrations has been assumed to be
gamma for application to certain storage-treatment analysis (Di Toro and Small 1979; Small
and Di Toro 1979) but is usually taken to be lognormal for application to stream water
quality impacts (Di Toro 1984).
2.6 SUMMARY
Regression equations, simulation models and the probabilistic method are
analytical tools for predicting pollutant loads from urban runoff. The primary advantage of
2-7
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the regression and probabilistic methods is that both allow a relatively quick, simple, and
inexpensive screening of stormwater problems. This is particularly useful in the early stages
of the planning process (Hydroscience 1979).
Each method has its own particular advantages and disadvantages. The regression
methods are simple and include explanatory variables such as rainfall characteristics, traffic
counts, population density, etc. However, this method can yield only the mean for a given
set of inputs and their range of validity is generally limited. Implementation of simulation
models requires considerable resources, especially in terms of calibration and verification
data requirements, and are not recommended for screening type analysis.
The probabilistic method requires rainfall statistics (readily available) and known
mean concentrations of pollutants in the runoff (somewhat harder to obtain) as input
variables (plus streamflow statistics, if receiving water effects are to be investigated). The
output is generally a frequency distribution of a water quality parameter and can be used to
estimate return periods, i.e., the frequency that a given concentration level be equalled or
exceeded. The method includes some approximations if an analytical solutionis desired,
however, a few of which may be important in limited individual cases (Roesner and
Dendrou 1985).
Based on the foregoing, the probabilistic approach has been selected for
screening level analysis. Its success in EPA's NURP and NOAA's National Coastal
Discharge Inventory Program, as well as the Federal Highway Administration (FHWA)
Runoff Program, and its increasing use by state water quality planners and other
professionals at the local level further encourage its selection. Also, as noted earlier, it has
been exhaustively reviewed by EPA's Science Advisory Board, who found the method to be
technically sound.
Probabilistic methodology provides a method to deal with the inherent variablity of
the stormwater runoff process and its impacts on the receiving water. It also emphasizes
the impossibility of reliably predicting pollutant loads based on an indivdual storm event
explicitly quantifies the uncertainty involved in the estimation of individual site pollutant
runoff. These characteristics of the probabilistic approach provide a basis for evaluating the
confidence level of the decision involved.
2-8
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CHAPTER 3.0
THEORETICAL BASIS OF MODEL
3.1 INTRODUCTION
Urban runoff, stream hydrology, and related water quality processes are inherently
random phenomena. Thus, it is appropriate to analyze water quality effects using a
probabilistic framework. The three approaches, as described below use the probability
distribution of model inputs (e.g., stream discharge, runoff flow, and concentrations) to
estimate the probability distribution of the resulting concentrations in the stream. Thus, if
the level of concentration that impairs the use of the receiving water is known, this method
can be used to estimate the frequency with which that use will be impaired.
The probabilistic water quality analysis starts with a deterministic mass balance
model, i.e., a mathematical representation of the mixing of surface runoff with the river or
stream and uses three different approaches. These include (i) the Analjtical Derived
Distribution Approach, (ii) the Direct Numerical Solution Approach (Gaussian Quadrature
Method), and (iii) the Monte Carlo Simulation approach to compute the probability
distribution of the stream concentrations. Critical situations (e.g., low flows, high loads,
etc.) are considered, and given their proper weight as dictated by their observed frequency
of occurrence.
3.2 MODEL FRAMEWORK AND STATISTICAL CONCEPTS I
The conceptual formulation consists of a spatially lumped, event-averaged model.
All concentrations and flows are event mean values. Although urban runoff discharges
usually enter the receiving water at several locations, the model considers the aggregate of
flows and assumes complete mixing in the stream. Figure 3-1 shows a.n idealized
representation of urban runoff discharges entering the river. Downstream concentrations
are estimated using mass balance or a dilution model:
- Qs r , OR
~ ~ ~ " (3.1)
where:
Co = downstream concentration in the river [mg/1]
CR = pollutant concentration in the runoff [mg/1]
Cs = upstream concentration in the river [mg/1]
QR = runoff flow rate [m3/s]
Qs = upstream flow rate in the river [m3/s] i
t
The objective of the methodology presented here is to estimate the probability distribution
of GO, for specified probability distributions of the input parameters Qs, QR, CR, and Cs
as shown in Figure 3-1.
The three probabilistic methodologies presented below, consider Cs, Qs, CR, and
QR as random variables, each specified as a probability distribution. As described in the
3-1
-------
CT
£
LI-
'S
E
3
O
CT
S>
LL
O
Runoff Flow
cr
s.
"5
O
1
O
Runoff
Concentration Cc
Runoff
Concentration
CR
cr
£
LL
"5
E
3
O
i
River
Concentration
Co
FIGURE 3-1
SCHEMATIC REPRESENTATION OF WATER QUALITY EFFECTS
OF URBAN RUNOFF ON RIVERS AND STREAMS
3-2
-------
characterized as lognormal distributions. The first two methods presented below the
Dtaa
The use of probability distributions other than log normal has been proposed in the
n/?HeSent taghly, VariaWe Water quality P^eters. For example, £SZ
(1983) presented an approach to estimate water quality criteria violations due to combined
°WS " PearS°n Percendles- * site-specific data suggest that disSbuSS
appr°pmte' ** ^ aPProach' a Monte Quio simulation
A A- * 3 -and 3'4' *e A11*1?11'*1 Derived Distribution Approach for computing
the distabution of Co is presented Although this method gives approximate results, it is
amenable to hand calculation and hence very useful. A numerical Gaussian Quadrature
method is presented in Sections 3.5 and 3.6. Sections 3.7 and 3.8 present the Monte Carlo
bimulation Approach. Finally, a useful transformation of the probabilistic results into a
more meaningful, time-based recurrence interval is described in Section 3.9.
3.3 THE ANALYTICALLY DERIVED DISTRIBUTION APPROACH
The analytically derived distribution approach is an analytical procedure to estimate
the exceedance probability of the downstream concentration and is based on the assumption
that this concentration is lognormally distributed. It involves the computation of the first
two moments (i.e., the mean and variance) of the downstream concentration.
f 11 • assumption of downstream concentration follows from the
thefaS? observatlons- The &lution model (i.e., mass balance equation) can be written in
Cs(l-0)
0 =
QR
QR+Qs
(3.2)
(3.3)
where
0 = the runoff flow fraction or dilution [dimensionless].
th* / ? ^ 0 ^oe lo£normal>then' since CR and Cs are assumed to be lognormal,
the products CR 0 and Cs (1 - 0 )would also be lognormal. Further it is assumed that the
sums ot lognormal random variables are also approximately lognormal (Janos 1970)
These observations suggest that if 0 is approximately lognormal, then C0 should be
approximately lognormal, and the mean and variance suffice to determine its distribution
ihis approach is approximate because, in fact, 0 has an upper bound of 1 and is thus not'
truly lognormal. While 0 may be appropriately considered lognormal in many cases, this
assumption may introduce unreasonable errors in other cases, and one of the approaches
presented in Sections 3.5 through 3.9 must be used
3-3
-------
With the above assumptions, the arithmetic mean and variance of Co are given by:
(3.4)
can be estimated using:
Coa = exp [|lln(Co) +
where
(3.8).
(3.9)
= the downstream concentration exceeded with a probability ( 1 - a) [mg/1]
= the standard normal a quantile [dimensionless]
In order to use Equations 3.4 and 3.5, it is necessary to compute the moments (i.e.,
mean and variance) of 0. Warn and Brew (1980) suggest a numerical method. However,
for screening level analysis, it is useful to find approximate expressions for these moments
and to use the numerical techniques for the evaluation of the exact distribution of Co- The
runoff fraction Equation 3.3 can be expressed as:
3-4
-------
ri=
P
1+D i (3.10)
where
D=Qs
OR - (3.11)
If both Qs and QR are lognormal, D would be lognormal. Thius the logarithmic
(normal) mean and standard deviation of D can be estimated using:
jiin(D) = nta(Qs) - MQu) (3.12)
a2!n(D) = a2in(Qs) + a2hl(QR) - 2o-ln(Qs)aln(QR)pln(Qs,QR) (3. 13)
i
where Pin(Qs.QR) is the cross-correlation coefficient between ln(Qs) and ln(QR).
Using Equation 3. 10 the a-quantile of 0 can be computed (refer to Appendix A) using:
l+exp[jiln(D) - zaaln(D)j ; (3
where
Pr(0<0a) = a (3.15)
Since 0 is assumed to be lognormally distributed, ln(0) is normally distributed with
mean of fiin(0) and standard deviation of (7^(0). Using this assumption, the cci and 0.2
quantiles of 0 can be estimated using:
ln(0ai) = M-ln(0) + ZaiGln(0) , (3.16)
ln(0«2) = |Iin(0) + Za2 0]n(0) , (3.17)
Equations 3.16 and 3.17 imply: ;
_ [ln(0ai) - In(0a2)]
(3.18)
3-5
-------
In the original development (Di Toro 1984), zotl was set equal to 1.645 and z«2 was set
equal to -1.645 to force agreement at the 5% and 95% quantiles i.e. by estimating the 5th
and 95th percentiles of 0 using Equation 3.14. In many water quality impact evaluations,
obtaining a close approximation of the stream concentrations with 3-year recurrence
intervals (per EPA toxic criteria) is of more practical interest than the concentration levels
that occur at a weekly or monthly frequency. Results of the sensitivity analyses, discussed
in Chapter 5, indicate that estimating the mean and standard deviation of 0 using two
values in the tail (e.g., exceedance percentiles of 90 and 99.6 percent) results in better
estimates of stream concentrations in the 1 to 5 year range of recurrence intervals, for a
range of values of the input parameters.
Equations 3.18 and 3.19 can then be used to estimate the logarithmic mean and
standard deviation of the lognormal approximation of 0. The arithmetic mean and standard
deviation of 0 (required for Equations 3.4 and 3.5) can be computed as:
2
= exp[|iln(0) + 0.5 aln(0)] (3.20)
Thus, the steps required to compute the exceedance probabilities of Co are:
(3.21)
^^Br
(i) Analyze the streamflow and the runoff data to estimate the mean and
standard deviation of the lognormal distributions.
(ii) Estimate the first two moments of D using Equations 3.12 and 3.13.
(iii) Estimate the values of 0 for the selected percentiles using Equation 3. 14.
(iv) Estimate the log mean and log standard deviation of 0 using Equations 3.18
and 3.19 and the arithmetic mean and standard deviation using Equations
3.20 and 3.21 respectively.
(v) Estimate the arithmetic mean and standard deviation of Co using Equations
3.4 and 3.5.
(vi) Estimate the desired percentile of Co using Equation 3.9.
(vii) Estimate the mean recurrence interval using Equation 3.43 described in
Section 3.7.
In Appendix B, an example calculation is given. The accuracy of this approximate
procedure is examined in Chapter 5.
3-6
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ANALYTICALLY DERIVED DISTRIBUTION
above ^ f°Uowing assumPtions ^ inherent in the probabilistic methodology presented
i
• All flows (QR,Qs) and pollutant concentrations (CR,CS) are lognormaUv
distributed.
• The runoff and upstream concentration (CR and Cs) are assumed to be
uncprrelated with the runoff flow and streamflow (QR and Qs) respectively as
weU as with the parameters 0 and (1-0). ,**»'* y>
• The sum of CR0 and Cs( 1-0); i.e., CQ is assumed to be lognormally distributed.
• The dilution parameter 0 (refer to Equation 3.3) and (l-0),are both assumed to
be lognormally distributed.
In addition, the mathematical model (Equation 3.1) uses the following assumptions:
• The stream is assumed to be fully mixed both upgradient and downgradient of
the urban runoff discharges.
• The runoff from the urban area is aggregated into a single event mean flow and
concentration value.
]
3.5 THE DIRECT NUMERICAL SOLUTION !
• * * * probability of downstream concentration, Co , exceeding ia particular value of
interest, C*0, can be expressed as a multiple integral of the joint probability density
function over the values of flows and concentrations for which C0> C*0 . This requires an
Since Qs and QR are both lognormal, so also is D. Fon a fixed CQ* the equality:
1+D 1+D
(3.23)
defines a surface in CR, Cs, and D space; and the required probability is the integral of the
joint probability density function over this surface. Specifically, for a fixed D:
CR + CSD = CO(I+D)
(3-24)
3-7
-------
is a straight line in the CR, Cs plane as shown in Figure 3-2. Note that the inequality m
equation 3-22 is valid in the region to the right of the straight line given by equation 3-24.
This region can be divided into two parts Region A and Region B defined by:
Region A
0
Reion B
0 £ CR < °o
The above is valid for the entire range of values of D; i.e.,D can vary from [0,°°]. In
order to calculate the exceedance probability shown in equation 3-22, it is necessary to
integrate over the regions A and B. Thus:
Pr{C0>C;} =L(0CS1|^lf(cR'CS'D) dCR dcs ^ + L~Lfo"f(CR'Cs'D) dCR dCs ^ (3-25)
Region A Region B
where:
= Co* (1+D) - CSD (3-26)
Csi = CQ (1+^) (3.27)
and f(CR, Cs, D) is the joint probability density function for CR, Cs and D.
Since CR, Cs and D are assumed to be lognormal, a change of variables can be
made and the joint probability density function in Equation 3.25 can be expressed as a
trivariate gaussian. A convenient method of expressing this density is the product of the
marginal and conditional densities, (Anderson 1958), each of which is umvanate gaussian.
Thus:
where:
f(cR,cs,d) = f(d) f(cs I d) f(cR I cs,d)
CR = hi CR
Cs = hi Cs
d = lnD
(3.28)
(3.29)
(3.30)
(3.31)
3-8
-------
FIGURE 3-2
REGIONS OF INTEGRATION FOR THE JOINT PROBABILITY DENSITY FUNCTION
3-9
-------
f(d)-N[d;n(d),a(d)] (3.32)
f(csld) = N[cs; ^(csld), a(csld)] (3.33)
f(CRlcs,d) = N[cR; ^(cRlcs,d), a(cRlcs,d)] (3.34)
Note Equations 3.32, 3.33 and 3.34 are the univariate gaussian densities with the
appropriate conditional logarithmic (normal) means and standard deviations. Equation
3.25 can now be written as:
Pr{C0>C0} = ]J(d) {_ fCcsldfCcRlc&d) dcR dcsdd +
j J(d) j^ f(csld) £f(cRlcs,d) dcRdcs } dd (3 35)
Equation 3.35 can be simplified using the following relationships:
rf(crlcs,d)dcr=l
j—
f f(ctlcs,d)=Pr{Cr>CR1}=QR
Jac-fu
This results in the following equation:
Pr{C0>Co} = J J(d) J^1 «csld) QR dcsdd + £f(dXS dd (3 36)
n and Qs are evaluated using the following equations:
? _ 0* rln(CRi) - |i(cRlcs, d)n
-^^ [ a(cRlcs,d) J (3.37)
a(csld) (3.38)
where Q*(z*) = Pr(z>z*), the probability that a zero mean, unit variance gaussian random
variable exceeds z*. This function can be computed using approximate expressions
(Abramowitz and Stegun 1954).
The remaining integrals can be evaluated numerically after a transformation of the
variable as discussed in Appendix A. The integral is solved using standard gaussian
quadrature formulae.
3-10
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3.6 KEY ASSUMPTIONS IN THE DIRECT NUMERICAL SOLUTON
The numerical methodology presented above is based on a number of assumptions.
These assumptions are listed below:
• The stream is assumed to be fully mixed both upgradient and downgradient of the
urban runoff discharges.
• The runoff from the urban area is aggregated into a single event mean flow and
concentration value.
• All flows (QR,QS) and pollutant concentrations (CR,CS) are lognormally
distributed.
3.7 THE MONTE CARLO SIMULATION APPROACH j
A commonly used approach for analyzing the uncertainty in model outputs due to
uncertainty in the model inputs is the Monte Carlo simulation approach. This approach is
conceptually simple, general in its applicability and asymptotically exact, but not suitable
for computationally intensive models. The application of this method to the dilution model
is described below.
Given a set of deterministic values for each of the input parameters, QS, QR, CS
and CR, the dilution model Equation 3.1 computes the downstream concentration Co
Application of the Monte Carlo simulation procedure requires that at least one of the input
variables, QS, QR, CS and CR, be uncertain and the uncertainty represented by a
cumulative probability distribution. The method involves the repeated generation of
pseudo-random values of the uncertain input variable(s) (drawn from the specified
distribution and within the range of any imposed bounds) and the application of the model
using these values to generate a series of model results, i.e., values of Co. These results
are then statistically analyzed to yield the cumulative probability distribution of the model
response. Thus, the various steps for the application of the Monte Carlo simulation
technique involve:
i) Selection of representative cumulative probability distribution functions for the
model input parameters.
ii) Generation of pseudo-random numbers from the distributions selected in (i). These
values represent a possible set of values for the input variables.
iii) Application of the dilution model to compute the downgradient concentrations.
iv) Repeated application of steps (ii) and (iii). ;
v) Presentation of the series of downstream concentrations generated in step (iii) as a
cumulative probability distribution function (CDF).
3-11
-------
The random number generators included in the software allows the user to select one of the
following distributions for the input parameters:
i) Normal
ii) Lognormal
iii) Uniform
iv) Exponential
v) Empirical
The first two distributions require the user to specifiy the mean and the standard
deviation. The third requires minimum and maximum values. The fourth distribution
requires only one parameter - the mean of the distribution. Finally for the empirical
distribution, the user is required to input the coordinates of the cumulative probability
distribution function which is subsequently treated as a piece-wise linear curve.
3.8 KEY ASSUMPTIONS OF THE MONTE CARLO SIMULATION APPROACH
The Monte Carlo simulation method does not use any assumptions other than those
included in the mathematical model (Equation 3.1). These are:
• The stream is assumed to be fully mixed both upgradient and downgradient of the
urban runoff discharges.
• The runoff from urban discharges is aggregated into a single event mean flow and
concentration value.
3.9 RECURRENCE INTERVAL CALCULATION
The fundamental result of the statistical analysis described above is the derived
cumulative probability distribution of event mean concentrations downstream of the urban
area. This distribution is given by:
(3.39)
where
Coa = the concentration that is exceeded with probability 1-cc
Za = the standard normal quantile.
One way of interpreting the probability corresponding to a given concentration level
( i.e., Pr{Q£C*o} ) is the long term average fraction of events that a stream event mean
concentration equals or exceeds the specified level. For example, a probability of 0. 1 0
would specify that, on average, one in ten events have a downstream event mean
concentration equal to or greater than the specified value.
For purposes of evaluation and interpretation, it is useful to transform the basic
probability statistic into a more meaningful or intuitive form. By combining the percent of
storms that cause various concentrations to be exceeded with the average number of storms
per year, a time-based recurrence interval may be established as described below.
3-12
-------
Tf P / r >r f*$ te^ly) on the marginal distribution of random variables.
If Pr{CoJC*0} is the probability of a value of magnitude C*0 being equaled or exceeded in
a given time period, then the mean recurrence interval (MRI) defined as the reciprocal
of probability is the average number of time periods between exceedances.
Thus annual recurrence is the reciprocal of the probability that an event
concentration of a given magnitude, C*0, is equaled or exceeded in a year. The statement
or the problem is:
= Pr{max(C0,i, C0,2,..., C0)N) > C*0} , (3.4o)
where
Co"18* = the maximum event concentration in a year [mg/1]
N = the number of events in a year.
Assuming that event concentrations are independent and identically distributed with
a known distribution, such as lognormal, the above equation becomes:
(3
Or the mean recurrence interval in years is:
MRI = 1
1 • PR{CO>CQ}N c\ Af)\
\~>-^-£)
A first order approximation to this is given by:
MRI = -
PR{Co>C0}N (343)
The mean recurrence interval in years as defined by the above equation Is a convenient and
meaningful way to interpret the basic probability results. In Chapter 4 the use and
interpretation of recurrence interval is described and example results are given
3-13
-------
-------
CHAPTER 4.0
APPLICATION OF METHODOLOGY
4.1 INTRODUCTION
This chapter presents an example application of the three probabilistic
methodologies described in the previous chapter. A step-by-step description of the
Analytically Derived Distributions Approach that is amenable to hand calculations is
K^fnH^Af ?e,C°mparfd.with ** results obtained using to Direct Numerical
Method and the Monte Carlo Simulation Technique. Since these mtthods are not
conducive to hand calculations, a Fortran 77 PC-based computer program described in
Chapter 6 was used. A comparison of the results obtained using the three methods is
presented in Chapters.
4.2 DATA NEEDS AND SOURCES
The input data consist of the parameters of the probability distributions for
streamflow and concentration of the pollutant to be analyzed, at a location immediately
upstream of the urban area. Similar data are required for the urban runoff discharges in
this case based on storm event averages for runoff flow and concentration. If computations
are performed with upstream concentrations of zero, the output would reflect the
downstream concentrations due to urban runoff discharges.
Streamflow characteristics may be estimated by analysis of USGS daily streamflow
records for an appropriate flow gaging station in the study area. Alternatively, estimates
tor typical regional values, suitable for preliminary estimates, can be derived from
information presented in the final NURP report (WCC 1983).
Statistical parameters for upstream concentration may be computed from local
monitoring data or STORET data, if the available period of record is sufficiently long.
Statistical parameters for pollutant concentrations in urban runoff may be computed
by: (i) using data obtained from local monitoring programs, (ii) the infomation presented
in the final NURP report, or (iii) by a combination of the two sources of data.
Statistical parameters for the urban runoff flows may be computed based on
analyzing data from a local monitoring program. However, if such information is not
available, an alternative is to estimate a runoff coefficient for the urban area being
examined. This would then be combined with the long-term rainfall record for the local area
to denve estimates of long-term statistics of runoff flows. Details of this method are
presented in WCC (1983).
For the Derived Distribution Approach and the Direct Numerical Method
computations are performed using log transforms of the source data; however results
evaluations, and interpretations are most appropriately presented as the relevant arithmetic
values. Note that there are specific relationships between arithmetic and logarithmic
parameters, so that transpositions to the desired forms can be made quite readily These
relationships are presented in Table 4-1. Note that for the two-parameter lognormal
distributions used in the methodology, specifying values for any two of the arithmetic or
logarithmic parameters defines all others.
4-1
-------
TABUE4-1
INPUT DATA REQUIREMENTS AND RELATIONSHIPS
FOR LOGNORMAL DISTRIBUTIONS
Arithmetic
Input
Parameter
Median
x
Mean
Wx>
Std.
Dev.
0(x)
Coef.
Var.
V(x)
Logarithmic
Log
Mean
^ln(x)
Log
S.D.
Obi(x)
Strcamflow Qs
Stream Concentration Cs
Runoff How QR
Runoff Concentration CR
Qs
OR
n(CR)
v(Qs)
o(Cs) v(Cs)
0(OjO V(0jl)
o(CR) v(CR)
MQs)
Oin(CR)
RELATIONSHIPS BETWEEN STATISTICAL PROPERTIES
ARITHMETIC AND LOGNORMAL TRANSFORMS
Arithmetic
x
Hx
0X
CTx
Vx
X
DEFINITION OF TERMS
Random Variable
Mean
Variance
Standard Deviation
Coefficient of Variation
Median
Logarithmic
lux
(not used)
(not used)
M =
-1
4-2
-------
When the underlying population of a variable is presumed to be lognormally
distributed, the appropriate way to analyze data drawn from a limited s;ample of the overall
population is to convert observed values to their corresponding logarithmic values, and to
perform the standard statistical analyses (determination of mean and standard deviation) on
the log transforms of the data. Arithmetic statistical parameters can then be computed from
the log-mean and log-standard deviation using the relationship shown in Table 4-1. The
presumption is that the best estimator of the value of interest (e.g., the mean) is provided
by this procedure. It should be recognized that estimates of the mean determined this way
will differ from values determined from a simple averaging of the observed data, especially
where sample size is small. For lognormal distributions, estimates of the mean using the
two approaches will converge to a common value as the number of observations increase.
4.3 COMPUTATION PROCEDURE USING THE ANALYTICALLY DERIVED
DISTRIBUTION APPROACH
The basic approach is summarized on Figure 4-1 and consists of the following four
steps:
Step 1: Compute Statistical Parameters of Inputs Assuming Lognormal Distribution
Analyze local data or use NURP summaries to define the parameters of the
lognormal distributions for each of the four input parameters. Published summary
characteristics are usually in terms of an arithmetic mean (or median) sind standard deviation
(or coefficient of variation). When the input parameters are estimated from analysis of site-
specific data, the log transforms of individual observations are used to estimate the log-
mean and log-standard deviation. In either case, the approximate formulae listed in Table
4-1 may be used to compute the remaining statistical parameters.
Step 2: Compute the Parameters of the Flow Ratio Og/O£ and the Dilution Factor
Estimate the statistical properties of the dilution factor (0) defined by Equation 3.3.
For this it is necessary to compute the statistical properties of the flow ratio (parameter D
defined by Equation 3.11) using Equations 3-12 and 3-13. Equation 3.14 is then used to
estimate the values of (0) for the selected percentiles. The values of za in Equation 3.14
are obtained from a Standard Normal Probability table for the corresponding value of a .
A few values of a and za are given below:
a 1 - a za
95 % 5 % z95= 1.645
5 % 95 % z.05= -1.645
50 % 50 % z.50= 0
84.13% 15.87 % z84= 1.0
15.87 % 84.13 % z 16 = -1.0
90.0% 10.0 % z9o = 1.282
99.6% 0.4 % z996= 2.652
4-3
-------
UPSTREAM
OS
Time
URBAN RUNOFF
QR
Probability
Time
Probability
DOWNSTREAM
DILUTION
Probability
Z
2
oc
Note: (Q indicates computation step as described in
procadure description
STREAM CONCENTRATION
Recurrence
Interval
FIGURE 4-1
SCHEMATIC OUTLINE OF PROBABILISTIC ANALYSIS METHOD
4-4
-------
These values are used to estimate the log-mean and log-standard deviation of
dilution (0) of the lognormal approximation of (0), using Equations 3.18 and 3.19 . The
arithmetic transforms are then computed using appropriate equations presented in Table 4-
1. Specifically, the arithmetic mean (|0.(0)) and standard deviation (o(0)) (Equations 3.20
and 3.21) are required in Step 3.
As discussed in Section 3.3, the probability distribution of the dilution function (0)
is not truly lognormal since it has an upper bound of 1 and a lower bound of 0. To ensure
that the lognormal approximation for this distribution yields the greatest accuracy the
estimates of instream concentration values in the tail of the distribution are selected. For
this example, exceedance percentiles (1 - a) of 10 and 0.4 percent are selected. The lower,
more frequently occurring stream concentrations will tend to be overestimated (perhaps
appreciably) by this choice. However, these values have little practicsil significance. The
sensitivity of projections to this approximation is examined further in Chapter 5.
Step 3: Compute the Statistical Parameters of the In-Stream Concentration
The probability distribution of in-stream concentration is computed from the
statistical parameters for flow, concentration, and dilution estimated in the previous steps,
using Equations 3.4 and 3.5. The coefficient of variation is calculated using Equation 3.8.
These statistics are used to derive the log-transformed statistical parameters using the
information presented in Table 4-1.
Step 4: Prepare Probability Distributions of Downgradient Concentrations and the
Recurrence Intervals
Once the appropriate statistical parameters have been computed, the expected
frequency at which a specific stream concentration of interest will occur can be determined
in one of two ways.
A probability distribution plot can be constructed on log probability paper, using the
estimated statistical parameters for the in-stream concentrations. Using this procedure, the
probability of occurrence of the concentration of interest can be scaled directly from the
plot. An example of this approach is illustrated by Figure 4-2.
Alternatively, the concentration that will not be exceeded at some specific frequency
(or probability) can be calculated using Equation 3.9. Similarly, this equation can be used
to estimate the stream concentration for a given probability of exceedaace by using (1-a)
for the calculation.
The probability relationship shown on Figure 4-2, or specific-percentiles calculated
by Equation 3.9, represents the distribution of stream concentrations during precipitation
related wet periods, i.e., when urban runoff takes place. The concentration shown is the
result of both upstream contributions and urban runoff discharges. The distribution of
stream concentrations will be lognormal when computed by the moments approximation
illustrated here. As indicated in the sensitivity analyses presented later, this is an
approximation of the actual stream concentration distribution.
The probability of exceeding a specified concentration developed under the previous
step may be converted to a mean recurrence interval (MRI) by taking the reciprocal of the
probability value. However, because runoff flow and concentration parameters are based
only on conditions when a runoff event is in progress, the recurrence interval computed
above is in terms of events. For example, a 1.0 percent exceedance probability (P = 0.010)
4-5
-------
100-TT
99.9 99 95 90 80 50 20 10 5
PERCENT EQUAL OR GREATER
0.1
A lognormal distribution plots as a straight line on
LOG-PROBABILITY graph paper
Only two of the points shown are required to define the line
The median is plotted
at the 50 % probability
X = EXP ( Mln(X))
The 16* and 84* percentile (±l
-------
has a mean recurrence interval of 1/0.010 = 100 storm events. On average, the
concentration corresponding to this probability will be exceeded once in every 100 storm
events.
It is often desirable, however, to estimate the recurrence interval in time units, to aid
in evaluating the significance of the conditions produced by urban runoff. This conversion
can be made using the average number of storm events per year determined from analysis
of rainfall records as described in Chapter 3. As a first order approximation, Equation 3.43
may be used. For example, if the local area has an average of 100 storms per year, then
the selected concentration that has a 1.0 percent exceedance probability (1 in 100 storm
events) will be exceeded on an average of once per year.
This first order approximation is a simplification of the recurrence pattern that is
likely to result in natural systems. Multiple exceedances would be expected in some years,
and none in others, with the clusters of exceedance events recurring at some interval greater
than the MRI of 1 year produced by the first order approximation. If this is kept in mind
when interpreting and comparing results, the MRI can be used to provide a relevant basis
for evaluating the results.
4.4 ILLUSTRATIVE EXAMPLE OF ANALYSIS
A step-by-step application of the Analytically Derived Distribution Approach
method is presented in Appendix C to further assist the user in understanding this
methodology. This illustrative example also indicates how results of this analysis may be
interpreted to guide evaluations of the significance of urban runoff water quality impacts
and decisions on control actions.
Appendix C includes a set of input data used for the example analysis, the
computations and interim results. The sample examines the effect of urban runoff pollutant
sources by setting upstream concentration at zero. A comparison between these results and
results obtained using the Direct Numerical Approach and the Monte Carlo Simulation
technique for a variety of input conditions is presented in Section 4.5.
The selected input parameters for the example have been developed based on the
following assumptions concerning the original sources of data:
• Urban Runoff Quality: Copper is selected for the analysis. Owing to limited local
monitoring data the best estimate of mean and variability of copper concentrations is
based on the NURP estimate (WCC 1983) of the quality characteristics of the
median urban site.
• Streamflow: It is assumed that there is no USGS flow gage available immediately
upstream of the urban area to estimate the streamflow characteristics. However,
analysis of flow records for appropriate gaged streams in the area support an
estimate of the average streamflow of 1.25 cfs/square mile of drainage area, with a
coefficient of variation based on daily flow values of 1.25. Total drainage area
upstream of the urban area contributing runoff to the stream i.s 100 sq.rni. Thus,
the average upstream flow is (100 sq. mi. * 1.25 cfs/sq. mi.) 125 cfs. This value
is used as an input for the computations.
4-7
-------
• Runoff Flow: Assume that the urban area under consideration has an area of 1
sq.rni., a runoff coefficient (Rv) of 0.25, and that long-term rainfall records for a
I1™!!26 have "^ following characteristics (based on statistical analysis using the
SYNOP program WCC (1988).
Coefficient of
Parameter Units Mean Variation
Storm average intensity [in/hr] 0.075 135
Storm volumes [inch] 0.40 1.50
Storm durations [hour] 7.0 120
Storm intervals [hour] 87.6 LOO
Using the above data, the average runoff rate for the 1 square mile of urban area is
computed using the mean storm intensity and the runoff coefficient
(0.075) (0.25) (645.33) = 12 cfs / sq. mi.
where the factor 645.33 converts inches per hour to cfs / square mile.
The coefficient of variation of runoff flow (QR) is assigned a value of 1.15 which is
85% of the coefficient of variation for the storm average intensity.
These values are listed as the input parameters on Appendix C. The subsequent
computations to determine the probability of resulting in-stream concentrations during
storm events are shown on Appendix C. Figure 4-3 is a plot of the output., The estimated
probabilities of in-stream concentrations are then converted to average recurrence intervals
to provide a more understandable basis for interpreting and evaluating results. Figure 4-4
presents the average recurrence interval associated with specific stream concentrations that
will be produced during storm runoff periods.
4.5 COMPARISON OF THE THREE PROBABILISTIC METHODOLOGIES
All three models were run using the data presented in Section 4.4. The results from
each model are presented in the form of the CDF of downstream concentrations. In
addition.simulations were made using a wide range of possible values for the inputs.
Results of these simulations are presented in tabular form.
Figure 4-5a shows the CDF for downstream concentration resulting from
simulations conducted using all three probabilistic models. At higher cumulative
probabilities (greater than 90%) all three methods result in approximately the same
downstream concentrations. This is clearly shown in Figures 4-5 a and b. The latter
presents only the highest 10% of the downstream concentrations. At lower probabilities
the Analytically Derived Distribution Method deviates from the other two solutions. This
is due to the choice of the 90 and 99.6 percent quantiles for estimating the parameters of the
assumed log normd distribution for the dilution. These percentiles were selected since the
intent here is to estimate downstream concentrations with return periods of the order of 3
years or so that generally correspond to concentrations above the 90th percentile (see
Section 3.3). Note that by varying these percentiles, the Analytically Derived Distribution
Approach can be forced to agree with other sections of the CDF.
4-8
-------
100
99.9
99
95 90 80
PERCENT EQUAL OR GREATER
Figure 4-3. Probability Distribution of Pollutant Concentrations
During Storm Runoff Events
4-9
-------
1000-p
.1 1
Mean Recurrence Interval
10
(YEARS )
100
Figure 4-4. Recurrence Intervals for Stream Concentrations
4-10
-------
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A series of simulations were conducted using (i) two different upstream
concentrations, (ii) four different ratios of runoff flow to streamflow, and (iii) two different
ratios of coefficients of variation of upstreamflow to runoff flow so that the three methods
could be compared over a wide range of possible inputs. Table 4-2 presents the parameters
for the model input distributions. Since the solution of the model is primarily dependent
upon the ratio of runoff flow to upstream flow, only the upstream flow was varied.
Results from the simulations are presented in Table 4-3.
A comparison of Run 1 to Run 2 shows the effect of increasing the upstream
concentration from essentially zero to one half of the runoff concentration. The maximum
difference in the downstream concentrations between the three methods was 7.8 mg/1 at the
99 percentile value. This is less than about 15% of the mean concentration for the three
methods and is within the accuracy range for a screening level methodology. For the other
percentiles, the maximum difference between methods is even less.
Comparison of Runs 3 and 4 show the effect of changing the ratio of upstream flow
to runoff flow from 10 to 100 using a ratio of coefficient of variation of 2. Runs 5 and 6
use the same flow ratios but have a ratio of coefficients of variation of 3. In general, the
three methods produce essentially identical results using the above inputs. However, for
the smaller flow ratio with a higher ratio of coefficient of variation (Run 5) the Analytically
Derived Distribution starts to deviate from the other two methods. For percentiles from 90
to 99.5 the Analytically Derived Distribution Approach over estimates the concentration by
14% to 30% relative to the Monte Carlo solution. The larger differences occur at the lower
percentiles. The Direct Numerical Solution and the Monte Carlo Method are usually within
10% of each other.
Runs 5 through 8 show the effect of changing flow ratio only. For these runs the
flow ratio varies from 1 to 1000. The results from these runs indicate that as the flow ratio
decreases the Analytically Derived Distribution Approach predicts higher concentrations
relative to the other two methods. If the runoff flow equals the upstream flow, the
Analytically Derived Approach can overestimate the concentration by over 40% relative to
the Monte Carlo Method and 25% relative to the Direct Numerical Solution Approach. The
Direct Numerical Solution Approach deviates from the Monte Carlo Method solution at
these flows also. For Run 7 it predicts concentrations about 15% greater than the Monte
Carlo Method.
Thus, for most sets of input parameters, the three methods estimate approximately
the same answer. However, when the upstream flow and runoff flow are of about the
same magnitude, the Analytically Derived Distribution Approach generally yields higher
estimates of concentration relative to the other two methods.
4.6 INTERPRETATION AND EVALUATION OF RESULTS
The significance of a particular magnitude/frequency of stream concentration caused
by urban runoff can be evaluated by comparing it with concentrations that are relevant for
the beneficial uses of the water body. If, for example, protection of aquatic life is the
predominant beneficial use of the urban stream, the concentrations thai: result from urban
runoff may be compared with a stream target concentration associated ,with appropriate
water quality criteria or standards.
Figures 4-6 and 4-7, which repeat the probability and recurrence interval plots
developed for the example analysis, illustrate how analysis results may be used for NFS
planning evaluation.
4-13
-------
Table 4-2
inputs Used in Sensitivity Analysis of the PDM Model
Run CS
Mean SD
1 0.01
2 20.4
3 0.01
4 0.01
5 0.01
6 0.01
7 0.01
8 0.01
0.
12
0.
0.
0.
0.
0.
0.
01
.24
01
01
01
01
01
01
CR
Mean SD
40.
40.
40.
40.
40.
40.
40.
40.
8
8
8
8
8
8
8
8
24
24
24
24
24
24
24
24
.48
.48
.48
.48
.48
.48
.48
.48
QS
Mean SD
125
125
10.0
100.
10.0
100.
1.0
1000.
156
156
15.
150.
30.
300.
3.
3000.
QR
Mean SD
12
12
1.
1.
1.
1.
1.
1.
.0
.0
00
00
00
00
00
00
13.8
13.8
0.75
0.75
0.75
0.75
0.75
0.75
4-14
-------
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Figure 4-6. Use of the Methodology to Evaluate the Water Quality
Impact of an Urban Runoff Control Program (I)
0.
4-16
-------
CO
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If the runoff concentrations used in the analysis were for copper.the projected
stream concentrations can be compared with the EPA toxicity criteria for copper, by
showing this value on the concentration axis. The acute toxicity value is selected as most
appropriate for comparison with the effect produced by intermittent short duration
exposures. If the total hardness of the receiving stream is 150 mg/1, the copper criterion
value is 26 |ig/l, and a horizontal line is drawn on the plot at this location. This is
considered to reflect a safe level when it is exceeded no more frequently than once in 3
years. This aspect of the target is designated on the plot by a vertical line locating this
frequency of occurrence, at 3 years on the MRI plot. For the number of storms per year
used in the example, this corresponds to an exceedence probability of 0.33 percent on the
probability plot.
In this case, the stream concentrations of copper caused by untreated urban runoff
discharges exceed the "target" to a significant extent. The target concentration is exceeded
at a recurrence interval of about 0.3 years (every 100 days on average), and at a 3-year
recurrence, the stream concentrations produced during runoff events will be about double
the target.
The water quality effects of proposed alternative NFS control programs can be
assessed by repeating the analysis using revised input values to represent the characteristics
of the urban stormwater runoff. To illustrate, it is assumed that an NFS control program
that might be implemented could effect a 60 percent reduction in the mean copper
concentration in the urban runoff. Results of the analysis are plotted on the" same figure to
provide a graphic display of the degree of mitigation of water quality impacts that such a
program is projected to produce. In this example, the 3-year concentration is reduced to
about 20 jJ.g/1, safely below the target. The target concentration of 26 |ig/l is now
projected to be exceeded not more than once every 7 years on average.
Such an approach is useful in the evaluation of the significance of urban runoff and
for identifying potential water quality effects from the implementation of control measures.
There are two reasons why the use of a probabilistic approach is important. First, because
of the stochastic nature of the urban runoff flow system, virtually any target concentration
will be exceeded at some frequency, however small. Second, from a practical point of
view, there are limits to the capabilities of controls. If the receiving water total hardness in
the example was 50 mg/1, the criteria concentration would be 9 (ig/1 rather than 26 |j,g/l, and
the improvement produced by the control would fall far short of meeting the targets.
Receiving water impacts may be significantly reduced, but will never be totally eliminated.
Therefore it is necessary to evaluate the degree of water quality improvement associated
with different control alternatives and to implement control measures to attain water quality
standards.
4-18
-------
CHAPTER 5.0
5.1
INTRODUCTION
5-2 VERIFICATION O r specific input parameters.
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5-1
-------
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5-2
-------
TABLE 5-1. RAPID CITY VERIFICATION TEST
Item
Stream
Flow
(CFS)
Runoff
Flow
(CFS)
Runoff
Cone.
CR
Upstream
Cone.
Cs
Calculated
Downstream
Cone.
Qs
OR
TSS
COD
TP
Pb
TSS
COD
TP
Pb
TSS
COD
TP
Pb
Mean
104
17.5
3689
219
2198
382
325
30
206
19
822
58
500
73
Median
90.5
12.6
2760
195
1435
253
158
25
100
4
523
47
293
35
Coefficient
of Variation
0.57
0.96
0.89
0.52
1.16
1.13
1.80
0.66
1.80
4.70
1.21
0.74
1.38
1.79
RAPID CITY — STORM AVERAGES
FLOW DISTRIBUTIONS
10*
10° '»" it'll 1—i i t . • . I
PROBABILITY
Figure 5-2. Probability Distribution of Stream
and Runoff Flows
5-3
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5-4
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excellent match between the data and the calculated distribution supports the
methodology used.
• Figures 5-4,5-5, and 5-6 present results for Total P, COD, and Lead, respectively.
In each case, the model results compare well with the data.
The results and comparisons presented above are based on an available "sample" of
storm events and are representative of the conditions which prevailed during the monitoring
activities. As such, this analysis is appropriately a "test" of the analysis procedure, and
does not necessarily reflect the projected long term conditions.
The above verification tests confirm the validity of the computational methods.
These results are essentially a test of the methodology because the data used represent a
limited sample of overall conditions drawn from monitoring results. Such conditions may,
and often do, represent a biased representation of typical long term conditions. The
favorable comparisons, however, do provide confidence that the long term projections used
to guide decisions will be sensitive to how well the long term input characteristics can be
defined.
5.3 VERIFICATION OF IMPLIED EFFECT ON BENEFICIAL USE
A second basis for determining the reliability of the methodology, in identifying the
potential of urban runoff discharges to cause significant impairment of a receiving water
body, is to compare the expected impact levels with observed biological conditions. This
requires (i) sufficiently accurate estimates of input parameters, (ii) accurate estimate of in-
stream concentrations, and (iii) the resultant exposure concentrations and the biological
effects are accurately evaluated.
The information available to examine this issue is more tenuous than that available
for checking the computation procedures of the methodology. Three of the NURP projects
examined the effects on aquatic life in streams receiving runoff from monitored sites:
• Bellevue, WA, concluded that whatever adverse effects were observed were
attributable to habitat impacts (streambed scour and deposition) as opposed to
chemical toxicity. For this project, heavy metal concentrations in the monitored
urban runoff sites were typical of the average for all urban sites. The screening
analysis projections under these conditions did not indicate the potential for a water
quality problem.
• Tampa, FL, conducted extensive bioassay tests but failed to show any adverse
effect of pollutants in urban runoff. When the screening analysis was implemented
using site concentrations representative of Tampa monitoring results, a toxicity
problem was not predicted.
• Rapid City, SD, conducted extensive biological testing of phytoplankton,
periphyton, benthic organisms, and fish in the stream receiving urban runoff from
the entire city. Observable adverse effects were not shown to; exist nor were any
significant incremental changes demonstrated as the river picked up an increasing
component of urban runoff. A healthy cold water fishery was reported to exist in
the stream. The screening analysis indicated that a significant impairment would
not be anticipated, even though strearnflows tend to be relatively low and monitored
lead concentrations in urban runoff are at the higher end of the range for urban
sites.
5-5 :
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1
i
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I
_ 10 co
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or
a.
w
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-------
5.4 SENSITIVITY ANALYSES
This section presents a series of comparisons between the results obtained under
different sets of conditions. The intent is to provide information on the sensitivity of model
results to certain choices and assumptions that a user might have to consider due to
uncertainty concerning the input parameters. The following issues are addressed:
• Analytically Derived Distribution Approach versus the Direct Numerical
Approach(quadrature)
i
• Effect of correlation between stream and runoff flows ]
• Effect of nonzero upstream concentrations !
In Section 4.5 the three probablistic methods were compared to each other. A conclusion
from this comparison was that the results obtained using the Direct Numerical Approach
approximate the results of the Monte Carlo Method. The following sensitivity analysis
section presents results obtained using the Analytically Derived Distribution Approach and
the Direct Numerical Approach.
5.4.1 Analytically Derived Distribution Approach vs. Direct Numerical Approach
This section examines the accuracy of the approximate solution by presenting a
comparison between the Derived Distribution Approach and the Direct Numerical
Approach. Results are compared for a variety of inputs. Conclusions are presented on the
applicability of the approximate model.
The Derived Distribution Approach offers a number of advantages over the Direct
Numerical Approach. The method is simple and hence enables the user to better
understand the methodology. A more practical attribute of the method is the speed and ease
of use for preliminary screening analyses. It can, for example, be set up in spreadsheet
format. This is particularly advantageous when a large number of different conditions are to
be examined, e.g., for sensitivity tests or for screening different alternatives. Where the
numerical solution computes the probability and return period for a specified stream
concentration, the moments approximation additionally allows the user to work in the
reverse direction. The ability to estimate the stream concentration that will occur at a user-
specified probability or return period proves to be quite useful in many applications.
It has been emphasized that the Derived Distribution Method provides an
approximation, and that its projections tend to be conservative. Even allowing for the fact
that normal good practice would dictate that final results be confirmed by a more exact
computation, it will be useful for the user to develop a sense of the accuracy of the
approximate solutions developed by this method. This section compares results obtained
by the use of the two computation methods.
An example computation using the Derived Distribution approximation was
presented in Chapter 4, and results were summarized on Figures 4-3 and 4-4. The two
computation methods are compared for exactly the same input conditions by performing
the analysis using the computer program described in Chapter 6. Results are presented and
compared with the Derived Distribution approximation in Figure 5-7 which shows the
probability distributions, and on Figure 5-8 which shows the mean recurrence intervals.
5-7
-------
I
100-F
o
1—
UJ
o
o
o
1
GC.
CO
1 DerivedDistributlon Approach
2 Direct Numerical Solution J
99.9 99 95 90 80 50 20 10 5
PERCENT EQUAL OR GREATER
Figure 5-7. Comparison of Results from the Derived Distribution
and the Direct Numerical Solution Approachs.
Probability Distributions for Selected Input Conditions.
i
5-8
-------
100-
-------
Note on Figure 5-7 that the distribution of stream concentrations computed using
the numerical method is not log normal. The straight line log normal approximation
produced by the Derived Distribution method matches the actual distribution quite well at
the upper tail (higher percentiles), but deviates appreciably at lower percentiles. This
particular pattern is the result of the selection of the 90th and 99.6th percentiles of the
dilution factor (0) in developing its lognormal approximation. Other input data
combinations would modify the match slightly as would the selection of other percentiles
for the lognormal approximation of (0). Sensitivity tests discussed below indicate that the
selected percentiles will provide satisfactory matches of the distributions over a wide range
of input conditions.
Although the departure of the lognormal approximation over the lower range of
probabilities may appear to be a cause for concern, as a practical matter, for the usual water
quality impact evaluation it has no effect whatever on any decisions that would be guided
by the analysis. Inspection of the probability plot indicates that the moments approximation
overestimates the median (50%) stream concentration by 100 percent. But this only
represents a concentration of 6 |ig/l rather than 3 jig/1. The absolute magnitude of the
difference (3 micrograms per liter) is quite small and within the overall accuracy of the
method. Further the 50th percentile event has a recurrence interval of about a week in a
situation where decisions will be based on recurrence intervals of several years.
The significance of this deviation in terms of its effect on the evaluation of impacts
is better shown on Figure 5-8, which compares the recurrence interval projections
produced by the two procedures. It emphasizes the close match between the Derived
Distribution approximation and the numerical method at MRIs ranging from several months
to decades.
The match can be expected to vary depending on the unique combination of input
conditions. The relative magnitude of the flows has an important influence, as does both
the magnitude and ratio of the coefficients of variation of the inputs. For an assessment of
the relationship between outputs of the two computation methods, repetitive analyses were
performed, using combinations of input parameters that cover the range of input values that
can be reasonably expected to cover most practical situations. The influence of flow rates
can be represented by the ratio of mean streamflow to runoff flow (QS / QR), and the range
of interest from 1 to 1000. Combinations of coefficients of variation were examined over
the range of flow ratios, using values ranging from 0.5 to 3 combined in several different
ratios.
Computations of resulting stream concentrations were made using both methods,
and the stream concentrations projected to result at a 3-year mean recurrence interval were
selected for comparison. The ratio between this concentration computed by the Derived
Distribution approximation to that computed by the computer program using the Direct
Numerical solution was determined. Results are summarized by Figure 5-9.
The dashed curve shows the comparison that results for conditions likely to be
encountered fairly often, coefficients of variation of about 2 for streamflow, of about 1 for
runoff flow, and about 0.75 for runoff concentrations. The solid curves define the
approximate envelope covering the range of conditions tested,
This sensitivity analysis indicates that the Derived Distribution approximation keyed
to the high percentiles of (0) provides a rather accurate estimate of 3-year stream
concentrations, certainly suitable for screening or planning level analyses. Over much of
the range of flow conditions, the approximation will provide results within 10% for most .^
of the commonly encountered input combinations. Results will virtually always be expected (III
5-10
-------
CO
o
-------
to be within 30% , the upper bound being encountered when coefficients of variation and
their ratios are high. The Derived Distribution method's estimates of 3-year stream
concentrations is seen to be generally conservative, though under some conditions it may
underestimate by a few percent.
5.4.2 Correlation between Streamflow and Runoff Flow
This section investigates the effect of a strong positive correlation between
Streamflow and runoff flow. A positive correlation implies that higher storm runoff events
tend to occur when streamflows are at their higher values. This would tend to reduce the
stream concentrations that would result from urban runoff. It is important to note that data
on correlation coefficients are generally not available.
The Direct Numerical Method was used for computations covering the same range
of input conditions examined in the preceding sensitivity analysis. For each combination of
input conditions, the projected 10-year MRI concentration was determined for values of
correlation coefficient P ranging from 0 to 1. A long recurrence interval was selected for
this sensitivity analysis because it will tend to magnify the differences.
Figure 5-10 shows the variation of normalized stream concentrations (normalized
with the concentration for the case of zero correlation) as a function of the correlation
coefficient. The effect of flow correlation is substantial for large streams with a ratio of
mean stream flow to mean runoff greater than 25, and minor for small streams with a
corresponding ratio of less than unity. How variability, reflected by the ratio of stream to
runoff coefficients of variation, has relatively minor effect, as illustrated by the width of the
bands for each of the flow ratios.
Although Figure 5-10, indicates that flow correlations could significantly influence
the magnitude of 10-year MRI stream concentrations, these results should be interpreted by
considering the following. Streams with small flows relative to urban storm flows would
reflect headwater segments and would most likely have high correlations. However, such
streams are relatively insensitive to flow correlation effects on stream concentrations. On
the other hand, the large streams which show the most dramatic influence are likely to
represent situations in which flow correlations are small or nonexistent.
When these considerations are taken into account, the general assessment is that
flow correlation is unlikely to influence the 10-year MRI stream concentrations by more
than about 10 or 15%. Therefore, unless site-specific data are available to establish a local
value for flow correlation, the assumption of a zero correlation is appropriate. Projections
may be slightly conservative, but in most cases by no more than 10 percent or so.
5.4.3 Effect of Upstream Concentrations
The influence of upstream concentrations of a pollutant on water quality
projections, and on the mitigation of impacts resulting from control measures is illustrated
by the following three figures. The influence of significant upstream concentrations of a
pollutant on quality below the stormwater discharge is shown by Figure 5-11. The solid
curve applies for input conditions which are identical to those used for the example problem
presented earlier in Chapter 3, where upstream concentration is assigned a value of zero.
The dashed curve reflects the stream concentration distribution when the mean upstream
(background) concentration is 50% of the mean runoff concentration. The same variability
is assumed for the upstream concentrations as for the runoff concentrations.
5-12
-------
1.0
RATIO
OF
MEANFLOWS
(OS/OR)
0.67
25
100
P= CORRELATION COEFFICIENT BETWEEN
STREAM FLOW AND RUNOFF FLOW
Figure 5-10. Effect of stream/runoff flow correlation
5-13
-------
i
0.10 1.00 10.00
Mean Recurrence Interval (yr)
100.00
Rgure 5-11. Effect of Upstream Concentration on Mean Recurrence Interval
5-14
-------
'0
The significance of this effect can be assessed by considering the degree of
improvement in stream concentrations that result from a control action that produces a 50%
reduction in the mean runoff concentration and no change in variability. Figure 5-12
illustrates the projected water quality impact of the control action for die case where
upstream concentration is zero. The 50% reduction in urban runoff concentration would
reduce the 3 year MRI concentration to 26 |ig/l, and meet the desired objective. However,
when the mean background concentration is close to the target concentration concentration,
a 50% reduction in urban runoff produces little improvement in overall stream conditions as
illustrated by Figure 5-13.
The practical significance of background concentrations (or those due to point
sources or other upstream nonpoint sources) will vary with the particular combination of
input conditions that characterize the local area. Stream and runoff flow relationships and
variabilities will influence results, however, the factor of major importance will be
background concentration levels relative to the target concentration and the runoff
concentration.
5.5 SUMMARY AND DISCUSSION
The Derived Distribution Method has been shown to produce sufficiently accurate
and usually conservative estimates of stream quality at recurrence intervals greater than a
year, when compared with the results of the Direct Numerical Solution and the Monte Carlo
Approach. The sensitivity analysis provides a basis for estimating the approximate
magnitude of the degree to which 3-year recurrence concentrations are overestimated by the
approximate method.
The magnitude and variability of upstream concentrations of pollutants have not
been analyzed in detail. However, as the sensitivity analysis indicates, they can have a
significant influence on control decisions. Where the upstream concentration of a pollutant
is comparable to the target concentration, substantial reductions in urban runoff discharages
may result in only marginal improvements in overall water quality and use impairment.
A high positive correlation between runoff flow and streamflow could result in
significantly lower in-stream concentrations. This implies that runoff discharges enter
when stream flows are in the higher range of possible values. This results in a high
dilution and lower in-stream concentrations. There is, at present, insufficient information
available to estimate typical values for this correlation. However, the data (rainfall and
streamflow) required to estimate this parameter are available and it would be useful to
analyze them. In the absence of this information, the assumption that the flows are
uncorrelated provides conservative projections. The sensitivity analysis performed
suggests that the assumption of no correlation probably overestimates 3-year stream
concentrations by no more than 10-15 percent. This assumption is probably approximately
correct for situations with large drainage area ratios (DARs), i.e., where the headwaters of
the receiving stream are a considerable distance upstream of the urban area. In these larger
streams, flow correlations would have the greatest effect. A more critical situation can be
expected to occur in smaller urban streams with low DARs, where high correlations would
be expected. However, in such cases runoff flows dominate conditions to the extent that
the degree of flow correlation has little effect on stream concentrations.
The two analytical portions of the probabilistic dilution model presented embody the
assumption that flows and concentrations are lognormal and independent. A large body of
i
5-15
-------
1000
H - 1
i 1 1 i
Upstream Concentration C = 0 jug/I
— Cp = 40.8 ftg/l (untreated)
CR = 20.4 ng/\ (50% control)
3 year MR!
0.01
-J 1—II I 1111
-4—I I I I I I
-H 1-
-> t I-1 I-
II
H—I I I I II
II
0.10 1.00 10.00
Mean Recurrence Interval (yr)
100.00
Rgure 5-12. Effect of Control with No Background Contribution
5-16
-------
1000-;
-»-»-(-•
Upstream Concentration C = 20 fig/}
— CR = 40.8 jt*g/l (untreated)
CR = 20.4 jug/l (50% control)
en
0.01
-f—W-
1 - 1- 1 +-H H
3 year MR!
0.10 1.00 10.00
Mean Recurrence Interval (yr)
Rgure 5-13. Effect of Control with High Background Contribution
5-17
-------
data gives strong support to the appropriateness of this assumption in the case of urban
runoff. For upstream concentrations and flows, nonzero correlations may be more likely
but the examination of a sufficiently large number of data sets will be required before the
significance of this assumption can be determined. Limited data review suggests that an
assumption of lognormality is probably reasonable for upstream concentrations in most
cases, but further investigation in this area would be useful. If, as in the case of the
example analysis, upstream concentrations are set at zero to examine the effect of only the
runoff discharge, then the issue of concentration lognormality and correlation with
streamflow becomes moot.
The main issue relating to the assumption of lognormality of the inputs is in the case
of the stream flow. Because of the presence of base flow in many streams, the current 2-
parameter log normal distribution will tend to underestimate the lowest flows, and thus to
overstate the magnitude of infrequently occurring stream concentrations. Some preliminary
screening analyses on a few stream records suggest that this could be resolved by
employing a 3-parameter log normal distribution, in which a minimum flow is also
specified. However, with the current analysis, it should be recognized that output from the
computation tends to produce conservative estimates for cases where the base flow
component is significant.
The analytical methodology could be adapted to accept input parameters that are
other than log normal. However, the computational structure is considerably simplified
when all distributions are log normal, and the analysis is simple and easy to apply. In
cases where input parameters are known to have distributions other than lognormal, the
Monte Carlo technique may be more appropriate to use.
The basic approach and the specific procedures presented are useful for evaluating
the significance of intermittent, variable pollutant discharges, and thus for providing
guidance for rational and cost-effective water quality management decisions. Note that
continuous point source discharges from publically owned treatment works (POTW) or
industrial facilities also produce variable receiving water quality responses, and can be
analyzed by essentially the same probabilistic methodology. The use of a probabilistic
approach can provide an opportunity for evaluating both point and nonpoint source
pollutant discharges using a similar framework, and hence evaluating point/nonpoint
tradeoff control options on the basis of the effect on receiving water quality.
5-18
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CHAPTER 6.0
USER'S GUIDE
This chapter provides guidance for using the Probabilistic Dilution Model (PDM) to
estimate the in-stream concentrations and recurrence intervals. The theoretical basis of the model is
described in detail in Chapter 3. A discussion of model application is given in Chapters 4 and 5.
This chapter is organized into five sections. Section 6.1 discusses the model acquisition
and installation procedure. Section 6.2 presents the overall structure of the model. Common
blocks and parameters used by the model are described in Section 6.3. A discussion of the batch
input data file and interactive input is given in Section 6.4. An example application of the model is
presented in Section 6.5. A list of the subroutines and the variables used in the model are
included in Appendices C and D, respectively.
6.1 MODEL ACQUISITION AND INSTALLATION PROCEDURES
6.1.1. Model Acquisition •
The PDM model along with the example input and output can be obtained by contacting:
Mr. Dennis Athayde ,
Nonpoint Source Branch
Criteria and Standards Division
U.S. E.P.A.
Waterside Mall, WH-554
Washington, D.C. 20460
The code is available on either 360K or 1.2M byte 5-1/4" - inch floppies for IBM PC-compatible
machines, using DOS.
6.1.2. Model Installation
The source code for the model is contained in a file named PDM.FOR. This file has been
compiled using the Ryan-McFarland FORTRAN compiler, and the resulting executable code,
PDM.EXE, is also included. This executable version of the model requires a co-processor, a
8087, 80287 or 80387. If one is unavailable, the program needs to be recompiled using a
compiler compatible with the computer running PDM. It is recommended that a backup copy of
the original floppy be made before running the model. Consult the DOS manual for instructions on
copying disks. If the model is run from a hard disk, then a directory should be created on the hard
disk using the MKDIR (MD) command (consult your DOS manual for details on using this
command). Once the directory has been created copy all of the files from the floppy into the
directory. *J
The interactive portion of the model requires that the device driver ANSIS YS,included
with DOS, be installed. This device driver is installed by including the command:
i
DEVICE=ANSI.S YS
in the CONFIG.S YS file. After installing the device driver the computer needs to be rebooted. '
(consult you DOS manual for instructions on altering the CONFIG.SYS file).
6-1
-------
To run the model, put the backup copy of the model disk in Drive A or access the directory
on the hard disk containing the model and type PDM. The model will prompt you for the
information it needs to run. If the model is to be run in batch mode, make sure that the input file is
in the same directory as the model (if using a hard disk) or on the floppy in Drive A (if using a
floppy disk).
If changes are made to any of the parameter statements in the code described in Section 6.3,
the entire code must be recompiled and linked. The code has been developed using a Ryan
McFarland version 2.0 compiler and a PLINK86 linker. The code requires the /b option when
compliling. When compiling, all FORTRAN and INCLUDE files (described in Section 6.3) must
be in the same directory and accessible to the user's compiler.
6.1.3. Testing the Model
Input and output files for the example problem discussed in Chapter 4 have been included
with the model for both the analytic and Monte Carlo options. This example problem should be
run and the results compared to the output file supplied with the model to ensure proper installation
of the model.
6.2. MODEL STRUCTURE
The structure of the model is shown in Figure 6-1. The structure of the analytic model
(shown Figure 6-2.) follows the steps presented in Section 3.3 for computing the exceedance
probabilities. The model is divided into three components: input (either batch or interactive),
output and computational subroutines. Subroutine descriptions are provided in Appendix C, and
FORTRAN variables are defined in Appendix D. Subroutines used for low level control as part of
the interactive preprocessor (e.g., calculating the length of strings, calculating screen coordinates,
etc.) are not discussed or included in Appendix C or D.
6.3. COMMON BLOCKS AND PARAMETER STATEMENTS
Variables are passed between subroutines in the model through the use of common blocks
and as arguments to subroutine calls. The computational portion of the analytic model uses a total
of six common blocks, each containing a related set of parameters. These common blocks are
contained in a file named PDM.CMN which is accessed by the code through the use of INCLUDE
statements located at the beginning of a subroutine. The common blocks and a description of the
parameters they contain are given below.
Common Block Description
GENERAL General run data
QSCMN Stream flow and concentration
data input by the user
QRCMN Runoff flow and concentration
data input by the user
CONC Concentration data used by the
numerical integration routines
6-2
-------
SELECT MODEL
ANALYTICAL
MONTE CARLO
SELECT INPUT MODE
INTERACTIVE
BATCH
READ
DATA
ECHO
INPUT
CALL
COMPUTATIONAL
ROUTINES
OUTPUT
RESULTS
Figure 6-1 Flow Chart of the PDM Model
6-3
-------
Batch
/interactive\
Interactive
Read input
file
N. or batch /
N. input,/
Interactive
input
i
Convert input values from
arithmetic space to log space
JL
Calculate first two moments of
dilution (eqs. 3.'12 & 3.13)
1
Calculate the user input percentite values
of runoff fraction (eq. 3.14)
Calculate the first two moments
of dilution factor (eqs. 3.18 & 3.19)
Convert moments
to arithmetic space
±
Calculate the first two moments of downstream
concentration in arithmetic space (eqs. 3.4 & 3.5)
Convert downstream concentration
moments to log space
Calculate exceedance
percentiles (eq. 3.9)
i
Calculate return period
t
Write results of
Method of Moments
solution
IIIIIIIIIMIIIIIlltlE
METHOD
OF
MOMENTS
SOLUTION
FIGURE 6-2 FLOWCHART OF THE ANALYTICAL RDM MODEL
6-4
i>
-------
Calculate roots and weights
for Gauss-Lagendre Quadrature
for dilution integral
I
iiiummiimiii
Evaluate upper bound
of Cs integral
Calculate roots and weights
for Gauss-Legendre Quadrature
forQ integral
Do J=1, ORDER J
Evaluate integrand in Cs integral
Evaluate C_ integral
i
Evaluate integrand in
dilution integral
Output results
of Gaussian Quadrature
GAUSSIAN
QUADRATURE
I SOLUTION
FIGURE 6-2 FLOWCHART OF THE ANALYTICAL PDM MODEL (concluded)
6-5
-------
QSLCMN
QRLCMN
Statistics of stream flow and
concentration calculated by the model
Statistics of runoff flow and
concentration calculated by the model
i
Li addition to the common blocks, the file PDM.CMN contains one parameter statement
defining the value of two parameters. These parameters are used to dimension arrays required for
calculating percentiles and return periods. The parameters and their assigned values are defined as
follows:
Parameter Name
ECXMAX
NPERCT
File Value Description
PDM.CMN 10 Maximum number of sets of
exceedances and return periods that
can be calculated using die numerical
integration routines
PDM.CMN 10 Maximum number of exceedances
and return periods that can be
calculated using the method of
moments
These values are assigned in the file PDM.CMN and can be changed if necessary. Note the
program has to be recompiled and linked if the values are changed.
The interactive preprocessor uses two additional common blocks and four parameter
statements. The common blocks and the include files containing them are listed below.
Common Block
SCREEN
File
SCREEN.CMN
Description
Number of lines in program title
LINEST
SCREEN.CMN
ASCII codes used to define horizontal
and vertical lines
Parameter statements used by the interactive preprocessor and the INCLUDE files
containing them follow.
Parameter
Name
HTYPE
VTYPE
KIN
File
SCREEN.PAR
SCREEN.PAR
USRFILES
Parameter
Value
1
2
5
Description
Indicates that single lines are
to be used for horizontal lines
Indicates that double lines are
to be used for vertical lines
Unit number used to read
from the keyboard
fi-6
-------
KOUT USRFILES 6 Unit number used to write to
the screen
CHR1WT USRFILES FALSE Usecl to indicate if first
character in output formats
should be a blank or not
FXISTS CMPLRERR 2013 Compiler error number
indicating that a file exists
i
6.4. MODEL INPUT
Two sets of data are used by the PDM model, data for the analytical approaches and data
for the Monte Carlo Approach. For each case data can be input in one of two ways (i) read from a
file or (ii) entered interactively from the keyboard.
6.4.1 Analytical Model Data Requirement^
Although the data requirements for the model are minimal, for ease of presentation it has
been divided into three groups: general run data, flow and concentration data, and exceedence
data. The formats necessary when using the batch input file are discussed below and shown in
Table 6-1. When the model is run in the interactive mode, the model will prompt the user for
input and no further guidance is required.
In addition to the data required by the model, comment lines can be included in a data file.
Comments are identified by the presence of three asterisks (***) as the first three nonblank
characters on a line. This is illustrated in the example input file presented in Table 6-2. Blank
lines in a data file are treated as comment lines. Any system of units may be used for flow and
concentration, but units must be consistent (i.e., all flows in cfs, all concentrations in mg/1).
The general run data consist of five parameters: the correlation between streamflow and
runoff flow, the two quantiles used to estimate the mean and standard deviation of the dilution
parameter, the number of storms per year, and the order of the gaussian quadrature approximation.
The formats for entering these data are presented in Table 6-1. Note that the order of the gaussian
quadrature approximation (ORDER) is restricted to 4, 5, 6, 10, 15, 20, 60, 104, and 256.
The exceedance probabilities (used in the Derived Distribution Appproach) for which the
concentrations are to be calculated are input next The format is presented in Table 6-1 A
maximum of NPERCT (defined in the INCLUDE file PDM.CMN) values can be read in.
i
Flow and concentration data are input next using the formats presented in Table 6-1. Data
associated with the stream are input first followed by data for the runoff.
The Direct Numerical Method calculates the exceedance probability for a given
concentration. The concentrations used in calculating the exceedance probabilities are input next.
The model reads in the concentrations in groups of three, a minimum concentration, a maximum
concentration and an increment of concentration. The model calculates an exceedance probability at
the minimum concentration and at the user-defined increments between the minimum and
maximum concentrations. (For example, if the user read in 1,100, 10 the model would calculate
the exceedance probability for the concentrations 1,11, 21, 31, 41, 51, 61,71, 81 and 91) In all
ECXMAX (defined in the INCLUDE file PDM.CMN) sets can be read in. Table 6-1 presents the
formats for input.
6-7
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LINE
1
2
3
4
5
6
TABLE 6-1
INPUT FORMATS FOR ANALYTIC PDM MODEL
PARAMETER
RHO, ALPHA1, ALPHA 2, NST, ORDER
PROBLS(I)
MQS, CVQS, MCS, CVCS
MQR, CVQR, MCR, CVCR
LCONC(I), HCONC(I), ICONC(I)
END
FORMAT
3F10.0, 2110
8F10.0
4F10.0
4F10.0
3F10.0
VARIABLES UNITS
RHO
ALPHA1,
ALPHA2
NST
ORDER
PROBLS(I) fraction
MQS
CVQS
MCS
CVCS
MQR
CVQR
MCR
CVCR
LCONC
L3/T
—
M/L3
—
L3/T
—
M/L3
—
M/L3
DEFINITION
Correlation between stream flow and runoff
Z-score of quantiles used to estimate mean and
standard deviation of dilution of the lognormal
distribution
Number of storms per year
Order of gaussian quadrature approximation,
must be one of the following: 4, 5, 6, 10, 15, 20,
60, 104, 256
Exceedance probability for which concentration and
return period are to be calculated using the Method of
Moments
Mean streamflow
Coefficient of variation of streamflow
Mean upstream concentration
Coefficient of variation of upstream concentration
Mean runoff flow
Coefficient of variation of runoff flow
Mean runoff concentration
Coefficient of variation of runoff concentration
Lowest concentration for which return period is
desired in the Gaussian Quadrature Approximation
6-8
-------
TABLE 6-1
INPUT FORMATS FOR ANALYTIC PDM MODEL (concluded)
VARIABLES UNITS
HCONC M/L3
ICONC
END
M/L3
DEFINITION
Highest concentration for which return period is
desired in the Gaussian Quadrature Approximation
Increment of concentration for which return period
is defined in the Gaussian Quadrature Approximation
The word END marking the end of the input file
6-9
-------
Table 6-2
Sample Input file PDM1.IN for Analytic PDM Models
****
***
***
*** GENERAL DATA
TEST DATA SET FOR PDM MODEL
TEST DATA IS FROM USERS MANUAL
*** RHO
0.0
***
***
.9998
0.200
***
***
***
ALPHA1
1.282
ALPHA2
2.652
NST
100
ORDER
15
PROBABILITY DATA
.9900 .9800 .9500
0.100
,9000
.8000
.7000
.6000
STREAM FLOW AND CONCENTRATION DATA
MQS CVQS MCS CVCS
125.0 1.25 0.010 0.1
***
***
***
***
***
RUNOFF
MQR
12.
LCONC
10.
.5
FLOW AND
CVQR
1.15
HCONC
100.
10.
CONCENTRATION DATA
MCR
40.8
I CONG
10.
.5
CVCR
0.60
END
6-10
-------
The last line in the data file should contain the letters 'END1. This signifies the end of all
data.
6.4.2 Monte Carlo Data Requirements
This section describes the format of the data for the Monte Carlo simulation approach.
Information presented in this section is further illustrated by the sample input file shown in Table
6-3.
The Monte Carlo input file consists of data lines and two types of general utility lines. The
first type of utility line is the coment line indicated by the presence of three asterisks ('***'), (also
see Section 6.4.1), as the first nonblank characters in the line. These lines are ignored and are
provided to allow the user to type in comments, table headings, and other information useful in
making the input file more understandable. Comment lines may be inserted anywhere in the data
set. The second type of utility line is the END line used to mark the end of specific data groups.
These lines are indicated by the word 'END' in the first three columns of the input line, and should
be used only where specified in the following discussion.
Monte Carlo input data are comprised of five data groups: (1) simulation control
parameters, (2) input distribution parameters, (3) empirical distribution data, (4) output options,
and (5) correlated variable input. Data are read sequentially starting with Data Group 1 and ending
with Data Group 5. Specific formats for each Data Group are shown in Tables 6-4 through 6-8
and are discussed below.
Data Group 1 —Simulation Control Parameters - This data group consists of two lines of
data describing simulation options. Formats for these data are shown in Table 6-4. The first line
contains the (alphanumeric) title for the run and is used to label the output. The second line
contains the number of Monte Carlo runs to be used in the simulation, and idle confidence level
(percent) for estimating the confidence bounds for the quantile to be estimated.
Data Group 2 — Input Distribution Parameters - This data group, shown in Table 6-5,
consists of one line of data for each model variable (CS, CR, QS, QR, CO). The first entry on
each line is a label, used to identify the random variable. The first two characters in the label must
correspond to the labels shown in Table 6-4. The next 18 characters are user specified. The
remaining data on these lines consist of parameters of the frequency distribution for the selected
variables. After a data line is provided for each desired random variable, an END card must be
supplied to mark the end of this data group. If data for any variable is not included in the input file
it will be taken as a constant with a value of zero. Note that by setting the distribution flag VAR
(5) to 0, the user can specify a constant value for a model input. In this case, the mean value of the
variable (VAR (1)) will be used in the simulations. This option allows the user to vary the
variables to be randomly generated without extensive modification of the input file.
Data Group 3 — Empirical Distribution Data - This data group, shown in Table 6-6 contains
the piecewise linear descriptions of cumulative frequency distributions for empirically distributed
variables specified in Data Group 2 (by assigning a value of 5 to VAR(5)). The first data line for
each distribution contains the number of data pairs NDAT used to describe the cumulative
distribution and the name of the variable. This is followed by NDAT data lines, each containing
(1) a value of the variable and (2) the corresponding cumulative probability for the specified value.
These data pair lines must be input in ascending order of cumulative probaisility. If no variables
have empirical distributions, this data group is not necessary. No END card is required for this
data group.
6-11
-------
Table 6-3
Sample Output File PDM1MC.OUT
EXAMPLE 1 FOUND IN PDM MANUAL
***
*** CONTROL PARAMETERS
***
2000
***
*** MONTE CARLO INPUTS
PARAMETER NAME MEAN
QS
QR
CS
CR
END
***
***
QS
QR
CS
CR
CO
END
***
END
***
125.
12.0
0.01
40.8
EMPIRICAL DISTRIBUTION
OUTPUT CONTROL
NO CDF
NO CDF
NO CDF
NO CDF
NO CDF
CORRELATION
END INPUT
STD DEV.
156.25
13.80
0.01
24.48
MIN
0.0
0.0
0.0
0.0
MAX
10000.
5000.
1.0
5000.
DIST TYPE
2
2
2
2
***
WRITE
WRITE
WRITE
WRITE
WRITE
i
6-12
-------
TABLE 6-4
FORMAT FOR READING SIMULATION CONTROL PARAMETERS (DATA GROUP 1)
LINE PARAMETER FORMAT
1 TITLE A80
2 NRUNS, PALPH 15, F10.0
VARIABLES DEFINITION
TITLE Run Title.
NRUNS The number of Monte Carlo runs.
i
PALPH The confidence level 1 - PALPH (percent) confidence
bounds on quantiles. If a value is not specified,
90% confidence bounds are used.
6-13
-------
TABLE 6-5
FORMAT FOR READING DISTRIBUTION PARAMETERS FOR
INPUT VARIABLES (DATA GROUP 2)
1*
PARAMETER
PNAME, VAR(l), VAR(2), VAR(3), VAR(4), VAR(5)
FORMAT
A20
5F10.0
VARIABLES
PNAME
VAR(l)
VAR(2)
VAR(3)
VAR(4)
VAR(5)
DEFINmON
A label of up to 20 characters identifying the variable to be randomly
generated. The first two characters must be one of the following:
CS - upstream concentration
CR - runoff concentration
QS - upstream flow
QR - upstream flow
The mean of the distribution (or the most frequent value for triangular
distributions).
The standard deviation of the distribution.
The minimum value for the random variable.
The maximum value for the random variable.
A flag specifying the distribution type of the random variable:
0 = Constant
1 = Normal
2 = Log-normal
3 = Exponential
4 = Uniform
5 = Empirical distribution, to be supplied
in Data Group 3, Table 6-6
6 = Triangular
i
* Data lines repeated for each input variable.
The end of this Data Group is marked by an END card.
6-14
-------
TABLE 6-6
FORMAT FOR READING EMPIRICAL DISTRIBUTION DATA (DATA GROUP 3)*
JNE
1
2a
PARAMETER
NDAT, NAME
DIST(U), DIST(L2)
FORMAT
15, A20
2F10.0
VARIABLES
NDAT
NAME
DIST(I,2)
DEFINITION
Number of data pairs used to describe the
piecewise linear cumulative distribution.
The name of the variable corresponding to the
empirical distubution. (CS, CR, QS, QR)
The value of the variable corresponding to
the quantile for data pair I.
The cumulative probability associated with
the quantile DIST(I,1).
a This line is repeated NDAT times. >
* Data Group repeated for each variable having an empirical distribution (VAR(5) = 5 in Data Group 2).
6-15
-------
TABLE 6-7
FORMAT FOR READING OUTPUT OPTIONS (DATA GROUP 4)
LINE PARAMETER
1* (SNAME(I), 1=13)
FORMAT
3A20
VARIABLES
SNAME(l)
SNAME(2)
SNAME(3)
DEFINITION
A label used to identify which variable
is to be statistically summarized. First two
characters must be from the list shown in
Table 6-5 or a 'CO' can be used to indicate
downstream concentration.
A flag that indicates if cumulative
distributions should be plotted for the
variable SNAME(l). This option is
selected by putting "CDF" here. Putting
anything else will result in the CDF not
being plotted (i.e., NO COF).
A flag that indicates if values of the
variable are to be written out for each
Monte Carlo run (selected by putting
"WRITE" here).
€
* Data lines repeated for each desired output variable. The end of this Data Group is marked
by an END card.
6-16
-------
TABLE 6-8 i
FORMAT FOR READING CORRELATED VARIABLE INPUT (DATA GROUP 5)
LINE PARAMETER FORMAT
1* PNAME(l), PNAME(2), CORR(1,2) 2A20, F10.0
VARIABLES DEFINITION
PNAME(l) A 1- to 20-character label identifying the
first correlated variable. Distribution
type must be normal or log normal. First two
characters must be CS, CR, QS, or QR.
PNAME(2) A 1-to 20-character label identifying the
second correlated variable. Distribution
type must be normal or log normal. First two
characters must be CS, CR, QS, QR or 'CO1,
the latter indicating downstream contamination.
CORR(1,2) The value of the correlation coefficient
for variables 1 and 2. Must be between -1 and 1.
* Data line repeated for each pair of correlated variables.
The end of this data group is marked by an END card.
6-17
-------
Data Group 4 - Output Options - The Monte Carlo simulation technique requires the
application of the model for each combination of input parameters that is randomly generated. This
process may be repeated several hundred times, resulting in several hundred values of model
responses. The post-processor included in the model analyzes these concentration values to
estimate (i) the mean, coefficient of variation, skewness coefficient, and kurtosis, (ii) the CDF of
the output concentration, (iii) selected percentile values, and (iv) confidence intervals for the
estimated percentile values. In addition, the post-processor includes a plotting subroutine that can
be used to develop printer plots of the CDF. The results from these analysis are written to a file
PDM.OUT.
This data group specifies the statistical output options for each variable to be written out.
Data formats are shown in Table 6-7. This data group consists of one line for each output variable
containing: (1) a character label up to 20 characters long identifying the output variable, (2) a flag
indicating if a cumulative distribution should be plotted for this variable (selected by supplying the
letters "CDF" here), and (3) a flag indicating if values of the variable random-generated are to
written out for each Monte Carlo run (selected by supplying the word "WRITE" here). If
"WRITE" option is chosen for any variable the results are written in file PDM.VAR. The labels
used to identify variables must correspond to these specified in Table 6-5 through 6-7. A statistical
summary table will be printed out for all variables selected in this data group. An END card is
supplied to mark the end of this Data Group after a data line is supplied for each output variable.
. . Data Group 5 - Correlated Input Variables - This data group, shown in Table 6-8, is used
to indicate which of the input variables specified in Data Group 2 are correlated. Note that only
variables with normal or log-normal distributions can be correlated. One line of data is provided
for each pair of correlated variables. The first two entries on this line are labels identifying the two
variables that are correlated. The third entry on these data lines is the value of the correlation
coefficient. After a data line is supplied for each correlated pair of variables, an END card must be
provided to mark the end of the data group. Note that an END card for this data group is always
required even if no variables are correlated.
6.5. EXAMPLE APPLICATION
This section presents an example application that demonstrates the capabilities of the model.
The input file necessary to run this example in the batch mode and the output file it produces are
included on the floppy disk. Details of the example are discussed in Section 4.4 and Appendix B.
The input file for this example is called PDM1.IN for the analytic mode. An example input file for
the Monte Carlo option is also included called PDM1MC.IN.
Table 6-2 shows the input file for the analytical approach, Table 6-3 for the Monte Carlo
Approach. The corresponding results are presented in Tables 6-9 and 6-10, respectively.
6-18
-------
Table 6-9
Sample Output File PDM1.OUT
RECEIVING WATER IMPACT ANALYSIS
****************************************************
PROBABILITY DISTRIBUTION AND RETURN PERIOD
OF
RECEIVING WATER CONCENTRATIONS
Developed by
Woodward-Clyde Consultants
500 12th St., Oakland, CA. 94607
(415)893-3600
****************************************************
GENERAL MODEL RUN INFORMATION
Correlation between streamflow and runoff .000
First Quantile for forced straight line agreemen 1.282
Second Quantile for forced straight line agreeme 2.652
Number of storms per year 100
Order of gaussian quadrature 15
CONCENTRATIONS ESTIMATED AT THE FOLLOWING EXCEEDANCE PROBABILITIES
1.000 .990 .980 .950 .900 .800 .700
.200 .100
FLOW RATE CONCENTRATION
COEFFICIENT ' COEFFICIENT
OF OF
MEAN VARIATION MEAN VARIATION
.600
Upstream of Discharge
Runoff
125.00
12.00
1.25
1.15
.01
40.80
.10
.60
6-19
-------
Table 6-9 (continued)
Sample Output File PDM1.0UT
i
Stream concentrations percent exceedance and
return period calculated for
Lowest Highest Increment
10.000
.500
100.000
10.000
10.000
.500
Downstream Concentration Statistics
Mean - 8.3 Standard Deviation =
7.8
Coef. of Var. =
,93
ESTIMATES OF CONCENTRATIONS
USING THE METHOD OF MOMENTS APPROXIMATION
EXCEEDANCE
PROBABILITY
.000200
.010000
.020000
.050000
.100000
.200000
.300000
.400000
.800000
.900000
CONCENTRATION
100.0873
38.3399
30.9056
22.3661
16.7790
11.8460
9.2162
7.4381
3.1307
2.2103
RETURN PERIOD
YEARS
50.007
1.000
.500
.200
.100
.050
.033
.025
.012
.011
i
6-20
-------
Table 6-9 (concluded)
Sample Output File PDM1.OUT
ESTIMATES OF EXCEEDENCE PROBABILITY
USING GAUSSIAN QUADRATURE SOLUTION
PROBABILITY
CONCENTRATION
.000181
.000314
.000563
.001046
.002031
.004146
.009007
.021181
.055567
.175212
.175212
.187099
.200028
.214122
.229524
.246403
.264958
.285426
.308090
.333296
.361466
.393124
.428933
.469739
.516640
.571092
.635050
.711138
.802649
.911458
100.0000
90.0000
80.0000
70.0000
60.0000
50.0000
40.0000
30.0000
20.0000
10.0000
10.0000
9.5000
9.0000
8.5000
" 8.0000
7.5000
7.0000
6.5000
6.0000
5.5000
5.0000
4.5000,
4.0000
3.5000
3.0000
2.5000
2.0000
1.5000
1.0000
.5000
RETURN PERIOD
i YEARS
55.382
31.87-5
17.774
9.557
4.924
2.412
1.110
.472
.180
i .057
J . 057
.053
.050
.047
.044
.041 .
.038
.035
.032
.030
,028
.025
.023
.021
. 019
.018
.016
.014
.012
.011
6-21
-------
Table 6-10
Sample Output File PDM1MC.OUT
HONTE CARLO
HONTE CARLO ANALYSIS OF PROBABILISTIC DILUTION MODEL
DEVELOPED BY
PHIL HINEART
ATUL SALHOTRA
WOODWARD-CLYDE CONSULTANTS
OAKLAND, CA
FOR
U.S. ENVIRONMENTAL PROTECTION AGENCY
ENVIRONMENTAL RESEARCH LABORATORY
ATHENS, GA 30613
JULY 1989
EXAMPLE 1 FOUND IN PDH MANUAL
INPUT DATA
NUMBER OF MONTE CARLO RUNS = 2000
NUMBER OF MODEL INPUT PARAMETERS
BEING VARIED = 4
CONFIDENCE LEVEL FOR PERCENTILES = 90.00 %
-------
NOTATIONS
Co = downstream concentration in the river [mg/1]
Q)a = the downstream concentration exceeded with a probability (1 - a) [mg/1]
CR = pollutant concentration in the runoff [mg/1]
CR = natural logarithm of CR
Cs = upstream concentration in the river [mg/1]
cs = natural logarithm of Cs
d = natural logarithm of D
DP = mean rainfall event intensity [Tj
DR = mean runoff event duration [T]
f (cs Id) = conditional density of cs given d
f(CR, Cs, D) = the joint probability density function for CR, Cs, and D
f(CR!CS ,d) = conditional density of CR given cs and d j
f (d) = marginal probability density of d
i
MRI = the mean recursion interval [years] •
QP = mean rainfall event intensity [L3/T] !
QR = mean runoff event flow rate [L3/T]
Qs = upstream flow rate in the river [m3/s] !
Rv = average runoff coefficient [dimensionless]
VR = mean runoff event volume [L3] ,
VP = mean rainfall event volume [L3]
I
Za = the standard normal a quantile j
n(Co) = coefficient of variation of Co in the arithmetic space
(i(x) = arithmetic mean of the variable x
M-in (Q)) = mean of co in the logarithmic (normal) space
N-l
-------
NOTATIONS (concluded)
InCQs.QiO = the cross-correlation coefficient between ln(Qs) and ln(QR)
S(x) = standard deviation of the variable x
Sin (Q)) = standard deviation of Co in the logarithmic (normal) space
0 = the runoff flow fraction or dilution [dimensionless]
i
N-2
-------
Appendix A
MATHEMATICAL DETAILS OF THE NUMERICAL PROCEDURE
A-l
-------
APPENDIX Al
DERIVATION OF Pr { 0 < 0*}
il
The runoff fraction is defined as:
_ __ = __
QS 1+D i + ed
where: D = QR/QS and d= hi (D), which is a normal random variable. Thus, the
a-quantile, 0a, is such that:
(A-l)
a=Pr {0<0a}=pr
+ ed
(A-2)
0a
since taking reciprocals inverts the inequality and taking logs of each side does not
Normalizing the probability results in:
d
a
r s(d)
0a
s(d)
(A.3)
The left hand side of the inequality is standard normal, za. Solving inside the brackets for
In
s(d) = ^(D) - zaSi(D)
and:
1
l+exp[n1(D)-zaSJ(D)
(A-4)
(A-5)
where Za is the standard normal a-quantile. The reason the derivation is straightforward is
that 0 is a monotonic function of d which is easily invertable.
il
A-2
-------
APPENDIX A2
EVALUATION OF THE INTEGRALS DEFINING Pr { C0 > Co •}
Equation 3.36 is stated below:
Pr{Co>Co} =L.f(d) {£CS1f(csld)QRdcs
A number of methods have been tried for the numerical computation of these integrals. The
most successful is based on an inverse probability transformation which removes the
probability density from the integral and results in a finite range of integration. If the
integrals are written using the lognormal probability density function:
where f is the lognormal probability density function. Define new variables of integration,
y and x, by the relationships: ,
y = [oCsf(CsID)dCs=Ps(Cs) .., (A.8)
X = |oDf(D)dD=PD(D) (A.9)
Then: I
dy = f(CsID) dQ (A-10)
dx=f(D)dD ' ! ' (A-n)
The limits of integration need to be transformed also. ;
For D = 0 ;
I
1
[°f(D)dD = 0 • (-A-121
JO i \r\ 1**J
For D ='
£f(D)dD = l | (/
A-3
-------
For Cs = 0
ForCs =
Jo°QRf(CsID)dCs = 0
JoCsi f(CslD) dCs = Pr (Cs £Csi) = Ps (Csi)
(A-14)
(A-15)
Expressions for Cs and D are also needed. These can be written as inverse functions of the
form P'1 (y) = Cs (y) and P4 (x) = D(x).
With these substitutions, equation A-7 can be rearranged to give:
Q)>Co}
dx
(A-16)
which can now be evaluated using standard guassian quadrature formulas. It. should be
noted that QR* is now a function of Cs(y) and D(X) and Qs* is now a function of D(x).
The probability, Ps( Csi ), and the inverse functions, Cs(y) and D(x), are computed from
the standard normal cumulative distribution function:
P(z*) =Pr { z < z* } = 1 - Q* (z*)
and its inverse. That is, if p = P(z*), then the inverse function, P-i, satisfies the
relationship:
(A-17)
Thus:
(A-18)
Ps(Csi) = Pr {Cs£Csi} = 1 -Pr (Cs>CSi) = 1 -Qs
The inverse functions for Cs(y) and D(x) can be found using equation A-18:
In Cs(y) -p. (Csld)
S(Csld)
Therefore
and similarly
Cs(y) = exp
s(csld) ]
(A-20)
(A-21)
A-4
-------
; (A-22)
Gauss-Legendre Quadrature was used to integrate equation A-16. In order to use this
method the limits of integration need to be in the interval [-1,1]. The roots of the
Legendre Polynomials can be transformed to the interval defined by the actual limits using
the following formula:
v. = b + a . b-a x.
1 2 2 ' (A-23)
where a and b are the upper and lower limits of integration, and the xi' s are the roots of the
Legendre Polynomials. For the D integral,, the roots are tranformed using:
"l 2 2 ' i (A-24)
, I '
and for the Cs integral the roots are transformed using • f
yy=~^(1+Xi)V- | (A-25)
With these transformations the integrands in equation A-16 can be evaluated at the
Legendre Polynomial roots. The D integral integrand is:
^ -* l S(csidi) (A-26)
where DI is evaluated as:
A = D (xi) = exp [n(d) + S(d) P'1 (xi)] ; (A-27)
where the conditional moments are given in Table A1.
The Cs integral integrand is:
Qk[Cs(yij),D(xi)] =
^ In [C^(l+Di) - DI Cs(yy)] - |J, (CR!CS, d)
S(cRlcs,d) ! (A-28)
1
with the conditional moments evaluated at Cs = In (CsCyy)) and d = InjDj. The integrals
can now be solved using standard Gaussian Quadrature.
A-5
-------
i
-------
Appendix B
EXAMPLE APPLICATION OF THE METHOD OF MOMENTS METHOD
B-l
-------
I
III
-------
FIGURE B-l
EXAMPLE APPLICATION OF THE METHOD OF MOMENTS METHOD
Input Data
Stream Flow (Qs)
Runoff Flow (QR)
Runoff Conc.(CR)
Upstream Cone. (Cs)
Correlation of Qs and QR
r (Qs, OR)
Unit
cfs
cfs
^gA
UgA
Mean
125
12
40.8
0
0
1
Coefficient of Variation
1.25
1.15
0.60
-
-
STEP 1: Compute statistical parameters of inputs assuming lognormal distributions.
• For runoff concentration (CR):
Arithmetic
Mean |i(CR) = (given) = 40.80
Coef. Var. n(CR) = (given) = 0.60
Std. Dev. S(CR) = |i(CR) • n(CR) = (40.8) • ((0.6) = 24.48
Median (QO = ntCR)/Vl+n2(CR) = 40.8/Vl+(0.6)2 =34.99
Logarithmic
Log Mean MOD = ln(CR) = In (34.99) = 3.555
Log Std. Dev. Sin(CR) = Vlnd+n^CR) = Vln(l+(0.;6)2) = 0.555
• Repeat these computations for the other input parameters, and tabulate
results for convenience in the following steps:
Arithmetic Logarithmic
Mean Median Std.Dev. Coef.Var. Mean Std.Dev.
Runoff Conc.(CR) 40.80 34.986 24.48 0.60 3.555 0.555
Runoff Flow (QR) 12.00 7.874 13.80 1.15 2.064 0.918
Streamflow (Qs) 125.00 78.087 156.25 1.25 4.358 0.970
B-2
-------
i
-------
STEP 2:
(a) Compute standard deviation and the log mean of the flow ratio Qs/QR = D:
Sln(D) =s2h((^)+S2in((^) - 2r
The first two terms are taken from the table just prepared (and squared). Since, for
this example, flows are uncorrelated (r =0), the third term drops out.
Sin (D) = V(0.970)2 + (0.918)2 = 1.336
= 4.358 - 2.064 = 2.294
i
(b) Compute the 90th and 99.6th percentiles of the actual distribution of the dilution
factor (0) using the appropriate values of z for the selected percentiles:
Z90 = L2S2 Z.996 - 2'652
1
l+exp[nln(D)-zaSln(D)]
Substituting the appropriate values gives:
090 = °'358 0 996
(c) Compute log standard deviation and log mean of lognorrml approximation of
the distribution of the dilution factor (0):
Log std.dev. Sln(0) = to(099-6 > • ln(09° } = 0.565
6 mv>v Z99.6 - Z9Q
Log mean 11^(0) = ln(099.e ) - Z99.6Sin(0) = -1.750
B-3
-------
(d) Compute arithmetic transforms using equations in Table 4-1, and tabulate for
convenience:
Arithmetic
Logarithmic
Mean Median Std.Dev. Coef.Var. Mean Std.Dev.
Dilution Factor (0) 0.2038 0.1738 0.1249 0.61 -1.749811 0.564562
STEP 3: Compute the statistical parameters of the resulting in-stream concentration (Co).
(a) Compute the arithmetic mean concentration using previously tabulated values:
ntO) ) = (rr(CR )• n*0) + [ rxjCs ) • ( 1 - n*0) ) ]
= (40.80-0.2038)+ [0] = 8.32
(b) Compute the standard deviation:
s(Co ) = Vs2(0)
s2(CR ) [s2(0)'+n2(0)] + s2(Cs ) [s2(0) + (1- 01(0)2]
x
= 0.0156 • (40.80 - 0) + 599.27 • (0.0156+0.0415) + [ 0 ]
= 7.76
(c) Compute and tabulate other statistical parameters of stream concentration for
use in subsequent analyses:
il
il
Arithmetic
Logarithmic
Mean Median Std.Dev. Coef.Var. Mean Std.Dev.
Stream Cone. (Co) 8.32 6.08 7.76 0.93 1.805 0.791
B-4
-------
STEP 4:
Use the statistical parameters of stream concentration computed in the previous step to
construct a graphical display (or table) summarizing: (a) the probability distribution and (b)
the recurrence interval.
(a) Probability plot using log probability paper (see Figure 4-3)
• Plot the median (QJ = 6.08) at the 50% plotting position.
• From a table of standard normal distributions (any text on statistics)
select a set of probabilities and their corresponding z scores. For
example:
i
Probability (a) 2a.
0.50 (50%) 0
0.159 (16%) -1.000
0.841 (84%) +1.000
0.050 ( 5%) -1.645
0.950 (95%) +1.645,
• For the selected probability values, estimate the corresponding
downgradient percentile concentration using Equation 3.9 shown below:
Qa = exp [^hl(Q)) + zaSln(Co)]
Prob. z Cn
Log mean =1.805128 5% -1.645 1.65
Log sigma = 0.791337 16% -1.000 2.75
84% +1.000 13.43
95% +1.645 22.36
B-5
-------
i
(b) Plot the Mean Recurrence Interval (see Figure 4-4).
• Rearrange Equation 3.9 (used in previous step) to:
= ln(Q).)-Mfa(Co)
3 Sin(Cb)
• Select a series of values for stream concentration (Co), take the natural
(base e) log, and enter in the equation with the log mean and log sigma
(determined in Step 3), to compute the value of Za corresponding with the
selected concentration.
• From a standard normal table, determine the probability (a) associated
with za. Note that standard tables are usually set up so that a reflects the
probability equal to or less than.
• To compute the mean occurence interval (MRI) at which the' selected
concentration will be equalled or exceeded:
exceedance probability = 1 - a
and
MRI =
(exceedence prob.) • (no.storms / yr)
For example, given the stream concentration distribution from Step 3 and
100 storms per year:
Exceedance Prob.
Q)a
100
80
60
50
40
30
20
Za
3.538
3.257
2.893
2.663
2.381
2.017
1.505
fl-a)
0.000201
0.000564
0.001909
0.003876
0.008645
0.021852
0.066220
MRI
(years)
49.7
17.7
5.2
2.6
1.2
0.5
0.15
B-6
-------
Figure B-2(a) presents the cumulative probability distribution of the downstream
concentrations. Figure B-2(b) presents the average recurrence interval associated with
specific stream concentrations that will be produced during storm runoff periods.
B-7
-------
100
.1
99.9 99 95 90 80 50 20 10 5
PERCENT EQUAL OR GREATER
0.1
Figure B-2(a). Probability Distribution of Pollutant Concentrations
During Storm Runoff Events
1000-F
o°
100--
o o
« d
o a
o
Q
i
.1 i 10
Mean Recurrence Interval ( YEARS )
Figure B-2(b). Recurrence Intervals for Stream Concentrations
100
i
B-8
-------
Appendix C
LIST OF SUBROUTINES INCLUDED IN THE PDM MODEL
C-l
-------
TABLE C-1
LIST OF SUBROUTINES INCLUDED IN THE ANALYTIC PDM MODEL
SUBROUTINE
Input/Output Routines
READF
COMRD
OPENF
B3-D
OUTPUT
SCREEN
GENERL
CONCIN
CONCN2
EXCEED
INTACT
READR
ERR
DPDM
Computational Routines
CALLED BY
DPDM
READF
READF, ECHO
DPDM
DPDM
DPDM
DPDM
DPDM
DPDM
DPDM
GENERL, CONCIN,
EXCEED
INTACT
INTACT
MAIN
DESCRIPTION
Reads data from a file in batch mode
Searches for data lines in input files
Opens input and output files
Echos input data to output file
Writes results to output file
Writes status screen to terminal
Interactively reads general run data ,
Interactively reads flow and
concentration data
Interactively reads concentration data used
by Gaussian Quadrature Method
Interactively reads exceedance data
Interactively inputs a table of values
Reads input data from the screen
Prints an error message to the screen when
interactively inputting data
Main calling routine for analytical models
i
TRNLOG DPDM
Transforms mean and standard deviation
from the arithmetic space to log space
C-2
-------
TABLE C-1
LIST OF SUBROUTINES INCLUDED IN THE ANALYTIC RDM MODEL (concluded)
SUBROUTINE
CALLED BY
DESCRIPTION
MOMENT
Function INVNRM
NUMSUL
DPDM
MOMENT
DPDM
GAUSU3
DG4USS
NUMSOL
GAUSLG
Function NRMPRB
Function QSTARR
GAUSLG
Calculates exceedance probabilities using
the method of moments
Approximates the inverse of the normal
(gaussian) probability distribution
function
Calculates the exceedance probability using
direct numerical interaction of joint
probability function using Gaussian
Quadrature
Integrates an arbitrary real function as
a single variable over a finite interval
using Gaussian-Legenclre Quadrature
Computer roots and weight fractures for
the Gauss-Legendre Quadrature.
Transforms the integrate interval to
[-1,1]
Approximates the normal probability
function
Function QR (see Chapter 3)
C-3
-------
8910148Aef / CON-1
TABLE C-2
LIST OF SUBROUTINES INCLUDED IN THE MONTE CARLO PDM MODEL
i
Subroutine
BMPCHR
CENTER
COMRD
CQNOUT
DECOMP
Called By
NAMFLX
OPENF
READM
MAIN
INITMC
Description
Capitalizes character variables.
Centers a character variable.
Reads and checks if input lines contain comments,
data, or end of file.
Reads output options in Monte Carlo mode.
Decomposes correlation matrix into coefficient
EMPCAL
RANDOM
ERR VARBLE, CONOUT
ERRCHK COMRD
EXPRN
FRQPLT
FRQTAB
INCOR
INIT
INITMC
RANDOM
OUTPOR
OUTPOR
MAIN
MAIN
MAIN
INTRO
MAIN
matrix required to generate correlated random
numbers.
Generates empirically distributed random
numbers by interpolating linearly from an input
empirical distribution.
Prints error message to screen.
Writes error messages when unexpected end-of-
file is encountered in input data file.
Generates exponentially-distributed random
numbers.
Writes plots of cumulative frequency
distributions.
Writes tabulated cumulative frequency
distributions.
Reads in correlative matrix.
Initializes Monte Carlo variables.
Initializes statistical summation arrays,
reorganizes Monte-Carlo input arrays to account
for constant variables, and performs other
miscellaneous Monte-Carlo intializations.
Writes initial screen.
C-4
-------
89lOl48Aef / CON-2
TABLE C-2 (Continued)
LIST OF SUBROUTINES INCLUDED IN THE MONTE CARLO PDM MODEL
Subroutine
Called Bv
Descripfjnn
LFTJUS
LNGSTR
MCECHO
MTPV
NAMFIX
NAMFIX
READM
RANDOM
NAMFIX
NMB
OJTFOR
OUTPUT
PDM
QUES
QUESST
QUES2
RANDOM
READA
READM
READM
RANDOM
OUTPUT
MAIN
MAIN
MAIN
OPENF
MAIN
MAIN
CONOUT
MAIN
READR
Left-justifies character variables (i.e., removes
blanks from the left-side of character strings).
Finds the length of character variables (i.e., the
number of non-blank characters).
Writes Monte-Carlo input data to the output file.
Multiplies a vector of uncorrelated variables by a
coefficient matrix to reform a vector of correlated
variables.
Left-justifies and capitalizes character variables.
Generates normally-distributed random numbers.
I
Writes frequency tables and plots.
Writes out statistical summaries of Monte-Carlo-
runs to the output file.
Monte Carlo computational routine.
Asks user whether input in batch or interactive.
Asks user whether to overwrite a file.
Asks user which model to run.
Generates a vector of random numbers from
specified distributions.
Reads character variable from screen.
Reads Monte-Carlo input data from a user-
specified input file unit number.
VARBLE Reads real number from screen.
C-5
-------
8910148Aef / CON-3
TABLE C-2 (Concluded)
LIST OF SUBROUTINES INCLUDED IN THE MONTE CARLO PDM MODEL
i
Subroutine Called Bv
REMP MAIN
STATIS
STOUT
UNIF
VARBLE
MAIN
OUTPUT
RANDOM.NMB,
EXPRN
MAIN
Description
Reads empirical distributions for the Monte Carlo
model in interactive mode.
Performs summations required to compute
statistical moments for random model inputs and
model outputs over all Monte-Carlo runs.
Computes statistical moments (mean, standard
deviation, skewness, kurtosis, correlations,
minimum and maximum) from summations
computed by STATIS. Statistics are then written
out to the Monte-Carlo output file.
Generates uniform random numbers ranging
between 0 and 1.
Reads in Monte Carlo distribution data while in
interactive mode.
C-6
-------
Appendix D
LIST OF VARIABLES INCLUDED IN THE PDM MODEL
D-l
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Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES
Variable
A
A
A
A
AMN
AMX
ANS
ARC
AV
B2
B
Type*
Array (DP)
Scalar (C)
Scalar (R)
Scalar (R)
Scalar (R)
Array (R)
Scalar (C)
Scalar (R)
Scalar (R)
Scalar (R)
Array (DP)
Description
Array to be multiplied
by a vector
Character string to
be left-justified and
capitalized
Minimum value for a
distribution
Triangular distribution
parameter
Minimum value of an
output data set
Maximum value of an
output data set
Character string to
be capitalized
Value used to calculate
log normal distribution
parameters
Intermediate value used
to generate normal
random numbers
Triangular distribution
parameter
Decomposed correlation
matrix
Subroutine
or Function
MTPV
LFTJUS
NAMFIX
LNGSTR
TRANSB
RANDOM
OUTFOR
FRQTAB
OUTFOR
BMPCHR
MAIN
TRNLOG
NMB
RANDOM
RANDOM
DECOMP
Designation**
I
M
M
M
I
L
L
I
L
L
L
M
L
O
M
Array (R)
Vector to be multiplied
by an array
MTPV
D-8
i
-------
APPF-2
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
B
BBT
C
CN
CD
CORR
CORR
CR
CS
CVAR
CV
DECOM
DIST
Type*
Scalar (R)
Array (DP)
Scalar (R)
Array (R)
Scalar (R)
. Array (DP)
Scalar (R)
Scalar (R)
Scalar (R)
Scalar (R)
Scalar (R)
Array (DP)
Array (R)
Description
Maximum value for a
distribution
Correlation matrix for
Monte Carlo inputs
Triangular distribution
parameter
Array of random
correlated numbers
Downstream
concentration
Array of correlation
terms for summary
output variables
Correlation
coefficient
Runoff concentration
Upstream concentration
Coefficient of variation
Coefficient of
variation
Decomposed correlation
matrix for Monte Carlo
inputs
Array storing empirical
distributions
Subroutine
or Function
TRANSB
READM
INITMC
STATIS
MCECHO
DECOMP
RANDOM
RANDOM
TRANSM
MAIN, PDM
,
INITMC
STATIS
OUTPUT
STOUT
READM
].
MAIN, PDM
MAIN, PDM
STOUT
OUITOR
FRQTAB
INITMC
!
1
READM
RANDOM
MCECHO
EMPCAL
Designation**
I
O
I
L
I
M
L
M
I
O
M
M
I
M
L
I
I
L
L
I
O
O
I
I
I
D-9
-------
APPF-3
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
FACTOR
FATAL
FN
EC
FXM
GR
G
HH
H
ICDF
ICHAR
ICH
ICOUNT
••••
Type*
Scalar (R)
Scalar (L)
Scalar (R)
Scalar (R)
Scalar (R)
Array (C)
Scalar (R)
Scalar (DP)
Scalar (DP)
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (I)
Subroutine
Description or Function
Power of ten values
on the X-axis of
cumulative distribution
plots
Variable indicating
error in input data
The number of values
A randomly-generated
cumulative probability
Triangular distribution
parameter
Grid line contents
Intermediate value used
to generate normal
random numbers
Element of the
decomposed correlation
matrix
Element of the
decomposed correlation
matrix
Counter for cumulative
distributions
Counter for number of
characters in a line
of data
The number of characters
in a character string
Counter for correlated
variables
-rr-To
OUTFOR
FRQPLT
COMRD
ERRCHK
STOUT
EMPCAL
RANDOM
RANDOM
FRQPLT.
NMB
DECOMP
DECOMP
OUTPUT
OUTFOR
COMRD
BMPCHR
MCECHO
Designation**
L
I
L
I
L
L
L
L
L
M
L
L
L
I
L
L
L
^^^
C
(I
-------
APPF-4
Table D-2. i
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
IERROR
IERR
IE
IHL
IJ
IN2
INDEX
IOUT2
IOUT
IPERC
IPL
IRUN
ISCR
Type*
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (C)
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (C)
Scalar (C)
Scalar (I)
Scalar (I)
Description
Flag indicating error
condition
Input error flag
Counter for empirical
distributions
"-" character
Do loop counter
Monte Carlo input file
unit number
Counter for empirical
distributions
Output file unit number
for results of each
Monte Carlo run
Output file unit number
for the summary output
file
"%" character
"+" character
Counter for number of
Monte Carlo runs
Unit number for screen
input and output
Subroutine
or Function
ERRCHK
READM
RANDOM
MCECHO
COMRD
TRANSM
READM
MCECHO
FRQPLT
FRQPLT
READM
RANDOM
OUTFOR
EMPCAL
STATIS
OU1TUT
STOUT
FRQPLT
FRQPLT
STATIS
i
READM
INITMC
COMRD
ERRCHK
DECOMP
Designation**
I
L
L
I
L
O
L
L
L
L
I
L
L
I
I
I
I
L
L
I
I
I
I
I
I
n-ii
-------
APPF-5
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable Type*
ISEED Scalar (I)
Description
Seed for uniform
Subroutine
or Function
RANDOM
random number generator EXPRN
ISTAR Scalar (C)
ISTRT Scalar (I)
I Scalar (I)
ITYPE Scalar (I)
IVL Scalar (C)
IWDTH Scalar (I)
IWRTX Scalar (I)
"*" character
Counter for the first
non-blank character in
a character string
Do loop counter
Flag indicating whether
variables are trans-
formed to or from
Johnson SB space
"!" character
The length of a
character string
Format for label on
X-axis of cumulative
EMPCAL
FRQPLT
LFTJUS
READM
INITMC
RANDOM
STATIS
OUTPUT
MCECHO
COMRD
LFTJUS
NAMFIX
MTPV
EXPRN
DECOMP
OUTFOR
FRQPLT
FRQTAB
STOUT
EMPCAL
TRANSM
TRANSB
FRQPLT
LFTJUS
NAMFIX
OUTFOR
FRQPLT
Designation**
0
M
I
L
L
L
L
L
L
L
L
L
L
L
L
M
L
L
L
L
L
L
L
I
L
L
L
L
L
IWRTY
Scalar (I)
probability plots
Y-axis format index
FRQPLT
n-i?
-------
APPF-6
DESCRIPTION,
Table D-2.
!
TYPE, AND DESIGNATION OF VARIABLES USED
BY
• MONTE CARLO ROUTINES (Continued)
: — .
Variable Type*
IX Scalar (I)
IY Scalar (I)
JE Scalar (I)
JJ Scalar (I)
J Scalar (I)
• .
.
K Scalar (I)
Kl Scalar (I)
KFILE Scalar (I)
KK1 Scalar (I)
Description
The value of the random
number used
Do loop counter
Do loop counter
Counter for correlated
input variables
Do loop counter
Counter for correlated
input variables
Intermediate integer
used in generating
uniform random numbers
Input file unit number
Counter used to write
Subroutine
or Function
UNIF
NMB
FRQPLT
FRQPLT
READM
MCECHO
READM
READM
INITMC
STATIS
MCECHO
DECOMP
OUTiFOR
FRQPLT
FRQTAB
STOUT
READM
INITMC
MTPV
TRANS M
DECOMP
OUTPOR
FRQPLT
UNIF
•
COMRD
MCECHO
Designation**
M
M
L
L
;L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
M
L
L
KK2
Scalar (I)
names of correlated
variables
Counter used to write
names of correlated
variables
D-13
MCECHO
-------
APPF-7
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued) ^i
Variable
KK
KOUNT
LO
LI
L2
L3
L4
L5
L6
L7
L8
LARR
LA
LC
LE
LF
LINE
LI
LL
Type*
Scalar (I)
Scalar (I)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Array (I)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (C)
Description
Counter for correlated
input variables
Counter for correlated
variables
"0" character
"1" character
"2" character
"3" character
"4" character
"5" character
"6" character
"7" character
"8" character
Array storing array
addresses for random
input variables
"A" character
"C" character
"E" character
"F" character
Temporary storage of
the contents of input
data lines
"I" character
"L" character
Subroutine
or Function
READM
MCECHO
MCECHO
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
READM
INITMC
MCECHO
FRQPLT
FRQPLT
FRQPLT
FRQPLT
READM
MCECHO
COMRD
FRQPLT
FRQPLT
Designation**
L
L
L
L
L
L
L
L
i
L m
L
L
M
M
I
L
L
L
L
L
L
O
L
41
n-14
-------
APPF-8
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
LM
LNMEAN
LNSTD
LN
LQ
LR
L
LT
LU
LV
LY
LYY
MCMAX
MCVAR
MEAN
Type*
Scalar (C)
Scalar (R)
Scalar (R)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (I)
Scalar (C)
Scalar (C)
Scalar (C)
Scalar (I)
Scalar (C)
Scalar (I)
Scalar (I)
Scalar (R)
Subroutine
Description or Function
"M" character
The mean of log-trans-
formed data
The standard deviation
of log-transformed
data
"N" character
"Q" character
"R" character
Index for an element
of the decomposed
correlation matrix
"T" character
"U" character
"V" character
Interval number on Y-
axis
"Y" character
Maximum possible number
of random input
variables
Number of random input
variables
The arithmetic mean of
a variable
FRQPLT
TRNLOG
TRNLOG
FRQPLT
FRQPLT
FRQPLT
DECOMP
FRQPLT
FRQPLT
FRQPLT
FRQPLT
FRQPLT
READM
INITMC
MCECHO
EMPCAL
READM
INITMC
MCECHO
TRNLOG
OUTFOR
FRQTAB
Designation**
L
0
O
L
L
L
L
L
L
L
L
L
I
I
I
I
O
M
L
I
L
I
D-15
-------
APPF-9
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
i
Variable Type*
MESAGE Scalar (C)
MVAR Scalar (I)
MVARX Scalar (I)
Description
Error message to be
written to the terminal
screen
The number of random
input variables
The maximum allowed
number of input
variables
Subroutine
or Function
COMRD
ERRCHK
RANDOM
TRANSM
RANDOM
TRANSM
Designation**
L
I
I
I
I
I
NAME1 Scalar (C)
NAME2 Scalar (C)
NCARLO Scalar (I)
NCDF
NDAT
NDAT
Scalar (I)
NCMAX Scalar (I)
Array (I)
Scalar (I)
NDAGE Scalar (I)
Label corresponding READM
to the first correlated
variable
Label corresponding READM
to the second correlated
variable
The number of random
input variables
The number of cumulative
distribution plots
Maximum number of
variables for which
cumulative distributions
can be plotted
Number of values in
empirical distributions
MTPV
DECOMP
READM
STATIS
READM
STATIS
OUTPUT
OUTFOR
READM
RANDOM
MCECHO
I
L
L
L
I
I
I
I
O
I
I
Number of values in EMPCAL
empirical distributions
Number of values in READM
empirical distributions
i
D-16
-------
APPF-10
NMAX
NRUN
N
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
'
Variable Type*
NDIS Array (C)
NEMP Scalar (I)
NEVAR Scalar (I)
Description
Probability distribution
names
Maximum data pairs
allowed to input to
empirical distributions
The number of empiri-
cally-distributed input
Subroutine
or Function
MCECHO
READM
RANDOM
MCECHO
EMPCAL
READM
MCECHO
Designation**
L
I
I
I
I
L
I
Scalar (I)
NRMAX Scalar (I)
NRUNS Scalar (I)
Scalar (I)
Scalar (I)
NUMPLT Scalar (I)
NVAL Scalar (I)
variables
Maximum possible number READM
of variables for which INITMC
summary statistics can STATES
be printed OUTPUT
MCECHO
STOUT
Maximum number of Monte READM
Carlo simulations STATES
allowed OUTPUT
OUTFOR
The number of Monte OUTPUT
Carlo runs STOUT
The number of Monte READM
Carlo runs MCECHO
The dimensions of the MTPV
decomposed correlation DECOMP
matrix
Plot number FRQPLT
The number of values OUTFOR
in the data set to be FRQTAB
plotted
D-17
I
I
I
I
I
I
O
I
I
L
-------
APPF-11
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
NVAR
NWRITE
NY
OUT2
OUTMSG
OUT
PALPH
PC
PCC
PCCM1
PCZERO
Type*
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (I)
Scalar (C)
Scalar (I)
Scalar (R)
Array (R)
Array (R)
Array (R)
Scalar (R)
Subroutine
Description or Function
Number of summary
output variables
The number of variables
written to an output
file for each Monte
Carlo run
The number of intervals
on the Y-axis
Output file unit number
for results of each
Monte Carlo run
Error message written
to the terminal screen
Monte Carlo summary
output file unit number
Probability associated
with quantile confidence
bounds
Percent of time in
intervals
Exceedance probabilities
Cumulative probabilities
Cumulative probability
for the first interval
READM
INITMC
STATIS
OUTPUT
MCECHO
STOUT
STATIS
FRQPLT
READM
MCECHO
ERRCHK
READM
MCECHO
READM
OUTPUT
MCECHO
OUTFOR
FRQTAB
OUTFOR
FRQPLT
FRQTAB
OUTFOR
FRQPLT
FRQTAB
OUTFOR
OUTFOR
FRQTAB
Designation**
O
I
I
I
I
I
L
L
I
I
L
I
I
O
I
I
I
I
L
I
I
L
I
I
L
L
I
il
D-18
-------
APPF-12
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable Type'
Description
Subroutine
or Function Designation**
PERC50 Array (R)
PERC80 Array (R)
PERC90 Array (R)
PERC95 Array (R)
PLTOPT Scalar (I)
PLTTYP Scalar (I)
PNAME Array (C)
POWER Scalar (R)
PP Scalar (DP)
P Scalar (DP)
PXMAX Scalar (R)
PYMAX Scalar (R)
OR
OS
R
Scalar (R)
Scalar (R)
Array (R)
The 50th percentile OUTFOR
and confidence bounds FRQTAB
The 80th percentile OUTFOR
and confidence bounds FRQTAB
The 90th percentile OUTFOR
and confidence bounds FRQTAB
The 95th percentile OUTFOR
and confidence bounds FRQTAB
Option flag for cumu- OUTPUT
lative distribution OUTFOR
plots FRQPLT
Flag for type of plot FRQPLT
Input labels used to READM
flag ramdom input INITMC
variables MCECHO
Power of 10 for X-axis OUTFOR
Intermediate value used DECOMp
to decompose the
correlation matrix
Intermediate value used DECOMP
to decompose the
correlation matrix ;
Maximum value of data set OUTFOR
Maximum probability FRQPLT
to be plotted
Runoff flow MAIN, PDM
Upstream concentration MAIN, PE|M
Array of random values
generated from various
distributions
D-19
RANDOM
TRANSM
L
I
L
I
L
I
L
I
L
I
I
O
M
I
L
L
L
L
I
I
O
O
-------
APPF-13
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
RCV
RMEAN
RMN
RVAL
SD
SLOPE
SNAME
STAT
STATUS
STD
SUM2
SUM
T
Type*
Scalar (R)
Scalar (R)
Scalar (R)
Array (R)
Scalar (R)
Scalar (R)
Array (C)
Array (DP)
Scalar (L)
Scalar (R)
Scalar (R)
Scalar (R)
Scalar (R)
Subroutine
Description or Function
Coefficient of variation
Mean value of data set
Mean value
Array containing sorted
results of Monte Carlo
runs
Standard deviation of
data set
Slope used to inter-
polate values from
empirical distributions
Input labels used to
flag summary output
variables
Array of summary statis-
tics for output
variables
Variable indicating
that the end of the
input file has been
encountered
The standard deviation
of a variable
Summation term
Summation term
Probability associated
MCECHO
OUTFOR
MCECHO
OUTFOR
OUTFOR
FROTAB
EMPCAL
READM
STATIS
OUTPUT
MCECHO
STOUT
INITMC
STATIS
OUTPUT
STOUT
READM
COMRD
TRNLOG
STOUT
OUTFOR
OUTFOR
OUTFOR
Designation**
L
L
L
I
L
I
L
0
I
I
I
I
M
M
I
M
L
O
I
L
L
L
L
€'
with two-tailed
confidence bounds
D-20
-------
APPF-14
VAR
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable Type*
TABOPT Scalar (I)
TEMP Scalar (R)
TEROFL Scalar (C)
TITLE Scalar (C)
TITLE Scalar (C)
Ul Scalar (R)
U2 Scalar (R)
U Scalar (R)
VALUE Scalar (R)
Subroutine
Description or Function
Option flag for cumu-
lative distribution
tables
Temporary variable used
in sorting data
Unit number for plot
output
The title of the run
Title for cumulative
distribution plots
Intermediate value used
to generate normal
random numbers
Intermediate value used
to generate normal
random numbers
Intermediate value used
to generate normal
random numbers
The value interpolated
OUTPUT
OUTFOR
OUTFOR
OUTFOR
FRQPLT
FRQTAB
READM
MCECHO
OUTPUT
OUTFOR
FRQPLT
NMB
NMB
NMB
EMPCAL
Designation**
L
I
L
I
I
I
L
I
L
I
I
M
M
M
O
Array (R)
Scalar (R)
from empirical
distributions
Array storing distri-
bution parameters for
random input variables
Intermediate value used
to generate normal
random numbers
D-21
READM
INITMC
RANDOM
MCECHO
TRANSM
NMBi
O
M
I
I
I
M
-------
APPF-15
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Continued)
Variable
W
X
X
X
XCDF
XLAB
XLAB
XLIM
XMAX
XMIN
XM
XNAME
XNVAL
Type
Scalar
Array
Scalar
Scalar
Array
Array
Array
Array
Scalar
Scalar
Scalar
Scalar
Scalar
*
(R)
(R)
(R)
(R)
(R)
(R)
(R)
(R)
(R)
(R)
(R)
(C)
(R)
Description
Width of triangular
distribution base
Array storing values
of summary output
variables
Value in Johnson SB
space
Value of cumulative
probability
Array storing values
for plotting cumulative
distributions
Label for X-axis
of plot
Values for X-axis
Values corresponding to
exceedance probabilities
Maximum value
Minimum value
Triangular distribution
parameter
Label for the X-axis
of the plot
Number of values in
Subroutine
or Function
RANDOM
NMB
STATIS
TRANSB
EXPRN
STATIS
OUTPUT
OUTFOR
FRQPLT
OUTFOR
FRQTAB
OUTFOR
FRQTAB
OUTFOR
FRQTAB
RANDOM
OUTFOR
FRQPLT
OUTFOR
Designation**
L
M
I
O
M
M
I
L
I
L
I
L
I
L
I
L
I
I
L
data set
D-22
-------
APPF-16
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Concluded)
Variable Type*
YMAX Scalar (R)
Description
Maximum probability
to be plotted
Subroutine
or Function
OUTPOR
FRQPLT
Designation**
L
I
Scalar (R)
Scalar (R)
A value transformed TRANSB
from Johnson SB space
to normal space '.
Standard normal variate OUTJFOR
corresponding to per-
centile confidence level
O
Array Dimensions
MCMAX Scalar (I)
NMAX Scalar (I)
NCMAX Scalar (I)
NRMAX Scalar (I)
NEMP Scalar (I)
Monte Carlo Inputs
BBT Array (DP)
DIST Array (DP)
Maximum possible number of Main
random input variables i
Maximum possible number of Main
variables for which summary
statistics can be printed i
Maximum number of variables Main
for which cumulative dis- \
tributions can be plotted
Maximum number of Monte Carlo Main
simulations allowed
IN2
Scalar (I)
Maximum number of data pairs
(empirical distribution,
probability value) allowed to
input the empirical distribution
Correlation matrix for
inputs
Array storing empirical
distributions
Monte Carlo input file
D-23
Main
READM, Random
Main
READM, INITMC
Main
READM, Random
Main
-------
APPF-17
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Concluded)
Variable
Type'
Description
Subroutine
or Function Designation**
IN2 Scalar (I)
IOUT Scalar (I)
IOUT2 Scalar (I)
ISCR Scalar (I)
MCVAR Scalar (I)
NDAT Array (I)
Monte Carlo input file
unit number
Monte Carlo summary output
file unit number
Output file unit number for
results of each Monte Carlo run
Unit number for screen input
output
Number of random input
variables
Number of values in
empirical distributions
NRUNS Scalar (I) Number of Monte Carlo runs
NVAR Scalar (I)
PALPH Scalar (I)
PNAME Array (C)
SNAME Array (C)
VAR Array (R)
Number of summary output
variables
Probability associated with
quantile confidence bounds
Input labels used to flag
random input variables
Input labels used to flag
summary output variables
Array storing distribution
parameters for random
input variables
Main
READM
Main
READM, OUTPUT
Main
STATIS
Main
READM, INITMC
Main
READM, INITMC
RANDOM
Main
READM, RANDOM
Main
READM, OUTPUT
Main
Main
READM, OUTPUT
Main
READM, INITMC
Main
READM, OUTPUT
Main
READM, INITMC,
RANDOM
D-24
tl
-------
APPF-18
Table D-2.
DESCRIPTION, TYPE, AND DESIGNATION OF VARIABLES USED BY
MONTE CARLO ROUTINES (Concluded)
Variable Type*
Description
Subroutine
or Function Designation**
Storage Arrays
CN Array (R)
CORR Array (DP)
DEOOM Array (I)
IRUN Scalar (I)
IVAR Scalar (I)
LARR Scalar (I)
RMC Array (R)
STAT Array (DP)
XCDF Array (R)
XMC
Array (CR)
Temporary array of correlated
random numbers
Array of- correlation terms for
summary output variables
Decomposed correlation matrix
for Monte Carlo inputs
Do loop counter for
Monte Carlo runs
Do loop counter for variable
number
Array storing array addresses
for random input variables
Array of randomly-generated
numbers
Array of summary statistics
for output variables
Array storing values of
selected variables for
plotting cumulative
distributions
Array storing values of
summary output values
Main
RANDOM
Main
STATIS, OUTPUT
1
Main L
INITMC, RANDOM
Main
STATIS
Main
Main
READM, INITMC
Main
RANDOM
Main
STATIS, OUTPUT
Main
STATIS, OUTPUT
Main
STATIS
*DP = Double Precision
I = Integer
R = Real
C = Character
L = Logical
**I,O,M,L, = Input, Output, Modified, Local
0-25
*U.S. Government Printing Office: J.989-625-505
-------
i
-------
-------
------- |