&EPA
United States
Environmental Protection
Agency
Office of Water
Planning and Standards
Washington, D.C. 20460
EPA-440/5-79-015
Water
Quantitative Techniques
For the Assessment of
Lake Quality , <
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This report has been reviewed by the
clean lakes program staff, EPA. and
approved for publication. Approval
does not signify that the contents
necessarily reflect the views and
policies ofthe U.S. EPA, nor does
mention of trade names or commercial
products constitute endorsement or
recommenthtion for use. The statements
contained herein are ascribed solely
to the author.
EPA Report No. EPA-440/5-79-015
Contract No. W-1485-NHEX
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QUANTITATIVE TECHNIQUES
FOR THE ASSESSMENT
OF LAKE QUALITY
by
Kenneth H. Reckhow
Department of Resource Development
Michigan State University
East Lansing, Michigan
U.S. Environmental Protection Agency
Office of Water Planning and Standards
January 1979
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PREFACE
This manual provided a unique opportunity for me to discuss, in
one volume, several related topics of research, including lake modeling,
data analysis, and treatment of uncertainty. I am grateful to the
Department of Resource Development, the Agricultural Experiment Station
of Michigan State University, and to the Inland Lakes Management Unit
of the Michigan Department of Natural Resources, for support and en-
couragement during this past year. I would also like to acknowledge
the assistance of the following people from the Department of Resource
Development, in the preparation of this document: Patty McKenna who
typed draft versions of this report, Linda Boyer who typed the final
copy, Steven Willis who edited the final version, and Paul Schneider
who prepared much of the graphical material. Finally, I would like
to thank Albert Massey, Richard Mikula, and Howard Wandell of the
Inland Lakes Management Unit for helpful suggestions and criticisms of
the draft report.
Kenneth H. Reckhow
East Lansing, Michigan
Noveiider, 1978
11
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TABLE OF CONTENTS
Page
PREFACE ii
CHAPTER
1. INTRODUCTION 1
2. DATA ANALYSIS AND SAMPLING DESIGN 7
2.1 Introduction 7
2.2 Data Analysis 8
2.2.1 Mean and Median 8
2.2.2 Standard Deviation and Variance 9
2.2.3 Standard Error of the Mean 12
2.2.4 Correlation 12
2.2.5 Regression 18
2.2.6 Student’s t 21
2.3 Sampling Design 24
2.3.1 Introduction . 24
2.3.2 Sampling Design Relationships 26
2.3.3 Estimation of Factors Important in Sampling
Design 33
2.3.4 Some Specific Issues 34
2.3.5 Annual Phosphorus Budget 35
2.3.6 Sun ary and Conclusions 41
3. PHOSPHORUS MODELS 42
3.1 Introduction 42
3.2 Historical and Theoretical Development: A Brief
Review 42
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CHAPTER Page
3.3 Empirical Models 47
3.4 Error Analysis 56
3.5 A Case Study Comparison of Four Models 63
3.6 Special Lake Types 73
3.7 Conclusions 78
Appendix 79
Uncertainty Analysis 85
4. LAKE CLASSIFICATION 89
4.1 Introduction 89
4.2 Trophic State Criteria 90
4.3 Phosphorus Loading Criteria 96
4.3.1 Early Development 96
4.3.2 Recent Work by Vollenweider . . . . 103
4.3.3 A Lake Classification Scheme Proposed by
Uttormark and Wall 107
4.3.4 Uncertainty Analysis Applied to
Vollenweider’s Loading Criterion - A
Graphical Approach 110
4.3.5 Lake Quality Discriminant Analysis . . . 120
4.4 Sun nary 127
Appendix 130
Derivation and Explanation of the Procedure Used to
Apply Uncertainty Analysis to Vollenweider’s
Loading Criterion 130
Lake Quality Discriminant Functions 135
5. FINAL COMMENTS 137
REFERENCES 140
STATISTICAL SY! OLS 144
SY1 OLS FOR LAKE MODELS 145
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LIST OF TABLES
TABLE Page
2.1 Total Phosphorus Concentration (mg/i) 8
2.2 Dissolved oxygen Concentration (mg/i) 11
2.3 Phosphorus Concentration (mg/i) in Two Hypothetical
Lakes 22
2.4 Estimated Conditions in a Hypothetical Lake 30
2.5 Initial Uncertainty Estimates for Lake Winnipesaukee
Phosphorus Loading 40
3.1 Model Proposed by Reckhow (1977) for Oxic Lakes with
< 50 rn/yr 52
3.2 Model Proposed by Walker (1977) 52
3.3 A Comparison of Empirical Models 54
3.4 Data for Lake Charlevoix (EPA-NES, 1975) 64
3.5 Comparison of the Predictions of the Four Models for
Lake Charlevoix 74
3A.l Models Developed by Reckhow (1977) 80
3A.2 Statistics for 95 North Temperate Lakes
(Reckhow, 1977) 81
3A.3 Statistics for the Three Model Classes 82
3A.4 Parameter Statistics for the Reckhow Models . . . 83
3A.5 Parameter Statistics for the Walker Model . . . . 84
3A.6 Prediction Error Contribution from Parameter Un-
certainty for the Model Proposed by Reckhow for
Oxic Lakes With z/r < 50 rn/yr 84
4.1 Trophic State Vs. Chlorophy II c (From EPA-NES 1974a) . 91
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TABLE Page
4.2 EPA-NES Trophic State Delineation (From EPA-NES 1974a) . 91
4.3 Carison’s Trophic State Index 95
4.4 Permissible Loading Levels for Total Nitrogen and Total
Phosphorus (Biochemically Active) (g/m 2 -yr) (after
Vollenweider, 1968 98
4.5 Point System for Lake Condition Index (From Uttormark
and Wall, 1975) 108
4.6 Suggested Management Strategies Associated with the
Four Quadrants in Figure 4.5 (From Uttormark and Wall,
1975) 111
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LIST OF FIGURES
FIGURE Page
1.1 Factors Affecting Lake Quality (From Dillon, 1974) . 2
2.1 A Normal Distribution and A Skewed Distribution . . . 10
2.2 The Relationship of Sunii er Average Chlorophyll to
Spring Total Phosphorus (From Dillon and Rigler, 1974). . 14
2.3 The Relationship Between Median Total Phosphorus and
Mean Chlorophyll ct in Phosphorus Limited Lakes . . . 15
2.4 The Relationship Between Median Total Phosphorus and
Mean Secchi Disc Depths in Phosphorus Limited Lakes . 16
2.5 V versus X 17
2.6 Sampling Design Uncertainty Versus Cost 28
3.1 A Comparison of the Kirchner and Dillon Model for R
with Chapra’s Constant Apparent Settling Velocity
Approach 50
3.2 Lake Ontario: Probability of Standard (P=.045 mg/i)
Violation 60
4.1 Voilenweider’s Phosphorus Loading Criterion (1975),
L versus q 5 (z/T) 99
4.2 Dillon’s Phosphorus Loading Criterion, With Lines of
Constant Phosphorus Concentration (in ig/l) Distin-
guishing Trophic States (From Dillon, 1975) 100
4.3 The Larsen-Mercier Phosphorus Loading Criterion
(From Larsen and Mercier, 1975) 102
4.4 Vollenweider’s Phosphorus Loading Criterion (1976),
L versus q 5 , z, and T 106
4.5 Uttormark’s and Wall’s Lake Classification Scheme
(From Uttormark and Wall, 1975) 109
4.6 Uncertainty Estimates for the Loading Criteria on
Vollenweider’s 1975 Axes of L Versus q (From Reckhow,
l978d) 115
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FIGURE Page
4.7 y Versus% (From Reckhow, 1978d) 116
4.8 The Probability of Oligotrophic Classification as a
Function of Phosphorus Loading, Model Uncertainty, and
Phosphorus Loading Uncertainty (From Reckhow, 1978d). . 117
4.9 The Probability of Mesotrophic Classification as a
Function of Phosphorus Loading, Model Uncertainty, and
Phosphorus Loading Uncertainty (From Reckhow, 1978d). . 118
4.10 The Probability of Eutrophic Classification as a Func-
tion of Phosphorus Loading, Model Uncertainty, and
Phosphorus Loading Uncertainty (From Reckhow, l978d). . 119
4.11 Walker’s Loading Criterion Assessing the Probabilities
of Trophic Classification (From Walker, 1977) 123
4.12 The Oxic-Anoxic Discriminant Function (From Reckhow,
1978a) 125
4.13 Lake Sammamish and Lake Washington: Probability of
Oxic Conditions (P 0 1 )vs. Annual Areal Phosphorus
Loading (L) (From Reckhow, 1978a) 128
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CHAPTER 1
INTRODUCTION
Increases in wealth, mobility and scientific understanding have
corresponded with an increase in our ability to influence the quality
of the environment and in changes in our perception of environmental
quality. The result has been a greater awareness of linkages between
human activities and environmental degradation and an intensification
of efforts devoted to environmental management. This has been, and is
being,demonstrated in the field of lake management. Motivated in part
by the identified association between cultural activities and nutrient
enrichment, work on state and federal programs (such as those funded
under Sections 208 and 314 of P.L. 92-500) may provide insight on
success ful lake management strategies.
Fundamental to the development of policies for lake management
is the determination of the likely causal relationships among the lake
and lake basin characteristics. This manual focuses on the role of
phosphorus in lake eutrophication. Figure 1.1 (Dillon, 1974) qualita-
tively presents the major causal linkages among the phosphorus-related
components. Phosphorus was selected for this analysis because it is
generally considered the most manageable of the major nutrients.
Further, as Sawyer (1971) notes:
The phosphorus content of domestic and certain industrial
wastewaters has been closely scrutinized for at least three
1
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GEOLOGY
LAND USE
PRECIPITATION
POPULATION
DENSiTY
HYDROLOGIC
BUDGET
NATU RAL
PHOSPHORUS -*
LOAD
ARTIFICIAL
PHOSPHORUS-*
LOAD
SPRING
-* PHOSPHORUS
CONCENTRAT ION
in lake
V
N)
LAKE
MORPHOMETRY
CHLOROPHYLL a
SECCHI DISC —4— CONCENTRATION
(summer avg.) (summer avg.)
Figure 1.1
Factors Affecting Lake Quality (From Dillon,
1974)
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—3—
reasons. First, wastewater treatment methods to remove phos-
phorus from the effluent have been known for several years.
Second, a large part of the phosphorus in domestic wastewaters
and essentially all phosphorus in some industrial wastes is
contributed by synthetic detergents. Third, phosphorus limita-
tion in lakes and streams seems to be the only known means to
control the nitrogen-fixing blue-green algae. Thus, phosphorus
removal must be part of any plan to control cultural eutrophica-
tion.
In order to successfully design a wastewater treatment plant for
phosphorus removal, one needs to quantify the inputs to the proposed
plant and use mathematical equations, or models, from the sanitary
engineering literature to specify the design. Clearly, little faith
would be put in a design that resulted solely from intuitive judgment
with no consideration of the quantitative information available.
Similarly, the evaluation of strategies for lake management can and
should reflect consideration of quantitative information on lake in-
puts (of phosphorus, for instance) together with mathematical equa-
tions, or models, that relate inputs to lake quality. Here, too, it
would not be wise to rely solely on intuitive judgment. However,
personal judgment of people experienced in the management of the system
of interest (a lake, for example) can be quite important in the design
of a data gathering program,in the interpretation of the data, and in
the interpretation of a model’s prediction. This is discussed in Chapter 2.
Additional justification for the use of all available sources of
input information, particularly those that are quantitative in nature,
comes from the realization that all decisions imply a weighing of
costs, benefits, and other characteristics affected by the decision.
Whatever the adopted strategy, the implication is that the resultant
quality associated with that strategy is somehow optimal given the
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costs, benefits and constraints. Thus even though a decision maker may
resist quantification or cost-benefit comparisons because it does not
seem reasonable to him/her, any management strategy selected has this
quantification or weighing of factors implicitly written in. It seems
more reasonable to attempt to quantify all relevant factors, weigh or
relate them (through a model, perhaps) and thus explicitly consider
this quantitative information as one input to the decision-making proc-
ess.
This is not to be taken as blanket endorsement of quantification
and modeling, however. It is critical that an appropriate model be
selected (assuming an appropriate model can be found) for the intend-
ed purpose, and that the individual applying the model has a clear
understanding of the model’s limitations. Lake models exist over a
wide range of complexity and cost of application. As a rule, the
highly complex theoretically-based models are intended to study inter-
actions among ecosystem components and are not meant for general man-
agement application. Their values lie in increasing knowledge of
the chemical, physical, and biological interactions. However, their
cost is prohibitive and their accuracy questionable (for the short
time frame changes they are designed to examine) for general use. On
the other hand, simple empirical models, like those discussed in the
following chapters, are designed to assess the quality impact of a
single component. They require relatively little data and are easily
employed using a “desk-top” calculator. However, since they are
empirical they cannot be applied with confidence beyond the range of
conditions in the lakes used to establish the model’s parameters. For
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instance, if the data set used to develop the model contained lakes with
a mean depth range of five to sixty meters, then the model results y
not hold for lakes with mean depths outside this range (and as is
shown in Chapter 3, the confidence limits widen considerably within
this range as the extremes are approached). Another example of this
empirical model limitation is noted in Chapter 3 for the Dillon-Rigler-
Kirchner model. This model was developed from a phosphorus-poor data
set from the Canadian Shield, and thus it must be applied with caution
in nutrient—enriched lakes.
It should be the modeler’s responsibility to clearly document the
proper use and limitation on the use of his/her model. For an empir-
ical model, this documentation should include a statement of all
limitations (geographic, trophic, geomorphologic, etc.) associated with
the data set used to develop the model. In addition, any biases noted
within the range of application should also be indicated. Another im-
portant but generally neglected statement of information about the model
concerns the type of issues and decisions appropriate for model applica-
tion and the value of the information provided by the model toward
this application. As noted above, a variety of lake models exist,
and a particular model or type of model may be justified as most suit-
able for a given application. The modeler will enhance the value of
the model as a management tool if a clear statement is provided dis-
cussing appropriate application.
A measure of the value of the information provided by the model
may be obtained from an explicit consideration of uncertainty. Model
error can be combined with input variable error for the assignment of
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confidence limits to the prediction using first-order analysis. This
concept has not previously been considered in the application of models
for lake management, so it is particularly important here. Prediction
error provides a direct measure of model prediction information; as the
confidence limits associated with a prediction become smaller, a de-
cision-maker should assign more weight to the information provided by
the model.
When the prediction uncertainty is calculated, it may easily be
contined with a probabilistic water quality standard (e.g., a 90%
chance that the phosphorus concentration is less than .025 mg/i) to
assess the likelihood of a standard violation. Few standards are
presently stated in a probabilistic form. However, since neither
mathematical nor conceptual projections of future conditions can be
made with certainty, a standard expressed as a probability of violation
is consistent with our ability to characterize the system. An example
illustrating how a probabilistic standard facilitates decision making
is presented in Chapter 3.
The remainder of this manual outlines quantitative techniques
that can be used to evaluate strategies for lake management. Chapter 2
presents a nunter of coiiiiionly employed statistical techniques for the
design of sampling programs and for data analysis. Chapter 3 discusses
the simple empirical models for phosphorus, and Chapter 4 outlines
several schemes proposed for lake classification. Examples are pre-
sented to illustrate the use of the techniques outlined in those chap-
ters. In conclusion, Chapter 5 briefly suggests additional research
that may lead to refinements of the techniques described herein.
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CHAPTER 2
DATA ANALYSIS AND SAMPLING DESIGN
2.1 Introduction
Data analysis and statistics are becoming more comon in water
quality studies. Often statistics (e.g., the mean) are used to surrinar-
ize the information from a set of data, and thus it is important to
interpret these statistics properly. Statistical tests may be used
to evaluate or compare data sets, and thus we need to know how to
apply the tests and use the results. These topics are addressed in
this chapter.
Oneareaof major importance in water quality studies in which
statistical analysis has not been used to full advantage is the design
of sampling programs. All too often, the selection of water quality
sampling sites and times is based on convenience (e.g., sample once
every two weeks at an easily-reached site), and no consideration is
given to the value of the information gathered. As we shall see, samp-
ling programs should reflect the variance (uncertainty) of the popula-
tions sampled and the cost of sampling. Convenience, if it does enter
into the design, is a cost factor, but it must often be balanced
against a reduction in the value of the data gathered. This topic is
considered later in this chapter.
—7-
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2.2 Data Analysis
2.2.1 Mean and Median
The mean, or average value, is defined as the sum of a set of data
divided by the number of elements:
mean = i = (x 1 + x 2 + . . . + x ) = x 1 (2.1)
n
where = the sum of the elements x. from i=l to i=n.
i=l 1
The median is the middle value (or the average between the two
middle values if the data set contains an even nunter of data points)
in a set of data when the data are ranked from lowest to highest (or
vice versa).
An example provides a good comparison of the information presented
by each of these statistics representing the “central tendency” of a
set of data. Table 2.1 presents the total phosphorus concentration
taken in a lake at one site at five different depths (top to bottom).
TABLE 2.1
TOTAL PHOSPHORUS CONCENTRATION (mg/i)
Data Median Mean
.020 .022 .040
• 022
.021
.042
.095
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-g —
The median, or middle value is .022 mg/i, while the mean is .040 mg/i.
This points out an important distinction between these two statistics.
The mean is influenced by extreme values (e.g., .095 mg/i) while the
median is not. To view this in a general sense, look at Figure 2.1.
The horizontal axis presents numerical values (such as the data in
Table 2.1), and the vertical axis presents the frequency of occurrence
of each numerical value in the data set. Two curves that represent the
distribution of numerical values in hypothetical sampling programs are
displayed. When the data are distributed “normally” (a bell-shaped
curve), the median and the mean are equal. However, when the data are
skewed (some extreme values exist), the mean will be influenced by the
extreme values in the skewed distribution and will differ from the medi-
an.* It is quite important to realize that this distinction exists
between the mean and the median. For this reason, it is informative
to present additional information on a data set, reflecting the spread
(variance) and the skewness. In addition, when examining a report in
which only the mean or median is presented, be aware that your ability
to determine the characteristics of the original data (or the popula-
tion) is limited.
2.2.2 Standard Deviation and Variance
The standard deviation describes the spread in a set of data.
Mathematically it is estimated by:
*A commonly observed skewed distribution is the log-normal, which
looks similar to the skewed distribution in Figure 2.1. It is called
log-normal because a logarithmic transformation of all data results in
a normal distribution. The mean of the logarithmically-transformed
data equals the geometric mean of the original data set.
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NUMERICAL VALUE
>-
C-)
w
a
w
L&
-J
Figure 2.1
A Normal Distribution and A Skewed Distribution
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—11—
r 1½
I . n _21
standard deviation = s = (x. - x) (2.2)
[ i=1 ]
The variance is simply the standard deviation squared.
As an example, these statistics will be estimated for the data in
Table 2.2, which presents dissolved oxygen data for the epilimnion of
a lake. Note that the mean and the median are nearly identical as the
set of data in Table 2.2 has no extreme values.
TABLE 2.2
DISSOLVED OXYGEN CONCENTRATION (mg/i)
Standard
Data
Mean
Deviation
8.3
8.2
0.33
7.9
7.9
8.0
Median
Variance
8.7
8.1
8.1
0.11
8.6
A good way to describe this data set is: “approximately uniformly
distributed with a mean of 8.2 mg/i +0.33 mg/i.” Therefore, a good
presentation of data should always include the standard deviation
(or another measure of data “spread”), and graphical presentation
should include error bars (-4- one standard deviation) on mean values.
If a data set is normally distributed, about 2/3 of the data fall
within +s of the mean.
The standard deviation or the variance provides a good measure
of the value of the information in a set of data. Specifically, the
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inverse of the variance has been equated to “information value” (M. G.
Kendall and A. Stuart, 1967). Alternately, the variance is an estimate
of uncertainty, a concept that is mentioned in Chapter 1 and used to
assess the impact of lake management strategies in Chapter 3 and
Chapter 4.
2.2.3 Standard Error of the Mean
The standard error of the mean expresses the uncertainty, or
conversely the confidence, in the estimate of the mean value for a
set of data. It is estimated by:
= (2.3)
x
Note that the relationship equates the standard error of the mean to
the standard deviation divided by the square root of the number of
samples. This means that our uncertainty in the estimate of the mean
is less than the uncertainty in the data set (represented by the
standard deviation). Further, the larger the number of samples, the
better (less error) we can estimate the mean.
2.2.4 Correlation
Correlation may be defined (statistically) as a measure of linear
association. It is expressed in terms of a correlation coefficient,
r, which may be calculated by:
n
z (x. — x)(y. — y)
— 1=1 1 = covariance (x,y ) (2.4)
r n s(x) s(y)
(x i) 2 ( -
i=l i=l
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—13—
r ranges between +1 and -1, and the higher the absolute value of r, the
stronger the linear association. Figures 2.2 - 2.4 present plots of
bivariate data, all exhibiting relatively high values of the correlation
coefficient. Note the difference in slope between the positive coeffi-
cient in Figure 2.3 and the negative coefficient in Figure 2.4, and the
difference in the strength of the association between Figure 2.2 (r .95)
and Figure 2.3 (r = .78).
In the application of Equation 2.4 or the evaluation of a correla-
tion coefficient and its associated bivariate plot, there are a few con-
cepts that must be kept in mind.
1. The correlation coefficient evaluates a linear (straight line)
relationship. Often, a transformation (e.g., logarithmic)
will turn what appears to be a nonlinear relation into a
linear one. However, the correlation coefficient will not
reflect an intrinsically nonlinear relationship.
2. The correlation coefficient is likely to be biased towards
a higher absolute value if it is applied to a set of data
that is not normally distributed. For example, Equation 2.4
“sees” the set of data in Figure 2.5 as basically 4 points
(one large point at the origin) with a fairly strong linear
association. As a result, each of the three extreme points
on the right has more weight in determining the correlation
coefficient than does any one point in the bunch at the lower
left. Thus, error in the point at the extreme right can
significantly affect the correlation coefficient. A trans-
formation of x and y, that results in a normal distribution
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-14-
1000
E
E
c,l
-j
-J
>- .
0
cD
c
0
-J
= 0
C-,
w
=
ci 1.0
0 Values from Sokamoto (1966)
Literature values compiled
by Dillon and Rigler (1974)
(r:.95 forodoto)
.1 10 100 1000
TOTAL PHOSPHORUS mgm 3
Figure 2.2 The Relationship of Suniner Average Chlorophyll to Spring
Total Phosphorus (Dillon and Rigler, 1974)
0
0
A
A
A
0
0
0
0
0
-------
- 0 0
1.0 10.0
r:0.78
:0.846 10910 ChI—2.354
o Oligotrophic
I I I I I III
100.0
MEAN CHLOROPHYLL (pg/I)
Figure 2.3 The Relationship Between Median Total Phosphorus and Mean Chlorophyll a in Phosphorus
C l)
0
=
0
(I)
0
=
0
10.00
1.00
0.10
0.01
0.00 I
c i
ci
ci
ci
0 00
ci
ci
ci
-J
0
w
ci
ci
‘ci
ci
A ° °
0
O•1
ci Eutrophic
t Mesotrophic
I I It
I I I I II II__I_
1000.0
Limited Lakes
-------
10.0
l.0
0.I
0.01
0.001
a
a
MEAN SECCHI DISC DEPTHS (inches)
Figure 2.4 The Relationship Between Median Total Phosphorus and Mean Secchi Disc Depths in
Phosphorus Limited Lakes
r:—O.82
log 10 1 ’total 0.8 18 1.307 lO9 SD
C’,
0
0
C,)
0
0
-J
1—
0
LU
0
0
0
a
a
0
00
00
a
a
A
0
Eutrophic
Mesot rophic
01 igotrophic
a
a
0
a
a
a
a
0
—
0
10.0 100.0
1000.0
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—17—
B /1
Y
S
S•
S
.‘ .
x
Figure 2.5 V versus X
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-18-
of data along the x—axis and along the y—axis, leads to the
desired weighting of the data points in determining the
correlation coefficient.*
2.2.5 Regression Analysis
There are many situations in which it is desirable to know, in
addition to the strength of the linear association between two vari-
ables, the turictional relation between the variables (defined as a
dependent variable, y, and an independent variable, x). In the simplest
approach, one can fit, by eye, a line through a set of points on a
bivariate plot (y versus x), such as in Figure 2.5. The equation of
this line may then be calculated from its intercept and slope, and it
may be expressed as:
y=a+bx (2.5)
However, this does not provide an “optimal” fit (optimal accord-
ing to a mathematical criterion) to the data points. For that pur-
pose, the technique, least squares regression, has been developed.
Least squares regression specifies that the line of best fit is that
line which minimizes the sum of squared deviations (or standard error)
from the line. Regression equations may be simple (y versus x), mul-
tiple (y versus x 1 , x 2 , . . .), or nonlinear. Regression programs are
available both in computer packages and pocket calculators, so none
of the equations for calculation of the regression coefficients will
be presented here.
*Generaljzjng, a normal distribution of data is desirable for most
data analysis techniques.
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Two measures of “goodness of fit” of a regression equation are
the coefficient of determination, and the standard error of the
estimate, Sm• R 2 is a measure of the fraction of the variation in
y “explained” by the regression equation. It is also the square of
the correlation coefficient between the predicted and the observed
values. Sm may be estimated by:
Sm = pre : 1 obs ) (2.6)
where:
‘pred. = predicted V value
‘obs. = observed Y value
d.f. = degrees of freedom
Both and R 2 may provide biased estimates of the goodness of
fit if the data are not normally distributed. For example, if R 2
and Sm were estimated from the untransformed data in Figure 2.5,
R 2 would be higher, and lower, than their “true” values. The
regression between y and x in Figure 2.5 should occur under variable
transformations (perhaps logarithmic) that yield approximately norm-
ally distributed variables (as in Figures 2.3 and 2.4, for example).
is required for the estimation of “confidence limits” on a
regression line. Confidence limits on a simple regression line
are estimated by:
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(x -
Confidence limits ‘pred. ± tSm V + E(x - x) 2 (2.7)
where:
t = student’s t (discussed in next section)
= mean of value of x
n = number of data points for x, y
x 0 = a particular value of x
Solution of Equation 2.7 for a range of x 0 -values will provide con-
fidence limits for the line at the desired significance level (speci-
fied through t). These confidence limits may be interpreted such
that the true line will fall within pred ± the confidence limits
at a frequency specified by the significance level (e.g., 95% of the
time, 70% of the time, etc.).
Often confused with the confidence limits on the line are the
confidence limits on a prediction , or the “prediction interval.” The
prediction interval is estimated by:
Prediction interval = pred ± tSm i/i + + X) (2.8)
The prediction interval is generally much larger than the confidence
limits, as it reflects the variability in a single data point. Pre-
diction intervals are used to estimate prediction uncertainty for the
nodels discussed in Chapters 3 and 4.
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-21—
Regression analysis is an extremely valuable tool in describing
variable relationships and in developing predictive equations. How-
ever,,a few final words of caution are in order:
1. Multicollinearity, or correlation among independent variables,
can affect the regression coefficients. Thus, cause-effect
conclusions are difficult when multicollinearity is signifi-
cant.
2. Regression equations are empirical, and thus they should not
be applied outside the range of conditions that exist in the
model development data set.
3. Significant errors in the independent variables can bias
the model parameters.
4. When developing or using a regression equation, it is a good
rule to check for normally distributed variables ( required
if confidence limits or prediction intervals are to be used).
If the variables are not normally distributed, transform
the variables to achieve a normal distribution. Be wary of
the interpretation of a regression equation developed from
variables that are not normally distributed.
2.2.6 Student’s t
Student’s t, presented in Equations 2.7 and 2.8, is a useful
statistic for significance tests with small samples. Most statistics
books and some calculators provide values of t as a function of the
nuriter of degrees of freedom and the specified significance level.
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-22-
A particularly common use of student’s t is the t-test for the com-
parison of two groups of data. For example, to test the “null”
hypothesis that there is no significant difference between the mean
values of phosphorus concentration in two lakes, consider the hypo-
thetical data in Table 2.3.
TABLE 2.3
PHOSPHORUS CONCENTRATION (mg/i) IN TWO HYPOTHETICAL LAKES
Lake A Lake B
.010 .015
.012 .017
.015 .013
.006 .011
.007 .015
.010 .011
.010 .016
.013 .014
.017 .019
.009 .009
. 011 . 010
.0109 mean .0136
.120 sum .150
Student’s t is calculated from:
x -x
A B (2.9)
+
V ( A)(nB)
where:
XA ,XB = the means for groups A and B
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-23-
= the nunter of samples in groups A and B
s 2 = the pooled within - group variance
(calculated below, for the example)
For the problem posed, first calculate the sums of squares (ss) within
each group:
S 5 A EXA (z) 2 (.010)2 + (.012)2 + . . + (.011)2 - ( .120)2
SSA = 1.049 x 10
SSB = EXB ( ExB) 2 = (.015)2 + (.017)2 + . . . + (.01o) - ( .150)2
SSB = 9.855 x
The pooled within - group variance is:
2 SSA+SSB
S = (nA-i) + (nB-i)
2 - 1.049 x 10 + 9.855 x l0
S — 10 + 10
= 1.017 x
Thus:
t = . 0109 - .0136
fl.017 x
t =- 1.985
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-24-
This value of t has (nA_ i) + (nB_ i), or 20, degrees of freedom. Con-
sulting a t-table, we find that for 20 degrees of freedom, this value
of t is significant at the 90% level, but not quite at the 95% level.
Thus, depending upon the acceptable level of significance (i.e., the
risk of being wrong in one direction), we can reject (at 90%) or
accept (at 95%) the null hypothesis (that there is no significant
difference in phosphorus concentration between the lakes).
In conclusion, some words of caution on the t-test are in order:
1. The samples should be random (or relatively random, see the
next section on sampling design).
2. The population (e.g., all possible phosphorus concentration
values in the lake of study) should be normally distributed.
A transformation (e.g., logarithmic) may be necessary.
3. The t-test, as applied, is based on the assumption of equal
variance for the groups (lakes) compared.
2.3 Sampling Design
2.3.1 Introduction
The gathering of water quality samples is a task that requires
careful planning, or design. Sampling should always be undertaken
with an objective in mind, whether the sampling provides Hbaselinehl
data, nutrient flux from a particular source, or a nutrient budget for
a lake. In general terms, the objective may usually be stated as:
minimize the uncertainty in the estimate of the characteristic(s)
of interest subject to a budget (cost) constraint.*
*Convey . e1y, sampling may be undertaken to estimate the character-
istic(s) of interest to a specified level of precision (uncertainty)
at a minimum cost.
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-25-
Data collection provides a limited amount of information (a sample)
about a characteristic. The sample is designed to best estimate “true”
values or “population” values, of the characteristic. For example,
if the characteristic is phosphorus concentration in a lake, there
are population values for the mean and standard deviation for phosphorus
concentration. A data collection program may be undertaken, and this
can provide “sample” estimates of the mean and standard deviation.
Techniques presented here and elsewhere (Snedecor and Cochran, 1967)
may then be used to estimate the reliability of the “sample” estimates
as “population” values, using, for example, the standard error of the
mean.
What does sampling design, under an objective like that stated
above, do for water quality assessment? Three advantages are:
1. Sampling becomes more efficient. Samples should be gathered
only if they lead to a reduction in uncertainty. Thus, char-
acteristics should not be sampled if the uncertainty in the
estimates of these characteristics cannot be reduced further
given the budget constraint. In addition, efficient sampling
means that trade-offs should be made among different sampl-
ing schemes to achieve the most cost-effective design.
2. Sampling design stimulates thinking about the issue of concern,
and it leads to a good definition of informational needs.
3. Sampling design leads to an explicit quantification of the
uncertainty in the estimate of the characteristic(s) sampled.
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-26—
It is shown in Chapters 3 and 4 that an estimate of uncertainty
is quite valuable to both modelers and planners, as it quanti—
fies the value of the information contained in the gathered
data.
2.3.2 Sampling Design Relationships
It has already been stated that sampling design is a function of
uncertainty (precision) and cost. In addition to those two factors,
though, sampling design depends upon the variance (or the magnitude
and variability) of a characteristic. For example, if the character-
istic to be sampled is the phosphorus concentration in a lake, then
the primary factors needed for the design of a sampling program are the
cost of sampling, the precision expected from each proposed design, and
an estimate of the variance in phosphorus concentration. The variance
is a population statistic, and it exists largely because of natural
fluctuations, or variations, in concentration throughout a lake.
Sampling design relationships are often presented on the basis
of random design. Most water quality sampling, however, is systematic
(e.g., samples are taken every two weeks, or at the top, middle, and
bottom). However, random sampling design equations may be used for
systematic sampling if there is no bias introduced by incomplete
design, and if there is no periodic variation in the characteristic
measured. Use is further justified if the systematic sampling begins
with a random “start” (e.g., begins on a randomly selected day or at
a randomly selected site or depth, with systematic sampling from that
point on).
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—27-
Random sampling is characterized by the following equation:
22
_tS (2.10)
d
where:
n = number of samples
t = student’s t
= population variance estimate
d = precision, or uncertainty, of sampling design.
This equation states that the number of samples taken should be increased
if greater precision is desired, or if the population variance becomes
larger. Sample number is related to cost, perhaps as:
C(n) c 0 + c 1 n (2.11)
where cost of sampling involves a fixed cost plus a variable cost
dependent upon the number of samples taken. Thus, for different sampl-
ing programs, a plot of sampling cost versus uncertainty (precision)
can be constructed (Figure 2.6). “Optimal” sampling programs are
those lying on the curve, as these programs minimize uncertainty for
a given cost (or vice versa).
For many types of problems, sampling can be more efficient when
the design is based on the fact that a population often contains strata
with characteristics of different magnitudes and variabilities. For
example, stratified lakes often have different conditions in the
hypolimnion and epilimnion. In addition, the nutrient flux to a
-------
>-
I-
w
C)
=,
sampting program
A
N)
03
COST
Figure 2.6 Sampling Design Uncertainty Versus Cost
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-29-
lake can vary significantly from tributary to tributary. In these situ-
ations, sampling is more efficient when sample numbers are allocated
according to stratified random sampling design. Then, within each
stratum, sampling is random or systematic with a random start. Sampl-
ing is allocated in stratified random sampling design according to:
w 1 (c.v.) (xi)
i = i (2.12)
n w (c.v.) (x 1 )
where:
n = total number of samples
= number of samples in stratum i
x = magnitude (mean) of characteristic x in stratum i
w. = a weight reflecting the size (number of units, for
example) of stratum i
(c.v.) = coefficient of variation (standard deviation divided
by the mean) of characteristic x in stratum i.
If sampling cost may be estimated by:
C = c 0 + (2.13)
then
(2.14)
In order to apply Equation 2.12 or 2.14, a relationship is needed
for the total number of samples n. Two equations are available, de-
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-30-
pending upon whether precision cost is fixed beforehand. If pre-
cision is fixed (at d), and cost may be estimated according to Equa-
tion 2.13, then (Cochran, 1963):
2 2 1 (2.15)
d /t
If cost is fixed, then (Cochran, 1963):
(c -
fl i(c.v.) x 1 )v ) (2.16)
As an example, Table 2.4 presents figures for a hypothetical
lake (suimier conditions). These figures are based on limited existing
information or informed judgment. The problem is to design a sampling
program to estimate the mean phosphorus concentration to within ±.005
mg/i at the 95% level. Assume that costs of epilimnion and hypolimnion
sampling are equivalent.
TABLE 2.4
ESTIMATED CONDITIONS IN A HYPOTHETICAL LAKE
Characteristic
Epi
1 imnion
Hypol
imnion
mean phosphorus
concentration, x.
(mg/i)
.020
.070
s (mg/i)
.010
.050
(c.v.)
.50
.71
volume (m 3 )
x io6
i
x io6
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—31—
To solve this problem, one must first note that epilimnion and
hypolimnion volumes are different. Thus, one must “weight” the
epilimnion concentration by 5/6 and the hypolimnion volume by 1/6,
since the epilimnion volume is five times the hypolininion volume.
Therefore, the total number of samples to be taken are (using Equation
2.15):
= [ (5/6)(.50)(.020)+ (l/6)(.71)(.o7o)]2
(.005) /2
n = 2.76 x l0 44.18
6.25 x io6
n 45 samples
Equation 2.12 may be used to allocate the samples between the
hypolimnion and the epilimnion:
- ( 5/6)(.50)(.020 )
r (5/6)(.50)(.020) + (l/6)(.7l)(.070)
where ne = number of samples to be taken in the epilimnion.
Thus, 50% of the samples (22-23 samples) are to be taken in the
epilimnion, and 50% of the samples are to be taken in the hypolimnion.
As a check, the precision (standard error of the mean) of this sampl-
ing design is estimated by (Equation 2.3):
-------
—32—
x n
For the epilimnion:
s = = .0021 mg/i
e
For the hypolimnion:
s = = .0104 mg/i
For the entire lake:
s [ (5/6) C .0021 ) ]2 + [ (i/6)(.0104)]2
= 3.06 x io6 + 3.06 x io-6
s = .0025 mg/i
x
At the 95% level, t 2, and precision, d, is:
d = ts_
x
d = .005 mg/i,
which corresponds to the desired precision.
In suninary, the composition of the stratified random sampling
design equations leads to the following general conclusions concern-
ing stratified sampling. A larger sample should be taken in a stratum
if the stratum is:
-------
-33—
1. more variable (c.v.)
2. larger (w, x)
3. less costly to sample (c)
2.3.3 Estimation of Factors Important in Sampling Design
At this point, it should be apparent that before the sampling
design relations can be applied, estimates are needed for the magnitude
and variability, or variance, of the characteristic(s) sampled. These
estimates may be, and often are, “rough,” but as long as they are repre-
sentative and thoughtfully selected, they will be useful in guiding
sampling design, using the equations above. If one is concerned that
these estimates are highly uncertain, it might be useful to calculate
designs using a range of these “prior” estimates, in order to test the
“sensitivity” of the design to the estimated values..
Cochran (1963) identifies four approaches for the prior estimation
of magnitude and variability, or variance:
1. Use existing information on the same “population” or a
similar population (e.g., existing data on lake concentra-
tion, or nutrient inflows).
2. Use informed judgment, or an educated guess.
3. Take the sample in two steps. Use the first-step sampling
results to estimate the needed factors for best design of
the second step. Use data from both steps to estimate
the final precision of the characteristic(s) sampled.
-------
-34-
4. Perform a “pilot study” on a “convenient” or “meaningful”
subsample, and use the results to estimate the needed
factors. Here the results of the pilot study generally can-
not be used in the calculation of the final precision, because
the pilot sample often is not representative of the entire
population to be sampled. This possible non-representative-
ness must also be taken into account when the pilot survey
results are used to estimate magnitude and variability, or
variance. A modification might be necessary if it is thought
that the pilot survey provided an overestimate or underesti-
mate of the population values.
2.3.4 Some Specific Issues
Experience in limnology and data analysis is invaluable in the
design and interpretation of sampling programs. Specifically, this
experience may be useful in making adjustments to the design and
analysis so that the results best represent the characteristics of
interest. Many situations may benefit from this experience; a few
examples are presented below.
Some lakes are irregularly shaped or contain concentration gradi-
ents due to major point sources. In those cases, stratified sampling
may be undertaken with additional strata in the horizontal direction.
In addition, nutrient concentrations and nutrient flux estimates
are dependent on hydrology. Severe storms can change lake nutrient
concentrations and can introduce a significant fraction of the total
annual nutrient loading to a lake. Unusual hydrologic years (very
“wet” or very “dry” years) can also significantly affect nutrient
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-35-
loading and lake quality. Thus, conditions observed during the period-
of-sampling may not generally be representative of conditions over time
or of “average” conditions. Hydrology (and, of course, land use
changes) must be taken into account.
2.3.5 Annual Phosphorus Budget
Sampling design can be a particularly useful exercise in the
measurement of nutrient flux to a lake. Application of the sampling
design equations, or of the concepts important in sampling design, can
lead to efficient sampling programs based on explicit trade-offs among
different sampling schemes. In addition, an estimate of the uncertainty
associated with gathered data on nutrient flux is valuable information
for use in the models and classification schemes in Chapters 3 and 4.
Lake phosphorus budget sampling design is discussed in consider-
able detail by Reckhow (1978c). This section will briefly summarize
some of the issues presented in that paper. Major sources of phos-
phorus considered were tributaries, sewage treatment plants, urban
runoff, precipitation, septic tanks - groundwater, and lake sediments.
For each source, the sampling design was based on an estimation tech-
nique, or model, that converted the gathered data to an annual phos-
phorus flux estimate.
Phosphorus flux in lake tributaries has been studied extensively,
and thus there is a substantial quantity of literature that may be
used for the estimation of the expected magnitude and variability of
that flux. The EPA National Eutrophication Survey is a good source
of data, and many of the EPA-NES streams have been classified by
land use (Omernik, 1977). In general, phosphorus concentration (in
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-36-
streams) decreases with flow in streams impacted by a sizeable point
source, and increases with flow in streams undisturbed by major point
sources. On that basis, phosphorus flux is probably best estimated
by multiplying average flow times the flow—weighted concentration or
by a regression equation of flux on flow. Since those calculations
of flux require information on flow, it is recommended that continuous
flow measurements be made, or that a regression equation (of flow on
precipitation and watershed characteristics) be used to provide flow
data. Regression equations like that described are available from
the U.S. Geological Survey. Sampling for concentration should be
allocated among tributaries using stratified random sampling, and it
should probably occur on 2-4 week intervals (with a random start, and
allocated according to seasonal flow variations). More frequent sam-
plingresults in auto-correlation among samples, and less frequent sam-
pling may result in considerable error. Finally, some consideration
should be given to sampling major storm events, as a large percentage
of the phosphorus loading may occur during those times.
Much data also exist on wastewater treatment plants, and again
the EPA—NES is a good source. Treatment plant data exhibit a distinct
diurnal cycle, so composite sampling is preferrable. Phosphorus flux
estimates may be made from flow-weighted concentration times flow
(continuous flow data should be available). Existing EPA-NES data
indicate that the average phosphorus concentration varies considerably
from plant to plant, while the coefficient of variation (the variabil-
ity) of phosphorus concentration generally lies between .3 and .5.
Sampling among plants should be based on stratified random design,
-------
-37-
while sampling over time should be based on randomsampling to reach a
desired or minimum precision.
Urban runoff sampling clearly must be geared to storm events.
Insufficient data exist to guide sampling designs in most situations.
Therefore, only some general recoiimendations can be made. Automatic
sampling may be most effective, since human response to a storm may
miss a portion of the “first flush.” Composite sampling for concen-
tration may be used to estimate flux, as average flow times average
concentration. Grab sampling can be used to fit an exponentially-
decaying concentration model (Marsalek, 1975), that may be used to
estimate flux with continuous flow data.
Existing data on phosphorus in bulk precipitation (precipitation
plus dry fallout) indicate considerable variability from year-to-year,
site—to—site, and storm-to—storm. Bulk precipitation phosphorus
results from industrial air pollution, bare agricultural fields, dirt
roads, etc. In many lakes, precipitation is a relatively minor source
of phosphorus. Thus, literature values for precipitation phosphorus
(Uttormark, et al., 1974) should probably be compared to the expected
flux of phosphorus (to the lake of study) from other sources before
a sampling program is undertaken for this source.
No satisfactory techniques have yet been developed to measure
phosphorus flux to a lake from septic tanks and groundwater. The
most common technique used is a soil retention coefficient, specific
to a soil type. However, a constant soil retention does not consider
the time-dependency of retention, the total volume of soil through
which phosphorus in solution must pass, and the loading of phosphorus
-------
-38-
to the soil. Probably a better technique at this time is a system
of Hseepage meters” in the shallow lake sediments and wells ininediately
onshore (Lee, 1977). The seepage meters are used to measure ground-
water flow (assumed to decrease exponentially with distance from shore),
and the wells are used to measure phosphorus concentration. Unfortun-
ately, insufficient data exist to design this program, but the concepts
of stratified random sampling (magnitude, variability, and cost) sug-
gest that sampling units should be most dense in areas with the great-
est density of septic tanks and in areas with soils of lowest retention
coefficients.
Finally, the lake sediments are another source of phosphorus
that is not well—defined. As a rule, the sediments are considered to
be a significant source only under anaerobic conditions. However,
studies indicate (Snow and DiGiano, 1976) that aerobic sediments often
release phosphorus also. Estimation techniques, such as a constant
daily release of phosphorus, or release proportional to the concentra-
tion gradient between the water column and the interstitial water,
have been proposed (Reckhow, 1978c). Experimental procedures have been
developed for both the laboratory and the field (Snow and DiGiano,
1976). It is suggested that “typical” release rates, presented in
Reckhow (l978c) and Snow and DiGiano (1976), be compared to expected
phosphorus flux from other sources, before a sampling program is
undertaken for the lake sediments.
As an example of lake phosphorus budget sampling design, the fol-
lowing analysis was conducted to guide the sampling of phosphorus flux
to Lake Winnipesaukee in New Hampshire (Reckhow and Rice, 1975; Reckhow,
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-39-
l978c). This analysis emphasized the concepts of stratified random
sampling (take a larger sample if more variable, larger, or costs
less); it did not consist of the explicit trade-offs and computations
that might be possible with the material presented above. Nonetheless,
it does show, in a general sense, how sampling design may develop.
Table 2.5 presents the mean, standard deviation, and precision
of existing estimates for phosphorus flux from the tributaries and
the wastewater treatment plants. Informed judgment yielded the magni-
tude and range estimates for the other three phosphorus sources; data
were deemed insufficient to specify these terms more precisely. This
table then provided the basis for general sampling design recommenda-
tions, summarized in the statements below:
1. Existing estimates of the phosphorus flux from tributaries
and sewage treatment plants may be sufficient (i.e., no
additional sampling necessary), if they were obtained with
an unbiased sampling design, and if significant changes
(land use, etc.) have not occurred.
2. Considerable sampling effort should be devoted to estimating
the mean and variance in phosphorus flux from septic tanks.
3. The other sources of phosphorus (sediments and precipitation)
should be investigated through the literature, but they may
not require sampling.
4. If tributary sampling is undertaken:
a) spatial coverage should be based on stratified random
-------
TABLE 2.5
INITIAL UNCERTAINTY ESTIMATES FOR LAKE
WINNIPESAUKEE PHOSPHORUS LOADING
Term
Prior
Estimates
of
Standard Error
of the Mean (%)
Estimated
Range
Magnitude
Coefficient
Variation
1.
Tributary Flux
16,000 lb P/yr
.65
+20
2.
Septic Tanks
4,000-30,000
lb P/yr
3.
Sewage Treatment
Plants
22,000 lb P/yr
.30
±10
4.
Precipitation
.
4,000-7,000
lb P/yr
5.
Sediment Release
700-7,000
lb P/yr
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-41-
sampling design (which may result in no sampling in the
smallest streams that are not culturally impacted).
b) temporal coverage should consist of a sampling interval
of 2-3 weeks, with sampling being more frequent during
high runoff months and less frequent during months of
low runoff.
2.3.6 Summary and Conclusions
In conclusion, a good sampling design requires the following
information:
1. Prior knowledge of the factors that affect the characteristic(s)
to be sampled (e.g., sources of phosphorus for a phosphorus
budget).
2. Some knowledge of the magnitude and variability of the char-
acteristic(s) to be sampled.
3. Pre—specified needs for the collected data. For example,
phosphorus flux data may be used for a year-of-sampling
estimate or for future predictions. Different designs and
estimation techniques may be appropriate for each of these
applications.
4. A knowledge of costs associated with sampling.
5. A model, or models, for estimation (when appropriate) that
is compatible with the chosen sampling design.
-------
CHAPTER 3
PHOSPHORUS MODELS
3.1 Introduction
The empirical phosphorus models discussed in this chapter are all
based on the assumption that a lake may be treated as a black box. This
means that for modeling purposes, internal lake mechanisms may be ig-
nored, and models are developed to quantify processes at the lake
interface only. These modeled processes are materials (phosphorus)
input and output, and the interfaces to the lake are the land, atmos-
phere, and lake sediments. This chapter contains a review of the form
of these black box models and an examination of proposed empirical
approaches. These empirical models are compared and examined for bias,
limitations, and sensitivities. Uncertainty analysis is then discussed
as a measure of model prediction information. Finally, application
of four of the models on a case study lake is used to underscore the
importance of uncertainty when the models are used in lake management.
The reader whose major concern is the application of these models may
prefer to skip Section 3.2 and focus on Sections 3.3-3.5.
3.2 Historical and Theoretical Development: A Brief Review
Vollenweider (1969) expressed the lake mass balance on phosphorus
in the following form:
V M- jPV-QP (3.1)
-42-
-------
-43-
where, on an annual basis,
P = lake phosphorus concentration (mg/i)
V = lake volume (10 6 m 3 )
M = annual mass rate of phosphorus inflow to lake (lO 3 kg/yr)
Q = annual volume rate of water outflow from lake (10 6 in 3 /yr)
= sedimentation coefficient (yr ).
Equation 3.1 states that the change in phosphorus mass in the lake
(VdP) per unit time (dt) is equal to the mass input of phosphorus (M)
minus the mass output of phosphorus through the outflow (QP) minus
the net mass of phosphorus deposited in the sediments (aPV).This sediment
“sink” term for phosphorus is based on the assumption that the rate of
phosphorus deposition to the sediments is proportional to the total
mass of phosphorus in the lake (PV). Note that the lake and outflow
concentrations of phosphorus are assumed to be equivalent. Equation
3.1 has the time-dependent solution:
= L I 1 — e I + p° e (3.2)
oZ + zR
and the steady-state solution ( = o):
= zQ + ySj _ (3.3)
where
L = annual areal phosphorus loading (g/m 2 -yr)
z = lake mean depth (m)
T = hydraulic detention time (yr)
-------
-44-
t = averaging time length (yr)
P° = average lake phosphorus concentration at time
t = 0 (mg/i).
The phosphorus mass balance may also be expressed as:
V M-v 5 PA--QP (3.4)
where
A = lake surface (bottom) area (km 2 )
= apparent settling velocity (rn/yr).
Equation 3.4 is different from Equation 3.1 in that the sediment sink
term is treated as an areal sink. Thus, the rate of deposition of
phosphorus to the sediments is a function of the bottom (surface)
area. This results in the steady-state model:
= V + Z/T (3.5)
where
= = surface overflow rate (m/yr).
The major difference between the two steady-state models (Equations
3.3 and 3.5) is that one model is based on the assumption of a depth—
dependent settling velocity (cJz), while the other model is based on
the assumption of a constant settling velocity (v 5 ).
A third steady-state phosphorus model form results from the defi-
niton of the fraction of influent phosphorus retained in the lake:
-------
-45-
M - QP
R = M ° (3.6)
where P 0 = average outflow phosphorus concentration (mg/i).
Since M = LA and Q/A =
Lt
R _L- (z/T)P _ - o
p L LT
z
where
average influent phosphorus concentration (mg/l).
If it is assumed that P 0 = P (lake and outflow concentrations are
equal), then the third model becomes:
p = p = ! (1 - R ) (3.7)
The three basic model forms, from which the empirical models are esti-
mated, are Equations 3.3, 3.5, and 3.7. By setting Equation 3.3 equal
to Equation 3.7, it may be shown that:
— o — 1 (38)
p — l/t + a — 1 + 1/Ta
In a similar manner, the relationship between R and v (Equations 3.5
and 3.7) is:
V V
— S — S 3
pV 5 +Z/tv 5 +q 5
-------
where q 5 = Z/T.
To understand the appropriate criteria that must be considered in
selecting the ubestu model form (or, more fundamentally, to determine
whether a “best model form” exists at all), we should examine the phos-
phorus mass balance in its most general form:
V = M 1 - - (M 1 , M 0 , P, Ca, Fe, Al, pH, °2’ T,
Z, Z/i, V. . . .) (3.10)
where
M 1 = annual mass influx of phosphorus
= annual mass efflux of phosphorus
= annual net flux of phosphorus to the sediments
The removal of phosphorus from a lake occurs through two pathways,
the outlet (represented by M 0 ) and the sediments (represented by ).
is a multidimensional variable; it is dependent upon the influent
and effluent phosphorus mass, lake geoniorphology and hydrology, the
dissolved oxygen concentration and the pH at the sediment-water inter-
face, major cations (Ca, Fe, and Al) that conbine with phosphorus and
transport it to or hold it in the sediments, and probably additional
mechanisms important on a limited scale. Thus, the expression of the
sediment removal term as a phosphorus retention coefficient, an
apparent settling velocity, or a sedimentation coefficient, with the
functional dependencies as expressed in Equations 3.1 - 3.9, is actually
a simplification of the true form. Since the three basic model forms
are simplified versions of reality, any discussion over the “best”
-------
-47-
form of the sediment sink term must be of an empirical, and not theo—
retical, nature, since a, v , and as functionally defined in Equa-
tions 3.1 - 3.9, are incomplete representations of 4.
3.3 Empirical Models
The first attempt to develop an empirical fit to one of the mass
balance models presented in the previous section was by Volienweider
(1975) for Equation 3.3. From an analysis of available data (not spec-
ified further), Vollenweider found that a could be estimated by:
a = in 5.5 - 0.85 in z (3.11)
(r 0.79)
or
a . (3.12)
Thus, based on Equation 3.12, Volleriweider implicitly advocated a con-
stant settling velocity (V 5 = 10 m/yr) empirical model.
Dillon and Rigler (1974) looked at Equation 3.7 in a paper en-
titled: “A Test of a Simple Nutrient Budget Model Predicting the
Phosphorus Concentration in Lake Water.” However, it must be empha-
sized that Dillon and Rigler did not test an empirical formulation for
R . but rather they examined through Equation 3.7, the frequently in-
voked assumption that lake and outflow concentrations are identical.
It has been shown (Reckhow, 1977) that this assumption is strongly
supported by available data.
Equations 3.8 and 3.9 suggest that R may be empirically estimated
as a function of z/t or . Kirchner and Dillon (1975) confirmed this
-------
-48-
in a model for developed from a data set of 14 Canadian Shield
lakes:
RKD = 0.426 exp (-0.271 z/T) + 0.574 exp (-0.00949 Z/r).*
(r = .94) (3.13)
This results in the “Dillon-Rigler” or “Dillon-Rigler-Kirchner” model:
P = 1. . (1 - R ) (3.14)
Dillon and Rigler (1975) proposed that the model expressed in Equation
3.14 be used to assess the impact of land use on lake quality.
Chapra (1975), in a comment on Equation 3.13, noted the relation-
ships between R and a (Equation 3.8), and R and v (Equation 3.9).
Using the Canadian Shield lake data, Chapra found that v = 16 rn/yr
provided the best fit to Equation 3.9. This model has recently been
applied by Chapra (1977) to predict present and future concentrations
of phosphorus in the Great Lakes. Chapra’s model and the Kirchner-
Dillon model are presented in Figure 3.1. Dillon and Kirchner (1975)
responded to Chapra with a statement equating the v and a models for
P. Actually, the implicit assumption behind the differential Equations
3.1 and 3.4 is that a (Equations 3.1 - 3.3) and v (Equations 3.4 and
3.5) are constants , not functions of limnological variables. Thus
despite Dillon’s and Kirchner’s contention that the models presented
in Equations 3.3 and 3.5 are equivalent, they are equivalent only if
v is re-expressed in Equation 3.4 as a linear function of z. Dillon
and Kirchner concluded their reply with a revised best-fit for v of
13.2 rn/yr. This change from Chapra’s estimate of 16 rn/yr was proposed
*exp(a) = ea = (2.71828...
-------
-49-
because Dillon and Kirchner rejected data from two “outlier” lakes in
Figure 3.1 based on suspected errors in data collection. It should be
noted that this conclusion on data errors resulted from comparisons
of observations with model predictions and not from known field or
laboratory problems (so it is not immediately clear which of the two
figures, 13.2 or 16, has more merit). These models may be expressed
as:
= 16 (Chapra) (3.15)
= 13.2 Zi (Dillon—Kirchner) (3.16)
In an essentially concurrent study, Larsen and Mercier (1975)
examined data collected through the EPA National Eutrophication Survey
(EPA-NES) for northeast and north central U.S. lakes. Using a data
set of 20 lakes, they found the best fit for phosphorus retention to
be:
RLM 1 + 1.12 (1) 0.49 (3.17)
or, more simply,
R = 1 . (r .94) (3.18)
LM 1+/lIT
This results in the Larsen-Mercier model:
P = (1 RLM) (3.19)
-------
1.0
0.8
0.6
R
Qi
C
0.4
0.2
0 240
z/r (m/yr)
Figure 3.1 A Comparison of the Kirchner and Dillon Model For Rp With Chapra’s Constant Apparent
Settling Velocity Approach
40 80 120 160 200
-------
-51-
Note that Equation 3.18 is identical to Equation 3.8 when a becomes
variable and is set equal to l/f
Finally, in a recent paper, Jones and Bachmann (1976) examined
data on a large number of north temperate lakes and fit Equation 3.3
with a = 0.65 yr and with a multiplicative factor of 0.84. Thus,
their model is:
0.84L
- z (0.65 + l/T) 3.20)
Much of the previous modeling work forms the basis for comprehen-
sive studies of the empirical models by Reckhow (1977) and Walker
(1977) on north temperate lakes. Reckhow noted shortcomings in some
of the empirical models, focusing on the incomplete nature of the
“sediment sink” term. Because of the incompleteness of that term, it
was thought that improvements in the models might result if lakes were
classified according to those processes, important in sedimentation,
that may not easily be modeled, or according to those variables that
produce a nonlinear response in the dependent variable (total phos-
phorus concentration). Then, models could be developed for each of
the lake classes. A model for one of these classes (oxic lakes, with
Z/T < 50 m/yr) is given in Table 3.1; additional statistics, discussed
later in this chapter are presented in the Appendix. The remaining
class models and a quasi-general model* are also presented in the
Appendix. Table 3.2 presents a general model proposed by Walker, while
additional model information is presented in the Appendix.
model is called “quasi-general” because lakes with high
phosphorus concentrations (P > .150 mg/l) and high qs-values (qs
200 m/yr) were excluded from the analysis. The development of this
model is discussed in the Appendix of Chapter 4.
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—52-
TABLE 3.1
MODEL PROPOSED BY RECKHOW (1977)
FOR OXIC LAKES WITH z/T < 50 rn/yr
Standard Error
Model
R 2
ofthe
Estimate*
—
— i8Z
L .876
+ 1.05 z/t e .0l 2 z/r
.123
*Based
on a logarithmic transformation of terms.
TABLE 3.2
MODEL PROPOSED BY WALKER (1977)
Model
2**
R
Standard Error
of the
Estimate**
_L r
- Li +
1
.
.824t 454
or
= in [ +
.906
.824T454]
.171
where P =
average influent phosphorus concentration (mg/i)
**Based on a logarithmic transformation of all terms. These sta-
tistics compare the predicted concentration with the observed outlet
concentration.
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—53-
Table 3.3 presents a comparison of the empirical models discussed
in this section. The evaluative criteria describe the appropriateness
of each model for a given application and the value of the information
provided by the model. Recall that empirical models should not be
applied outside the bounds of the data set used to construct the model
(without prior verification). This means, for example, that the Dillon-
Rigler-Kirchner model, which was fitted on a data set of phosphorus-
poor Canadian Shield lakes, was not corroborated, or verified, in the
literature for use outside that geographic region. Reckhow (1977) has
tested (or “verified”) this model on a large data set of north temperate
lakes, thus removing this geographic constraint. However, it was found
that the Dillon-Rigler-Kirchner model may yield biased predictions for
phosphorus—rich lakes (Reckhow, 1977). A similar, though less severe
phosphorus-rich-lake bias was found for the Larsen—Mercier model (see
Table 3.3).
In Table 3.3, the bounds of the data sets used to develop each
model are described, in part, under the “data base” and “known con-
straints” columns. The “known biases” and “comments” column entries
are taken from the study by Reckhow (1977). The uncertainty column
indicates whether or not the modeler provided model standard error
and parameter statistics so that error can be explicitly evaluated.
Entries such as “unspecified,” “unknown,” “not evaluated,” and “not
considered” must be taken as indicators of the need for cautious use
of the models or of the model ‘s restricted and indeterminate value in
application.
In conclusion, the differences among these models may appear
subtle up to this point. It remains to be argued, in the next section,
-------
TABLE 3.3: A COMPARISON OF EMPIRICAL MODELS
01
Model
Data Base
Known f
Constraints*
Known Biases*
1 on 51det tbo1
of Uncertainty
Coments
Constant V 5
i) Vollenweider
vs 10 rn/yr
unspecified
unknown
not evaluated
not considered
Based on the
range of ap-
parent settling
velocities ob-
served in natural
lakes (Reckhow,
1977). these
models will prob-
ably overestimate
P in lakes with
high z/T and under-
estimate P In
highly enriched lakes.
—
2) Chapra
v 5 • 16 rn/yr
14 CanadIan
Shield lakes
P < .015 mg/i
LT/z < .050 mg/l
not evaluated
not considered
—
3) Dillon & Kirchner
• 13.2 rn/yr
12 Canadian
Shield lakes
P < .015 mg/l
LT/z < .050 mg/l
not evaluated
not considered
Constant a
Jones & Bachmann
a — 0.65 yr
51 natural lakes,
primarily north
temperate
unknown
not evaluated
not considered
This model will
probably over-
estimate P in
shallow lakes
with high values
of z/T.
R - regression
1) KIrchner & Dillon
R
14 Canadian
Shield lakes
.
P < .015 mg/l
LT/z < .050 mg/i
significant
underestimation
of P In lakes with
LT/z ‘ .300 mg/i
not considered
RLM was found
(Reckhow, 1977)
to be less biased
than for phos-
phorus-enriched lakes.
2) Larsen & Mercier
RLM
20 north temper-
ate lakes, pri-
manly oligotrophlc
P < .012 mg/i
LT/z .025 mg/i
significant
underestimatIon
of P in lakes with
1hz > .300 mg/i
not considered
-------
TABLE 3.3 (continued): A COMPARISON OF EMPIRICAL MODELS
0 1
Model
Data Base
Known
Constraints*
1
Known Biases*
Consideration I
of Uncertainty
Co ents
Reckhow Models
47 north
temperate
lakes
P <.135mg/i
LT/z <.298mg/i
< 187. rn/yr
none significant
Parameter errors
and model error
are presented so
that the total
uncertainty may
be calculated.
This permits the
assignment of
confidence limits
to a prediction.
Linked to a
graphical error
analysis in
Chapter 4.
Smaller errors
than RKD and RLM
on independent
data set
(Reckhow, 1977).
Verification not
conclusive
(Reckhow, 1977).
Needs to be
verified.
1) QuasI-General
Model
2) Oxic,
z/T < 50 rn/yr
lakes
33 north
temperate
lakes
P < .060 mg/i
LT/z < .298 mg/l
none significant
3) z/T > 50 rn/yr
lakes
28 north
temperate
lakes
P < .135 mg/l
1hz < .178 mg/l
z < 13 m
T < 0.25 yr
none significant
4) Anoxic lakes
21 north
temperate
lakes
,017 mg/i < P
< .610 mg/l
.024 mg/l < LT/z
< .621 mg/l
none significant
Walker Model
105 north
temperate
lakes
similar to
those in
Table 3A.2
not evaluated
Prediction confi-
dence limits can
be estimated.
No major short-
comings known.
*Constraints or
biases that may significantly affect the value of the model for prediction purposes.
-------
-56—
that the selection of the best model for use (given the present state
of the art) is one that incorporates explicit consideration of uncer-
tainty. Aside from that, though, the choice of the appropriate model
from a set of simple, empirical models should be based on:
1. The data base used to fit the model.
a. Does it contain lakes similar to yours (similar in terms
of geography, climate, size, depth, trophic state, etc.)?
b. Is the data base large (lending more credibility to the
estimates of the model parameters)?
2. Previous experience with the model. Has the model been used
successfully by you and others on similar lakes?
3. Model Documentation. Are model use, misuse, and limitations
carefully outlined? The manual by Dillon and Rigler (1975)
is a good first cut at this, but it is somewhat weak on the
misuse and limitations. This is particularly important if
you have had little prior experience with these models.
3.4 Error Analysis
It has been stated earlier in this manual that quantification of
all uncertainty associated with the application of a model can provide
the decision maker with information that may be used to weigh the value
of model output. This section discusses the use of first-order uncer-
tainty analysis (Cornell, 1972) for the estimation of the confidence
limits associated with the prediction of one of the empirical models.
Error analysis, like statistical sampling design (Chapter 2), should
become a standard or routine step in comprehensive water quality man-
-------
—57—
agement. Detailed development of the relationships used in error
analysis is given in the Appendix to Chapter 3. In this section, the
importance of error or uncertainty in decision making is discussed
and illustrated through example.
Uncertainty is present in most water quality assessments, simply
because we cannot sample throughout time and space. Uncertainty is
also present in all water quality predictions, because predictions
imply the use of a model. Since a model is a simplification of reality,
it is imprecise. Clearly then, predictive information is more valu-
able if we know how precise it is. Yet, model prediction uncertainty
is generally not part of model documentation. As a result, we have no
information on the prediction uncertainty associated with the applica-
tion of most of the models discussed in the last section.
At present, only the models developed by Reckhow and the model
proposed by Walker are accompanied by the statistics needed to esti-
mate prediction uncertainty. The prediction uncertainty estimated
with these models may then be used to weigh the value of the model pre-
diction; the smaller the prediction uncertainty, the more precise the
prediction, and the greater the weight that may then be associated with
the prediction. This is useful information for a decision maker. The
output from the model must be considered as one contribution to the
lake management process. If the modeler can also provide the decision
maker with a quantitative assessment of the value of the model output
in the form of prediction uncertainty, then the decision maker will be
better able to use the model prediction wisely.
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-58-
The application of uncertainty analysis may also be viewed in the
following way. Uncertainty analysis provides confidence limits, or
an estimate of the level of statistical significance at which we can
discriminate the expected difference in outcomes of two strategies.
In other words, the model predictions for different phosphorus loadings
(different strategies) have confidence limits associated with them,
calculated from the uncertainty analysis. The extent of overlap of
these confidence limits determines the level of statistical signifi-
cance. The statistical significance level, in turn, quantifies our
ability to state that two strategies will have different outcomes
(discrimination). This analysis is done like a t-test, as described
in Chapter 2, with aninfinitenunber of degrees of freedom.
So that the estimation and application of uncertainty are under-
stood, an example is provided below. The model presented in Table
3.1 is used to predict the phosphorus concentration in Lake Ontario and
to assign confidence limits to this prediction. Specifically, this
approach is used to examine the likelihood of violation of a hypo-
thetical phosphorus standard for Lake Ontario. While few standards
are presently stated in a probabilistic form, the models •and the
phosphorus loading estimates are well suited for probabilistic state-
ments. The expression of water quality standards in probabilistic terms
is more realistic for both mathematical models and conceptual models,
since both reflect uncertainty in the behavior of a lake system. Given
uncertainty in the applied model and in the independent variables, the
probability of meeting a lake quality standard (for phosphorus) can be
calculated. This is done in Figure 3.2 for Lake Ontario (z = 89 m,
-------
-59
z/T = 11.3 rn/yr), assuming a total phosphorus standard of 0.045 mg/i,
and assuming the independent variable error exists in the phosphorus
loading estimate only. (The large volume should dampen out hydrologic
budget fluctuations.) Curves are presented in Figure 3.2 covering
a range of loading levels.
Let us now discuss how Figure 3.2 was developed and how it may be
interpreted. Total prediction error may be estimated from Equation
3.21 given below, under the following general conditions:
= [ e - 10109 e + (s 2 + s ] + s (3.21)
where
Sf = total prediction error (mg/i)
= predicted phosphorus concentration (mg/i)
s = parameter error contribution to the total error
(logarithmic form)
Sm = standard error of the estimatefor the n del
(logarithmic form)
= contribution to the error from the independent
variables, primarily phosphorus loading (mg/i)
First, estimates must be available for the parameter error and
model error terms. These are presented in Table 3.1, Table 3.2, and
in the Appendix, for the Reckhow and Walker models. Note that they are
given for the log-transformed model (for direct insertion into Equa-
tion 3.21), since these models were constructed under a logarithmic
transformation. Second, an estimate is needed for the independent
-------
.70
D
.2 j-
>
‘.- H
oQ-
0
0
0
.4 .8 1.2 1.6
L(g/m 2 -yr)
.013 .027 .040 .054
P(mg/l) (Model Prediction)
C
Figure 3.2 Lake Ontario: Probability of Standard (P=.045 mg/i) Violation
-------
-61-
variable (L, z, z/i,or T) error contribution. For most lakes (except
perhaps those with low values of ), this error contribution is a
function of the phosphorus loading only.* It is assumed here that the
independent variable error is a function of the phosphorus loading
only, and that it is proportional to the loading. This means that the
independent variable error contribution (to the total error) is pro-
portional to the phosphorus concentration (since the model is linear in
L). This can be written as:
s , = kPe (3.22a)
or iv = (s(L)) = (kL) (3.22b)
where B is defined in Table 3A.6 (the denominator of the Reckhow
model), and s(L) is the phosphorus loading error estimate.**
With these term definitions and the error terms for the Reckhow
model from Table 3.1 and the Appendix (the equation for s is solved
using Lake Ontario data for z and z/T; the result is s = 4.02 x lO_2),
Equation 3.23 may be written and used to estimate the total error in
milligrams per liter:
*When this is not true, the error propagation equation, presented
in the Chapter 3 Appendix, may be used to estimate the independent
variable error contribution from all varibles. When this is true,
the source and quality of the phosphorus loading data determines how
loading uncertainty is contined with model and parameter uncertainty;
see Section 3.5.
** j is important that the units be consistent. s 2 is placed
where it is in Equation 3.21 because it is expressed in 1 nits of ± mg/i.
If instead, this error contribution term is expressed as a log trans-
form, then Eq ation 3.21 should be rewritten so that s is added
directly to Sm and s in the exponent. iv
-------
-62-
= [ e - 10 1og e ± [ (4.02 lo_2)2 + (.123)2]½] 2
+ (kPe) 2 (3.23)
Then, assuming normally distributed errors, a phosphorus standard of
.045 mg/i, and a predicted phosphorus concentration (from the model)
of ‘e’ the probability of a standard violation may be estimated from
the value of Zn (the standard normal deviate) and a cumulative normal
frequency distribution table:
.045 — P
= e
fl ST
The curves in Equation 3.23 represent different values of hIkH (express-
ing the fractional loading error); the curve marked 0 indicates the ef-
fect of model and parameter error alone. In order to use this graph,
we must decide: 1) the likelihood of standard violation we are will-
ing to accept, and 2) the uncertainty associated with our estimate of
the phosphorus loading, L. Assume that we want a 10% likelihood of
violation of this standard, and that we think we can estimate future*
phosphorus loading to within ± 50% of its true value. Then, we look
across the graph, from .10 on the vertical axis, to the .50 curve,
and down to the L-axis. This corresponds to about .79 g/m 2 -yr on the
L-axis and about .0265 mg/i on the P-axis (the predicted concentration,
using the Reckhow oxic model from Table 3.1). Thus, to achieve a
*Future phosphorus loading predictions require.the additional un-
certainty term, S{v, in Equation 3.21. An explanation for this is pro-
vided in Section 3.5.
-------
-63-
90% likelihood of standard compliance (10% likelihood of violation),
when the future loading can be estimated to within + 50%, we recommend
a design loading of about .79 g/m 2 -yr, which has an expected phosphorus
concentration of about .0265 mg/i. The difference between the standard,
.045 mg/i, and the design concentration, .0265 mg/i, may be considered
an “engineering safety factor.” It exists because we have uncertainty
in the model and in the loading estimate, and because we desire a small
likelihood of standard violation. Note that if we design for a 50%
likelihood of standard violation, then the design concentration equals
the standard concentration.
This graphical approach has been used by Reckhow and Rice (1976,
1977) in the evaluation of lake management plans for a “208” Program
in New Hampshire. This method facilitated the understanding of uncer-
tainty, and it permitted the comparison of alternative strategies.
Most important, in this investigator’s view, it is “honest” in that it
clearly indicates that, in fact, we cannot be certain that a strategy
will or will not result in a standard violation.
3.5 A Case Study Comparison of Four Models
As further illustration of the usefulness of error analysis, four
models (Dillon-Rigler-Kirchner, Larsen-Mercier, Walker, and Reckhow
oxic) are applied below to estimate the total phosphorus concentration
in Lake Charlevoix in Michigan. This lake was randomly selected from
a group of oligotrophic lakes from the EPA National Eurtophication
Survey (EPA-NES). Data reported by the EPA-NES (1975) are given in
Table 3.4.
-------
-64-
TABLE 3.4
DATA FOR LAKE CHARLEVOIX
(EPA—NES, 1975)
z
=
16.76 m
L = 0.12
g/m 2 -yr
=
3.2 yrs
= LT/Z =
.02291
mg/i
z/t =
= 5.238
rn/yr
P = .007
(Fall mean and
mg/i
median)
The Diilon—Rigier-Kirchner model is (Equation 3.14):
(1 R )
z KD
where:
= 0.426 exp (—0.271 z/-r) + 0.574 exp (—0.00949 z/T)
Substi tuting:
RKD = 0.426 exp [ -0.271 (5.238)] + 0.574 exp [ -0.00949 (5.238)]
= .1030 + .5462
= .6492
Thus:
, Lr, 1 R
— z’ KD
= .02291 (1 - .6492)
= .0080 mg/i
-------
-65-
The Larsen-Mercier model is (Equation 3.19):
P = (1 - RLM)
where:
RLM = ÷ 1.12(1)0.49
Substi tuting:
1
RLM 1 + 1 12 1 \0.49
I\32)
= .6122
Thus:
P = (1 - RLM)
= .02291 (1 - .6122)
= .0089 mg/i.
Before the uncertainty analysis is applied in conjunction with
the Walker and Reckhow models, some discussion on the appropriateness
of the terms in Equation 3.21 for prediction error estimation is in
order. The error analysis procedure based on Equation 3.21 is most
appropriate when the error in the variables in the data set used to
develop the model contributes a small or negligible amount to the
total prediction error. However, estimates of phosphorus loading,
in particular, can be quite uncertain, as can the estimates of average
phosphorus concentration and hydraulic detention time for selected
lakes. Thus, it is likely that a non—negligible fraction of the model
standard error results from the errors in these variables (in the model
-------
—66-
development data set). In that case, two recommendations are made,
given the existing models and model statistics.
1. If a model is used to assess current conditions in a lake for
which the phosphorus input and hydraulic detention time are
direct measurements , the level of uncertainty in these vari-
ables probably is approximately the same as the average un-
certainty in these terms in the model development data set
(for the models considered here). In that case, the uncer-
tainty in these variables is incorporated in the model stan-
dard error, and the prediction confidence limits may be esti-
mated using Equation 3.21 with = 0 (k = 0 in Equation
3.23 and on Figure 3.2).
2. If a model is applied to a lake for which the independent
variables, particularly phosphorus loading, are indirect esti-
mates (e.g., literature export coefficients for phosphorus
loading, or direct measurements for future predictions),
then it is likely that the uncertainty in these estimates is
quite large. The compoient of the model standard error con-
tributed by the errors in the variables is probably small in
comparison. In this case, the best estimate of the total pre-
diction uncertainty would be provided by Equation 3.21, un-
modified.
Therefore, to illustrate the use of the Walker and Reckhow models,
calculations are presented under both conditions 1 and 2 above. Under
condition 1, the EPA—NES data in Table 3.4 may be used in an error
analysis with k=0. Then, for the purpose of example, we can use the
-------
-67-
same EPA—NES data under condition 2. That is, we will assume that the
EPA—NES loading data has an uncertainty of ± 50% for future predictions,
and then we can calculate the resultant prediction confidence limits
using Equation 3.21.
The Walker model may be applied using Table 3.2, Equation 3.21,
and Equation 3.22a. The model is (Table 3.2):
1
Z [ i +
Substituting:
P=.02291 [ 1
[ i + .824(3.2 454
= .02291 (.4171)
= .0096 mg/i
The parameter error contribution may be calculated from (Table 3A.5):
s = 0.001 y 4 .908 [ 4.49 + 1.44 (in T)2 +
0.032 in T)]
where:
Y = 1/(1 +
= .4171
Substituting:
s = 0.001 (.4l7l) (3.2).908 [ 449 + 1.44 (in
3.2)2 + 0.032 ln (3.2)]
-------
-68-
0.001 (3.027 io (2.875) (6.475)
5.635 x 10
The model error is presented in Table 3.2:
Sm .171.
Now, Equation 3.21, with = 0, may be used to estimate the pre-
diction uncertainty:
= [ Pe - 10 10g e ÷ [ 5.635 x i0 + (.i7i)2]
After substituting the predicted concentration e = .0096 mg/i) into
the above relation, we can estimate the confidence limits*:
Sf 2 = [ .0096 - 10109 (.0096) ÷ .173] 2
Sf = + .0047 mg/i
sf = - .0032 mg/i
This result may be expressed as:
.0064 mg/i < true < .0143 mg/i*
where true = the “true total phosphorus concentration (mg/i).
For condition 2, the independent variable error term may be esti-
mated from Equation 3.22a:
sj. , .50
*The confidence limits given here equal ÷ one standard error. This
means that about 68% of the time that the standard error is estimated,
the I*truefl value falls within +one standard error of the predicted value.
-------
—69-
The three error components may now be substituted into Equation 3.21,
to estimate the prediction uncertainty:
S f 2 = Ee - 10 1og e [ 5.635 x 10 + (.171)2] 2 + (.50 e 2
After the predicted concentration = .0096 mg/i) is substituted
into the above relation, the confidence limits* can be calculated:
= [ 0096 - 10 1og (.0096) + .173]2+ [ (.50)(.0096)]2
= (.0047)2 + (.0048)2
Sf = + .0067 mg/i
and
Sf 2 = [ .0096 - 0096) - .173] + [ (.50)(.0096)]2
= (.0032)2 + (.0048)2
Sf = - .0058 mg/i
The result may be expressed as:
.0038 mg/i < true < .0163 mg/1*
where true = the “true” total phosphorus concentration (mg/i).
*The confidence limits given here equal + one standard error.
This means that about 68% of the time that the standard error is esti-
mated the “true” value falls within +one standard error of the predicted
val ue.
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-70-
Using Table 3.1, Equation 3.21, Equation 3.22a and Table 3A.6, the
Reckhow oxic model is applied in a similar manner. The model is
(Table 3.1):
L
- l8 + 1.05 z/Te .012z/T
Substituting:
0.12
P ( 18)(16.76 ) + 1 Q\ .012(5.238)
10 + 16.76 •• e
— 0.12
— 11.273 + 5.857
- 0.12
— 17.130
= .0070 mg/i
The parameter error contribution may be calculated from the rela-
tionship for s in Table 3A.6. Substituting the values for z and Z/T
for Lake Charlevoix into this relationship, we get:
= 7.997 x l0
The model error is also available from Table 3.1:
= .123,
Now, Equation 3.21, with s 1 = 0, may be used to estimate the
prediction uncertainty;
-------
-71—
5f2 = - 10 1og e + [ 7.997 x + (.123)2]½]
After substituting the predicted concentration e = .0070 mg/i) into
the above relation, we can estimate the confidence limits :*
= [ .0070 — 10 1og (.0070) + .126] 2
sf = + .0024
Sf = - .0018
This result may be expressed as:
.0052 mg/i < true < .0094 mg/i*
For condition 2, the independent variable error, stated earlier, is:
s , = .50
These three error components may now be substituted into Equation 3.21,
to estimate the prediction confidence limits* (for the prediction,
= .0070 mg/i):
= 0070 - 10 1og (.0070) + [ 7.997 x i0 + (.123)2]½1 2
+ [ (.50)(.0070)]2
*+ one standard error.
-------
-72-
= [ .0070 - io2 55 + .126] 2 + 1.225 x
sf = + .0042
and SI = - .0039
As in the analysis with the Walker model, this result may be expressed
as:
.0031 mg/i < true < .0112 mg/l**
It should be noted that inclusion of the parameter error contri-
bution, s , had no effect on the confidence limits. This occurs for any
lake with independent variable (L, z/r, and z) values well within the
range of values (maximum-minimum) as specified for this model development
data set (Table 3A.3).*
Thus, s can often be removed from the equation for the prediction
confidence limits with little or no effect. Presumably, one will be-
come more adept, with experience, at determining when s may be
neglected. At a minimum, though, use of the relationship without
provides a quick “first-cut” uncertainty estimate. Removing s , the
prediction uncertainty may be estimated from:
= [ - 10 1og e Sm] 2 + (3.24a)
the table in the Appendix for a definition of this range for
the Reckhow oxic model; the Walker model has a range similar to that
for the Reckhow general model.
**+ one standard error.
-------
-73—
For the Reckhow oxic model, when the independent variable error is
proportional to the phosphorus loading, this equation becomes:
S f 2 = [ - 10 log e ÷ .123 ]2 + (kPe) 2 (3.24b)
Equation 3.24b may thus be solved for the confidence limits for the
prediction, e’ when an estimate is provided for the loading error
proportionality constant, k (if loading error needs to be included).
In summary, Table 3.5, displays the output from the application
of the four models. All four models predict the phosphorus concentra-
tion relatively well in terms of absolute error (although
error varies from 0% to 37%). However, the major point made in Table
3.5 is that only two of the models provide prediction confidence limits.
Clearly, the information provided by the application of these two
models is more useful in decision making as a result of the uncertainty
analysis.
3.6 Special Lake Types
For certain types of lakes, the application of the empirical
phosphorus lake model is often not too successful. This occurs because
of special conditions (e.g., heavy aquatic weed growth) or violation
of the assumptions explicit or implicit in the models (e.g., no out-
flow), or perhaps because a lake is outside or near the bounds on the
data sets used to develop the models (e.g., extremely shallow lakes).
In this section, some suggestions are made for the treatment of these
lake types with a model, while a warning is issued against placing
too much emphasis on the model results (until these lake types are
more systematically studied).
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-74-
TABLE 3.5
COMPARISON OF THE PREDICTIONS OF
THE FOUR MODELS FOR LAKE CHARLEVOIX
Observed Total Phosphorus Concentration = .007 mg/i
(Fall mean and median)
Predicted Phosphorus Prediction
Model Concentration (mg/i) Confidence Limits*(mg/1 )
Dillon-Rigler-Kirchner .0080 -
Larsen-Mercier .0089
.0064 < P < .0143**
Walker .0096 true
.0038 < P < .0163***
true
.0052 < P < .0094**
Reckhow oxic .0070 true
.0031 < P < .0112***
true
*± one standard error
noadditiona1 loading error
*** additional loading error=± .5L
-------
—75—
Lakes with no measurable outflow (closed lakes) are not uncommon
but are not included in the data sets used to construct the models
discussed above. Strictly speaking, therefore, these models are not
applicable to this lake type. However, it is proposed here that we
might bound the expected phosphorus concentration with the prediction
from two models. First, the lower bound on phosphorus concentration
might be estimated by the constant settling velocity model proposed
by Chapra (Equation 3.5 with v 5 = 16 m/yr):
= 16 (3.25)
This model is a lower bound since it was developed from a differential
equation (3.4) that includes an outlet term, and thus it has a usinkhi
term that does not exist for closed lakes. v 5 = 16 rn/yr was selected
rather than V 5 = 13.2 rn/yr (Dillon and Kirchner, 1975) to further ex-
tend this lower bound. Note that since v 5 = 16 rn/yr was estimated
for the steady state form in Equation 3.5, for which there is no
explicit outlet term, Equation 3.25 y be applicable to closed lakes*
For an upper bound on the phosphorus concentration, consider the
differential Equation 3.4 without an outflow term:
= M - v 5 PA (3.26)
The steady state solution is:
(3.27a)
vs
*We cannot state this with any confidence, since closed lakes
were not in the model development data set.
-------
- 76—
Assume here that v = 13.2 rn/yr (13.2 as opposed to 16 for a con-
servative upper bound) represents the effective removal of phosphorus-
associated material from the lake water column. Then the upper bound
on the phosphorus concentration can be estimated from:
L
— 13.2
(3.27b)
The predicted phosphorus concentration in closed lakes may thus be
expressed as:
L
-------
-77-
a gap and prevent application of a black box model alone on a closed
lake.
A second lake type that needs brief mention is the extremely
shallow lake (z less than perhaps three meters). While the model de-
velopment data sets for the Reckhow, Walker, and Dillon-Rigler-Kirchner
models contain extremely shallow lakes, this investigator has found
that this lake type can exhibit unpredictable behavior (in terms of
total phosphorus concentration). It is possible that, because of the
depth to area ratio and the potential for mixing of the sediments,
phosphorus concentration is more variable in shallow lakes (than
in deep lakes). On the other hand, these same conditions may prevent
the development of anaerobic conditions (which, of course, can increase
phosphorus release from the sediments) and reduce concentration vari-
ability. Experience in the application of the lake models to this
lake type may be most useful here. With experience, the model user
may be able to determine factors that affect the success of the models,
and the user may then modify the prediction and confidence limits, if
needed. In the absence of this modification, the models must be applied
cautiously in extremely shallow lakes.
Finally, lakes with extensive aquatic weed growths are probably
not modeled well with the techniques proposed here. This should not
be surprising since aquatic weed growths in lakes are generally
localized, restrict mixing, and interact directly with the sediments.
These three conditions suggest violation of the assumptions behind the
models. At this point, due to lack of experience with aquatic
weed infested lakes, this investigator can offer no suggestions (other
than caution) for the use of lake models for this lake type.
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-78-
3.7 Conclusions
A final statement for this chapter on the phosphorus lake models
should probably be “apply the models with caution.” It has been shown
that black box models designed to predict the average annual phosphorus
concentration in a lake are relatively successful. Further, uncertainty
analysis was introduced as a technique that may be used to estimate con-
fidence limits for the model predictions. However, the models are
based on certain assumptions specified in this chapter, and the appli-
cation of the models is constrained by the limits of the model develop-
ment data sets. Hence, there is a need for careful consideration in
the selection of a model and careful use of the model in lake manage-
ment.
All the models require an estimate of the phosphorus loading.
For many applications, this term will not be available from direct
measurement, and thus the literature must be consulted for export co-
efficients relating land use to annual phosphorus flux. Four sources
of this information are reports by Uttormark et al. (1974), Omernik
(1976, 1977), and McElroy et al. (1976). Again, careful use is in
order. Confidence limits are provided for some of the loading terms,
but these apply only for the data gathered and regions studied. Thus,
one should “verify” the appropriateness of a particular relationship
or export coefficient before application. The ultimate value of a
quantitative technique, coefficient, or measurement depends in large
part on its uncertainty and applicability. Consideration of that
fact will enhance the utility of the material presented in this chap-
te r.
-------
APPENDIX
-79--
-------
TABLE 3A.l
MODELS DEVELOPED BY RECKHOW (1977)
Q
Class
Model
R 2
Standard Error
of the
Estimate*
Quasi-General
L
.915*
.128*
11.6 + 1.2 qs
Oxic lakes, Z/T
< 50
rn/yr
L
.876*
.123*
18z +
1O+z
l.O5(i’\el 2 ’T
\t/
Lakes with z/r
5O
rn/yr
L
.949
.0088
2.77z
+
‘ r1
Anoxic Lakes
I
.948*
.105*
T7z + 1.13z [ T
*Based on logarithmic transformation of terms.
-------
-81-
TABLE 3A.2
STATISTICS FOR 95 NORTH TEMPERATE LAKES (RECKHOw, 1977)
Geometric
Variable Minimum Mean Maximum
z (m) 0.78 7.8 100.0
T (yr) 0.003 0.47 57.6
z/r (m/yr) 0.53 16.5 575.0
LT/z (mg/i) .0075 .066 1.179
P (mg/i) .004 .033 .958
R -.08 .38* 95
Correlation Coefficients
Variable log ‘r log (z/r) log (Lt/z) log P R
log z .751 —.379 -.151 -.408 .508
iog T -.896 .119 -.249 .700
log z/T — .260 .074 - .639
log Lr/z .851 .049
logP -.405
R
*arjthmetjc mean
-------
-82-
z (m)
t (yr)
Z/t (rn/yr)
LT/Z (mg/i)
P (mg/i)
R
z (m)
t (yr)
Z/T (m/yr)
Lt/z (mg/i)
P (mg/i)
R
Oxic Lakes, z/t < 50 rn/yr (33 lakes )
0.73 18.2
0.065 2.57
0.96 7.1
.008 .040
.004 .013
.11 .56*
100.0
57.6
48.6
.298
060
.95
34.0
4.5
425.0
.621
.610
.56
TABLE 3A.3
STATISTICS FOR THE THREE MODEL CLASSES
Geometri c
Variable Minimum Mean Maximum
Lakes with z/T > 50 rn/yr (28 lakes)
13.4
0.85
4.05
0.003
0.028
0.252
51.8
147.0
575.0
.008
.047
.178
.007
.034
.135
-.08
.18*
37
Anoxic
1 .40
0.008
3.3
024
.017
-.03
z (m)
r (yr)
z/t (m/yr)
LT/Z (mg/i)
P (mg/i)
R
*arj thmeti c mean
Lakes (21 lakes)
6.9
0.52
13.3
.119
.088
.23*
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-83-
TABLE 3A.4
PARAMETER STATISTICS FOR THE RECKHOW MODELS
Standard
Parameter Deviation Correlations
Oxic, Z/T
18.0
10.0
1.05
0.12
50 rn/yr
2.41
6.56
9.27
5.45
lakes
—3
x 10
x
—2
x 10
x 1O
Lakes with z/t > 50 rn/yr
18.0
10.0
1.05
0.12
.538
-.274
.133
.028
.735
—. 351
2.77
1.05
-.650
.0011
.415
.889
2.77
1.24
1.05
9.27 x io . .2
.0011
4.37 x 10
Anoxic Lakes
0.17
1.13
2.61 x io_2
3.95 x
i.13
- .656
-------
-84-
TABLE 3A.5
PARAMETER STATISTICS FOR THE WALKER MODEL
Correlation Coeffi ci ent
Parameter Standard Deviation .546
.824 .067 -0.0064
.546 (= 1 — .454) .046
Prediction Error contribution from Parameter Uncertainty
s = 0.001 Y 4 t 9 ° 8 [ 4.49 + 1.44 (in T)2 + 0.032 in )]
whete Y = 1/(1 +
in = base e logarithm
TABLE 3A.6
PREDICTION ERROR CONTRIBUTION FROM PARAMETER UNCERTAINTY FOR THE
MODEL PROPOSED BY RECKHOW
FOR OXIC LAKES WITH z/T < 50 rn/yr
= [ (h-) (z/T)(eOl 2 ’t)(.O927)] 2 + [ ( )(1.O5(z/T) 2 e Ol 2 ZhhT)(.OO545)] 2
— 2( ) 2 (z/T) 3 (eOl 2 Z ’t) 2 (1.O5)(.35l)(.O927)(.OO545)
where
B = + 1.05
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-85-
Uncertainty Analysis
The technique used to estimate the prediction confidence limits
is derived from the Taylor series expansion, which provides an approxi-
mation for a function, f(x). For a bivariate relationship, the Taylor
series may be written as:
f(x) = f(x 0 ) + f(X)(x - x ) + 1 B 2 f(x ) - x )2 +... (3A.l)
If it may be assumed that the error (x - x 0 ) is small and the higher
derivatives of f(x) are not large (the relationship is not highly non-
linear in the region of interest), then Equation 3A.l reduces to:
f(x) - f(x 0 ) -(x - x ) (3A.2)
For a multivariate relationship, this becomes:
f(x. ,x. ,Xk )
f(xl x xk) - f(xIO x O xkO) Xj 3 (x - x 10 ) +
f(x. ,x. ,xk) f(x. ,x.,xk )
(x - x 0 ) + (xk - XkO)
(3A. 3)
Squared and expressed in standard error form, Equation 3A.3 may
be converted to the error propagation equation given below.
s(P) j l [ ( i) 2 s 2 (x ) + j +l 2 ] ½
(3A. 4)
s(x ) is the estimate of the standard deviation of x 1 , and (x x ) is
-------
-86-
the correlation between x. and x 3 . Berthouex (1975) and Cornell (1972)
discuss this error analysis approach for sanitary engineering and hydro-
logic applications, respectively.
Equation 3A.4 may be used to estimate the effect of model para-
meter error and input variable error. This was done for the Reckhow
oxic lakes (z/t < 50 m/yr) model, and the result is presented in Table
3A.6. These error terms are then contined with the standard error
of the model for an estimate of the prediction confidence limits. The
empirical models presented in Chpater 3 are linear (or nearly linear)
in L and approximately linear in z/t for most lakes. Since the vari-
ance in z is usually quite small, it is ignored. Therefore, Equation
3A.4 may be used with an empirical model (providing model and para-
meter error estimates) for all lakes within the model’s constraints,
except for those lakes with a highly non-linear relation between P and
z/T (over the estimated variance in Z/T for that lake). This non-
linearity should be checked before application of Equation 3A.4;
otherwise, Equation 3A.4 may significantly overestimate the total
error.
The model error,and the contributions to the total prediction un-
certainty from the parameter error and the independent variable uncer-
tainty,are normally contined in variance form:
s = s + s + s (3A.5)
where
ST = total error (logarithmic form)
s , = parameter error contribution to the total
error (log form)
-------
-87-
Sm = standard error of the estimate for the model
(log form)
= input variable error (log form) contribution
to total error
Sm for the models discussed herein are presented in Tables 3A.l and
3.3, and s may be calculated for each lake from Equation 3A.4 and the
statistics presented in Tables 3A.4 and 3A.5. s 1 results from uncer-
tainty in the phosphorus loading and in the hydraulic detention time.
would, therefore, be estimated using the error propagation equa-
tion for the model variables , just as the error propagation equation
was used for the model parameters to derive the relationship for s .
However, for most lakes, the major component of the variable error
results from phosphorus loading, and often the loading uncertainty is
estimated as proportional to the loading. That approach was adopted
for the applications presented in this chapter. Since the models dis-
cussed herein are linear in L, the loading uncertainty is proportional
to the phosphorus concentration. Since the loading uncertainty is
proportional to the concentration rather than to the log of the con-
centration, it is easier, for calculation purposes, to include this
uncertainty term in the form:
sj. , = kP
where
= untransformed uncertainty component due to
the phosphorus loading
k = a constant
-------
-88-
The model applied here was developed from a logarithmic transfor-
mation of terms. For this reason, Equation 3A.5 must refer to the log-
transformed model. The error terms are thus conbined in the following
way:
ST 2 = Sf 2 = - 10 1og P + {(s ) 2 + (Sm) 2 + (siV)2]] (3A.7)
For application purposes, this is rewritten:
S f 2 = - 10 1og p + ( 2 + 5 2y½ 1 2 + S (3A.8)
-------
CHAPTER 4
LAKE CLASSIFICATION
4.1 Introduction
The models presented in Chapter 3 may be used to predict the
phosphorus concentration in a lake. An alternative way to model or
quantify the effect of phosphorus (nutrient) loading or concentration
on lake quality is through lake classification criteria. Much of the
same information is conveyed through the application of either the
models or the classification criteria, but the information is presented
in different formats. The models may be used to estimate the concen-
tration of a single constituent (phosphorus) on a continuous scale.
Classification criteria, on the other hand, may be used to group
lakes into a small nunter of discrete quality (trophic) states. This
results in an easily interpretable ‘iake-labeling scheme which has
the advantages and disadvantages of any classification system designed
to reduce the dimensionality or discretize a set of continuous data.
Lake classification schemes actually include two distinct approaches.
The first, often identified with trophic state criteria, classify a
lake according to the current quality of the water column (the water
in the lake) using data from in-lake samples . These typically require
measurements of total phosphorus concentration, total nitrogen concen-
tration, chlorophyll a level, Secchi disk depth and/or other constitu-
ents as well (all taken within the lake). The second approach, gener-
-89-
-------
-90-
ally identified with nutrient (phosphorus) loading criteria, can be
used to predict the quality state of a lake from data on the phosphorus
loading and lake geomorphology (no in—lake measurements). Loading
criteria can be used to forecast the effect of changes in phosphorus
loading on lake quality. Further, they can be reformulated and used
to estimate the phosphorus carrying capacity associated with each
quality state. In summary, the trophic state criteria can provide a
multivariable index of present water quality, while the loading criteria
focus on a single nutrient but can be used to predict quality changes
over time and estimate carrying capacity.
This chapter presents discussions of both approaches. The em-
phasis is placed on loading criteria, however, because of the useful-
ness of that information for planning and management of lake quality.
Case studies are presented illustrating some applications.
4.2 Trophic State Criteria
A number of attempts have been made to establish a trophic state
criterion as a function of coniiionly measured water quality variables.
The EPA National Eutrophication Survey (1974a) has compared the work
of some investigators (Sakamoto, 1966; National Academy of Sciences,
1972; and Dobson et al., 1974) on chlorophyll a levels versus trophic
state. This is presented in Table 4.1. The EPA ’s own estimates of
values of chlorophyll a, total phosphorus, and Secchi disk depth
indicative of trophic states are presented in Table 4.2.
While there have been other attempts at single variable trophic
state criteria (or indices), all are relatively similar in approach
(see Table 4.1). More importantly, they represent subjective judgment,
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—91—
TABLE 4.1
TROPHIC STATE VS. CHLOROPHYLL a
(from EPA-NES 1974a)
Trophic
Condition
Sakamoto
Chloro
Academy
phyll a (i tg/l)
Dobson
EPA-NES
Oligotrophic
0.3-2.5
0-4
0-4.3
<7
Mesotrophic
1-15
4-10
4.3-8.8
7-12
Eutrophic
5—140
>10
>8.8
>12
TABLE 4.2
EPA-NES TROPHIC STATE DELINEATION
(from EPA-NES 1974a)
Trophic State
Chlorophyll a
( gIl)
Total Phosphorus
( g/l)
Secchi Disc
Depth (m)
Oligotrophic
<7
<10
>3.7
Mesotrophic
7-12
10-20
2.0-3.7
Eutrophic
>12
>20
<2.0
-------
—92-
and possibly limited geographic regions, so it is unlikely that uni-
versal agreement will rest on one approach. Therefore, the selection
of a univariate trophic state criterion should be based primarily on
personal acceptance and credibility.
More robust trophic state criteria or indices may be developed
with a multivariate approach. Shannon and Brezonik (1972) developed
a trophic index for Florida lakes composed of the variables: primary
production (PP, in mg. of carbon per cubic meter-hour), chlorophyll a
(CHA, in mg/rn 3 ), total organic nitrogen (TON, in mg/l as N), total
phosphorus (TP, in mg/i as P), Secchi disk transparency (SD, in meters),
specific conductance (COND, in 1 . 1 n 1o/cm) , and a cation ratio (CR, a
dimensionless ratio of (Na + K)/(Ca + Mg)). For lakes without ap-
preciable organic color, the trophic state index (TSI) was estimated
as:
TSI = 0.936 (1/SD) + 0.827 (COND) + 0.907 (TON)
+ 0.748 (TP) + 0.938 (PP) + 0.892 (CHA)
+ 0.579 (1/CR) + 4.76 (4.1)
A TSI of about 3 to 5 defines the transition zone between eutrophy
and mesotrophy, and a TSI of 1.2 to 1.3 separates the mesotrophic and
oligotrophic classes.
The index was developed using principal component analysis, and
the TSI is the first principal component. This technique may be used
to identify “comon elements” among variables, and the first principal
conponent is a linear combination of the variables that best describes
the most conunon element. When all of the variables in an analysis are
-------
-93—
thought to be good indicators of a concept called trophic state, then
it is reasonable to assume that the most coninon element extracted from
this set of variables (the first principal component) would be a good
index of trophic state. In fact, this component is more “robust” than
any one variable as an indicator of trophic state. This means that it
is less likely, than a single variable index, to misclassify a lake
based on an erroneous measurement. Incorrect data on one variable
can lead to misclassification based on that variable, but it may not
lead to misclassification if the classification criterion is based on
other variables (correctly measured) as well.
Despite the fact that a principal component trophic state index
has this desirable feature of robustness, the TSI proposed by Shannon
and Brezonik cannot be recomended for use on north temperate lakes.
The ISI was developed from a data base of Florida lakes only, and the
significant climatic (and thus thermal)difference between that area
and the north temperate region is likely to affect the index. Since
this effect is unclear, we are unable to interpret the TSI in north
temperate lakes. Equally important, most of the trophic variables
are log—normally distributed (see Chapter 2 for an explanation),
which means that the best estimate for the TSI should be made under a
logarithmic transformation for these variables. Without this trans-
formation (as in the case of Shannon’s and Brezonik’s TSI), the index
may be less precise. Therefore, while principal component analysis
is a useful technique for defining a trophic state index, the TSI
developed by Shannon and Brezonik is inappropriate for north temperate
lakes.
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-94-
A trophic state index has recently been proposed by Carlson (1977)
that may also be considered multivariate. Carison’s index may be
estimated from sunniler values of Secchi disk depth, sununer total phos-
phorus concentration, or sununer chlorophyll a concentration (or a
weighted contination of all three). Carison used regression analysis
to relate Secchi disk depth to total phosphorus concentration and to
chlorophyll a concentration. He then reasoned that a doubling of
biomass levels, or a halving of the Secchi disk depth, corresponds to
a change in trophic state. Carison arbitrarily assigned a TSI scale
of 0—100 to the three trophic variables, such that a change of 10
units in TSI corresponds to a halving of the Secchi disk depth and
a change in trophic state. The regression equations then were used
to relate the TSI to phosphorus and chlorophyll. Table 4.3 contains
the index values and variable relationships.
Carison’s TSI may be estimated from any of the three variables,
using Table 4.3. Carison felt that this was important as:
1. Secchi disk readings may be misleading as a trophic state
indicator in colored lakes or highly turbid (non-algal)
lakes.
2. Chlorophyll a may be the best indicator during the growing
season.
3. Phosphorus may not be a good indicator in non-phosphorus
limited lakes.
Thus different variables may be used depending upon the season, lake,
and availability and quality of data. While Carlson suggests that
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-95-
TABLE 4.3
CARLSON’S TROPHIC STATE INDEX
Surface
Surface
Secchi
TSI disk (m)
phosphorus
(mg/rn 3 )
ch1oroph 11
(mg/mi)
0 64 0.75 0.04
10 32 1.5 0.12
20 16 3 0.34
30 8 6 0.94
40 4 12 2.6
50 2 24 6.4
60 1 48 20
70 0.5 96 56
80 0.25 192 154
90 0.12 384 427
100 0.062 768 1,183
-------
-96-
the variable that the index is based on be selected on a pragmatic
basis, he reconinends that consideration be given to chlorophyll in
the summer and to phosphorus in the fall, winter, and spring.
A comparison of Carison’s TSI and the indices proposed in Tables
4.1 and 4.2 (under the assumption that a lake with Carison’s TSI < 40
is oligotrophic and a lake with Carlson’s TSI > 50 is eutrophic) shows
a great deal of similarity, which should be expected since the
relationships among Secchi disk depth, chlorophyll levels, and phos-
phorus levels are well established and understood. Thus Carlson’s
major contribution is not new insight on trophic state divisions,
nor a multivariate index*, but rather the expression of the index on
a 0—100 scale. This effectively eliminates the subjective labeling
of oligotrophic, mesotrophic, and eutrophic states, and substitutes
a scale easily interpretable by the layman. This advantage may be
removed, however, by the tendency of researchers to mentally convert
Carison’s 1ST value back to one of the three standard trophic states
for ease of interpretation. In sunulary, then, the selection of a
trophic state index from among those given in Tables 4.1 - 4.3 should
probably be made on the basis of familiarity by the users, since no
single index conveys appreciably more information than any of the
others.
4.3 Phosphorus Loading Criteria
4.3.1 Early Development
The development of nutrient loading criteria, in particular
phosphorus loading criteria for lakes, centers around Vollenweider’s
*The EPA’s index in Table 4.2 can be considered multivariate, since
the trophic state may be based on any or all of the three variables.
-------
-97-
work. Vollenweider’s initial work on this topic (1968) was derived
from the well-documented inverse relationship between mean depth and
various measures of quality or biomass. Based on available data and
his subjective judgment on lake quality, Vollenweider estimated “per-
missible” and “dangerous” (2x permissible) annual areal nutrient
loading levels as a function of mean depth. These loading criteria
are contained in Table 4.4.
However, it became apparent that hydraulic flushing rates also
influence the capacity of a lake to accept nutrient loading. Lack of
consideration of this factor may result in an underestimation of the
trophic state for lakes with high flushing rates (low detention times).
Therefore, Vollenweider (1973) modified his original work, and he se-
lected overflow rate (z/r) as the best determinant (along with areal
nutrient loading) of lake quality across a wide range of lake classes.
This time, he again subjectively established “permissible” and “dan-
gerous” loading levels (as a function of z/t and L). Despite this
subjectivity, these loading criteria are attractive because they may
be easily applied to assess the susceptibility of a lake to phosphorus
input. As a result, they have been used extensively. A plot of this
second loading criterion is given in Figure 4.1.
At about the same time Vollenweider developed his revised load-
ing criterion, Dillon (1975) also recognized the importance of deten-
tion time or flushing rate. As a result of analyses performed on two
Canadian Shield Lakes, Dillon proposed a phosphorus loading criterion
graph with LT(l - R ) on one axis and z on the other axis. Dillon’s
phosphorus loading criterion is given in Figure 4.2. This criterion
-------
-98-
TABLE 4.4
PERMISSIBLE LOADING LEVELS FOR TOTAL NITROGEN
AND TOTAL PHOSPHORUS (BIOCHEMICALLY ACTIVE)*
(g/m 2 —yr)
Mean Depth
up to
Permissible Loading, Dangerous Loading,
up to in excess of
N P N P
5 m 1.0 0.07 2.0 0.13
10 m 1.5 0.10 3.0 0.20
50 m 4.0 0.25 8.0 0.50
100 m 6.0 0.40 12.0 0.80
150 m 7.5 0.50 15.0 1.00
200 m 9.0 0.60 18.0 1.20
*After Vollenweider (1968).
-------
—99-
1.0
c J
0.I
0.OIo, 100
z/t Im/yr)
Figure 4.1 Vollenweider’s Phosphorus Loading Criterion (1975),
L versus (Z/T)
g TP/m 2 y
I0
1.0 10
-------
,c J
—E
— , ,.
-100-
I0
l.0
0.I
.0I 1000
Figure 4.2 Dillon’s Phosphorus Loading Criterion, With Lines of
Constant Phosphorus Concentration (in pg/i) Distinguishing
Trophic States (From Dillon, 1975)
to too
z(m)
-------
-101-
accounted for the effect of detention time, and it resulted in trophic
state delineation by lines of constant phosphorus concentration. This
becomes apparent when it is noted that the definition of the outlet
phosphorus concentration is:
— R ) (4.2)
Thus Dillon’s criterion is actually a plot of this definition
for the outflow concentration. However, application of this criterion
is possible only with an estimate for R [ available later using the
empirical model of Kirchner and Dillon (1975)] independent of phos-
phorus budget considerations; otherwise application of Dillon’s
criterion requires as input the variable it is designed to estimate
(or classify), the total phosphorus concentration.
Recently, Larsen and Mercier (1975) in conjunction with the EPA—
NES (EPA National Eutrophication Survey), proposed another variation
of the phosphorus loading criterion graph. They plotted mean influent
phosphorus concentration versus the measured phosphorus retention co-
efficient, and distinguished trophic states using lines of constant
phosphorus concentration. Their approach is presented in Figure 4.3.
This criterion plot illustrates the interplay between the lake basin
impact (influent phosphorus concentration) on the lake quality, and
the lake’s capacity to absorb that impact (the phosphorus retention
coefficient). Like Dillon’s plot, the Larsen—Mercier graph is a
direct result of Equation 4.2, and thus is subject to the same com-
ments. (Larsen and Mercier have also proposed an empirical relation
for Rn.) However, Larsen and Mercier felt that their graph was more
-------
-102-
1000
100
I—
I0
5
0 1.0
The Larsen—Mercier Phosphorus Loading Criterion (From
Larsen and Mercier, 1975)
0.2 0.4 0.6 0.8
Rp
Figure 4.3
-------
-103-
informative, since it separates the effects of the two major deter-
minants of lake phosphorus concentration (irifluent concentration and
retention coefficient).
In summary, with the exception of Vollenweider’s first loading
criterion (z vs. L) which did not explicitly consider variations in
lake flushing rate (and thus may be misleading, particularly for lakes
with high flushing rates), the phosphorus loading criteria discussed
in this section predict trophic state about equally well for a set of
lakes. The choice of which of these three to use, may, therefore, be
a function of preferred form or familiarity. However, before select-
ing one of these three loading criteria, Sections 4.3.2, 4.3.4, and
4.3.5 should be read for more recent developments. Specifically, con-
sideration of an additional variable, consideration of uncertainty, and
development of “optimal,” objective loading criteria are discussed.
While not in common use yet, these new phosphorus loading criteria
represent improvements in the development and application of this tech-
nique, and thus they should be given serious consideration.
4.3.2 Recent Work By Vollenweider
Vollenweider (1976) recently proposed new phosphorus loading
criteria that build upon his previous work and the work of others.
Most of the loading criteria discussed in the last section are used
to examine the relation between phosphorus loading and trophic state
as a function of mean depth (z), hydraulic detention time (T), or
surface overflow rate (q 5 ), individually. In this recent paper,
Vollenweider proposes that, in general, refinements in trophic state
-------
-104-
assessment can be made if any two of these variables (z, r, and q 5 ) are
considered together.
Vollenweider empirically derived a model to estimate ‘criticai”
loading (the transitional loading between the oligotrophic and meso-
trophic states):
L =P z(’ ) (43)
where
L = critical phosphorus loading (g/m 2 -yr)
= critical phosphorus concentration (mg/i)
z = lake mean depth (m)
r = hydraulic detention time (yr)
This is equivalent to:
L = P q (1 + /z/q 5 ] (4.4)
where
q 5 = z/’r = surface overflow rate (m/yr)
Further, if Equation 4.3 is rearranged so that phosphorus concentration
is on the left side of the equation, then:
P = Lt( 1 \= , ( 1 (4.5)
Z\i +/ / 1 fl\l ÷/ J
where
= P = average influent phosphorus concentration (mg/i)
-------
-105-
With additional rearranging of terms in Equation 4.5, we can derive
the Larsen—Mercier phosphorus model from Chapter 3:
P= - - (1- 1 (4.6)
Z \ l+1/v’ /
Vollenweider used Equations 4.3 and 4.4 to construct his new graphical
loading criterion. It is interesting to note that these equations are
equivalent in form to the Larsen-Mercier model, which was used to con-
struct the Larsen-Mercier phosphorus loading criterion.*
Vollenweider’s new phosphorus loading criterion, developed using
either Equation 4.3 or 4.4 with a critical phosphorus concentration of
.010 mg/l, is presented in Figure 4.4. Note that it is plotted on the
same axes, L vs. q 5 , used for Vollenweider’s second criterion (Figure
4.2). The difference between these two criteria is that Figure 4.4
may be used to estimate the additional effect of mean depth or hydraulic
detention time (trhough a series of lines representing the critical
loading). Vollenweider provides the following example for the appli-
cation of Figure 4.4. For Lake Ontario, with z = 84 m, T = 7.9 yr,
and 10.6 m/yr, start from q 5 = 10.6 on the horizontal axis. Pro-
ceed vertically up the imaginary = 10.6 line to a point, within the
grid of z and r lines, that best represents z 84 m and T 7.9 yr.
Then, from that point proceed horizontally to the vertical axis and
read the critical loading value. For Lake Ontario, critical loading
(the transition between oligotrophic and mesotrophic) is 350-400
*Thus, Vollenweider, unlike Larsen and Mercier, is able to examine
the effect of both qs and z because he “removed” qs from P and ex-
pressed loading as an area] term (L). However, it is unclear which
approach is more informative.
-------
- I I I I VIII !
I U I I 11111 I 1 1 11 1 11
I I lull
1000
100
10
E
E
C -,
-J
0
-J
-J
C-)
I-
C-)
Figure 4.4 Vollenweider’s Phosphorus Loading Criterion (1976), L versus q 5 , z, and r
I I I I 11111
I I I 111111
I 10 100
HYDRAULIC LOAD
qs m/y
IN
C
0
E
500
200
100
50
20
10
(ni)
I I I 111111
/0
(years,
/
p
/
/
/
/
p
/
/
I
/
/
/
/
/
/
/
Lc 10(211j(t+Jf)
/
/
1__ _ I I I liii
-------
-107-
mg/m 2 -yr. Eutrophic loading, according to Vollenweider, is then
simply double the critical loading.
This approach represents a restatement of the loading criteria
discussed in the last section. By graphically including q 5 and either
z or T in the trophic state assessment, an easy-to-use procedure was
developed that appears to be sensitive to changes in both and z, or
both q 5 and t (which, if the procedure is valid, should make the analysis
more robust). Future applications of this criterion will serve to
check its usefulness and reliability in trophic state assessment.
4.3.3 A Lake Classification Scheme Proposed by Uttormark and Wall
Uttormark and Wall (1975) proposed a subjective “Lake Condition
Index,’ LCI, that actually is a trophic state index, but it is included
in this section because of the applications suggested by Uttormark and
Wall, when the LCI is coupled with phosphorus loading information.
Briefly, the LCI is constructed from easily detected measures of eutro-
phication. Penalty points are assigned according to Table 4.5. The
sum of the penalty points is the LCI; higher values indicate poorer
quality. Uttormark and Wall provide more detailed tables in their
report for specifying the points more precisely.
Uttormark and Wall plotted lakes on a graph of LCI (which is used
to estimate present lake quality) versus areal phosphorus loading (in
g/m 2 —yr, and which may be considered an estimate of potential or
future quality). They reasoned that the position of a lake on this
plot, shown in Figure 4.5, would indicate the type of management strategy
that would be most effective for that lake. Figure 4.5 may be divided
-------
-108-
TABLE 4.5
POINT SYSTEM FOR LAKE CONDITION INDEX*
Parameter Points
Hypolimnetic dissolved oxygen 0 - 6
Transparency 0 - 4
Fishkills 0 — 4
Use impairment (extent of
macrophyte or algal growths) 0 - 9
Total 0 - 23
*From Uttormark and Wall (1975)
-------
00
1’O
0
E °
D
o 0
o 0
_ o o
-J
U)
0
o 0
a-
U,
o
0
Satisfactory Unsatisfactory
Lake Condition Parameter
Figure 4. 5 Uttormark’s and Wall’s Lake Classification Scheme (From Uttormark and Wall, 1975)
-------
-110-
into four quadrants, and the management strategies appropriate to each
quadrant are given in Table 4.6.
Uttormark and Wall caution that the position of a lake on this
curve can be affected by many factors, and thus they recommend that
this analysis should be carried out on a limited geographic region
(e.g., a state). When a nuither of lakes from a state are plotted on
Figure 4.5, Table 4.6 might be used to guide state policy for lake
management, particularly if many lQkes fall in one quadrant. The
procedure was tested on selected Wisconsin lakes (natural lakes, with
inlet and outlet, seasonally stratified, and with no known point sources
of nutrients) for which phosphorus inputs were estimated from the
literature. Results were mixed, and Uttormark and Wall concluded that
the concepts were reasonable but refinements were needed.
In suninary, it must be noted that the lack of success in the
trial of this technique may be due to a nunter of factors, including
error in the phosphorus loading estimates, the validity of the assump-
tion that LCI represents current conditions and loading represents
potential or future conditions, and the subjectivity (and possible bias)
of the LCI estimation procedure. The overall technique is attractive
because of its simplicity. However, since it has not yet been suc-
cessfully tested, it will not be reconi’riended here.
4.3.4 Uncertainty Analysis Applied to Vollenweider’s Loading
Criterion - A Graphical Approach
In Chapter 3, the concept of uncertainty was introduced and a
procedure was developed for the estimation of confidence limits on
a model prediction. The same idea is equally important in this chap-
-------
—111—
TABLE 4.6
SUGGESTED MANAGEMENT STRATEGIES ASSOCIATED
WITH THE FOUR QUADRANTS IN FIGURE 4•5*
Set
Phosphorus
Loading
Lake Condition
General Water Quality
Management Grouping
a
Low
Satisfactory
No present danger
b
Low
Unsatisfactory
Renewal desirable; long-
term benefits may be pos—
sible without extensive
nutrient abatement
c
High
Satisfactory
Prompt protection needed;
degradation may be
imminent
d
High
Unsatisfactory
Problem lakes; renova-
tion desirable but last-
ing improvement may
require extensive nutri-
ent abatement
*From Uttormark and Wall (1975)
-------
-112-
ter. That is, there is uncertainty in the estimates of L, z, r, and
There is also uncertainty in the subjective estimates of trophic state
transition lines (loading criteria). As in Chapter 3, there is clearly
a need here to estimate this uncertainty,and to use this total uncer-
tainty estimate as a weight associated with a prediction or trophic
state assessment.
A graphical technique has been developed (Reckhow, l978d), and
is presented below, that contines the uncertainty in phosphorus load-
ing (uncertainty in the other variable, is assumed to be negligible)
with the uncertainty in a model used to develop a loading criterion on
Vollenweider’s criterion axes (L vs. q 5 ). These contined uncertainties
represent an estimate of total prediction uncertainty, which is expressed
here as the likelihood (probability) that a lake will fall in each of
the three trophic classes. Assigning a probability to a predicted
classification is more consistent with our current knowledge (or igno-
rance), since we cannot, in fact, state with certainty what the future
classification of a lake will be.
As applied in Chapter 3, the uncertainty analysis should, under
certain conditions, include a term for loading error. Under other con-
ditions, this additional loading error term would not be necessary.
The following guidelines, presented first in Chapter 3, describe the
conditions necessary for each error analysis approach.
1. If the model is used to assess current conditions in a lake
for which the phosphorus input and hydraulic detention time
are direct measurements , the level of uncertainty in these vari-
ables probably is approximately the same as the average uncer-
tainty in these terms in the model development data set. In
-------
-113—
that case, the uncertainty in these variables is incorporated
in the model standard error, and the trophic state probabilities
may be estimated using Equations A.8 and A.9 with = 0, or
using the k = 0 line on Figures 4.8—4.10.
2. If the model is applied to a lake for which the independent
variables, particularly phosphorus loading, are indirect esti-
mates (e.g., literature export coefficients for phosphorus
loading, or direct measurements for future predictions), then
it is likely that the uncertainty in these estimates is quite
large. The component of the model standard error contributed
by the errors in the variables is probably small in compari-
son. In this case, the best estimate of the total prediction
uncertainty would be provided by Equations A.8-A.9, unmodified,
or by Figures 4.8-4.10 (with k = the fractional loading uncer-
tainty) according to the approach described below.
The procedure, outlined and illustrated below, is entirely graph-
ical, but may also be solved mathematically. At most, the only infor-
mation required to estimate the trophic state probabilities are L,
and the uncertainty in L. The steps are:
1. For the lake of interest, estimate q 5 , L, and, when necessary,
the uncertainty in L (expressed as a fraction of L). The
uncertainty in L should be estimated such that about 70% of
the time the actual loading will fall within L + the uncer-
tainty in L. One approach might be to estimate the maximum
and minimum phosphorus loadings expected, and then assign
-------
-114-
these loadings as L + 2 L and L - 2 SL. respectively (i.e.,
assume that the 95% confidence limits approximately represent
the range). As in Chapter 3, consider Lake Charlevoix
(Michigan), with data (USEPA, 1975):
L = 0.12 g/m 2 -yr
q 5 = 5.24 m/yr*
SL = .50 L = kL = (uncertainty in L, see Chapter 3).
If desired, plot these data on Figure 4.6, with error bars
representing the uncertainty in L. Figure 4.6 is a loading
criterion like that proposed by Vollenweider (1975), with
solid lines representing the “dangerous” and “permissible”
loading levels, and dashed and dotted lines representing the
uncertainty in the dangerous and permissible lines.
2. Using Figure 4.7, estimate y from q 5 . (See the Appendix for
a definition of y.) For Lake Charievoix, y = .056. Multiply
y times L. (This equals P, the predicted lake phosphorus
concentration.) For Lake Charlevoix, yL = P = .0067 mg/i.
3. Using Figures 4.8, 4.9, and 4.10, the estimate of yL, and
the fractional uncertainty, k, in L (if applicable): find
the probability that the lake of interest should be classified
as oligotrophic, mesotrophic, and eutrophic. This was done
for Lake Charlevoix and:
*Note that i may be easily estimated, since q 5 Q/A, where Q
average annual volumetric water flow to lake, and A = lake surface area.
-------
-115-
I0.0
I.0
L
(g/m 2 — yr)
0.I
Figure 4.6
.01
I.0
2.0 5.0 tO 20
ci (m/yr)
50 100 200
Uncertainty Estimates for the Loading Criteria on
Vollenweider’s 1975 Axes of L Versus (From Reckhow,
1978d)
-------
-fl6-
.08
.06
.04
y
.02 5. 10.
y
.03
.02
.0t
0
15. 20. 25.
q s(m/yr)
50. tOO. 150.
q s(m/yr)
200.
Figure 4.7
y Versus q 5 (From Reckhow, 1978d)
-------
-117—
1.00
.90
.80
.70
.60
.50
.40
.30
.20
.10
0
0
C-)
I .-
U)
U)
0
C -)
C-)
0
0
0
-
0
0
Figure 4.8
yL:P
(predicted phosphorus concentration, in mgII)
The Probability of Oligotrophic Classification as a
Function of Phosphorus Loading, Model Uncertainty, and
Phosphorus Loading Uncertainty (From Reckhow, 1978d)
.005 .010 .015 .020 .025
.030
-------
1.00
.90
.80
0
C-,.
U.,
D
c-’ .60
C-,
0.
0
0•
In
a)
.40
.10
0
.035
yL:P
(predicted phosphorus concentration, in mg/I)
Figure 4.9 The Probability of Mesotrophic Classification as a Function
of Phosphorus Loading, Model Uncertainty, and Phosphorus
Loading Uncertainty (From Reckhow, 1978d)
.005 .010 .015 .020 .025 .030
-------
- Il9-
(.00
.90
.80
.70
.60
.50
.40
.30
.20
.10
0
0
0
C-,
U)
U,
0
(-)
C-,
0
w
0
0
0
a-
Figure 4.10
yL=P
(predicted phosphorus concentration, in mg/I)
The Probability of Eutrophic Classification as a Function
of Phosphorus Loading, Model Uncertainty, and Phosphorus
Loading Uncertainty (From Reckhow, l978d)
.005 .0(0 .015 .020 .025 .030
035
-------
-120-
a. when L is treated as a measured loading, as part of the
National Euthrophication Survey, there is no need for an
additional loading error term. Thus k = 0, and:
probability oligotrophic = .95
probability mesotrophic .05
probability eutrophic 0
b. when L is treated as a loading estimated from the litera-
ture , or a measured loading used for future prediction , the
estimated loading uncertainty must be considered in addi-
tion to the model uncertainty. Here, k = .50, and:
probability oligotrophic .80
probability mesotrophic .20
probability eutrophic 0
Note that the probabilities must sum to one.
This procedure is simple, yet, as stated above, it presents a
relatively realistic appraisal of the current state of knowledge in
trophic state assessment. Refinements to the procedure (discussed in
the Appendix), particularly in the area of loading uncertainty estima-
tion, are necessary and should further improve the usefulness of un-
certainty analysis.
4.3.5 Lake Quality Discriminant Analysis
Some of the phosphorus loading criteria discussed above rely on
subjective judgement for trophic state assessment, and thus probably
do not make “optimaP use of the information contained in the limnolog-
ical variables (L, z, q 5 , and t) for classifying lakes. In addition,
-------
-121-
not all of the above criteria are functionally dependent upon at least
two of the geomorphology/hydrology variables q 5 , z, and , and thus,
again, these criteria may not make best use of the information avail-
able.* In this section, two lake classification schemes are proposed
that represent an attempt to address those expressed concerns. The tech-
nique of discriminant analysis was used to construct an optimal classifi-
cation function. Discriminant analysis may be considered as analogous
to regression analysis (Chapter 2), except that the dependent variable
is discrete instead of continuous. For example, regression analysis
may be used to develop a predictive equation for phosphorus concentra-
tion, which can assume a continuous set of values, whereas discriminant
analysis may be used to develop a predictive equation for trophic state,
which, as defined here, can assume only three values (oligotrophic,
mesotrophic, and eutrophic). For the loading criteria described below,
discriminant analysis was used to develop a trophic state classification
scheme and a function, or graph, for classifying a lake as either oxic
or anoxic. Some details on the development of these criteria are pre-
sented in the Appendix.
Walker (1977) used discriminant analysis on a data set of about
100 north temperate lakes in order to develop a predictive function,
composed of L, q 5 , and , that best classified a lake into one of the
trophic classes. Many of his lakes were from the EPA National Eutrophi-
cation Survey, and he relied upon the subjective judgment of these in-
vestigators for the initial assignment of trophic states for his data.
*Thjs of course, depends upon whether average lake quality is
determined by two or more of these variables. Casual conclusions on
that statement cannot be made at present.
-------
-122-
This can lead to inconsistencies in the data, since different people
often have different definitions of the trophic state transition points.
Nonetheless, using the discriminant analysis technique, Walker developed
the graph in Figure 4.11, which may be used to predict the probability,
or likelihood, that a lake belongs in each of the three trophic states.
This probability is a function of the variable x, where:
x = L Eq 5 (1 + .824T .454)] 815 (4.7)
Note that x is quite similar in composition to Walker’s phosphorus
model, presented in Chapter 3.
Walker provides additional graphs that modify the probabilities
according to the uncertainty in the independent variables. However,
he notes that the technique used to derive Equation 4.7 incorporates
the uncertainties in the model-development data set into the probabil-
ities estimated using Figure 4.11. Thus, Figure 4.11 may be used with-
out additional uncertainty estimates for lakes for which the data
collection and assessment techniques are similar to those in Walker’s
model—development data set (primarily National Eutrophication Survey
iakes).* This is true for the Lake Charlevoix data, for example.
Thus, application of Walker’s loading criterion yields the following
predictions for Lake Charievoix (t = 3.2 yrs):**
*Note that this is the same criterion used for the error analysis
discussed in Chapter 3 and Section 4.3.4.
**The major reason for the difference between the probabilities
estimated by Walker’S criterion and those estimated by Reckhow’S un-
certainty analysis approach in section 4.3.4 is expressed in the differ-
ence in phosphorus concentration predictions (.0067 mg/i for Reckhow’s
approach and .0096 mg/i for Walker’s model in Chapter 3). Since the
oligotrophic/meSOtrOPhic transition point is often expressed as a phos-
phorus concentration of .010 mg/l, it is understandable that a slight
difference in predicted phosphorus concentration at those levels can
lead to a large difference in trophic state probabilities.
-------
i.0
F—
08
C -,
U-
(I)
0.6
0.4
0.2
0.0
Figure 4.11
LOG 10 [ X L [
l+.824t 454 )]_.815]
Walker’s Loading Criterion Assessing the Probabilities of Trophic Classification
(From Walker, 1977)
U-
0
>-
I—
-J
aD
aD
0
a:
-J
N)
—2.4 -2.2 -2.0 —1.8 -1.6 —1.4 —1.2 —1.0 —0.8 -0.6
-------
-124-
log x = - 1.82
probability oligotrophic .67
probability mesotrophic .33
probability eutrophic 0
Reckhow (1978a) also used discriminant analysis to construct phos-
phorus loading criteria, but in addition, he attempted to remove some
subjectivity from the classification procedure by basing his technique
on well—defined dissolved oxygen states. Specifically, lakes were
classified according to oxic or anoxic conditions in the hypolimnion,
such that a single anoxic measurement resulted in an anoxic classifica-
tion. The oxic—anoxic transition was selected because it is well-rec-
ognized as a water quality transition point (in part because it is
associated with phosphorus adsorption or release from the lake sediments).
The resultant loading criterion was based on 55 north temperate lakes
with:
z > 3.0 m
T > 0.25 yr
1 .0 rn/yr < z/t < 50.0 rn/yr
This oxic-anoxic discriminant function may be expressed or applied
in two ways. A lake may be plotted on the graph in Figure 4.12, which
is a plot of the numerator versus the denominator of the discrirninant func-
tion (d.f.) given below in Equation 4.8.
d f — lO 468 LL 95 4 8
— 2.58 1.85
z
-------
-125—
2.8 3.6 4.4 5.2 6.0 6.8 7.6
Iog [ z/ L85(z)2. 5 8]
AOXIC Ling/m 2 —yr
B POSSIBLY ANOXIC, BUT UNCERTAIN z in in
C = AtJOXIC z/tin rn/yr
I I I
—I— -— -
A
C A
F-
U,
0 i
(0
j.
0
C
C
C
C
B CC C C
C
C B C
AC
CCC
B
66
6.0
5,4
4.8
4.2
3.6
3.0
2.4
1.8
A
A
A
A
A A
A A
A A
A
A
A
A
A
A
I I I I I I I I I I I
Figure 4.12 The Oxic—Anoxic Discriminant Function (From Reckhow, 1978a)
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-126-
The lakes that were used to develop this discriminant function are plotted
in Figure 4.12. Six of those lakes could not confidently be classified
as either oxic or anoxic, and thus they help to define a transition
zone” between d.f. = .20 and d.f. .60. Outside of this transition
zone, the likelihood of anoxic, or oxic, conditions increases with dis-
tance from the transition zone. Expressed in another way, a value for
d.f. of less than .20 implies oxic conditions, while a value for d.f.
of more than .60 suggests anoxic conditions.
Another way the oxic-anoxic loading criterion may be expressed is
in a probability statement like Walker’s criterion. In this case, the
probability that a lake will have oxic conditions in the hypolirunion
is:
p 1
OX C z 2 49 L 2 °° q 78 ÷ 1
The anoxic probability is, of course, one minus the oxic probability.
As with Walker’s criterion, Equation 4.9 has incorporated within it
the uncertainty in the independent variables from the model-development
data set (primarily National Eutrophication Survey data). Thus, calcu-
lation of the oxic-anoxic probabilities with that level of uncertainty
does not require additional analysis beyond Equation 4.9, if the data
for the application lake were gathered and reduced in a manner similar
to that for the National Eutrophication Survey data. Since this is
the case for Lake Charlevoix (z 16.8 m), the probability of oxic con-
ditions, in consideration of uncertainty in L, q 5 , and z, is:
-------
—127—
p 1
oxic - loS(l 6 . 8 r 2 . 49 (.l 2 ) 2 .oo( 5 . 24 rL 78 + 1
p = • 94
oxi C
anoxic = .06
The oxic probabilities were assessed, using Equation 4.9, for
Lakes Washington and Sammamish before and after nutrient diversion
projects. After these diversions, noticeable improvements in water
quality were observed in Lake Washington but not in Lake Sammarnish.
Figure 4.13 is a plot of the oxic probabilities, and it represents
evidence supporting one reason why different results developed from
the diversion projects. While the post-diversion phosphorus loading
resulted in a high likelihood of oxic conditions in Lake Washington,
the likelihood of oxic conditions associated with the post-diversion
phosphorus loading in Lake Saniiiamish was only about 0.5. This implies
that phosphorus loading must be reduced more, to substantially de-
crease the likelihood of anoxic conditions and improve water quality.
Thus, Equation 4.9 may be used to assess the impact of changes in
phosphorus loading on water quality.
4.4 Sumary
A nunter of trophic state classification schemes have been evalu-
ated, and it was found that most approaches convey essentially the
same information. The selection of the best or most appropriate
index, for a given application, probably should be a function of
familiarity, perceived credibility, ease of understanding, and robust-
-------
Lake Sammamish and Lake Washington:
Annual Area] Phosphorus Loading (L).
(.2
L(g/m 2 —yr)
Probability of Oxic Conditions (P ) vs.
(From Reckhow, 1978a) OXIC
Post—diversion
Lake Scinmomish
— — — Lake Washing/on
(.00
90
.80
.70
.60
.50
.40
.302
Pre —di version
0
0
Figure 4.13
Post—diversion
‘S . . ,’
“S . ,.
—a
t\)
.4 .6 .8 1,0
Pre —diversion
‘.4
(.6
1.8
2.0 2.2
-------
-129-
ness under unusual limnological conditions. The index proposed by
Carison is suggested here as the best choice on the basis of the latter
two points.
Trophic state indicators have limited value in lake watershed man-
agement and planning, however, since they evaluate only current con-
ditions. Nutrient, particularly phosphorus, loading criteria, are re-
quired for that purpose. Therefore, several of the early loading
criteria were evaluated, and while these criteria are useful and often
successful, they can be improved. Improvements that were suggested,
and that have led to the development of new loading criteria, deal with
optirnality, objectivity, and uncertainty. In particular, recent pro-
posals by Reckhow and Walker have focused on consideration of these
concepts in the construction of new phosphorus loading criteria. These
new criteria should generally convey more reliable information on the
effect of phosphorus loading than earlier criteria. Refinements,
particularly in the area of uncertainty analysis, should further im-
prove the usefulness of these approaches.
As a final note, it should be stated that the loading criteria
discussed here are no more or no less accurate than the phosphorus
models discussed in Chapter 3. In fact, as pointed out in this chap-
ter, many of the loading criteria are simply graphical representations
of a phosphorus model. They are expressed in graphical form for con-
venience and for display purposes, but they convey no more information
than the phosphorus models. Thus the same warning given in Chaper 3
is applicable here; apply the loading criteria with caution and a good
understanding of the documented uses, and limitations, of each technique.
-------
APPENDIX
DERIVATION AND EXPLANATION OF THE PROCEDURE USED TO APPLY
UNCERTAINTY ANALYSIS TO VOLLENWEIDER’S LOADING CRITERION
In order to estimate the uncertainty associated with the applica-
tion of phosphorus loading criteria on Vollenweider’s criterion axes
(L vs. q 5 ), a model is needed that provides the equation of the trophic
state transition lines. It was assumed here that trophic states are
determined by the total phosphorus concentration in a lake. Further,
it was assumed that the transition points are those proposed by the
EPA (1974) in Table 4.2. These are:
Total Phosphorus
Trophic State Concentration (mg/i )
oligotrophic < .010
mesotrophic .010 - .020
eutrophic .O2O
A model was developed, for north temperate lakes, to predict the
phosphorus concentration at the transition points. The uncertainty
associated with this model, which is one component of the total pre-
diction uncertainty, was then used to construct error bars (expressed
as a function of uncertainty in L) on the loading criterion, Figure 4.6.
This is intended to graphically illustrate the degree of uncertainty
in the model that specifies where, on this graph of L versus q 5 , the
-130-
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-131—
trophic state transition lines, associated with phosphorus concentra-
tions of .010 mg/i and .020 mg/i, must lie.
The model-development data set lakes are characterized by:
Variable Minimum Maximum
P .004 mg/i .135 mg/i
L .07 g/ni 2 -yr 31.4 g/rn 2 -yr
1.23 rn/yr 187. rn/yr
The lakes are assumed to be completely mixed, and the phosphorus con-
centration represents an “average” value (no distinction among spring,
growing-season, or annual averages).
The model used to predict the position of the trophic state transi-
tion lines is of the form:
v 5 q (A.l)
where
V 5 = apparent settling velocity (m/yr).
Using weighted least squares regression, v was found to be a linear
function of
v 5 = 11.6 + 0.2 q 5 (A.2)
Therefore, the phosphorus model becomes:
P 116 12 qYL (A.3)
-------
-132—
where
y = 11(11.6 + 1.2 q 5 ) (A.4)
The model standard error, was estimated from:
Sm = (A.5)
where
ob = observed phosphorus concentration (mg/i)
Pe = estimated (predicted) phosphorus concentration (mg/i)
n = 47 lakes
For the log-transformed model, Sm = .128. In addition, the parameter
error contribution to the total uncertainty was found to be negligible
in comparison to s , and thus it was dropped from further consideration.
Therefore, the error lines in Figure 4,6 represent the standard
error of the model and were estimated in the following way. The
equation for the trophic state transition lines is:*
L (11.6 + 1.2 q 5 ) (P) (A.6)
where
P = .010 mg/l, .020 mg/i.
The equation for the error lines then becomes* (after modifications,
necessary because the model was estimated in the log-transformed state):
*An approximation, since the regression of x on y does not equal
the regression of y on x.
-------
-133-
L = (11.6 + 1.2 q) ( 10 1og P + Sm) (A.7)
where
P = .010 mg/i, .020 mg/i
Sm = .128
Figures 4.8, 4.9, and 4.10 were constructed by summing the squared
values of the uncertainty estimates for:
1. the model (the standard error of the estimate), 5 m•
2. the total uncertainty component from the fractional uncertainty
in phosphorus loading, SL.
The independent variable (L q ) uncertainty was assumed to be a func-
tion of loading uncertainty alone. Therefore, the total prediction un-
certainty was estimated as:
s 1 2 = S 2 + (A.8)
The curves in Figures 4.8 - 4.10 were calculated, assuming normally-
distributed errors, using the standard normal deviate, z :
p _p
= t.s. e (A.9)
n
where
= .010 mg/l, .020 mg/l; the phosphorus concentrations
associated with the trophic state transitions.
= estimated phosphorus concentration, using Equation A.3.
-------
-134-
Note that the probabilities were estimated by assuming that the model
error is associated with e’ not (i.e., the transition lines are
“tru&’ values, with no uncertainty, while the estimated lake concen-
tration has model and loading uncertainty associated with it).
In summary, two comments must be made on the limitations on the
use of this procedure:
1. Application lakes should have characteristics not unlike those
of the model development data set discussed earlier.
2. This must be considered an introduction to the use of un-
certainty analysis in lake quality assessment. Since it is
a “first-cut,” there are no established procedures for esti-
mating loading uncertainty, and there is even confusion as to the
meaning of loading uncertainty. As expressed here, the loading un-
certainty is determined such that almost 70% of the time that the load-
ing uncertainty is estimated, the “true” loading (under steady-
state conditions) will lie within the estimated loading +
the loading uncertainty. Clearly, this is difficult to esti-
mate, given natural fluctuations, sampling limitations, human
error, and human inexperience.* Thus, more work and experi-
ence are needed before a successful uncertainty analysis
approach is developed that has the same meaning for both
the model developer and the model user.
*An additional problem develops when the phosphorus loading un-
certainty nears + 100% and is assumed to be normally distributed. When
this occurs, there is a nonzero probability that the lake phosphorus
concentration will be less than zero. For this reason, some of the
lines in Figures 4.8 - 4.10 are shorter (horizontally) than others, and
for this reason more work is needed on the distributional aspects of
the loading uncertainty.
-------
LAKE QUALITY DISCRIMINANT FUNCTIONS
Discriminant analysis is a useful procedure for constructing a
predictive relationship for a dependent variable expressed in a small
number of discrete states. A heuristic analogy was drawn in Section
4.3.5 between discriminant analysis and regression analysis. That is,
regression analysis is most often used to develop predictive equations
for a continuous dependent variable (like phosphorus concentration),
whereas discriminant analysis may be used to develop predictive rela-
tions for a discretized dependent variable (like trophic state).
For the oxic-anoxic discriminant function, the following discrim—
inant function was presented:
4.68 1.95
d.f. 2 . 58 qL 1 . 85 (A.lO)
This function is actually the first principal component, and it maxi-
mizes the explained variance or discrimination between classes. Since
it is a single component, it maximizes the group separation on a
single axis.
In order to be consistent with other graphical classification
criteria, a plot of this function is made, in Figure 4.12, with the
log of the numerator on the y-axis and the log of the denominator on
the x—axis. A transition zone is defined, with the help of the ‘un-
certain” lakes, to convey the notion that the transition from oxic
to anoxic conditions (or vice versa) cannot be specified with certainty.
-135-
-------
-136-
The same information found in the discriminant function may be
re-expressed in terms of classification functions which lead to clas-
sification probabilities. The classification function, cf.,, is con-
verted to a classification probability estimate using:
p 1 (A.ll)
oxic + i
with
anoxic = - poxic (A. 12)
When the classification function is substituted into Equation A.ll,
the oxic probability becomes:
poxic io 5 z 249 L 2 °° q -1.78 + (A.13)
-------
CUAPTER 5
FINAL COMMENTS
At this point, it seems appropriate to offer some final observa-
tions on the application and development of quantitative techniques
like those discussed in this manual. First, some thoughts on applica-
tions are presented below:
1. The estimation of data uncertainty and distribution variance
probably creates uneasiness in the minds of many users. None-
theless, the importance of these estimates should be clear
from the discussion in this manual. Further, as more investi-
gators think about these terms, develop appropriate estimates,
and present them in the literature, this task should become
easier.
2. The models need to be considered within the broad context of
lake management. That is, the models merely provide informa-
tion on the impact of watershed characteristics and activities
on lake quality. This information is just one input to the
analysis of lake management strategies. We need to know, in
addition, the costs and benefits associated with suggested
management strategies for the lake of study. Further, with
explicit consideration of uncertainty, we should try to assess
the risk of overestimation and underestimation of water quality
—137-
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-138-
goals. It is important, therefore, not to rely too heavily on
the lake models and projected lake trophic states for the de-
termination of policy, since policy is not simply a technical
I s sue.
Finally, the list below presents proposed research to improve the
usefulness of simple lake models coupled with uncertainty analysis:
1. A better procedure is needed for the treatment of phosphorus
loading error. One approach would involve the removal of
the loading error contribution, attributed to the averageu
loading error in the model development data set, from the
model standard error.* Then, all applications of this approach
would require this revised model standard error and an esti-
mate of the loading error for the application lake.
2. Another way to eliminate the problem of phosphorus loading
error would be to develop a model that bypasses phosphorus
loading. That is, phosphorus loading is actually an ‘9nter—
vening variable t ’ between the decision variable, phosphorus
concentration, and the control variables, watershed character-
istics and activities. When enough data are available on
watershed land use for several lakes, modeling work should
probably be undertaken to directly relate the watershed to
the lake quality. If these models can be successfully devel-
oped, they should be easier to apply since data may be avail-
*Thjs would also necessitate re-estimation of the model coefficients
to correct any bias attributed to the variable error.
-------
-l 39—
able from land use maps, and since independent variable error
could be relatively small.
3. This author is currently involved in research on the follow-
ing topics in lake modeling:
a. A procedure is being developed (Chapra and Reckhow, 1978)
that expresses Vollenweider’s 1976 trophic state criterion
in probabilistic terms.
b. Work is underway (Reckhow, 1978e) to develop a model that
relates nutrient loading, lake hydrology, and lake geo-
morphology to the probability of a lake water quality
standard violation. Many lake water quality standards are
not expressed in terms of the traditional model dependent
variables, like nutrient concentrations. Therefore, this
research defines a more common water quality standard
variable, minimum dissolved oxygen level, as the dependent
variable. Thus, the proposed model will predict the likeli-
hood of a standard violation for dissolved oxygen, as a
function of the independent variables listed above.
-------
REFERENCES
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12. Dillon, P. J. “The Phosphorus Budget of Cameron Lake, Ontario:
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14. Jones, J. R. and Bachmann, R. W. “Prediction of Phosphorus and
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16. Kirchner, W. B. and Dillon, P. J. “An Empirical Method of Esti-
mating the Retention of Phosphorus in Lakes.” Water Resources
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17. Larsen, D. P. and Mercier, H. T. “Lake Phosphorus Loading Graphs:
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18. Lee, David R. “A Device for Measuring Seepage Flux in Lakes and
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19. Marsalek, J. “Sampling Techniques in Urban Runoff Quality Studies.”
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20. McElroy, A. D.; Chiu, S. Y.; Nebgen, J. W.; Aleti, A.; and Bennett,
F. W. Loading Functions for Assessment of Water Pollution
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22. Omerriik, J. M. Nonpoint Source - Stream Nutrient Level Relation-
ships: A Nationwide Study . U.S.E.P.A. Ecological Research
Series, EPA—600/3-77-l05, Corvallis, Oregon, 1977.
23. _____. The Influence of Lake Use on Stream Nutrient Levels . U.S.
Environmental Protection Agency, EPA-600/3-76-01 4, Corvallis,
Oregon, 1976.
24. Piontelli, R. and Tonolli, V. “The Time of Retention of Lacustrine
Waters in Relation to the Phenomena of Enrichment in Intro-
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25. Rawson, D. S. “Morphometry as a Dominant Factor in the Productivity
of Large Lakes.” Verh. Internat. Verein. Limnol . 12 (1955):
164—175.
26. Reckhow, Kenneth H. “Phosphorus Models for Lake Management.” Ph.D.
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27. _____. “Lake Quality Discriminant Analysis.” Water Resources
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cations, Limitations, and Uncertainty.” To be published in
Perspectives on Aquatic Ecosystem Modeling , Ann Arbor, Michigan:
Ann Arbor Science, 1978b.
29. _____. “Lake Nutrient Budget Sampling Design.” Paper presented
at the 1978 American Water Resources Symposium on water Quality
Monitoring Programs (to be published in a Symposium Proceed-
ings), 1978c.
30. _____. “Uncertainty Analysis Applied to Vollenweider’s Phosphorus
Loading Criterion - A Graphical Approach.” Journ. Water Poll.
Control Fed . (in press) 1978d.
31. _____. “A Simple Model to Assess the Likelihood of a Lake Water
Quality Standard Violation: l978e. (Unpublished manuscript.)
32. Reckhow, K. H. and Harbert Rice. 208 Modeling Approach . Report
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33. _____. Nutrient Budget and Error Analysis for Lake Winnipesaukee
and Allied Lakes . Prepared for the New Hampshire Lakes Region
Planning Commission, Meredith, New Hampshire, 1976.
34. _____. Water Quality Modeling for the Lakes Region 208 Project .
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35. Sakamoto, M. “Primary Production by Phytoplankton Community in
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A Multivariate Approach.” Journ. San Eng’g. Div. ASCE 98, No.
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38. Snedecor, G. W. and Cochran, W. 6. Statistical Methods . Ames, Iowa:
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39. Snow, Phillip D. and DiGiano, Francis A. Mathematical Modeling of
Phosphorus Exchange Between Sediments and Overlying Water in
Shallow Eutrophic Lakes . Report No. Env. E. 54-76-3.
Amherst, Mass.: Dept. of Civil Engineering, University of
Massachusetts, 1976.
40. U.S.E.P.A. “The Relationships of Phosphorus and Nitrogen to the
Trophic State of Northeast and North-Central Lakes and
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41. U.S.E.P.A. “Report on Lake Charlevoix.” National Eutrophication
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Environmental Protection Agency, EPA-660/3-74—020, Washington,
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44. Vollenweider, Richard A. “Advances in Defining Critical Loading
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STATISTICAL SYMBOLS
Syntol Definition
x sample mean
s sample standard deviation
‘pred. predicted Y value
‘obs. observed Y value
d.f. degrees of freedom
t student’s t
n nuither of samples
x 0 a particular value of x
the mean for group i
the nunter of samples in group i
sample of variance
d precision, or uncertainty, of sampling design
S - standard error of the mean
r correlation coefficient
R 2 coefficient of determination
Sm standard error of the estimate
C(n) cost of sampling
C 0 fixed cost of sampling
C. cost of sampling stratum i
c.v. coefficient of variation
sampling design weighting factor for stratum i
-144-
-------
SYMBOLS FOR LAKE MODELS
Syntol Definition Units
P lake phosphorus concentration mg/i
V lake volume 10 6 m 3
M annual mass rate of phosphorus inflow to
lake 10 kg/yr
L annual areal phosphorus loading g/m 2 -yr
z lake mean depth rn
T hydraulic detention time yr
sedimentation coefficient yr
Q annual volume rate of water inflow to lake l0 6 rn 3 /yr
P 0 average lake phosphorus concentration to
time t = 0 mg/i
A lake surface (bottom) area km 2
v 5 apparent settling velocity rn/yr
surface overflow rate rn/yr
R fraction of influent phosphorus retained
p inalake
P 0 average outflow phosphorus concentration mg/i
average influent phosphorus concentration mg/i
M 1 annual mass influx of phosphorus
annual mass effiux of phosphorus
annual net flux of phosphorus to the sediments
RKD Kirchner and Dillon empirical estimate for R
RLM Larsen and Mercier empirical estimate for R
- 145-
-------
-146-
Synbol Definition Units
Sf total prediction error mg/i
sj . , input variable error (untransformed) contribu-
tion to total error mg/i
ST total error (logarithmic form)
s parameter error contribution to the total
p error (log form)
Sm standard error of the estimate for the model
(log form)
s input variable error (log form) contribution to
total error
predicted phosphorus concentration mg/i
1 ’true the “true” total phosphorus concentration mg/i
k a constant
L critical phosphorus loading g/m 2 -yr
PC critical phosphorus concentration mg/i
ob observed phosphorus concentration mg/i
log base 10 logarithm
in base e logarithm
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