Comparisons Of Models Predicting Ambient Lake Phosphorus Concentrations Working Paper No. 704 September 1978


United States	Environmental Monitoring
Environmental Protection	and Support Laboratory
Agency	P.O. Box 15027
Las Vegas NV 89114
Research and Development	
EPA Comparisons of Working
Models Predicting Paper 704
Ambient Lake
Phosphorus
Concentrations

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COMPARISONS OF MODELS PREDICTING
AMBIENT LAKE PHOSPHORUS CONCENTRATIONS
WORKING PAPER NO. 704

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COMPARISONS OF MODELS PREDICTING
AMBIENT LAKE PHOSPHORUS CONCENTRATIONS
by
Stephen C. Hern, Victor W. Lambou
and Llewellyn R. Williams
Environmental Monitoring and Support Laboratory
Las Vegas, Nevada 89114
Working Paper No. 704
NATIONAL EUTROPHICATION SURVEY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
September 1978

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FOREWORD
The National Eutrophication Survey was initiated in 1972 in response to
an Administration commitment to investigate the nationwide threat of acceler-
ated eutrophication to freshwater lakes and reservoirs. The Survey was
designed to develop, in conjunction with State environmental agencies, infor-
mation on nutrient sources, concentrations, and impact on selected freshwater
lakes as a basis for formulating comprehensive and coordinated national,
regional, and State management practices relating to point source discharge
reduction and nonpoint source pollution abatement in lake watersheds.
The Survey collected physical, chemical, and biological data from 815
lakes and reservoirs throughout the contiguous United States. To date, the
Survey has yielded more than two million data points. In-depth analyses are
being made to advance the rationale and data base for refinement of nutrient
water quality criteria for the Nation's freshwater lakes.
ii

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ABSTRACT
The Vollenweider, Dillon, arid Larsen/Mercier models for predicting ambient
lake phosphorus concentrations and classifying lakes by trophic state are com-
pared in this report. The Dillon and Larsen/Mercier models gave comparable
results in ranking 39 lakes relative to known ambient phosphorus concentrations.
The Vollenweider model, which does not include a phosphorus retention capacity
component, was unable to achieve the high rank correlations found with the
other models.
Trophic state predictions from the phosphorus loading models are compared
with National Eutrophication Survey lake report designations. Disagreements
of 14, 18, and 25 percent, respectively, were found with the Dillon, Larsen/
Mercier, and Vollenweider concepts.
iii

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COMPARISONS OF MODELS PREDICTING
AMBIENT LAKE PHOSPHORUS CONCENTRATIONS
The phosphorus loading - mean depth - relationship formulated by Vollenweider
(1968) has been widely accepted and used to indicate the degree of eutrophy of
lakes and evaluate the level of phosphorus loading to lakes. Dillon (1975)
pointed out that there has been too little thought and criticism given to the
limitations of the model by people using it. Subsequently, modifications
of the basic mass balance equation have been derived to predict mean ambient
lake phosphorus concentrations at equilibrium. Dillon (1975) utilizes phos-
phorus areal loading (L), the retention coefficient for phosphorus (R), the
hydraulic flushing rate (P), and mean depth (Z) in a plot of the form
L (1-R)
p	versus Z
to estimate trophic state. Vollenweider (1975) revised his original
formula to include T , hydraulic residence time, so that areal phosphorus
loading (L) is plotted against mean depth (Z) divided by T . Larsen and
Mercier (1976) provide an alternative (to the prior loading concepts) which
avoids the criticism of Edmondson (1970) that the effect of an increasing
phosphorus load upon a lake depends, in part, upon whether that increase re-
sults from increases in influent flows, concentrations, or both. The Larsen/
Mercier formula plots mean tributary phosphorus concentration against
phosphorus retention coefficient, called R experimental, computed in the same
way as Dillon's R, i.e.,
r = *1 _ Total phosphorus leaving
Total phosphorus entering
It is interesting to note that in applying the above-mentioned formulas,
each of the authors have selected to use levels of 10 and 20 yg/1 of ambient
lake phosphorus to divide lakes into the three standard trophic classi-
fications -- oligotrophic, mesotrophic, and eutrophic.
To compare the three models, we selected 39 lakes sampled during 1973
by the National Eutrophication Survey (U.S. Environmental Protection Agency,
1975) which represented the entire range of water transparency as measured
by Secchi disk. Table 1 demonstrates the variety of lakes selected.
1

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TABLE 1. THE NUMERICAL AVERAGE AND RANGE OF MEAN CHEMICAL, PHYSICAL AND
BIOLOGICAL CHARACTERISTICS OF 39 LAKES, SELECTED FROM THOSE
SAMPLED DURING 1973 BY THE NATIONAL EUTROPHICATION SURVEY
PARAMETER
MEAN
RANGE

Surface Area (km2)
40.75
0.23
263.05
Drainage Area (km2)
3502.6
4.3
38850.0
Mean Depth (m)
7.5
0.9
21.0
Maximum Depth (m)
21.6
1.5
57.8
Volume (m3 x 106)
379.28
0.45
2608.00
Hydraulic Retention Time (days)
251
1
2446
Secchi Disk (cm)
177.8
15.2
563.9
Total Phosphorus (pg/liter)
147
5
1120
Chlorophyll a^ (yig/liter)
53.9
1.4
456.6
The data used in this report are from the various individual National
Eutrophication Survey (NES) lake reports for the lakes listed in Table 2,
e.g., Report on Lake Lulu (EPA, 1976). Similarly, NES lake reports provide
data for Lake Mead (EPA, 1977a) and Flaming Gorge Reservoir (EPA, 1977b)
included as examples of the application of the formulas.
Solution analyses based on 20 pg/1 were employed to place the three
formulas on an equivalent basis by dividing the appropriate theoretical minimum
eutrophic "loading" rate for a given lake (i.e., that which would produce an
ambient lake concentration of 20 yg/1) into the actual "loading" rate
determined for that lake. Hereafter, this ratio will be referred to as the
"trophic ratio". Trophic ratios which exceed or equal 1.0 represent eutrophic
loadings, whereas trophic ratios extending from 0.5 to less than 1.0 represent
mesotrophic loadings and trophic ratios below 0.5 represent oligotrophic
loadings, regardless of the formula employed.
Table 2 lists the 39 lakes used in this study, ranked in descending
order by total phosphorus concentration, and gives the mean total phosphorus
concentration and mean Secchi disk value for each lake. In addition, the
trophic states given in the individual NES lake reports are listed along with
the trophic ratios calculated for the Larsen/Mercier, Dillon, and Vollenweider
models and the trophic states predicted by the ratios. The trophic states
indicated in the NES reports were based largely upon lake mean total
phosphorus, chlorophyll a_, Secchi depth, hypolimnetic dissolved oxygen values
and phytoplankton data (Allum et al., 1977).
Spearman rank correlation coefficients (rs) were calculated for each
trophic ratio against measured mean ambient phosphorus concentrations with
the following results: Larsen/Mercier (.94), Dillon (.92), and Vollenweider
(.82). The Larsen/Mercier model provided the best estimation of the
relative rank of the lakes based on ambient phosphorus concentrations. It
was followed closely by the Dillon model. Both of these models take into
consideration the phosphorus retention capacity of lakes. These models are
2

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TABLE 2. COMPARISON OF LAKE AMBIENT PHOSPHORUS PREDICTION MODELS AND TROPHIC CLASSIFICATIONS WITH
ACTUAL AMBIENT LAKE CONDITIONS. THE 39 LAKES COMPARED ARE RANKED IN DESCENDING ORDER
BY MEAN SUMMER AMBIENT PHOSPHORUS CONCENTRATION



MEAN








MEAN
AMBIENT








AMBIENT
SECCHI
NES
LARSEN/KERCIER
DILLON
V0LLENWE1DER

LAKE NAME
TOTAL-P
DEPTH
TROPHIC
TROPHIC TROPHIC
TROPHIC
TROPHIC
TROPHIC
TROPHIC
RANK
(STATE)
(pq/1)
(cm)
STATE*
RATIO* STATE*
RATIO*
STATE*
RATIO*
STATE*
1
Lake Lulu









(Fla.)
1120
23
E
259.90 - E
A

75.62
- E
2
Sloeurn Lake









(111.)
882
20
E
61.77 - E
74.00
- E
29.42
- E
3
Lake Hancock









(Fla.)
608
30
E
79.90 - E
A

A

4
A11igator Lake









(Fla.)
429
58
E
38.92 - E
A

A

5
Fox Lake









(in.)
322
23
E
10.81 - E
3.40
- E
12.66
- E
6
Highland (Silver) Lake









(111.)
258
20
E
14.61 - E
15.38
- E
5.43
- E
7
Horseshoe Lake









(111.)
256
25
E
11.81 - E
12.00
- E
1.96
- E
8
Kill en Pond









(Del.)
216
66
E
5.58 - E
5.67
- E
9.25
- E
9
Lake Loramie









(Ohio)
204
15
E
18.57 - E
18.67
- E
5.02
- E
10
Crab Orchard Lake









(111.)
184
58
E
7.40 - E
7.33
- E
7.42
- E
11
Duhernal Lake









(N.J.)
179
61
E
3.17 - E
3.33
- E
18.92
- E
* E = eutrophic, M = mesotrophic, 0 = oligotrophia and a = insufficient data to make estimate of
trophic ratio.
(Continued)

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TABLE 2. COMPARISON OF LAKE AMBIENT PHOSPHORUS PREDICTION MODELS AND TROPHIC CLASSIFICATIONS WITH
ACTUAL AMBIENT LAKE CONDITIONS. THE 39 LAKES COMPARED ARE RANKED IN DESCENDING ORDER
BY MEAN SUMMER AMBIENT PHOSPHORUS CONCENTRATION (Continued)



MEAN









MEAN
AMBIENT









AMBIENT
SECCHI
NES
LARSEN/MERCIER
DILLON
VOLLENWEIDER

LAKE NAME
TOTAL -P
DEPTH
TROPHIC
TROPHIC
TROPHIC
TROPHIC
TROPHIC
TROPHIC
TROPHIC
RANK
(STATE)
(wq/D
(cm)
STATE*
RATIO*
STATE*
RATIO*
STATE*
RATIO*
STATE *
12
Lake Charleston










(111.)
164
23
E
8.57
- E
7.50 -
E
15.91
- E
13
Lake Apopka










(Fla.)
161
29
E
19.17
- E
22.00 -
E
3.94
- E
14
Marsh Lake










(Ind.)
115
127
E
5.80
- E
5.92 -
E
4.54
- E
15
Saluda Lake










(S.C.)
73
68
M
1 .84
- E
1.88 -
E
5.07
- E
16
Arkabutla Reservoir










(Miss.)
58
64
E
11 .28
- E
11.39 -
E
2.58
- E
17
Barren River Reservoir










(Ky.)
49
123
E
3.35
- E
2.32 -
E
2.10
- E
18
Lake Chesdin










(Va.)
40
120
E
2.20
- E
2.21 -
E
2.24
- E
19
Lay Lake










(Ala.)
39
104
E
4.84
- E
4.20 -
E
5.72
- E
20
Cherokee Lake










(Tenn.)
37
141
E
2.28
- E
2.43 -
E
3.61
- E
21
Hickory Lake










(N.C.)
34
114
E
2.04
- E
1 .74 -
E
3.13
- E
22
Walter F. George










Reservoir (Ga.)
30
no
E
3.12
- E
3.25 -
E
3.44
- E
* E = eutrophic, M = mesotrophic, 0 = oligotrophic and a = insufficient data to make estimate of
trophic ratio.
(Continued)

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TABLE 2. COMPARISON OF LAKE AMBIENT PHOSPHORUS PREDICTION MODELS AND TROPHIC CLASSIFICATIONS WITH
ACTUAL AMBIENT LAKE CONDITIONS. THE 39 LAKES COMPARED ARE RANKED IN DESCENDING ORDER
BY MEAN SUMMER AMBIENT PHOSPHORUS CONCENTRATION (Continued)
RANK
LAKE NAME
(STATE)
MEAN
AMBIENT
TOTAL-P
(uq/1)
MEAN
AMBIENT
SECCHI
DEPTH
(cm)
NES
TROPHIC
STATE *
LARSEN/MERCIER
TROPHIC TROPHIC
RATIO* STATE*
DILLON
TROPHIC TROPHIC
RATIO* STATE*
VOLLENWEIDER
TROPHIC TROPHIC
RATIO* STATE*
23
Moultrie Lake









(s.c.)
25
134
E
1 .51
- E
1.55 -
E
1.70 - E
24
Lake Hopatcong









(N.J.)
25
231
E
1 .32
- E
1 .00 -
E
0.34 - 0
25
Tims Ford Reservoir









(Tenn.)
25
240
E
2.22
- E
2.70 -
E
1.11 - E
26
Lake Minnehaha









(Fla.)
22
140
E
2.11
- E
A

A
27
Murray Lake









(S.C.)
20
218
E
1 .56
- E
1.56 -
E
1.49 - E
28
Chatuge Lake









(Ga.)
17
310
M
1 .06
- E
1.10 -
E
0.56 - M
29
Liberty Reservoir









(Md.)
15
381
M
0.71
- M
0.76 -
M
1.80 - E
30
Wanaque Reservoir









(N.J.)
14
467
M
1 .31
- E
0.96 -
M
0.48 - 0
31
Maxinkuckee Lake









(Ind.)
14
221
M
1.20
- E
1.33 -
E
0.75 - M
32
Dale Hollow Reservoir









(Ky.)
13
318
M
0.49
- 0
0.47 -
0
0.49 - 0
* E
= eutrophic, M = mesotrophic, 0 =
trophic ratio.
oligotrophic and
A = insufficient
data to
make
estimate of
(Conti nued)

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TABLE 2. COMPARISON OF LAKE AMBIENT PHOSPHORUS PREDICTION MODELS AND TROPHIC CLASSIFICATIONS WITH
ACTUAL AMBIENT LAKE CONDITIONS. THE 39 LAKES COMPARED ARE RANKED IN DESCENDING ORDER
BY MEAN SUMMER AMBIENT PHOSPHORUS CONCENTRATION (Continued)
LAKE NAME
(STATE)
RANK
33	Lake Wallenpaupack
(Penn.)
34	Martin Lake
(Ala.)
35	John W. Flannagan
Reservoir (Va.)
36	Deep Creek Lake
(Md.)
37	Summersville
' (W.Va.)
38	Harveys Lake
(Pervn.)
39	Tygart Reservoir
(W.Va.)
MEAN
AMBIENT
TOTAL-P
(wq/1)
MEAN
AMBIENT
SECCHI
DEPTH
(cm)
NES
TROPHIC
STATE*
DILLON
VOLLENWEIDER
LARSEN/MERCIER 		 	
TROPHIC TROPHIC TROPHIC TROPHIC TROPHIC TROPHIC
RATIO* STATE* RATIO* STATE* RATIO* STATE*
13
434
M
1.10
- E
1.12 -
E
0.50
- M
13
230
M
1 .04
- E
0.96 -
M
1 .35
- E
11
366
M
0.66
- M
0.69 -
M
1 .92
- E
11
366
M
0.39
- 0
0.38 -
0
0.34
- 0
10
549
M
0.83
- M
0.83 -
M
1 .28
- E
9
564
M
0.55
- M
0.59 -
M
0.54
- M
5
320
M
0.95
- M
0.94 -
M
2.02
- E
* E = eutrophic, M = mesotrophic, 0 = oligotrophic and a = insufficient data to make estimate of
trophic ratio.

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very similar, as areal phosphorus loading (L) divided by mean depth (Z)
approximates mean tributary concentration. The major difference between the
models is the flushing rate (P) employed by Dillon.
The Vollenweider model, which does not contain a phosphorus retention
capacity element, produced the poorest estimate of the ambient phosphorus
concentration.
Comparison of NES trophic state assignments to various phosphorus
model predictions revealed a 14, 18, and 25 percent disagreement, respectively,
for the Dillon, Larsen/Mercier, and Vollenweider concepts. The Dillon and
Larsen/Mercier models predicted the same trophic state in 33 of 35 lakes. The
only exceptions were Martin Lake and Wanaque Reservoir where the trophic
ratios, although very close, lay on opposite sides of the somewhat arbitrary
borderline. Nine of the 35 Vollenweider trophic state predictions differed
from the Dillon model predictions, while 8 differed from the Larsen/Mercier
model estimates. Generally, the Dillon and Larsen/Mercier models not only
predicted the same trophic state but their respective trophic ratios were
quite similar. By comparison, the Vollenweider model porvided less consistent
trophic state results and greater variation in trophic ratios.
Wanaque Reservoir was classified as eutrophic, mesotrophic, and oligo-
trophy by the models. However, the Dillon and Larsen/Mercier trophic
ratios were actually very close (0.96 and 1.13, respectively).
Even though all of the models produced relatively high Spearman rank
correlation coefficients, these models must be used with caution as the com-
parisons given in Table 3 illustrate. For the two western reservoirs, the
Vollenweider model predicted a much higher eutrophic loading rate than the
other models. Both of these reservoirs have high phosphorus retention capacities
(associated with high suspended sediment deposition) -- Lake Mead (0.93),
and Flaming Gorge Reservoir (0.82). These examples reinforce the concept
that the Dillon and the Larsen/Mercier models give comparable results. However,
the Larsen/Mercier model requires less information [(P) flushing rate and
(Z) mean depth are not required] and produced the highest Spearman rank correla-
tion coefficient (rs). Also, the Dillon model required mean depth information
which is neither uniformily available nor necessarily accurate, without an
extensive bathymetric survey. The mean depth of the lake is the only independ-
ent variable which sets the level of the theoretical eutrophic "loading" rate
in the Dillon model. In the Larsen/Mercier model the independent variable
which controls the theoretical eutrophic loading rate is the phosphorus
retention capacity.
7

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TABLE 3. COMPARISON OF THREE MODELS TO PREDICT MEAN AMBIENT LAKE PHOSPHORUS CONCENTRATIONS
AT EQUILIBRIUM IN TWO RESERVOIRS
CALCULATED
MODEL
VALUE
TROPHIC
STATE LEVELS
EUTROPHIC	OLIGOTROPHY
TROPHIC
RATIO
PREDICTED
TROPHIC
STATE
oo
Lake Mead, Nev./Ariz.
Vollenweider 6.23 g/m2/yr
Larsen/Mercier 372 yg/1
Dillon	1.47 g/m2
Flaming Gorge, Wyo./Utah
Vollenweider 1.35 g/m2/yr
Larsen/Mercier 93.5 wg/1
Dillon	0.41 g/m2
0.78 g/m2/yr
298.5	yg/1
1.18 g/m2
0.76 g/m2/yr
152.6	yg/1
0.68 g/m2
0.39 g/m2/yr
149.5 yg/1
0.59 g/m2
0.38 g/m2/yr
76.9 yg/1
0.34 g/m2
7.98
1 .29
1 .24
Eutrophic
Eutrophic
Eutrophic
1.77 Eutrophic
0.60 Mesotrophic
0.60 Mesotrophic

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LITERATURE CITED
Allum, M.O., R.E. Glessner, and O.H. Gakstatter. 1977. An evaluation of the
National Eutrophication Survey data. Working Paper No. 900. Assessment and
Criteria Development Division, Corvallis Environmental Research Laboratory,
Corvallis, Oregon. 75 p.
Dillon, P.J. 1975. The phosphorus budget of Cameron Lake, Ontario: The
importance of flushing rate to the degree of eutrophy of lakes. Limnol.
Oceanogr. 20:28-39.
Edmondson, W.T. 1970. Book review (Water Management Research). Limnol.
Oceanogr. 15:169-170.
Larsen, D.P. and H.T. Mercier. 1976. Phosphorus retention capacity of
lakes. J. Fish. Res. Board Can. 33:1742-1750.
U.S. Environmental Protection Agency. 1975. National Eutrophication Survey
Methods 1973 - 1976. Working Paper No. 175. Environmental Monitoring and
Support Laboratory, Las Vegas, Nevada, and Corvallis Environmental Research
Laboratory, Corvallis, Oregon. 91 p.
U.S. Environmental Protection Agency. 1976. Report on Lake Lulu, Florida.
Working Paper No. 263. Environmental Monitoring and Support Laboratory, Las
Vegas, Nevada, and Corvallis Environmental Research Laboratory, Corvallis,
Oregon. 30 p.
U.S. Environmental Protection Agency. 1977a. Report on Lake Mead, Nevada
and Arizona. Working Paper No. 808. Environmental Monitoring and Support
Laboratory, Las Vegas, Nevada, and Corvallis Environmental Research
Laboratory, Corvallis, Oregon. 91 p.
U.S. Environmental Protection Agency. 1977b. Report on Flaming Gorge
Reservoir, Utah and Wyoming. Working Paper No. 885. Environmental Monitoring
and Support Laboratory, Las Vegas, Nevada, and Corvallis Environmental Research
Laboratory, Corvallis, Oregon. 67 p.
Vol 1enweider, R.A. 1968. Scientific fundamentals of the eutrophication of
lakes and flowing waters, with particular reference to nitrogen and phosphorus
as factors in eutrophication. Organ. Econ. Coop. Dev., Paris. Tech. Rep.
DAS/CSI/68.27:159 p.
Vollenweider, R.A. 1975. Input - output models with special reference to
phosphorus loading concept in limnology. Schweiz. Z. Hydro!. 37:53-83.
9

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