Empirical
Phosphorus Conc. (mg/L) 
Trophic State 

<0.010 
Oligotrophic 
Suitable for waterbased recreation and propagation of cold water fisheries, such as trout. Very high clarity and aesthetically pleasing. Excellent as a drinking water source. 
0.010  0.020 
Mesotrophic 
Suitable for waterbased recreation but often not for cold water fisheries. Clarity less than oligotrophic lake. 
0.020  0.050 
Eutrophic 
Reduction in aesthetic properties diminishes overall enjoyment from body contact recreation. Generally very productive for warm water fisheries. High TOC and algal tastes & odors make these waters less desirable as a water supply. 
> 0.050 
Hypereutrophic 
A typical "oldaged" lake in advanced succession. Some fisheries, but high levels of sedimentation and algae or macrophyte growth may be diminishing open water surface area. Generally, unsuitable for drinking water supply. 
This model is based on a simple mass balance with terms for loading (W), settling, and outflow. There is no spatial, or temporal resolution.
Dividing
both sides by the surface area (A_{s}) gives:
where,
H is the lake depth, L is the areal loading (W/A_{s}) and q_{s}
is the overflow rate (Q/A_{s}).
At steady state (dP/dt =0), the solution becomes:
Based on data from 47 northern
temperate lakes included in EPA's National Eutrophication Survey, the settling
velocity (in m/yr) was found to be an empirical function of the overflow rate[2]:
so
substituting this into the steady state model above, we get:
where:
P =
mean annual total phosphorus concentration (gP/m^{3} or mgP/L)
L =
mean annual areal phosphorus loading (gP/m^{2}yr)
q_{s}
= mean annual areal water loading or overflow rate (m/yr) = Q/A_{s}
This
model was developed from lakes with phosphorus concentrations in the range of
0.0040.135 mg/L, phosphorus loadings of 0.0731.4 gP/m^{2}yr, and
overflow rates of 0.75187 m/yr. It
should not be used for lakes whose characteristics are outside of this
range. When used properly, the log
transform of the model has an estimated error (s_{mlog}) of 0.128. This value was determined from comparison of
observed and predicted phosphorus concentrations in the 47 lakes. Therefore, considering error, the model can
be written as:
From Chapra (pg 538)
from: Reckhow, 1979
q_{s} = Q/A_{s}
If
Q is not directly measurable from inflow or outflow, then it can be estimated
from:
Q = (A_{d} x r) + (A_{s}
x Pr)
where: 
q_{s =} 
areal
water loading (m/yr) 

Q = 
inflow
water volume to lake (m^{3}/yr) 

A_{d} = 
watershed
area (land surface) (m^{2}) 

A_{s} = 
lake
surface (m^{2}) 

r = 
total
annual unit runoff (m/yr) 

Pr = 
mean
annual net precipitation (m/yr) 
Data
Collection
· Determine total drainage area (A_{d}) from a GIS database, or USGS maps, using a polar planimeter, or cut paper with squares.
· Estimate the surface area of the lake (A_{s}). This may also be done by GIS or planimetry using a USGS map, or the cut paper method.
· Estimate annual runoff (r) which is usually expressed in meters/year. This information is generally available from the USGS.
· Determine average annual net precipitation (Pr), also expressed as meters/year. This information can usually be obtained from the USGS or the US Weather Service.
W = (Ec_{f} x Area_{f})
+ (Ec_{ag} x Area_{ag}) + (Ec_{u} x Area_{u}) +
(Ec_{a} x A_{s})
+ (Ec_{st} x #capitayrs x [1S.R.]) +
PSI
where: 
Ec_{f} = 
export
coefficient for forest land (kg/hayr) 

Ec_{ag} = 
export
coefficient for agricultural land (kg/hayr) 

Ec_{u} = 
export
coefficient for urban area (kg/hayr) 

Ec_{a} = 
export
coefficient for atmospheric input (kg/hayr) 

Ec_{st} = 
export
coefficient to septic systems impacting the lake (kg/(capitayr)yr) 

Area_{f} = 
area[3]
of forested land (ha) 

Area_{ag} = 
area
of agricultural land (ha) 

Area_{u} = 
area
of urban land (ha) 

A_{s} = 
surface
area of lake (ha) 

#capitayrs 
number
of capitayears in watershed serviced by septic tank impacting the lake 

S.R. = 
soil
retention coefficient (dimensionless) 

PSI = 
point
source input (kg/yr) 
Data
Collection
· Estimate land use drainage areas (forested, agricultural, urban). This information may be available from local planning agencies, otherwise it may be obtained from GIS data. For future projections, high and low estimates are needed for assessment of uncertainty
· Choose Export Coefficients for each category. Ranges should be selected for the major sources (often all but precipitation). Choice depends on characteristics of watershed as compared to those previously studied, for which there already exists export coefficients. Other factors may play a role such as the use of phosphate detergents (will impact Ec_{st}).
Some Nonspecific Phosphorus
Export Coefficients
Source 
Symbol 
Units 
High 
Midrange 
Low 
Agricultural 
Ec_{ag} 
kg/(hayr) 
3.0 
0.41.7 
0.10 

Ec_{f} 
kg/(hayr) 
0.45 
0.150.3 
0.02 
Precipitation 
Ec_{a} 
kg/(hayr) 
0.60 
0.200.50 
0.15 
Urban 
Ec_{u} 
kg/(hayr) 
5.0 
0.83.0 
0.50 
Input
to septic tanks 
Ec_{st} 
kg/(capitayr) 
1.8 
0.40.9 
0.3 
· Estimate SR: This is a number between 0 and 1 that indicates how well the soil and associated plants take up phosphorus. When it is low more of the phosphorus reaches the lake. Factors to consider include:
· phosphorus adsorption capacity
· natural drainage
· permeability
· slope
· Estimate number of capitayears on septic systems impacting lake: This requires some judgment, but usually a strip of about 20200 m wide surrounding the lake is considered the zone of influence. All septic systems within this zone would be counted in the following calculation:
Total
# of capitayears 
= 
average
# of persons per living unit 
X 
#
days spent at unit per year /360 
X 
#
of living units within zone of influence 
· Estimate Point source inputs: possibly from NPDES permits
· Now determine high, low and most likely estimates of W using above equation. These are obtained from high, low and most likely estimates of the various input parameters (note that the low value of S.R. should go with the high estimate of W, and vice versa).
· From these three estimates of W, calculate the high, most likely and low estimates for annual areal phosphorus loading
L = W/A_{s}
This
requires that the model error be appropriately combined with the uncertainty
inherent in the model terms. This is
done on log transforms of the model results, using standard error propagation
techniques.
· positive and negative model errors are calculated separately and not presumed equal.
s_{m+} =
antilog[logP_{ml} + s_{mlog}]  P_{ml}
s_{m} =
antilog[logP_{ml}  s_{mlog}]  P_{ml}
s_{L+ }= (P_{(high)}
 P_{(ml)})/2
s_{L }= (P_{(ml)}
 P_{(low)})/2
s_{T+} = [(s_{m+})^{2}
+ (s_{L+})^{2}]^{0.5}
s_{T} = [(s_{m})^{2}
+ (s_{L})^{2}]^{0.5}
· The intervals are 55% for 1 prediction error, and 90% for 2 (based on a modification of the Chebyshev inequality).
55% confidence interval: 
(P_{(ml)}  s_{T})
< P < (P_{(ml)} + s_{T+}) 
90% confidence interval: 
(P_{(ml)}  2s_{T})
< P < (P_{(ml)} + 2s_{T+}) 
This
lake is located in the northern section of

A_{d} = 
87.41
x 10^{6} m^{2} 

A_{s} = 
38.4
x 10^{6} m^{2} 

r = 
0.2415 m/yr 

Pr = 
0.254 m/yr 
Q = (A_{d} x r) + (A_{s}
x Pr)
Q =
30.863 x 10^{6} m^{3}/yr
q_{s}
= Q/A_{s}
q_{s}
= 0.804 m/yr
Land
Use 
Area
(ha) 

Agricultural 
16 


8347 

Urban 
378 

· forested land: mostly coniferous
· agricultural land: limited, mostly grazing and pasture
· urban areas: mainly residential and recreational, all are on septic systems
·
septic systems: phosphorusbased detergents are
banned in
· precipitation: value will also be low because of low agricultural and industrial inputs in the watershed which contribute to airborne phosphorus
Phosphorus Export
Coefficients
Source 
Symbol 
Units 
High 
Most
Likely 
Low 
Agricultural 
Ec_{ag} 
kg/(hayr) 
1.3 
0.40 
0.20 

Ec_{f} 
kg/(hayr) 
0.30 
0.20 
0.10 
Precipitation 
Ec_{a} 
kg/(hayr) 
0.50 
0.30 
0.15 
Urban 
Ec_{u} 
kg/(hayr) 
2.70 
0.90 
0.35 
Input
to septic tanks 
Ec_{st} 
kg/(capitayr) 
1.0 
0.6 
0.3 
·
Estimation of SR:
Soil Retention Coefficient
Symbol 
Units 
High 
Most Likely 
Low 
S.R. 
unitless 
0.50 
0.25 
0.05 
· Estimate number of capitayears on septic systems impacting lake: Only lakeside dwellings were counted:
Total
# of capitayears 
= 
3.5
persons/unit 
X 
60
days spent at unit per year /360 
X 
1000
living units within zone of influence 
Total
# of capitayears = 575.3
·
No point source inputs to
· Now determine high, low and most likely estimates of W
W = (Ec_{f} x Area_{f})
+ (Ec_{ag} x Area_{ag}) + (Ec_{u} x Area_{u}) +
(Ec_{a} x A_{s})
+ (Ec_{st} x #capitayrs x [1S.R.]) +
PSI
W_{(high)} = (0.30 x
8347) + (1.30 x 16) + (2.7 x 378) + (0.50 x 3840)
+ (1.0 x 575.3 x [10.05]) + 0
W_{(ml)} = (0.20 x
8347) + (0.40 x 16) + (0.90 x 378) + (0.30 x 3840)
+ (0.6 x 575.3 x [10.25]) + 0
W_{(low)} = (0.10 x
8347) + (0.20 x 16) + (0.35 x 378) + (0.15 x 3840)
+ (0.3 x 575.3 x [10.50]) + 0
· From these three estimates of W, calculate the high, most likely and low estimates for annual areal phosphorus loading
L = W/A_{s}
Summary of Results
Parameter 
High 
Most
Likely 
Low 
W 
6012.04
kg/yr 
3426.9
kg/yr 
1632.5
kg/yr 
L 
0.157
g/m^{2}yr 
0.089
g/m^{2}yr 
0.043
g/m^{2}yr 
P 
0.0125
mg/L 
0.0071
mg/L 
0.0034
mg/L 
· using the model, determine the three estimates of P:
for
results, see table above.
s_{m+} =
antilog[logP_{ml} + s_{mlog}]  P_{ml}
s_{m+} =
antilog[log0.0071 + 0.128]  0.0071
s_{m+} = 0.0024 mg/L
s_{m} =
antilog[logP_{ml}  s_{mlog}]  P_{ml}
s_{m} =
antilog[0.0071  0.128]  0.0071
s_{m} = 0.0015 mg/L
s_{L+ }= (P_{(high)}
 P_{(ml)})/2
s_{L+ }= (0.0125 
0.0071)/2
s_{L+ }= 0.0027 mg/L
s_{L }= (P_{(ml)}
 P_{(low)})/2
s_{L }= (0.0071 
0.0034)/2
s_{L }= 0.0019 mg/L
s_{T+} = [(s_{m+})^{2}
+ (s_{L+})^{2}]^{0.5}
s_{T+} = [(0.0024)^{2}
+ (0.0027)^{2}]^{0.5}
s_{T+} = 0.0036
s_{T} = [(s_{m})^{2}
+ (s_{L})^{2}]^{0.5}
s_{T} = [(0.0015)^{2}
+ (0.0019)^{2}]^{0.5}
s_{T} = 0.0024
55% confidence interval: 
0.0047 < P < 0.0107 
90% confidence interval: 
0.0023 < P < 0.0143 