U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2009–5199

National Water-Quality Assessment Program

Development and Application of Regression Models for 

Estimating Nutrient Concentrations in Streams of the 

Conterminous United States, 1992–2001

Development and Application of 

Regression Models for Estimating 

Nutrient Concentrations in Streams of the 

Conterminous United States, 1992–2001

By Norman E. Spahr, David K. Mueller, David M. Wolock, Kerie J. Hitt, and 

JoAnn M. Gronberg

National Water-Quality Assessment Program

Scientific Investigations Report 2009–5199

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt , Director

U.S. Geological Survey, Reston, Virginia: 2010

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Suggested citation:

Spahr, N.E., Mueller, D.K., and Wolock, D.W., 2010, Development and application of regression models for estimating 

nutrient concentrations in streams of the conterminous United States, 1992–2001: U.S. Geological Survey Scientific 

Investigations Report 2009-5199, 22 p.
iii

Foreword

The U.S. Geological Survey (USGS) is committed to providing the Nation with reliable scientific information that helps to enhance and protect the overall quality of life and that facilitates effective management of water, biological, energy, and mineral resources (http://www.usgs.gov/). Information on the 

Nation’s water resources is critical to ensuring long-term availability of water that is safe for drinking and 

recreation and is suitable for industry, irrigation, and fish and wildlife. Population growth and increasing 

demands for water make the availability of that water, measured in terms of quantity and quality, even 

more essential to the long-term sustainability of our communities and ecosystems.

The USGS implemented the National Water-Quality Assessment (NAWQA) Program in 1991 to support 

national, regional, State, and local information needs and decisions related to water-quality management 

and policy (http://water.usgs.gov/nawqa). The NAWQA Program is designed to answer: What is the quality of our Nation’s streams and ground water? How are conditions changing over time? How do natural 

features and human activities affect the quality of streams and ground water, and where are those effects 

most pronounced? By combining information on water chemistry, physical characteristics, stream habitat, 

and aquatic life, the NAWQA Program aims to provide science-based insights for current and emerging 

water issues and priorities. From 1991 to 2001, the NAWQA Program completed interdisciplinary assessments and established a baseline understanding of water-quality conditions in 51 of the Nation’s river 

basins and aquifers, referred to as Study Units (http://water.usgs.gov/nawqa/studyu.html). 

National and regional assessments are ongoing in the second decade (2001–2012) of the NAWQA 

Program as 42 of the 51 Study Units are selectively reassessed. These assessments extend the findings 

in the Study Units by determining water-quality status and trends at sites that have been consistently 

monitored for more than a decade, and filling critical gaps in characterizing the quality of surface water 

and ground water. For example, increased emphasis has been placed on assessing the quality of source 

water and finished water associated with many of the Nation’s largest community water systems. During 

the second decade, NAWQA is addressing five national priority topics that build an understanding of how 

natural features and human activities affect water quality, and establish links between sources of contaminants, the transport of those contaminants through the hydrologic system, and the potential effects of 

contaminants on humans and aquatic ecosystems. Included are studies on the fate of agricultural chemicals, effects of urbanization on stream ecosystems, bioaccumulation of mercury in stream ecosystems, 

effects of nutrient enrichment on aquatic ecosystems, and transport of contaminants to public-supply 

wells. In addition, national syntheses of information on pesticides, volatile organic compounds (VOCs), 

nutrients, trace elements, and aquatic ecology are continuing. 

The USGS aims to disseminate credible, timely, and relevant science information to address practical and 

effective water-resource management and strategies that protect and restore water quality. We hope this 

NAWQA publication will provide you with insights and information to meet your needs, and will foster 

increased citizen awareness and involvement in the protection and restoration of our Nation’s waters. 

The USGS recognizes that a national assessment by a single program cannot address all water-resource 

issues of interest. External coordination at all levels is critical for cost-effective management, regulation, 

and conservation of our Nation’s water resources. The NAWQA Program, therefore, depends on advice 

and information from other agencies—Federal, State, regional, interstate, Tribal, and local—as well as 

nongovernmental organizations, industry, academia, and other stakeholder groups. Your assistance and 

suggestions are greatly appreciated.

 Matthew C. Larsen

 Associate Director for Water

v

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Purpose and Scope ..............................................................................................................................2

Methods...........................................................................................................................................................2

Basin Characteristics...........................................................................................................................2

Correlation and Regression Analysis ................................................................................................2

Nationwide Prediction .........................................................................................................................3

Correlation between Nutrient Concentrations and Basin Characteristics..........................................3

Development of Models for Total Nitrogen ...............................................................................................3

Agricultural Sites ..................................................................................................................................5

Nonagricultural Sites ...........................................................................................................................6

Analysis of Residuals ...........................................................................................................................8

Nationwide Prediction of Total-Nitrogen Concentrations ....................................................................12

Development of Models for Total Phosphorus........................................................................................17

Comparison to Results from other Models..............................................................................................18

Summary and Conclusions.........................................................................................................................20

Acknowledgments.......................................................................................................................................20

References Cited..........................................................................................................................................21

Figures

1. Location of National Water-Quality Assessment sampling sites downstream 

 from agricultural and nonagricultural areas, as classified for use in the 

 development of total-nitrogen regression models..................................................................5

2. Comparison of observed and predicted flow-weighted mean annual 

 total-nitrogen concentrations at National Water-Quality Assessment 

 sampling sites downstream from agricultural areas..............................................................7

3. Comparison of observed and predicted flow-weighted mean annual 

 total-nitrogen concentrations at National Water-Quality Assessment 

 sampling sites downstream from nonagricultural areas .......................................................8

4. Geographic distribution of residuals for regression models of flow-

 weighted mean annual total-nitrogen concentrations at National 

 Water-Quality Assessment sampling sites downstream from agricultural 

 and nonagricultural areas...........................................................................................................9

5. Distributions of residuals for regression models of flow-weighted mean 

 annual total-nitrogen concentrations grouped by U.S. Environmental 

 Protection Agency nutrient ecoregion (U.S. Environmental Protection 

 Agency, 1998)...............................................................................................................................10

6. Distributions of residuals for regression models of flow-weighted mean 

 annual total-nitrogen concentrations grouped by quartile of basin size..........................11
vi

7. Predicted flow-weighted mean-annual concentrations of total nitrogen 

 for streams in the U.S. Environmental Protection Agency River Reach 

 File, 1992-2001..............................................................................................................................12

8. Probabilities that flow-weighted mean-annual concentrations of total 

 nitrogen in streams are greater than 1, 5, and 10 milligrams per liter...............................13

9. Probabilities that flow-weighted mean-annual concentrations of total 

 nitrogen in streams exceed the U.S. Environmental Protection Agency 

 nutrient-ecoregion criteria........................................................................................................14

10. Percentage of stream miles in each U.S. Environmental Protection Agency 

nutrient-ecoregion within which the probabilities of exceeding the regional 

total-nitrogen criteria are 50, 75, and 90 percent..................................................................15

11. Percentage of stream miles in each U.S. Environmental Protection Agency 

nutrient-ecoregion with a 90-percent probability of exceeding the regional 

total-nitrogen criteria with various reductions in fertilizer inputs......................................16

12. Comparison of observed and predicted total-phosphorus concentrations 

 and the distribution of residuals from the regression model ..............................................18

13. Comparison of prediction total-nitrogen concentrations from this report 

 and the SPARROW model (Alexander and others, 2008) .....................................................19

Tables

1. Correlation between flow-weighted annual mean total-nitrogen or totalphosphorus concentrations at National Water-Quality Assessment 

sampling sites and selected characteristics of the upstream basin...................................4

2. Explanatory variables in the model of log10 flow-weighted mean annual 

 total-nitrogen concentrations downstream from agricultural areas...................................6

3. Explanatory variables in the model of log10 flow-weighted mean annual 

 total-nitrogen concentrations downstream from nonagricultural areas ............................7

4. Explanatory variables in the model of log10 flow-weighted mean annual 

 total-phosphorus concentrations ............................................................................................17
vii

Conversion Factors

[Inch/Pound to SI]

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 4,047 square meter (m2

)

acre 0.4047 hectare (ha)

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Volume

ounce, fluid (fl. oz) 0.02957 liter (L) 

pint (pt) 0.4732 iter (L) 

quart (qt) 0.9464 liter (L) 

gallon (gal) 3.785 liter (L) 

gallon (gal) 0.003785 cubic meter (m3

) 

cubic inch (in3

) 0.01639 liter (L)

cubic foot (ft3

) 0.02832 cubic meter (m3) 

acre-foot (acre-ft) 1,233 cubic meter (m3

)

Flow rate

acre-foot per day (acre-ft/d) 0.01427 cubic meter per second (m3

/s)

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)

Mass

ounce, avoirdupois (oz) 28.35 gram (g) 

pound, avoirdupois (lb) 0.4536 kilogram (kg) 

Application rate

pounds per acre per year 

[(lb/acre)/yr]

 1.121 kilograms per hectare per year

 [(kg/ha)/yr]

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F=(1.8×°C)+32

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the “North American Vertical Datum of 1988 

(NAVD 88).”

Horizontal coordinate information is referenced to the “North American Datum of 1983 (NAD 83).”
viii

Altitude, as used in this report, refers to distance above the vertical datum.

Specific conductance is given in microSiemens per centimeter at 25 degrees Celsius 

(μS/cm at 25 °C).

Concentrations of chemical constituents in water are given either in milligrams per 

liter (mg/L) or micrograms per liter (μg/L).
Abstract

Data collected for the U.S. Geological Survey National 

Water-Quality Assessment program from 1992–2001 were 

used to investigate the relations between nutrient concentrations and nutrient sources, hydrology, and basin characteristics. Regression models were developed to estimate annual 

flow-weighted concentrations of total nitrogen and total 

phosphorus using explanatory variables derived from currently available national ancillary data. Different total-nitrogen 

regression models were used for agricultural (25 percent or 

more of basin area classified as agricultural land use) and 

nonagricultural basins. Atmospheric, fertilizer, and manure 

inputs of nitrogen, percent sand in soil, subsurface drainage, 

overland flow, mean annual precipitation, and percent undeveloped area were significant variables in the agricultural 

basin total nitrogen model. Significant explanatory variables in 

the nonagricultural total nitrogen model were total nonpointsource nitrogen input (sum of nitrogen from manure, fertilizer, 

and atmospheric deposition), population density, mean annual 

runoff, and percent base flow. 

The concentrations of nutrients derived from regression 

(CONDOR) models were applied to drainage basins associated with the U.S. Environmental Protection Agency (USEPA) 

River Reach File (RF1) to predict flow-weighted mean annual 

total nitrogen concentrations for the conterminous United 

States. The majority of stream miles in the Nation have predicted concentrations less than 5 milligrams per liter. Concentrations greater than 5 milligrams per liter were predicted for 

a broad area extending from Ohio to eastern Nebraska, areas 

spatially associated with greater application of fertilizer and 

manure. Probabilities that mean annual total-nitrogen concentrations exceed the USEPA regional nutrient criteria were 

determined by incorporating model prediction uncertainty. In 

all nutrient regions where criteria have been established, there 

is at least a 50 percent probability of exceeding the criteria in 

more than half of the stream miles.

Dividing calibration sites into agricultural and nonagricultural groups did not improve the explanatory capability for 

total phosphorus models. The group of explanatory variables 

that yielded the lowest model error for mean annual total 

phosphorus concentrations includes phosphorus input from 

manure, population density, amounts of range land and forest 

land, percent sand in soil, and percent base flow. However, the 

large unexplained variability and associated model error precluded the use of the total phosphorus model for nationwide 

extrapolations.

Introduction

During the past decade, nutrients have consistently been 

cited as one of the leading pollutants in rivers and streams 

(see, for example, U.S. Environmental Protection Agency, 

2002b and 2007). Nutrients are chemical compounds that 

contain nitrogen (N) or phosphorus (P). These compounds are 

essential to plant and animal nutrition, but in high concentrations they can be contaminants in water. Nutrient compounds 

are affected by chemical, biological, and hydrologic processes 

that can change their form and can transfer them to or from 

water, soil, biological organisms, and the atmosphere. 

In 1991 the U.S. Geological Survey (USGS) began 

full implementation of the National Water-Quality Assessment (NAWQA) program to assess the status and trends of 

the Nation’s surface and ground waters. The design of the 

NAWQA program is presented in Gilliom and others (1995). 

Nutrients are one of the primary synthesis topics of the 

NAWQA program.

Nutrients were investigated in streams and rivers sampled 

as part of the U.S. Geological Survey National Water-Quality 

Assessment (NAWQA) Program. Nutrient data were collected 

in 20 NAWQA study units during 1992–95, 16 study units 

during 1996-98, and 15 study units during 1999–2001. Mueller and Spahr (2005) compiled nutrient data collected during 

Cycle I (1992–2001) of the NAWQA program. To facilitate 

comparisons among sampling sites with variable sampling 

frequency, Mueller and Spahr (2006) computed daily loads 

by using regression models that relate constituent transport 

to streamflow and time. The specific method used for model 

calibration and load estimation is described by Runkel and 

others (2004). Mueller and Spahr (2006, page 6) used the 

Development and Application of Regression Models for 

Estimating Nutrient Concentrations in Streams of the 

Conterminous United States, 1992–2001

By Norman E. Spahr, David K. Mueller, David M. Wolock, Kerie J. Hitt, and JoAnn M. Gronberg
2 Development and Application of Regression Models for Nutrients in U.S. Streams

model results to compute mean annual loads, yields, and concentrations of total nitrogen and total phosphorus for NAWQA 

sampling sites. Sufficient data were available to calculate 

annual concentrations at all 481 sites for total nitrogen and at 

463 sites for total phosphorus. 

Purpose and Scope

A national comparison of nutrient conditions among 

stream sites was facilitated by modeling annual loads and 

concentrations (Mueller and Spahr, 2006). Nutrient concentrations, yields, and loads were compared to basin land-use 

characteristics and major nonpoint sources of nitrogen and 

phosphorus. This report extends the previous analysis and 

extrapolates results to basins where water-quality samples 

were not collected during Cycle I of the NAWQA program 

by development and application of regression models for 

estimating nutrient concentration in streams.

Methods

The flow-weighted mean concentrations of total nitrogen 

and total phosphorus as reported by Mueller and Spahr (2005 

and 2006) are considered to be the “observed” concentrations 

in this report. These concentrations are used in correlation and 

regression analyses, along with various basin characteristics, 

in order to estimate mean annual total nitrogen and phosphorus concentrations in streams throughout the conterminous 

United States.

Basin Characteristics

Characteristics of the drainage basins upstream from 

NAWQA sites were derived from data sets available for the 

entire conterminous United States. Atmospheric and manure 

nonpoint-source nutrient inputs were based on county level 

estimates as described by Ruddy and others (2006). Fertilizer 

nonpoint-source inputs were based on methodology described 

by Ruddy and others (2006) using updated data (J.M. Gronberg, U.S. Geological Survey, written commun., 2008). Population density was derived from the 1990 census and block 

group data (U.S. Bureau of the Census, 1991a and 1991b).

 Percent land use cover was determined from the 1992 

National Land Cover Data (NLCD) digital maps as revised by 

Nakagaki and Wolock (2005). Agriculture land use is the combined percentages of NLCD classes’ row crop, small grains, 

fallow, orchards and vineyard, and pasture and hay (NLCD 

classes 61, 62, 81, 82, 83, and 84). Undeveloped land use 

includes percentages of grasslands, shrub lands, all forested 

lands, as well as, open water, wetlands, tundra, and bare rock 

(NLCD classes 11, 12, 31, 32, 33, 41, 42, 43, 51, 71, 72, 91, 

and 92).

Percent sand in soil was obtained from the U.S. Department of Agriculture’s State Soil Geographic database (Natural 

Resources Conservation Service, 1994). Mean annual runoff 

(1950–1981) was determined from a map by Gebert and others (1987). Base flow index and Horton overland flow were 

obtained from online data sets (Wolock, 2003a and 2003b). 

Information on tile drains is from the National Resources 

Conservation Service (1995). Precipitation and evaporation 

(1961–90) are derived from a model developed by Daly and 

others (2001) 

Correlation and Regression Analysis

Nonparametric Spearman coefficients (Helsel and Hirsch, 

1993) were used to examine the various correlations between 

flow-weighted concentrations and basin characteristics. 

Results of the correlation analysis were used to select potential 

explanatory variables for regression models. During the correlation and regression calibration phase, nonpoint nutrient input 

terms (fertilizer, manure, and atmospheric deposition) represent the NAWQA high-intensity period for each sampling site 

as described by Mueller and Spahr (2005, ancillary data section). Other terms represent longer time periods as described 

in the section titled “Basin Characteristics.”

The multiple linear regression models were built in a 

stepwise process. The dependent variable in each model was 

the base-10 logarithm (log10) of the flow-weighted concentration. Initial explanatory variables included the various nonpoint-source inputs. Correlations between the residuals from 

these initial models and additional basin characteristics led 

to the selection of subsequent explanatory variables. In some 

cases, transformations (logarithmic, square root, and fourth 

root) of the explanatory variables were employed to achieve 

linearity, normality of residuals, and homogeneity of residual 

variance (Helsel and Hirsch, 1993). Partial residual plots were 

used to determine if selected transformations of explanatory 

variables were appropriate. With the addition of each new 

variable the adjusted R squared and the root mean square error 

were examined to evaluate the potential improvement gained 

by the addition of another variable. Explanatory variables were 

added to the model until no tangible improvements were seen 

or until subsequent variables were not significant at an alpha 

level of 0.05. Although somewhat subjective, this method balances model simplicity with the ability of the model to explain 

the variability of nutrient concentrations. All statistics, correlations, and regressions were analyzed using SAS statistical 

software (SAS Institute Inc., 1999).

To determine if the significance and form of a set of 

explanatory variables was dependent on a specific set of stations, one-third of the stations were withheld from model calibration and the regression coefficients were determined using 

the remaining two-thirds of the stations. This exercise was 

replicated 200 times, and the significance and form of each 

coefficient were compared among the calibrated models. If an 
Development of Models for Total Nitrogen 3

explanatory variable was not significant in a large percentage 

of these models or its coefficient was positive in some models 

and negative in others, then that variable was removed from 

consideration. 

Nationwide Prediction

The U.S. Environmental Protection Agency (USEPA) 

River Reach File version 1.0 (RF1), described by Nolan and 

others (2003) was used as the basis for predicting nutrient concentrations throughout the conterminous United States. RF1 is 

a georeferenced database of stream reaches that comprise the 

surface water drainage of the United States. Basin characteristics and nonpoint-source nutrient inputs were determined for 

basins upstream from each RF1 node (the point representing 

the downstream end of each reach). The previously calibrated 

regression equations then were used to predict concentrations 

at each node point. Retransformation bias was corrected using 

the smearing estimate of Duan (1983) as outlined by Helsel 

and Hirsch (1993).

Predicted concentrations were assumed to apply to the 

entire RF1 reach upstream from the node. Similar methodology 

was used in the NAWQA program to predict atrazine 

concentrations in stream water, as documented by Gilliom 

and others (2006), and dieldrin concentrations in fish tissue, 

as documented by Nowell and others (2006). In addition 

to predicting nutrient concentrations for each RF1 reach, 

the probability of a concentration being greater than some 

reference value can be determined. Because residuals from the 

regression equations are normally distributed, the probability 

of an estimated concentration being greater than some value, 

X, can be represented by the area under a normal curve above 

the value Z:

Z = C − X

Sc

(1)

where

Z is a standard normal deviate,

C is the concentration estimated from the 

regression model,

X is the reference concentration, and

SC is the standard error of C, equal to the 

prediction error of the regression model.

The primary nonpoint source inputs (fertilizer, manure, 

and atmospheric deposition) represent the entire NAWQA 

Cycle I period (1992–2001) for nationwide simulation. Therefore, the simulated concentrations represent the cycle one 

period of the NAWQA program. 

Correlation between Nutrient 

Concentrations and Basin 

Characteristics

Nutrient concentrations in stream water vary with 

upstream land-use practices that affect nutrient sources and 

transport of nutrients to stream channels. On a national scale 

there is great variability associated with stream nutrient concentrations and related basin characteristics. Spearman correlation coefficients between flow-weighted concentrations and 

various explanatory variables are listed in table 1. In addition 

to analyzing all sites together, sites were classified as agricultural (agricultural land use greater than or equal to 25 percent 

of the basin area) or nonagricultural (fig. 1). 

For some explanatory variables, such as percent sand in 

soil and percent area with subsurface drains, correlation with 

nutrient concentrations is greater in agricultural basins than in 

nonagricultural basins, indicating nutrient concentrations in 

streams might be responding to an interaction between land 

use and other basin characteristics. Nonpoint-source inputs of 

nitrogen and phosphorus from fertilizer are more highly correlated with stream nutrient concentrations in agricultural areas than 

nonagricultural areas, reflecting the greater application associated 

with agricultural land use. Population density is a surrogate 

for point sources of nutrients. Correlations between population density and stream nutrient concentrations are greater in 

nonagricultural areas which include many urban watersheds. 

Mueller and Spahr (2006) have shown that concentrations 

of all nutrients in streams increase as development in the basin 

increases. Percent of undeveloped land use has relatively high 

negative correlations with nutrient concentrations (table 1). 

Less development and lower percentage of agricultural land 

use corresponds with lower concentrations, and the magnitude 

of the correlation is greater for the agricultural than nonagricultural sites. Peterson and others (2004) describe higher 

phosphorus concentrations in basins with higher percentage of 

rangeland for NAWQA sites in Wyoming and this also appears 

to be the case for the national set of nonagricultural basins in 

this study.

Development of Models for 

Total Nitrogen

An initial regression model relating nitrogen concentrations to non-point sources (atmospheric, fertilizer, and manure 

nitrogen) for all sites explained only about 53 percent of the 

variability. Various basin characteristics were added to the 

all sites model based upon the correlation of residuals with 

additional ancillary data. Results from these regression models 

could not adequately explain the variability associated with 

different land use types and these models consistently underestimated concentrations for many agricultural basins. Finally, 
Explanatory variable Spearman correlation coefficient All Sites Agricultural Sites Nonagr

Total nitrogen Total phosphorus Total nitrogen Total phosphorus Total nitrogen 

n=462 n=440 n=220 n=214 n=242 

Atmospheric nitrogen input, kg/km2 0.35 -- 0.30 -- 0.20 

Fertilizer nitrogen input, kg/km2 0.71 -- 0.62 -- 0.54 

Manure nitrogen input, kg/km2 0.49 -- 0.28 -- 0.19 

Fertilizer phosphorus input, kg/km2 -- 0.44 -- 0.35 -- 

Manure phosphorus input, kg/km2 -- 0.34 -- 0.22 -- 

Population density, per km2 0.30 0.20 0.04 –0.05 0.50 

Undeveloped land use, percent of basin area –0.74 –0.50 –0.73 –0.50 –0.57 

Forest land use, percent of basin area –0.69 –0.63 –0.64 –0.47 –0.50 

Range land use, percent of basin area –0.09 –0.13 0.13 0.17 0.05 

Sand in soil, percent –0.38 –0.31 –0.32 –0.38 –0.11 

Subsurface drainage, percent of basin area 0.39 0.22 0.48 0.23 0.12 

Horton overland flow, percent of long-term 

mean streamflow 0.25 0.34 0.00 0.35 0.16 

Base flow index, percent of long-term mean 

streamflow

–0.38 –0.36 0.00 –0.25 –0.35 

Mean annual runoff 1951–90, inches –0.36 –0.39 –0.24 –0.15 –0.35 

Mean annual precipitation, inches –0.11 –0.14 –0.34 –0.10 –0.08 

Mean annual evaporation, inches 0.16 0.36 –0.09 0.18 0.26 

Table 1. Correlation between flow-weighted annual mean total-nitrogen or total-phosphorus concentrations at National Water-Quality Asssampling sites and selected characteristics of the upstream basin.[n, number of samples; kg/km2, kilograms per square kilometer; km2, square kilometer; bold type indicates absolute value of correlation is greater than 0.5; --, not app
Development of Models for Total Nitrogen 5

sub-setting sites into agricultural and nonagricultural groups 

led to regression models with acceptable explanatory capabilities. Sites where agricultural land use was at least 25 percent 

of the upstream basin area were considered agricultural. The 

geographic distribution of 220 agricultural sites and 242 

nonagricultural sites, based on this classification, is shown in 

figure 1. The variation in spatial density is a result of regional 

differences in arable land and also is influenced by the design 

of the NAWQA sampling network (Gilliom and others, 1995).

Agricultural Sites

Explanatory variables used in the concentrations of 

nutrients derived from regression (CONDOR) model for total 

nitrogen downstream from agricultural areas are listed in table 

2. Regression coefficients for these variables were significant 

(p-value < 0.01) in more than 92 percent of the 200 simulation 

tests where random selections of two-thirds of the sites were 

used for calibration of the regression model. Atmospheric 

nitrogen input and subsurface drainage were the variables 

that most frequently became nonsignificant in the tests. For 

the model calibrated using data from all agricultural sites, the 

root-mean-square error is 0.19, the adjusted R-squared value 

is 0.71, and the bias-correction factor used for de-transforming 

estimated total-nitrogen concentrations is 1.10.

As expected, all sources of nitrogen (atmospheric, fertilizer, and manure) in the model are positively correlated with 

total-nitrogen concentrations in streams. In general, total-nitrogen concentrations are higher downstream from areas with 

higher rates of atmospheric deposition, fertilizer application, 

or manure accumulation. The presence of subsurface drainage 

also is positively correlated with total-nitrogen concentrations, probably because installation of drainage tile shortcircuits the normal flow path and results in rapid movement 

of nitrogen to streams. The remaining explanatory variables 

in the model have negative coefficients; higher values of these 

basin characteristics generally are associated with lower totalnitrogen concentrations. Percent sand in soils is an indication 

of potential infiltration. Where percent sand is high, infiltration also is likely to be high, and there will be less direct 

runoff, and therefore less nitrogen transport, to streams. The 

negative coefficient on Horton overland flow seems counterintuitive, because greater overland flow could result in greater 

nitrogen transport. Overland flow occurs when precipitation 

exceeds infiltration capacity, and this is more likely on land 

that is less suitable for intensive agriculture, either because 

of soil properties or slope. Thus, the overland-flow variable 

in this model could be associated with basins where sources 

of nitrogen are limited, particularly in proximity to stream 

channels. Precipitation creating overland flow from areas with 

limited sources of nitrogen may have a dilution effect. The 

negative coefficient on precipitation indicates that areas with 

higher annual precipitation tend to have somewhat lower total 

nitrogen concentrations. This is hypothesized as a dilution 

effect. Runoff and streamflow were also tested as explanatory 

EXPLANATION

Nonagricultural sites

Agricultural sites

Study unit

Figure 1. Location of National Water-Quality Assessment sampling sites downstream from agricultural and 

nonagricultural areas, as classified for use in the development of total-nitrogen regression models.
6 Development and Application of Regression Models for Nutrients in U.S. Streams

variables and typically had negative coefficients, but precipitation was a more consistent predictor variable. Finally, the 

amount of undeveloped land use is negatively correlated with 

total-nitrogen concentrations. In basins with greater development (agricultural and urban areas), sources of nitrogen are 

more common and the probability of direct transport to stream 

channels is greater.

 A comparison of the observed and estimated total 

nitrogen at agricultural sites is shown in figure 2. The regression model generally performs well throughout the range of 

total nitrogen concentrations. Three sites (the Palouse River in 

Washington, Sacramento Slough in California, and the Niangua River in Missouri) have very low observed total-nitrogen 

concentrations, and the model did not fit these sites particularly well. These are the three sites with the lowest observed 

total nitrogen concentration values (fig. 2). These sites are not 

unique based on available national scale ancillary data other 

than the fact that they have low nitrogen concentrations for the 

given nonpoint source nitrogen inputs.

Nonagricultural Sites

Explanatory variables and regression coefficients in the 

CONDOR model for total-nitrogen concentrations downstream from nonagricultural areas are listed in table 3. In 

the simulation tests where two-thirds of the sites were randomly selected for model calibration, base flow index was 

nonsignificant eight times; other explanatory variables were 

always significant. For the model calibrated using data from 

all nonagricultural sites, the root-mean-square error is 0.25, 

the adjusted R-squared value is 0.62, and the bias-correction 

factor used for de-transforming estimated total-nitrogen concentrations is 1.20.

Models using manure, fertilizer, and atmospheric inputs 

as separate terms were not successful in explaining totalnitrogen concentrations in streams draining nonagricultural 

areas. Manure was typically a significant term but fertilizer 

inputs often were nonsignificant in the model development 

or subsequent model testing. Atmospheric deposition, which 

represents a much smaller nitrogen source, did not explain a 

sufficient amount of the variability in total-nitrogen concentrations. Combining the three nonpoint sources into one term 

resulted in a model that tested well, explained a significant 

proportion of the variability in total-nitrogen concentrations, 

and is useful in extrapolating results to areas where one of the 

three nonpoint-source terms may be dominant. 

The form of the nonagricultural model is very simple in 

that it includes only source terms and flow variables. Population density acts as a surrogate for urban nutrient sources, such 

as wastewater-treatment plants and septic systems. Totalnitrogen concentrations tend to be higher at sites downstream 

from areas where population density or other nitrogen sources 

(manure accumulation, fertilizer application, and atmospheric 

deposition of nitrogen) are greater. Conversely, sites with 

Explanatory variable (transformation) Regression 

coefficient Standard error p-value

Intercept 0.53537 0.16889 0.0021

Atmospheric nitrogen input, kg/km2

 (square root) 0.01437 0.00402 0.0004

Fertilizer nitrogen input, kg/km2

 (square root) 0.00429 0.00114 0.0002

Manure nitrogen input, kg/km2 (fourth root) 0.06193 0.00876 <0.0001

Sand in soil, percent (fourth root) –0.26667 0.04702 <0.0001

Subsurface drainage (1 if more than 5 percent of basin 

area, 0 otherwise) 0.14511 0.03897 0.0003

Horton overland flow, percent of long-term mean 

streamflow –0.01408 0.00310 <0.0001

Undeveloped land use, percent of basin area –0.00518 0.00112 <0.0001

Mean annual precipitation, inches –0.00536 0.00120 <0.0001

Table 2. Explanatory variables in the model of log10 flow-weighted mean annual total-nitrogen concentrations 

downstream from agricultural areas. 

[kg/km2, kilograms per square kilometer]
Development of Models for Total Nitrogen 7

greater runoff tend to have lower total-nitrogen concentrations, 

possibly due to dilution. Basins where a higher percentage of 

streamflow is generated during base-flow conditions also tend 

to have lower total-nitrogen concentrations. Although local 

conditions can vary, in general base flow might transport relatively less nitrogen than storm runoff. It should be noted that 

this model is applicable to areas with less than 25 percent 

agricultural land, where total-nitrogen concentrations in groundwater may not be elevated. Nolan and Hitt (2006) show that 

nitrate concentrations in shallow groundwater are directly 

correlated to agricultural land use (percent cropland and pastureland).

A comparison of observed and estimated total-nitrogen 

concentrations at nonagricultural sites is shown in figure 3. 

The regression fit for these sites is not as good as for the agricultural model, and less of the variation is explained. The site 

with the greatest observed nitrogen concentration in figure 3 

is Las Vegas Wash near Las Vegas, Nevada. This site receives 

a large contribution of nutrients from a single point source 

(wastewater-treatment plant). Consistent point-source data 

Explanatory variable (transformation) Regression coefficient Standard error p-value

Intercept –0.00625 0.12281 0.9595

Nitrogen from manure plus fertilizer plus atmospheric 

inputs, kg/km2

 (fourth root) 0.09353 0.01661 <0.0001

Population density, per km2

 (fourth root) 0.09782 0.01241 <0.0001

Mean annual runoff 1951–80, inches (log10) –0.55851 0.04065 <0.0001

Base flow index, percent of long-term mean streamflow –0.00469 0.00124 <0.0001

Table 3. Explanatory variables in the model of log10 flow-weighted mean annual total-nitrogen concentrations 

downstream from nonagricultural areas.

[kg/km2, kilograms per square kilometer; km2, square kilometer]

PREDICTED LOG10 TOTAL NITROGEN CONCENTRATION,

IN MILLIGRAMS PER LITER

OBSERVED LOG10 TOTAL NITROGEN CONCENTRATION,

IN MILLIGRAMS PER LITER

–0.5 0.0 0.5 1.0 1.5

–0.5

0.0

0.5

1.0

1.5

1:1 line

Niangua River

Sacramento Slough Palouse River

Figure 2. Comparison of observed and predicted flow-weighted mean 

annual total-nitrogen concentrations at National Water-Quality Assessment 

sampling sites downstream from agricultural areas.
8 Development and Application of Regression Models for Nutrients in U.S. Streams

were not available on a national scale, and none of the potential explanatory variables were useful in improving the estimation of total nitrogen at this site. Two other sites are seen 

above the scatter shown in figure 3 (Dry Fork of Roan Creek 

in Colorado and the Sabinal River in Texas). Streamflows can 

change very rapidly at these sites. During short-term high-flow 

events, large loads of nitrogen are exported from these basins. 

These high loads result in large flow-weighted total-nitrogen 

concentrations, and there is no term in the regression model to 

account for this type of event.

Analysis of Residuals

The geographic distributions of residuals from the totalnitrogen models for agricultural and nonagricultural sites 

are shown in figure 4. More sites fall into the plus or minus 

0.25 log units range for the agricultural model compared to 

the nonagricultural model as a result of the difference in the 

overall error between the models. There is no clear geographic 

pattern evident in figure 4, sites with low to moderate residual 

values are found in all geographic regions, along with the 

occasional sites where the model did not accurately estimate 

total nitrogen.

 An alternative way to investigate the regional pattern 

is shown in figure 5. Sites are classified by USEPA nutrient 

ecoregions (U.S. Environmental Protection Agency, 1998) 

and the distributions of residuals for sites in each ecoregion 

are shown by box plots. Due to the small number of sites in 

many nutrient ecoregions, the residuals for the agricultural 

and nonagricultural site models were grouped together. Eighty 

percent of the residuals for every nutrient ecoregion fall within 

plus or minus 0.5 log units. With the exception of ecoregions 

4 and 14, where the regression models tend to underestimate, 

and ecoregion 7, where the models tend to overestimate, more 

than 50 percent of the residuals are within plus or minus 

0.25 log units.

Basin area upstream from the sites used for model 

development ranged from 17 to 220,970 square kilometers 

(Mueller and Spahr, 2005). Sites were grouped into quartiles 

of upstream basin area (for example, sites downstream from 

the smallest 25 percent of basins are in quartile 1), and the 

distribution of regression model residuals for each quartile 

are shown in figure 6. The slightly better estimations for the 

agricultural model are evident by the smaller span of the box 

plots for the agricultural sites. The agricultural model estimates total nitrogen slightly more accurately for the largest 

basins (quartile 4). However, there are not large differences in 

the predictive capability of the models among different size 

basins.

PREDICTED LOG10 TOTAL NITROGEN CONCENTRATION,

IN MILLIGRAMS PER LITER

OBSERVED LOG10 TOTAL NITROGEN CONCENTRATION, 

IN MILLIGRAMS PER LITER

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

–1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

1:1 line

Las Vegas Wash

Dry Fork

Sabinal River

Figure 3. Comparison of observed and predicted flow-weighted mean 

annual total-nitrogen concentrations at National Water-Quality Assessment 

sampling sites downstream from nonagricultural areas.
Development of Models for Total Nitrogen 9

EXPLANATION

Log10 Residual

Greater than –0.25 through 0.0

Greater than 0.5

Greater than 0.25 through 0.5

Greater than –0.5 through –0.25

Greater than 0.0 through 0.25

Less than or equal to –0.5

Study Unit

AGRICULTURAL SITES

NONAGRICULTURAL SITES

Figure 4. Geographic distribution of residuals for regression models of flow-weighted mean annual totalnitrogen concentrations at National Water-Quality Assessment sampling sites downstream from agricultural and 

nonagricultural areas.
10 Development and Application of Regression Models for Nutrients in U.S. Streams

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 ALL SITES

LOG10 RESIDUALS

NUTRIENT REGIONS

12

88

12

13

63

27

28

73

11 61

5

3

23

462

43

28

EXPLANATION

NUMBER OF SITES

MEDIAN

25th PERCENTILE

75th PERCENTILE

90th PERCENTILE

10th PERCENTILE

Figure 5. Distributions of residuals for regression models of flow-weighted mean annual total-nitrogen 

concentrations grouped by U.S. Environmental Protection Agency nutrient ecoregion (U.S. Environmental 

Protection Agency, 1998). Ecoregions 12 and 13 have insufficient sites to determine percentile for box plots.
Development of Models for Total Nitrogen 11

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

QUARTILE 1 QUARTILE 2 QUARTILE 3 QUARTILE 4

AGRICULTURAL SITES

NONAGRICULTURAL SITES

ALL SITES

LOG10 RESIDUALS

61

55

46

69

60

56

75

40

242

220

40

EXPLANATION

NUMBER OF SITES

MEDIAN

25th PERCENTILE

75th PERCENTILE

90th PERCENTILE

10th PERCENTILE

Figure 6. Distributions of residuals for regression models of flow-weighted mean annual totalnitrogen concentrations grouped by quartile of basin size.
12 Development and Application of Regression Models for Nutrients in U.S. Streams

Nationwide Prediction of 

Total-Nitrogen Concentrations

The calibrated regression models were used to predict flow weighted total-nitrogen concentrations in streams 

throughout the conterminous United States (fig. 7). The 

values represent mean annual flow weighted concentrations 

for the Cycle I period (1992–2001) of the NAWQA program. 

Explanatory variables in the models were evaluated based on 

geographic-information-system data for basins upstream from 

each node in the RF1 file. Each basin was classified as either 

agricultural or nonagricultural so the appropriate regression model could be applied. Model applications produced a 

predicted mean-annual concentration of total nitrogen at each 

RF1 node. The concentration predicted at a node was assumed 

to occur in the entire reach upstream to the next node.

Predicted total-nitrogen concentrations for about 37 percent of the stream miles throughout the Nation were less than 

1 mg/L, and predictions were less than 5 mg/L for another 

54 percent of stream miles. Predicted concentrations less than 

0.5 mg/L occurred primarily in mountainous regions: the 

Rocky Mountain, Sierra Nevada, Cascade, and Coastal ranges 

in the West; and the Appalachian and Adirondack ranges in 

the East. Low concentrations also occurred in northern areas 

around the Great Lakes and Maine. Concentrations greater 

than 5 mg/L were predicted within a broad area extending 

from Ohio to eastern Nebraska. Agricultural application 

of fertilizer and manure is high throughout this area, and 

atmospheric deposition of nitrogen is high in the eastern part. 

Predicted concentrations also are high around urban areas 

in California, Arizona, Colorado, and Texas. The lack of a 

national data set for point sources of nutrients and the subsequent unavailability of a true point-source term in the regression equations results in an under representation of urban 

areas in the national prediction (the residuals from the calibration of the nonagricultural sites model showed a positive bias 

for urban basins).

Mean annual concentrations of total nitrogen exceeding 

the USEPA drinking-water standard for nitrate (10 mg/L; U.S. 

Environmental Protection Agency, 2003) were predicted for 

1 percent of stream miles. Because nitrate is a component of 

the total nitrogen concentration value, the nitrate concentration in water always will be less than the concentration of 

total nitrogen. Figure 7, therefore, indicates that widespread 

exceedence of the nitrate drinking water standard in streams is 

unlikely.

The predicted concentrations have an associated uncertainty as a result of variability in the observations that is 

unexplained by the regression models. This uncertainty can be 

used to estimate the probability that a predicted concentration 

exceeds a reference value or any other specified concentration 

(see equation 1). If the predicted value is equal to a reference 

concentration there is a 50 percent probability that the actual 

flow-weighted annual mean concentration in the stream reach 

is less than the reference value and a 50 percent probability 

that the actual concentration is greater than the reference concentration. To illustrate the uncertainty of the regression models 

EXPLANATION

Greater than 0.5 through 1

Greater than 5 through 10

Greater than 1 through 5

Greater than 10

0.0 through 0.5

TOTAL-NITROGEN CONCENTRATION

MILLIGRAMS PER LITER

Figure 7. Predicted flow-weighted mean-annual concentrations of total nitrogen for streams in the U.S. 

Environmental Protection Agency River Reach File, 1992–2001.
Nationwide Prediction of Total-Nitrogen Concentrations 13

 PROBABILITY (PERCENT)

THAT TOTAL NITROGEN IS GREATER

THAN 1 MILLIGRAM PER LITER

Greater than 75

Greater than 25 through 50

Less than or equal to 25

Greater than 50 through 75

 PROBABILITY (PERCENT) 

THAT TOTAL NITROGEN IS GREATER

THAN 5 MILLIGRAMS PER LITER

Greater than 75

Greater than 25 through 50

Less than or equal to 25

Greater than 50 through 75

 PROBABILITY (PERCENT) 

THAT TOTAL NITROGEN IS GREATER

THAN 10 MILLIGRAMS PER LITER

A

B

C

Greater than 75

Greater than 25 through 50

Less than or equal to 25

Greater than 50 through 75

Figure 8. Probabilities that flow-weighted mean-annual concentrations of total nitrogen in streams are greater 

than 1, 5, and 10 milligrams per liter.
14 Development and Application of Regression Models for Nutrients in U.S. Streams

in the predicted concentrations, probabilities of concentrations 

being greater than 1, 5, and 10 mg/L are shown in figure 8.

The reaches colored red in figure 8A show the extensive 

areas where the probability is at least 75 percent that totalnitrogen concentration exceeds 1 mg/L. Figure 8B shows far 

fewer areas where total-nitrogen concentration in stream water 

is likely to exceed 5 mg/L. The probability that the concentration exceeds 5 mg/L is at least 75 percent in many agricultural 

areas of the Midwest and near some urban areas in the West. 

In most other areas, the probability of total-nitrogen concentrations exceeding 5 mg/L is no more than 25 percent. The 

probability of total-nitrogen concentrations exceeding 10 mg/L 

is low everywhere except for isolated areas in the Midwest and 

California (fig. 8C).

The probabilities that mean-annual concentrations of total 

nitrogen in streams exceed USEPA regional nutrient criteria 

(U.S. Environmental Protection Agency, 2002a) are shown 

in figure 9. Each RF1 stream reach was associated with the 

appropriate USEPA nutrient ecoregion. The predicted concentration in the reach was compared to the regional criterion and 

the probability that the estimated concentration exceeded the 

criterion was computed, based on the regression-model prediction error. In the vast majority of streams, the total-nitrogen 

concentration is likely to exceed the criteria (probability at 

U.S. Environmental Protection Agency nutrient ecoregions (total nitrogen criteria, in milligrams per liter)

 8-Nutrient-Poor, Largely Glaciated Upper Midwest and Northeast (0.38)

 9-South eastern Temperate Forested Plains and Hills (0.69)

10-Texas-Louisiana Coastal and Mississippi Alluvial Plains (0.76)

11-Central and Eastern Forested Uplands (0.31)

12-Southern Coastal Plain (0.90)

13-Southern Florida Coastal Plain (cirteria not developed)

14-Eastern Coastal Plain (0.71)

 

 

 

 

 1-Willamette and Central Valleys (0.31)

 2-Western Forested Mountains (0.12)

 3-Xeric West (0.38)

 4-Great Plains Grass and Shrubland (0.56)

 5-South Central Cultivated Great Plains (0.88)

 6-Corn Belt and Northern Great Plains (2.18)

 7-Mostly Glaciated Dairy Region 0.54)

4

6 7

8

14

13

1

2

3

5

10

11

12

9

EXPLANATION

Greater than 90 through 100

Greater than 75 through 90

Greater than 25 through 50

Less than or equal to 25

Greater than 50 through 75

Figure 9. Probabilities that flow-weighted mean-annual concentrations of total nitrogen 

in streams exceed the U.S. Environmental Protection Agency nutrient-ecoregion criteria.
Nationwide Prediction of Total-Nitrogen Concentrations 15

least 50 percent). These exceedence probabilities are determined for mean annual concentrations; the probability of a 

shorter term exceedence is even greater. For many streams the 

probability of exceeding the regional criteria is more than 90 

percent. Streams where the probability of exceeding the criteria 

is low are located primarily in the northern-most States and the 

Southeast, but in most of these streams there is still a 25 to 50 

percent probability that these concentrations could exceed the 

regional criteria.

The percent of river miles likely to exceed the total 

nitrogen criteria in each nutrient ecoregion is shown in figure 10. 

Results are not available for nutrient region 13, Southern 

Florida Coastal Plain, because nutrient criteria are unavailable 

for this area. In all other nutrient regions, the probability of 

exceeding the criteria is at least 50 percent for more than half 

of the stream miles. With the exception of nutrient regions 8, 

12, and 14, there is a 90 percent probability that the total nitrogen criteria will be exceeded in more than 48 percent of the 

stream miles. In nutrient regions 3 and 7 the nitrogen criterion 

is likely to be exceeded in almost 80 percent of streams with a 

probability of at least 90 percent.

G r e a t e r t h a n 5 0 G r e a t e r t h a n 7 5 G r e a t e r t h a n 9 0

P R O B A B I L I T Y O F M E A N A N N U A L T O T A L - N I T R O G E N C O N C E N T R A T I O N E X C E E D I N G T H E R E G I O N A L C R I T E R I A

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

N U T R I E N T E C O R E G I O N

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

P E R C E N T O F R I V E R M I L E S I N N U T R I E N T R E G I O N

Figure 10. Percentage of stream miles in each U.S. Environmental Protection Agency nutrient-ecoregion 

within which the probabilities of exceeding the regional total-nitrogen criteria are 50, 75, and 90 percent 

(criteria not available for ecoregion 13).
16 Development and Application of Regression Models for Nutrients in U.S. Streams

The total-nitrogen models presented herein are statistical 

models based on correlations of nonpoint source inputs, basin 

characteristics, and land use. These models provide reasonable 

estimates of total-nitrogen concentrations in streams that do 

not have an abundance of measured data. The models can be 

used to provide broad regional estimates of how nitrogen concentrations may change with reductions in nonpoint sources 

of nitrogen. To investigate the regional response of stream 

nitrogen concentrations to reductions in nonpoint sources of 

nitrogen, fertilizer inputs were reduced by specified amounts 

and the percent of river miles with a 90-percent probability 

of exceeding the nutrient criteria was computed (fig. 11). It 

should be noted that the models are not being used to specifically infer what may change in any particular stream but, 

rather, what may occur on a broad regional scale. The results 

suggest a very small reduction in the number of stream miles 

with a 90-percent probability of exceeding the criteria, even 

when fertilizer inputs are reduced by 25 percent. 

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

0 5 1 5 2 5

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 4

N U T R I E N T E C O R E G I O N

P E R C E N T O F R I V E R M I L E S I N N U T R I E N T R E G I O N

PERCENT REDUCTION 

IN FERTILIZER INPUT

Figure 11. Percentage of stream miles in each U.S. Environmental Protection Agency nutrient-ecoregion with a 90-percent 

probability of exceeding the regional total-nitrogen criteria with various reductions in fertilizer inputs.
Development of Models for Total Phosphorus 17

Development of Models for 

Total Phosphorus

Attempts to develop national regression models to 

estimate total phosphorus concentrations had limited success. 

Unlike the total nitrogen models, division of sites into agricultural and nonagricultural groups did not improve the explanatory capability of the equations. The group of explanatory 

variables that yielded the lowest model error is listed in table 

4. For this model, calibrated using data from 440 NAWQA 

sites, the root-mean-square error is 0.36 and the adjusted 

R-squared value is 0.52.

Significant source terms include nonpoint-source inputs 

from manure and population density, which is acting as a surrogate for point-source inputs as well as other nonpoint urban 

sources. Nonpoint-source input of phosphorus from fertilizer 

was not a significant explanatory variable at the national scale, 

perhaps because phosphorus tends to adsorb to soil particles 

rather than dissolve in water contributing to streamflow or 

groundwater. Soil erosion and soil phosphorus also were not 

significant explanatory variables; however, range land use was 

significant and might be functioning as a surrogate for soil 

erosion. Other studies (Peterson and others, 2004; Mueller and 

Spahr, 2006) have shown a correlation between range land use 

and phosphorus concentrations, probably as a result of mineral 

and biological sources of phosphorus on range lands and transport of particulate phosphorus to streams along with sediment. 

In areas without rangeland, lower phosphorus concentrations 

are associated with areas of less development as indicated 

by the negative coefficient for the forested land-use variable. 

Percent sand also has a negative coefficient. More sand in 

soil can increase the infiltration rate, leading to less sediment 

and associated phosphorus transport to streams. Similar to the 

model for total nitrogen at nonagricultural sites, base-flow 

index was negatively correlated with total phosphorus. At sites 

with a low percentage of baseflow, there is a greater likelihood 

of runoff events, which transport sediment and associated 

particulate phosphorus to the stream. 

Observed compared to estimated mean-annual concentrations of total phosphorus and the distribution of the regression 

model residuals are shown in figure 12. The scatter about the 

unity line for total phosphorus is greater than the scatter for 

the total-nitrogen models (figs. 2 and 3). The residuals are 

well distributed about zero, but the range indicates a large 

amount of variability in total-phosphorus concentrations is not 

explained by the model. The degree of unexplained variability 

precludes the use of this model for nationwide extrapolation 

because the predicted values would have large uncertainty and 

the resulting geographic patterns of phosphorous concentrations could be misleading.

Explanatory variable (transformation) Regression 

coefficient Standard error p-value

Intercept –0.31618 0.16884 0.0618

Manure phosphorus input, kg/km2

 (fourth root) 0.08150 0.01431 <.0001

Population density, per km2

 (fourth root) 0.08482 0.01547 <.0001

Range land use, percent of basin area 0.00560 0.00096310 <.0001

Forest land use, percent of basin area –0.00731 0.00075089 <.0001

Sand in soil, percent (fourth root) –0.23044 0.06481 0.0004

Base flow index, percent of long-term mean streamflow –0.00701 0.00132 <.0001

Table 4. Explanatory variables in the model of log10 flow-weighted mean annual total-phosphorus 

concentrations.

[kg/km2, kilograms per square kilometer; km2, square kilometer]
18 Development and Application of Regression Models for Nutrients in U.S. Streams

Comparison to Results from 

other Models

 Smith and others (1997) calibrated statistically based 

models of total-nitrogen and total-phosphorus loads for 

streams in the conterminous United States using data from 

the USGS National Stream-Quality Accounting Network 

(NASQAN). Collectively, the models are referred to as a 

spatially referenced regression on watershed attributes (SPARROW) model of nutrient transport. The SPARROW model has 

since been used for numerous purposes, including estimation 

of natural nutrient background concentrations and comparison 

to USEPA criteria (Smith and others, 2003). Predictions of 

total-nitrogen load from a revised national SPARROW model 

(Alexander and others, 2008) were converted to concentrations 

using long-term mean runoff (R.B. Alexander, U.S. Geological Survey, written commun., 2008). These SPARROW -based 

concentrations were compared by river reach to predicted 

total-nitrogen concentrations from the CONDOR model. 

Nutrient ecoregions were chosen as a geographic framework to summarize the comparison of CONDOR and SPARROW 

model results (fig. 13). The geographic distribution of areas 

with high and low concentrations is similar for the predicted 

concentrations from the CONDOR and SPARROW models. 

Regions with high predicted concentrations using CONDOR 

(ecoregions 5, 6, and 7) also show high concentrations for 

SPARROW. Both models also predict low concentrations in 

the Western Forested Mountains and Nutrient-Poor, Largely 

Glaciated Upper Midwest and Northeast (ecoregions 2 and 8). 

SPARROW results for these regions are also among the lowest 

for the country.

Although the regional patterns are similar between the 

two models, SPARROW results are typically greater than 

CONDOR predictied concentrations in many areas, and the 

differences are greater in areas of higher concentrations. 

The largest differences are found in the western part of the 

Corn Belt (in ecoregion 6) and the eastern part of the South 

Central Cultivated Great Plains (ecoregion 5). Large relative 

differences were also found in areas of low concentrations. 

Predicted concentrations using CONDOR in ecoregions 2 and 

8 (Western Forested Mountains and Nutrient-Poor, Largely 

Glaciated Upper Midwest and Northeast) are less than concentrations derived from SPARROW. 

–3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0.0 0.5 1.0

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0.0

0.5

1.0

PREDICTED LOG10 TOTAL PHOSPHORUS CONCENTRATION,

IN MILLIGRAMS PER LITER

OBSERVED LOG10 TOTAL PHOSPHORUS CONCENTRATION,

IN MILLIGRAMS PER LITER

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

RESIDUALS

LOG10 UNITS

MEDIAN

25th PERCENTILE

75th PERCENTILE

90th PERCENTILE

10th PERCENTILE

1:1 line

Figure 12. Comparison of observed and predicted total-phosphorus concentrations and the distribution of 

residuals from the regression model.
Comparison to Results from other Models 19

There are differences between the model structures of 

CONDOR and SPARROW that could cause differences in the 

estimated nitrogen concentrations. The SPARROW model uses 

nonlinear regression to solve a mass balance of nutrient loads 

as a function of spatially referenced nutrient inputs, land-use 

delivery proportions, and instream decay characteristics along 

the river reach. CONDOR, in contrast, uses linear regression 

to relate nutrient concentrations measured at stream sites to the 

drainage basin characteristics (such as nutrient inputs, soil and 

hydrologic variables) of those sites. SPARROW uses data from 

the NASQAN network for the years 1975–2000 with a base 

year of 1992 to derive the loads used for model calibration, 

whereas CONDOR is based on concentration, streamflow, 

and load data collected during the 3- or 4- year high-intensity 

phase for sites within each of the Cycle I NAWQA study units 

during 1992–2001.

ALL 

REACHES

NUTRIENT ECOREGIONS

TOTAL NITROGEN CONCENTRATION,

 IN MILLIGRAMS PER LITER

CONDOR SPARROW

Insufficient number of reaches

 to compare for ecroregion 13

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14

U.S. Environmental Protection Agency nutrient ecoregions

 8-Nutrient-Poor, Largely Glaciated Upper Midwest and Northeast

 9-South eastern Temperate Forested Plains and Hills

10-Texas-Louisiana Coastal and Mississippi Alluvial Plains

11-Central and Eastern Forested Uplands

12-Southern Coastal Plain

13-Southern Florida Coastal Plain

14-Eastern Coastal Plain

 

 

 

 

 1-Willamette and Central Valleys

 2-Western Forested Mountains

 3-Xeric West

 4-Great Plains Grass and Shrubland

 5-South Central Cultivated Great Plains

 6-Corn Belt and Northern Great Plains

 7-Mostly Glaciated Dairy Region

EXPLANATION

MEDIAN

25th PERCENTILE

75th PERCENTILE

90th PERCENTILE

10th PERCENTILE

4

6 7

8

14

13

1

2

3

5

10

11

12

9

Figure 13. Comparison of prediction total-nitrogen concentrations from this report and the SPARROW model (Alexander and 

others, 2008).
20 Development and Application of Regression Models for Nutrients in U.S. Streams

Summary and Conclusions

Nutrients (compounds containing nitrogen or phosphorus) are one of the primary synthesis topics of the USGS 

NAWQA program. During the Cycle I (1992–2001) of the 

NAWQA program, water samples were collected at a national 

network of stream sites to quantify the occurrence and distribution of nitrogen and phosphorus. Extrapolation of these data 

to unsampled areas was accomplished through the use of statistical regression models relating flow-weighted mean annual 

concentrations to nonpoint sources, basin characteristics, and 

hydrologic conditions. Explanatory variables are limited to 

data that are nationally comprehensive. Higher resolution 

localized data were not applicable to this analysis due to the 

limited spatial extent. For example, point-source loading of 

nitrogen and phosphorus is unavailable in a nationally consistent form and thus could not be used to assist in the explanation of observed nutrient concentrations at a national scale.

The concentrations of nutrients derived from regression (CONDOR) models were used to predict annual flowweighted mean concentrations of total nitrogen, representative 

of the Cycle I period of the NAWQA program, for drainage 

basins associated with the USEPA River Reach File. Manure, 

fertilizer, and atmospheric inputs of nitrogen, percent sand in 

soil, subsurface drainage, overland flow, mean annual precipitation, and percent undeveloped area were significant variables 

in the agricultural basin total nitrogen model. Significant 

explanatory variables in the nonagricultural total nitrogen 

model were total nonpoint-source nitrogen input (sum of 

nitrogen from manure, fertilizer, and atmospheric deposition), 

population density, mean annual runoff, and percent base flow. 

The majority of stream miles in the Nation have predicted 

concentrations less than 5 mg/L. Concentrations greater than 5 

mg/L were predicted for a broad area extending from Ohio to 

eastern Nebraska, areas spatially associated with greater application of fertilizer and manure. Predicted concentrations also 

are high around urban areas in California, Arizona, Colorado, 

and Texas. The lack of a national data set for point sources of 

nutrients and the subsequent unavailability of a true pointsource term in the regression equations results in an under 

representation of urban areas in the national prediction. Mean 

annual concentrations of total nitrogen exceeding the USEPA 

drinking water standard for nitrate (10 mg/L) were predicted 

for less than 1 percent of stream miles. Because nitrate is a 

nitrogen compound, its concentration in water can be no more 

than the concentration of total nitrogen. Thus, widespread 

exceedence of the drinking water standard in streams is 

unlikely in most of the Nation.

Predicted concentrations have an associated uncertainty 

as a result of variability unexplained by the regression models. 

Probabilities that concentrations will exceed a given reference 

value can be determined by incorporating model uncertainty. 

The probabilities that mean-annual concentrations of total 

nitrogen in streams exceed USEPA regional nutrient criteria 

were determined for reaches in the RF1 River Reach File. In 

all nutrient regions where criteria have been established, there 

is at least a 50 percent probability of exceeding the criteria in 

more than half of the stream miles.

Dividing calibration sites into agricultural and nonagricultural groups did not improve the explanatory capability for 

total phosphorus models. The group of explanatory variables 

that yielded the lowest model error for mean annual total 

phosphorus concentrations includes phosphorus input from 

manure, population density, amounts of range land and forest 

land, percent sand in soil, and percent base flow. The national 

regression model to estimate total-phosphorus concentration 

had limited success. Available national-scale explanatory 

variables did not adequately explain annual total-phosphorus 

concentrations. The unexplained variability precluded the use 

of the model for nationwide prediction because the predicted 

values would have an unreasonably large uncertainty.

Acknowledgments

This report relies on water samples that were collected 

over a 10-year period by hydrologists and hydrologic technicians in 51 NAWQA study units and analyzed by technicians 

at the USGS National Water Quality Laboratory in Denver, 

Colorado. Without all their conscientious efforts, the data 

interpretation in this report would not have been possible.

Richard Moore, Wesley Stone, and Betty Palcsak of 

the U.S. Geological Survey provided valuable comments on 

the report. The authors also thank Chelsea Carr for editorial 

review and Mary Berger for manuscript preparation, layout, 

and illustration assistance.
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Spahr and others—Development and Application of Regression Models for Nutrients in U.S. Streams—Scientific Investigations Report 2009–5199

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