U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2018–5145

National Water Quality Program

Catchment-Level Estimates of Nitrogen and Phosphorus 

Agricultural Use from Commercial Fertilizer Sales for the 

Conterminous United States, 2012

Predicted nitrogen fertilizer use, 

metric tons (quantile)

EXPLANATION

State

County

0

0.0001 to 0.0003 (20th

0.0004 to 0.052 (40th)

0.05 to 0.46 (60th)

0.46 to 3.11 (80th)

Greater than 3.11 (100th)
Cover photographs:

Top: Map showing predicted 2012 nitrogen fertilizer use at the catchment scale (mertic tons), unconditional predictions.

Bottom: Photograph by Lynn Betts, U.S. Department of Agriculture, Natural Resources Conservation Service.
Catchment-Level Estimates of Nitrogen 

and Phosphorus Agricultural Use from 

Commercial Fertilizer Sales for the 

Conterminous United States, 2012

By Jana S. Stewart, Gregory E. Schwarz, John W. Brakebill, and 

Stephen D. Preston

National Water Quality Program

Scientific Investigations Report 2018–5145

U.S. Department of the Interior

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

DAVID BERNHARDT, Acting Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2019

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Suggested citation:

Stewart, J.S., Schwarz, G.E., Brakebill, J.W., and Preston, S.D., 2019, Catchment-level estimates of nitrogen and 

phosphorus agricultural use from commercial fertilizer sales for the conterminous United States, 2012: U.S. Geological 

Survey Scientific Investigations Report 2018–5145, 52 p., https://doi.org/10.3133/sir20185145.

ISSN 2328–0328 (online)
iii

Foreword

Sustaining the quality of the Nation’s water resources and the health of our diverse ecosystems 

depends on the availability of sound water-resources data and information to develop effective, 

science-based policies. Effective management of water resources also brings more certainty and 

efficiency to important economic sectors. Taken together, these actions lead to immediate and 

long-term economic, social, and environmental benefits that make a difference to the lives of 

the almost 400 million people projected to live in the United States by 2050. (https://water.usgs.

gov/nawqa/applications/).

In 1991, Congress established the National Water-Quality Assessment (NAWQA) to address 

where, when, why, and how the Nation's water quality has changed, or is likely to change 

in the future, in response to human activities and natural factors. Since 1991, NAWQA has 

been a leading source of scientific data and knowledge used by national, regional, state, and 

local agencies to develop science-based policies and management strategies to improve and 

protect water resources used for drinking water, recreation, irrigation, energy development, 

and ecosystem needs. Plans for the third decade of NAWQA (2013–23) address priority waterquality issues and science needs identified by NAWQA stakeholders (such as the Advisory 

Committee on Water Information) and the National Research Council. The plans are designed to 

meet increasing challenges related to population growth, increasing needs for clean water, and 

changing land use and weather patterns.

Federal, state, and local agencies have invested billions of dollars to reduce the amount of 

pollution entering rivers and streams that millions of Americans rely on for a variety of water 

needs and biota rely on for habitat. Understanding the sources and transport of pollution is 

crucial for designing strategies to improve water quality. Studies have indicated that estimates 

of total nitrogen and total phosphorus commercial fertilizer use are empirically important 

for estimating water-quality conditions in streams using models such as the U.S. Geological 

Survey’s (USGS) SPAtially Referenced Regressions On Watershed attributes model know as 

SPARROW (Preston and others, 2011). This report describes the methods and subsequent 

results of two models developed for estimating elemental nitrogen and phosphorus commercial 

fertilizer use on agricultural lands for the conterminous United States at the National 

Hydrography DatasetPlus (NHDPlus) catchment/county scale for the year 2012. The results of 

these models will prove useful for any water-quality models that estimate total-nitrogen and 

total-phosphorus loads to streams and for other studies needing fertilizer use estimates related 

to agricultural cropping practices in the United States.

The authors hope this publication will provide insights and information to meet water resource 

needs and will foster increased citizen awareness and involvement in the protection and 

restoration of the Nation’s waters. The information in this report is intended primarily for those 

interested or involved in resource management and protection, conservation, regulation, and 

policymaking at the regional and national level.

Dr. Donald W. Cline

Associate Director for Water

U.S. Geological Survey
iv

Acknowledgments

The authors thank Dr. Larry Oldham, Mississippi State University, for his assistance and 

feedback on model development. 

The authors thank Michael Wieczorek, Andrew LaMotte, Shannon Jackson, Sylvia Terziotti, 

and Nancy Baker for the development and support of providing model input data. The authors 

thank Michael Woodside, Neil Dubrovsky, and Anna Garcia for assistance and feedback in 

model development and David Anning, David Saad, Daniel Wise, Scott Ator, Anne Hoos, and 

Victor Roland for reviewing model results. The authors also thank and acknowledge the reviews 

provided by Nancy Baker, James Falcone, and Karen Ryberg.
v

Contents

Foreword ........................................................................................................................................................iii

Acknowledgments........................................................................................................................................iv

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

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

Purpose and Scope .......................................................................................................................................3

Methods...........................................................................................................................................................3

Approach................................................................................................................................................3

Spatial Framework................................................................................................................................4

Source Data...........................................................................................................................................6

Fertilizer Sales .............................................................................................................................6

Fertilizer Expenditures ................................................................................................................6

Cropland Data Layer....................................................................................................................6

Climate9

Nitrogen and Phosphorus From Manure .................................................................................9

Estimation Methodology......................................................................................................................9

Prediction Methodology ....................................................................................................................13

Unconditional Prediction..........................................................................................................13

Conditional Prediction...............................................................................................................14

Prediction Uncertainty..............................................................................................................15

Results ...........................................................................................................................................................16

Model Estimation ................................................................................................................................16

Nitrogen.......................................................................................................................................17

Phosphorus.................................................................................................................................20

Residual Analysis................................................................................................................................24

Model Predictions...............................................................................................................................25

Prediction Uncertainty Estimates ....................................................................................................25

Model Evaluation .........................................................................................................................................32

Summary and Conclusions.........................................................................................................................42

References Cited .........................................................................................................................................43

Appendix 1. Description of the Method Used to Assign County Origin of Sales For Reported 

Fertilizer Sales With No County Identifier..................................................................................45

Appendix 2. Summary of Information For Cropland Data Layer 2012 Crops by Crop Group Used in 

the 2012 Nitrogen and Phosphorus Fertilizer-Use Models......................................................47

Appendix 3. Accuracy of Crop-Group Classification in the 2012 Cropland Data Layer ...................49

Appendix 4. Derivation of the Best Linear Unbiased (Conditional) Estimates of Fertilizer Use......50

Appendix 5. Mississippi State University Extension Service Fertilizer Recommendations For 

Selected Crops Used to Estimate Total Nitrogen and Phosphate Fertilizer Applied to 

Crops in Mississippi Counties ......................................................................................................52
vi

Figures

1. Map showing catchment/county spatial unit used for estimating 2012 nitrogen 

and phosphorus fertilizer-use models.......................................................................................5

2. Map showing county-level 2012 nitrogen fertilizer sales data used to estimate 

nitrogen fertilizer use ...................................................................................................................7

3. Map showing county-level 2012 phosphorus fertilizer sales data used to estimate 

phosphorus fertilizer use.............................................................................................................8

4. Map showing Cropland Data Layer 2012 crop groups used for estimating 2012 

nitrogen and phosphorus fertilizer-use models.....................................................................10

5. Map showing studentized residuals for 2012 nitrogen fertilizer-use model.....................19

6. Graph showing observed 2012 nitrogen fertilizer sales compared to estimated 

2012 nitrogen fertilizer use summarized at the county level ...............................................20

7. Graph showing observed 2012 nitrogen fertilizer sales compared to estimated 

2012 nitrogen fertilizer use summarized at the state level...................................................21

8. Map showing studentized residuals for 2012 phosphorus fertilizer-use model...............22

9. Graph showing observed 2012 phosphorus fertilizer sales compared to estimated 

2012 phosphorus fertilizer use summarized at the county level .........................................23

10. Graph showing observed 2012 phosphorus fertilizer sales compared to estimated 

2012 phosphorus fertilizer use summarized at the state level ............................................23

11. Map showing predicted 2012 nitrogen fertilizer use at the catchment scale 

(metric tons), unconditional predictions.................................................................................26

12. Map showing predicted 2012 phosphorus fertilizer use at the catchment scale 

(metric tons), unconditional predictions.................................................................................27

13. Map showing predicted 2012 nitrogen fertilizer use at the catchment scale 

(metric tons per square kilometer of cropland), unconditional predictions .....................28

14. Map showing predicted 2012 phosphorus fertilizer use at the catchment scale 

(metric tons per square kilometer of cropland), unconditional predictions .....................29

15. Map showing ratio of predicted 2012 nitrogen fertilizer use to predicted 2012 

phosphorus fertilizer use, unconditional predictions ...........................................................30

16. Graph showing predicted 2012 nitrogen fertilizer use from the unconditional, 

spatially referenced method aggregated to the county level compared to 2012 

county predictions from Brakebill and Gronberg (2017) using the Gronberg and 

Spahr (2012) method...................................................................................................................33

17. Graph showing predicted 2012 phosphorus fertilizer use from the unconditional, 

spatially referenced method aggregated to the county level compared to 2012 

county predictions from Brakebill and Gronberg (2017) using the Gronberg and 

Spahr (2012) method...................................................................................................................33

18. Graph showing predicted 2012 nitrogen fertilizer use from the conditional, 

spatially referenced method aggregated to the county level compared to 2012 

county predictions from Brakebill and Gronberg (2017) using the Gronberg and 

Spahr (2012) method...................................................................................................................34

19. Graph showing predicted 2012 phosphorus fertilizer use from the conditional, 

spatially referenced method aggregated to the county level compared to 2012 

county predictions from Brakebill and Gronberg (2017) using the Gronberg and 

Spahr (2012) method...................................................................................................................35

20. Graph showing predicted 2012 nitrogen fertilizer use from the unconditional, 

spatially referenced method aggregated to the county level compared to 2012 

county estimates produced by the International Plant Nutrition Institute (2012)............35
vii

21. Graph showing predicted 2012 phosphorus fertilizer use from the unconditional, 

spatially referenced method aggregated to the county level compared to 2012 

county estimates produced by the International Plant Nutrition Institute (2012)............36

22. Graph showing U.S. Department of Agriculture, Economic Research Service 

survey data of 2010 nitrogen fertilizer use on corn and soybeans compared to 

unconditional predictions of 2012 nitrogen fertilizer use on corn and soybeans 

(metric tons) for selected states ..............................................................................................37

23. Graph showing U.S. Department of Agriculture, Economic Research Service 

survey data of 2010 phosphorus fertilizer use on corn and soybeans compared 

to unconditional predictions of 2012 phosphorus fertilizer use on corn and 

soybeans (metric tons) for selected states............................................................................37

24. Graph showing recommendation-based nitrogen fertilizer estimates applied 

to agricultural crops in Mississippi counties compared to unconditional 

predictions, 2012 nitrogen fertilizer use (metric tons)..........................................................38

25. Graph showing recommendation-based phosphate fertilizer estimates applied 

to agricultural crops in Mississippi counties compared to unconditional 

predictions, 2012 phosphate fertilizer use (metric tons)......................................................38

26. Map showing predicted 2012 nitrogen fertilizer use at the catchment scale, 

metric tons per square kilometer of cropland in the State of Mississippi, based 

on unconditional predictions from the spatially referenced model. ..................................39

27. Map showing predicted 2012 phosphate fertilizer use at the catchment scale, 

metric tons per square kilometer of cropland in the State of Mississippi, based 

on unconditional predictions from the spatially referenced model ...................................40

28. Map showing predicted 2012 fertilizer use, nitrogen to phosphorus ratio for 

catchments in the State of Mississippi, based on unconditional predictions from 

the spatially referenced model.................................................................................................41

Tables

1. Model input variables for estimating 2012 nitrogen and phosphorus fertilizer-use 

models ............................................................................................................................................4

2. Model coefficient estimates and statistics for the 2012 nitrogen and phosphorus 

fertilizer-use models of the conterminous United States.....................................................18

3. Residual variance and covariance estimates for the nitrogen and phosphorus 

fertilizer models...........................................................................................................................24

4. Average of catchment predicted nitrogen and phosphorus fertilizer use 

coefficient of variation, by hydrologic region and conterminous United States .............31

5. Statistical summary of model evaluation comparisons for 2012 nitrogen and 

phosphorus fertilizer-use models.............................................................................................32

1.1. Results of the regression of state estimates of the variance across counties 

of the quantity defined by the reported county share of total state fertilizer sales 

plus the product of the county share of state fertilizer expenditures and the 

state share of county assigned sales (S0c + sc (S* – 1)) on the state share of sales 

unassigned to a county (S*

) where * represents an individual state.................................46

2.1. Summary of information for Cropland Data Layer 2012 crops by crop group 

used in the 2012 nitrogen and phosphorus fertilizer-use models.......................................47

3.1. Accuracy of crop-group classification in the 2012 Cropland Data Layer.....................49

5.1. Mississippi State University Extension Service fertilizer recommendations for 

selected crops used to estimate total nitrogen and phosphate fertilizer applied to 

crops in Mississippi counties ...................................................................................................52
viii

Conversion Factors

International System of Units to U.S. customary units

Multiply By To obtain

Length

meter (m) 3.281 foot (ft)

kilometer (km) 0.6214 mile (mi)

Area

square meter (m2) 10.7639 square foot (ft2)

square kilometer (km2) 247.1 acre

Mass

kilogram per year (kg/yr) 2.205 pound avoirdupois per year (lb/yr)

metric ton (t) 2204.62 pound, avoirdupois (lb)

millimeter per year (mm/yr) 0.0393701 inch per year (in/yr)

Application rate

metric ton per square kilometer (t/km2) 8.92179122 pound per acre (lb/acre)

Datum

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

Abbreviations

AAPFCO Association of American Plant Food Control Officials

BLUE Best Linear Unbiased Estimator

COA Census of Agriculture

CDL Cropland Data Layer

IPNI International Plant Nutrition Institute

NAWQA National Water-Quality Assessment

NHDPlus National Hydrography Data Plus

NHDPlusV2 National Hydrography Data Plus version 2

P2O5 phosphate

R2 coefficient of determination

RMSE root mean square error

USDA U.S. Department of Agriculture

USGS U.S. Geological Survey
x
Catchment-Level Estimates of Nitrogen and Phosphorus 

Agricultural Use from Commercial Fertilizer Sales for the 

Conterminous United States, 2012

By Jana S. Stewart, Gregory E. Schwarz, John W. Brakebill, Stephen D. Preston

Abstract

Nutrient inputs from commercial agricultural fertilizer, 

particularly nitrogen and phosphorus, are important factors 

contributing to the degradation of surface-water quality and 

the alteration of aquatic ecosystems. Despite this importance, 

information about the application of fertilizer to agricultural 

land is not available in a consistent manner across the United 

States at a scale useful for regional water-quality assessment. 

To address this need, an approach is developed to relate 

commercial fertilizer sales to a set of explanatory variables 

using spatially referenced modeling methods. Spatially 

referenced modeling in this study refers to statistically relating 

fertilizer use, estimated from commercial fertilizer sales 

data, to spatially referenced data on watershed attributes. 

Separate models for nitrogen and phosphorus are developed 

to estimate elemental fertilizer use on agricultural lands for 

the conterminous United States at the National Hydrography 

Dataset Plus (NHDPlus) catchment scale for the year 2012. 

The approach builds on earlier efforts that use Association of 

American Plant Food Control Officials data on fertilizer sales 

to provide county-level estimates of nitrogen and phosphorus 

fertilizer use. The spatially referenced method improves on 

these efforts by allowing for varying nitrogen to phosphorus 

ratios at the catchment scale and expanding the set of variables 

used to allocate county-level sales data to the catchment scale. 

The models include catchment-level factors that are either 

primary determinants of fertilizer use, such as the acreage of 

different crop types, or measures reflecting the intensity of use, 

such as climate. Explanatory variables available only at the 

county scale, such as U.S. Department of Agriculture Census 

of Agriculture estimates of fertilizer expenditures, are included 

to improve the model predictions of elemental use. The 

nitrogen and phosphorus models explain more than 90 percent 

of the variation in elemental use at the state level, and the 

statistical approach allows for the estimation of uncertainty of 

predicted use in each catchment. The spatial patterns of model 

predictions reflect known agricultural cropping practices 

across the United States that transcend political boundaries, 

despite the county/state orientation of the fertilizer sales 

information. The results are expected to be useful for a variety 

of water-quality assessments that are intended to estimate 

nitrogen and phosphorus loads to streams. 

Introduction

Fertilizer is a principal source of nutrient contamination 

in U.S. streams and rivers, leading to eutrophic conditions 

that can affect the ecological health of aquatic environments, 

including the streams and estuaries into which they discharge 

(Alexander and others, 2008). To better understand the 

ecological consequences of this source of nutrients, models 

of water quality are created that relate the locational intensity 

of the fertilizer source to water-quality conditions at specific 

monitoring locations in the stream network. The identification 

of spatial referencing as an important feature for improving 

model accuracy (Smith and others, 1997) demonstrates a need 

to incorporate into these models highly spatially resolved 

fertilizer use information.

A principal source of information on fertilizer use 

is contained in the yearly estimates of county and state 

fertilizer sales, distinguished by elemental composition, 

produced by the Association of American Plant Food Control 

Officials (AAPFCO) (Association of American Plant Food 

Control Officials, 2015). Use of these data presents several 

complications. First, the data pertain to sales rather than use, 

with the understanding that sales made at a given location 

are commonly transported across county and state borders 

to the location of use. Second, the data are not uniformly 

available at a common scale, with many states, principally in 

the western United States, having only state-level information. 
2 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Third, most states reporting county sales data have some 

sales that are not assigned to any county; in some states, 

this class of sales is large relative to total state sales. Fourth, 

to make the data useful in a water-quality model, applying 

additional methods is necessary to allocate state- or countylevel fertilizer use to a much finer spatial scale. In a study by 

Ruddy and others (2006), modified by Gronberg and Spahr 

(2012), a method was developed to use the AAPFCO annual 

fertilizer sales data in combination with U.S. Department of 

Agriculture (USDA) Census of Agriculture (COA) fertilizer 

expenditure data (U.S. Department of Agriculture, National 

Agricultural Statistics Service, 2004) to obtain countylevel estimates of fertilizer use for nitrogen and phosphorus 

elemental components. This approach employed the following 

two fundamental assumptions: (1) at the state scale, fertilizer 

sales are equivalent to use, and (2) the ratio of nitrogen to 

phosphorus is constant across all counties in a state. Further 

allocation of the Gronberg and Spahr (2012) county estimates 

to individual catchments has been achieved using 30-meter 

land use information contained in the 2006 National Land 

Cover Dataset (Wieczorek and others, 2018; Fry and others, 

2011). The estimates from Ruddy and others (2006) have 

been determined to be empirically important in total nitrogen 

and total phosphorus water-quality models (Alexander and 

others, 2008).

The method described in this study uses spatially 

referenced statistical modeling methods to analyze the 

AAPFCO sales data to produce estimates of use at the 

catchment scale for the year 2012. Spatially referenced 

modeling refers to statistically relating observations, in this 

case, fertilizer sales, to spatially referenced data on watershed 

attributes (Smith and others, 1997). In so doing, the method 

directly relates the scale at which fertilizer data are compiled 

to the scale at which these data are employed in waterquality models. Multiple sources of ancillary information 

are incorporated into the analysis, including the USDA COA 

county fertilizer expenditure data, detailed 30-meter cropping 

pattern information, climate, and agricultural practices 

information (LaMotte, 2015; U.S. Department of Agriculture, 

National Agricultural Statistics Service, 2013; Wieczorek and 

others, 2018; Gronberg and Arnold, 2017a, Gronberg and 

Arnold, 2017b). A useful feature of the statistical approach 

is that it is possible to assess the uncertainty of the estimates, 

and to make use of observed fertilizer sales at larger scales, 

considered concordant with use, to improve the estimates of 

use at the smaller catchment scale, by relating fertilizer sales 

to a set of explanatory variables. For this study, catchment 

refers to the spatial unit defined by the local drainage area to 

each individual stream segment as defined by the National 

Hydrography Dataset Plus version 2 (NHDPlusV2) dataset 

(McKay and others, 2012; Moore and Dewald, 2016).

As with the approach by Ruddy and others (2006), the 

encompassing assumption used to implement the spatially 

referenced modeling approach is that at larger scales (such 

as states) fertilizer sales estimates reflect actual fertilizer 

application to the land surface, what the authors refer to as 

fertilizer “use”. Under this assumption and with the proviso 

that the model is specified to reflect factors affecting use 

rather than factors affecting sales, the estimation of the model 

with fertilizer sales data will yield a predictive model for 

use. In accordance with this understanding, the determinants 

of use are based primarily on the spatial extent of various 

crop types within a given region. The spatial extent and 

pattern of where fertilizer application, “use”, occurs on the 

land surface is determined by the crops grown on that land 

and information on cropping patterns. Thus, the specific 

crop type and area within a catchment is used in this study 

to define the spatial extent of fertilizer use and the type and 

amount of fertilizer applied within a catchment. These crop 

variables are referred to as extensive factors in this report. 

Crop nutrient requirements also vary based on geographically 

specific natural and anthropogenic factors that may influence 

the amount and intensity of fertilizer application. These 

factors may include fertilizer expenditures in the region, the 

prevalence of animal manure, and other climate conditions 

and are referred to as intensive factors in this study. In forming 

these relations, spatial referencing implies the interaction of 

the intensive factors, factors that do not depend on spatial 

extent (such as climate) with the extensive factors (such 

as cropland), is at the highest possible spatial resolution of 

the model—the intersection of individual stream segment 

catchment areas and counties. 

Two approaches to obtaining predictions are taken. 

First, is the unconditional approach where fertilizer use at the 

catchment scale is predicted based solely on the explanatory 

variables employed in the statistical analysis, with a correction 

for retransformation bias because of model error. Second, is 

the conditional approach where the observed fertilizer sales at 

the state level are apportioned to individual catchments based 

on the catchment’s share of state-level use as determined from 

the unconditional predictions. The final model predictions 

reflect relatively consistent elemental composition use patterns 

that transcend political boundaries, despite the county/state 

orientation of the fertilizer information. Because the method 

does not assume a fixed relation between nitrogen and 

phosphorus use, models for each being derived separately, 

predictions of the ratio of nitrogen to phosphorus use 

reproduce recognized patterns in farming practices. 
Methods 3

Model results are evaluated in four ways. First, 2012 

predictions derived from these models are compared to 

those generated by applying the Gronberg and Spahr 

(2012) method for the same 2012 period (Brakebill and 

Gronberg, 2017). Second, model predictions are compared 

to 2012 county-level estimates of nitrogen and phosphorus 

input, generated from AAPFCO fertilizer sales data by the 

International Plant Nutrition Institute (IPNI) (International 

Plant Nutrition Institute, 2012). Third, predictions are 

compared to USDA Economic Research Service survey data 

of reported nitrogen and phosphorus fertilizer use on corn 

and soybeans (U.S. Department of Agriculture, Economic 

Research Service, 2016) in selected states. Fourth, model 

predictions are compared to 2012 estimates of fertilizer use 

for Mississippi, generated from USDA, Farm Service Agency 

2012 crop acreage data and Mississippi Extension fertilizer 

recommendations (U.S. Department of Agriculture, Farm 

Service Agency, 2013; Oldham, 2012). 

Purpose and Scope

The purpose of this report is to describe the methods and 

subsequent results of two models developed for estimating 

elemental nitrogen and phosphorus commercial fertilizer use 

on agricultural lands for the conterminous United States at 

the catchment scale for the year 2012. The overall objective 

of this approach is to improve upon techniques currently used 

to estimate commercial fertilizer use on agricultural lands, 

including the estimated ratio of nitrogen to phosphorus applied 

to these lands (Gronberg and Spahr, 2012; Brakebill and 

Gronberg, 2017). The approach relates reported fertilizer sales 

to a set of explanatory variables using spatially referenced 

statistical modeling methods. The estimation of the nitrogen 

and phosphorus fertilizer-use models are described, and 

maps showing the predicted use at the catchment scale are 

presented. As validation of the approach, the nitrogen and 

phosphorus use predictions are compared to other estimates of 

fertilizer use at county and state scales.

Methods

Approach

Information on fertilizer use, the mass of nutrients 

(nitrogen or phosphorus) applied to agricultural crops across 

the conterminous United States, is largely unavailable at a 

scale needed for water-quality assessment. Previous studies 

were completed to address this issue and have determined 

that county-level estimates of fertilizer use (nitrogen and 

phosphorus) can be obtained, using existing datasets of 

county-level sales and county-level fertilizer expenditures 

(Ruddy and others, 2006; Gronberg and Spahr, 2012). In 

those studies, county-level fertilizer expenditure information 

(the annual dollar amount expended on commercial fertilizer 

products) was used to allocate county-level fertilizer sales 

(annual tonnage of elemental nitrogen and phosphorus 

fertilizer sold) (Association of American Plant Food Control 

Officials, 2015) to individual counties, with the assumption 

that at the state scale, fertilizer sales are equivalent to use. 

The method described in this study builds upon the 

earlier studies, with similar assumptions, but uses spatially 

referenced statistical modeling methods to relate fertilizer 

sales to a set of explanatory variables (such as crop type, 

crop acreage, fertilizer expenditures, climate factors, and 

agricultural practices) that are primary determinants of 

where and how much fertilizer mass is being used across the 

conterminous United States. The authors of this report propose 

(1) that the explanatory information will help allocate countylevel fertilizer sales to the catchment level (despite exports of 

fertilizer mass across county lines) and (2) will provide a basis 

for fertilizer use estimates to vary spatially across crop types 

(extensive factors) because of varying nutrient requirements 

and natural and anthropogenic conditions (intensity factors) 

that may affect the amount of fertilizer application. The 

spatially referenced modeling method also provides a means 

to allocate fertilizer use at finer scale (catchment/county 

units) than earlier county-level studies, allows for variation 

of the nitrogen phosphorus ratio by catchment, and provides 

a means to quantify statistical significance and uncertainty in 

the estimates. 

The fundamental spatial unit for modeling is the 

catchment/county (the intersection of NHDPlusV2 catchment 

and county boundaries) by which all explanatory variables 

are calculated from catchment-level and county-level datasets 

(table 1) (Moore and Dewald, 2016; LaMotte, 2016). The 

approach assumes that sales at some aggregate scale (such 

as state scale) approximate fertilizer use at that scale, and the 

model is specified with predictor variables that are assumed 

to be independent of net fertilizer exports. Under these 

assumptions, a fertilizer sales model estimated with countylevel data will predict fertilizer use.
4 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Table 1. Model input variables for estimating 2012 nitrogen and phosphorus fertilizer-use models.

[kg/yr, kilogram per year; PRISM, Parameter-elevation Regression on Independent Slopes Model; NHDPlus, National Hydrography Dataset Plus; 

mm/yr, millimeter per year; m2, square meter; NA, not applicable]

Variable name Variable 

type Description Units Source citation

Variable category—Agricultural practices

Nitrogen from 

manure

Intensive 2012 mean annual nitrogen from animal manure. 

Computed from county estimates of animal 

populations.

kg/yr Gronberg and Arnold, 2017

Phosphorus from 

manure

Intensive 2012 mean annual phosphorus from animal manure. 

Computed from county estimates of animal 

populations.

kg/yr Gronberg and Arnold, 2017

Variable category—Climate

Precipitation Intensive 2012 average annual precipitation in mm/yr based on 

PRISM attributed to NHDPlus version 2.1 reach 

catchments.

mm/yr Wieczorek and others, 2018

Actual 

evapotranspiration

Intensive Average annual actual evapotranspiration (2000–14) 

in mm/yr attributed to NHDPlus version 2.1 reach 

catchments. 

mm/yr Wieczorek and others, 2018

Variable category—Cropland

Crop group Extensive Area of crops grown based on the 2012 Cropland Data 

Layer, aggregated into five crop groups (appendix 2).

m2 U.S. Department of Agriculture, 

National Agricultural 

Statistics Service , 2013

Variable category—Fertilizer

County farm 

fertilizer 

expenditures

Intensive U.S. Department of Agriculture Census of Agriculture 

2012 county fertilizer expenditures. County estimates 

of dollar amounts spent on farm fertilizer products; 

replacement values for 2012 missing data from 2009 

and 2002 reported expenditures.

dollar/year LaMotte, 2015

Fertilizer price index NA U.S. Department of Agriculture Economic Research 

Service national price index used to normalize 

fertilizer expenditures from different years to an 

equivalent 1992 expenditure.

1992 = 100 U.S. Department of Agriculture, 

Economic Research Service, 

2016a

County fertilizer 

sales

Dependent Association of American Plant Food Control Officials 

2012 annual fertilizer sales data. Farm fertilizer 

products reported sold by nutrient content. Reported 

by individual states, in most cases at the county level.

kg/yr Association of American Plant 

Food Control Officials, 2015

Spatial Framework

The NHDPlusV2 (McKay and others, 2012) is a 

digital network of streams developed at 1:100000 scale with 

associated catchments. Catchments are polygons defined by 

30-meter digital elevation data that represent areas draining 

each stream segment (Moore and Dewald, 2016). The 

fundamental spatial unit of the modeling approach, which 

is the unit by which all explanatory data are defined, is the 

spatial intersection of the NHDPlusV2 catchment boundaries 

and county boundary polygons (LaMotte, 2016), referred to as 

catchment/county units (Moore and Dewald, 2016; LaMotte, 

2016). Numerical values representing each specific model 

variable are assigned to these units. This unit of analysis 

permits aggregation to either counties—the smallest regional 

unit for fertilizer sales data or catchments—the fundamental 

spatial unit for water-quality models. Approximately 

2.65 million NHDPlusV2 catchments are across the 

conterminous United States, which upon intersection with 

counties results in nearly 2.88 million catchment/county 

units (fig. 1).
Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000(McKay and others, 2012) and U.S. Census Bureau county boundaries 1:100,000 (LaMotte and others, 2016)

83°46' 83°44' 83°42' 83°40' 83°38' 83°36' 83°34' 83°32' 

32°46'32°44'32°42'32°40'32°38'

Catchment is the model mapping unit.It is a unique NHDPlusV2 catchmentthat may or may not span county boundaries.

Catchment/county is the model estimation unit. It isan NHDPlusV2 catchment within a single countyor a catchment that is split into two uniquespatial units because it spans two separate counties.

EXP

Estima

met

00.000.000.050.463.11

0 1 2 4 MILES

0 1 2 4 KILOMETERS

County 

Figure 1. Catchment/county spatial unit used for estimating 2012 nitrogen and phosphorus fertilizer-use models.
6 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Source Data

Development of the nitrogen and phosphorus fertilizer 

models utilizes six input datasets including (1) state reported 

commercial farm-fertilizer sales data, (2) reported commercial 

farm-fertilizer expenditure data, (3) crop-specific land cover, 

(4) mean annual precipitation, (5) evapotranspiration, and 

(6) nitrogen and phosphorus estimates from animal manure. 

The source data used in this study are listed in table 1. 

Variables serving as intensive variables in the model are 

logarithm transformed. To accommodate this transformation, 

any zero values for these variables are substituted with the 

corresponding minimum, nonzero value for that same variable. 

The dataset used to estimate catchment-level nitrogen and 

phosphorus fertilizer-use models is available for download as 

a U.S. Geological Survey (USGS) data release from the USGS 

ScienceBase Catalog (Stewart and others, 2019a).

Fertilizer Sales 

The principal source of fertilizer information is 

the AAPFCO commercial fertilizer sales data for 2012 

(Association of American Plant Food Control Officials, 

2015). These data are available for purchase, by elemental 

composition, for all states in the conterminous United States 

and by county for many states. During our analysis, the data 

for Wyoming consistently generated anomalous results and are 

deemed unreliable and excluded from model estimation. 

All fertilizer sold in a year is assumed to be applied in 

that same year. Negative sales in the data, which indicate 

fertilizer returns and account for less than 1 percent of total 

nitrogen or phosphorus elemental sales, are set to zero. 

Fertilizer sales data are converted from metric tons of product 

to metric tons of elemental nitrogen and phosphorus based on 

chemical composition data reported for each product. Each 

state reports the expected use of individual fertilizer products 

as being for farm or nonfarm use; only products identified 

as being for farm use are used in this study. Most states 

with county-level sales information include a component 

of sales that is not assigned to any county, with the share of 

this component to total state sales displaying considerable 

variation across states. 

To address this complication, the fertilizer sales data are 

adjusted as follows, prior to analysis. For states with more 

than 20 percent of elemental mass nitrogen plus phosphorus 

total sales of unknown county origin (coded as 998), the 

county attributed sales data are deemed unreliable, and all 

county-level sales numbers, for nitrogen and phosphorus, are 

aggregated to form a state-level reporting-unit sales estimate 

(figs. 2–3, respectively). This criterion affected the aggregation 

of county-level information for seven states. For the remaining 

states (those with less than 20 percent of total sales unassigned 

to any county), sales data for the county-level reporting unit 

is retained. In the case of counties with no reported sales, the 

zero-sales county data are aggregated into a single multicounty 

reporting unit within the state. If the state had unassigned 

sales, those sales are distributed to the reporting counties and 

multicounty units according to each unit’s share of cropland 

and USDA COA fertilizer expenditure (U.S. Department of 

Agriculture, National Agricultural Statistics Service, 2013; 

LaMotte, 2015) using a signal extraction method described 

in appendix 1. Counties (or multicounty) with no sales after 

allocation of county-unreported sales are excluded from 

model estimation. 

Fertilizer Expenditures

Fertilizer expenditures, reported by the USDA COA 

(LaMotte, 2015), present the dollar amount expended on 

commercial farm fertilizer products and soil conditioners 

by county in the year 2012. For 43 counties where census 

reporting requirements prevent the reporting of 2012 

expenditures, the most recent year of expenditures prior 

to 2012 are used (34 counties use 2007 expenditures and 

9 counties use 2002 expenditures). To ensure comparability of 

expenditure estimates across multiple years, all expenditures 

are normalized to a common base year, 1992, using the 

fertilizer price index (U.S. Department of Agriculture, 

Economic Research Service, 2016). An additional 52 counties 

contain little cropland and report no expenditures for any of 

the agricultural census, in which case expenditures are set 

to zero.

Cropland Data Layer

The spatial extent and distribution of crops differ across 

the United States, as do their nutrient (such as nitrogen and 

phosphorus) requirements. Understanding the location and 

spatial extent of crop types is important for understanding the 

amount and type of fertilizer application on those lands. The 

Cropland Data Layer (CDL) 2012 (fig. 4), maps parcels of 

land at 30-meter resolution classified into 1 of 108 different 

cultivated crop types, using a decision tree classifier in 

conjunction with medium resolution satellite imagery from a 

variety of sensors. These data are spatially intersected with the 

boundaries of the catchment/county data layer to determine 

the area of each crop grown within each catchment/county 

unit in the conterminous United States (U.S. Department of 

Agriculture, National Agricultural Statistics Service, 2013; 
EXPLANATION

County-level nitrogen fertilizer sales data, metric tons (quantile)

00.1 to 145 (20th)145.1 to 673 (40th)673.1 to 2,015 (60th)2,015.1 to 6,427 (80th)6,427.1 to 1,225,402 (100th)

States with state-level salesStates with county-level sales (no sales of unknown origin)States with county-level sales (< 20 percent sales of unknown county origin)States with county-level sales (> 20 percent sales of unknown county origin)

0 200 400 800 KILOMETERS0 200 400 800 MILES Base from U.S. Census Bureau county boundaries, 1:100,000

(LaMotte and others, 2016)

130° 120° 110° 100° 90° 80° 70° 

40°30°20°

Figure 2. County-level 2012 nitrogen fertilizer sales data used to estimate nitrogen fertilizer use (Association of American Plant Food Control Officials
Base from U.S. Census Bureau county boundaries, 1:100,000 (LaMotte and others, 2016) 130° 120° 110° 100° 90° 80° 70° 

40° 30° 20° 0 200 400 800 MILES 0 200 400 800 KILOMETERS EXPLANATION County-level phosphorus fertilizer sales data, metric tons (quantile) 00.1 to 19 (20th) 19.1 to 91 (40th) 91.1 to 274 (60th) 274.1 to 854 (80th) 854.1 to 175,073 (100th) States with state-level sales States with county-level sales (no sales of unknown origin) States with county-level sales (< 20 percent sales of unknown county origin) States with county-level sales (> 20 percent sales of unknown county origin) Figure 3. County-level 2012 phosphorus fertilizer sales data used to estimate phosphorus fertilizer use (Association of American Plant Food Control O
Methods 9

Wieczorek and others, 2018). Crops of similar types or with 

similar characteristics are grouped into the following five crop 

groups: (1) corn (corn, sweetcorn, popcorn, and double crops 

with corn), (2) grass-pasture-hay (grassland, pastureland, and 

nonalfalfa hay), (3) miscellaneous (81 fruit, vegetable, and 

other miscellaneous agricultural crops), (4) nitrogen fixing 

(soybeans and nitrogen-fixing cover crops of alfalfa, clover, 

and vetch), and (5) small grains (barley, durum wheat, millet, 

oats, rye, sorghum, speltz, spring wheat, triticale, winter 

wheat, and small grain double crops). Nitrogen-fixing chick 

peas, dry beans, lentils, peas, and peanuts were included 

with other miscellaneous crops and accounted for less than 

1 percent of all cropland. Fallow cropland was not included 

in the groupings and accounted for less than 4 percent of 

total cropland. Specific groupings of crop types are described 

in table 2.1, and the prevalence of the crop groups across 

the conterminous United States is shown in figure 4. The 

groupings are sufficiently broad that errors in classification 

among the groupings are relatively minor, with overall 

classification accuracy among agricultural land of nearly 

94 percent, as based on the CDL error matrices provided with 

the data (table 3.1). Evidence indicates, however, that the 

CDL classification is not highly accurate in differentiating 

agriculture from nonagriculture, particularly because 

of commission and omission errors associated with the 

classification of grass-pasture-hay (table 3.1). 

Climate

Climate (such as precipitation, temperature, 

evapotranspiration) plays an important role in crop production 

influencing plant growth, nutrient uptake, soil microorganisms, 

and other factors, thereby influencing the type and amount 

of commercial fertilizer needed for crops. Two climatic data 

layers are used in the models as intensive variables—2012 

mean annual precipitation and a long-term average of actual 

evapotranspiration (2000–14). Precipitation is derived from 

monthly rainfall estimates at various locations throughout 

the country, generalized to the land surface, and averaged 

across all months as one of the products of the Parameterelevation Regression on Independent Slopes Model (PRISM) 

dataset (McCabe and Wolock, 2011; Wieczorek and others, 

2018; Wolock and McCabe, 2018). Average annual actual 

evapotranspiration, in millimeters per year, is based on 

monthly estimates for 4-kilometer grid cells, derived from a 

water balance model (McCabe and Wolock, 2011; Wolock and 

McCabe, 2018), averaged during 2000ؘ–14, and apportioned to 

individual catchments (Wieczorek and others, 2018).

Nitrogen and Phosphorus From Manure

Animal manure is a recognized source of nitrogen 

and phosphorus and often utilized as a fertilizer source for 

pastureland, cropland, and hay production, thereby reducing 

the need for commercial fertilizer on some agricultural lands. 

County estimates of nitrogen and phosphorus from farm 

animal manure are derived from county animal population 

inventories compiled from the 2012 COA (LaMotte, 2015) 

and expected nutrient content in manure from each animal 

type (Ruddy and others, 2006). County-level farm animal 

population inventories are organized by animal categories 

of cattle, hogs, poultry, and other animals. The populations 

are multiplied by animal-specific estimates of nitrogen and 

phosphorus content in manure and summed by element 

to estimate total nitrogen and phosphorus from manure 

(Gronberg and Arnold, 2017a, Gronberg and Arnold, 2017b). 

County nutrient estimates are normalized by land area and 

apportioned to catchments based on the area of each county 

within a given catchment.

Estimation Methodology

The conceptual model of elemental fertilizer use is 

based on its observed relation with factors representing the 

extensivity and intensivity of use. In the model, the extensivity 

of use within a catchment/county is related to the land areas 

of a broad set of crop types that approximately scale with 

the extent of that type within each spatial unit. The intensity 

of use for each crop type within a spatial unit is related to 

several ancillary factors such as climate, the availability of 

commercial fertilizer substitutes like manure, and observed 

expenditures for total fertilizer. Relations between use and 

the crop type and intensity variables are quantified by a set 

of nonlinear model coefficients that must be statistically 

estimated. The coefficients are estimated through the 

application of nonlinear least-squares methods to minimize 

the sum of squared differences between observed county or 

state elemental sales and the predicted elemental use in each 

catchment/county, summed to the same level of aggregation as 

the sales observation (county or state). The model is spatially 

referenced because the interaction of variables explaining 

fertilizer use occurs at the fine scale of a catchment/county, 

whereas the data representing the dependent variable are at 

the coarser scale of a county or state. Differences between 

actual fertilizer use and fertilizer sales within a county or state, 

because of import or export, are quantified as uncertainty. 

Under specific statistical assumptions regarding these 

net exports, predictions from the model can be expected 

to correspond to elemental use, despite estimation with 

elemental sales data, and estimates can be made regarding the 

uncertainty of use.
Base from U.S. Census Bureau State boundaries, 1:500,000, 2014 130° 120° 110° 100° 90° 80° 70° 

40° 30° 20° EXPLANATION Crop groups used in fertilizer-use models Other land cover types Corn Developed land Fallow land Forest-shrubland-wetland Grass-pasture-hay Miscellaneous Nitrogen fixing Small grains Water Other nonagricultural land 0 200 400 800 MILES 0 200 400 800 KILOMETERS Figure 4. Cropland Data Layer 2012 crop groups used for estimating 2012 nitrogen and phosphorus fertilizer-use models.
Methods 11

Elemental fertilizer (either nitrogen or phosphorus) 

applied to a given crop-group k in catchment/county unit j, 

denoted Wjk, is assumed to be proportional to the area of the 

crop group, Ajk, with the rate of application specified to depend 

on a vector of spatially varying intensity variables, denoted 

Zjk, according to

W A jk  αk jk k   jk exp ' β Z (1)

where

αk is a crop-group-specific application factor;

Ajk is the area of the crop group;

βk is a crop-group-specific vector of coefficients, 

having the same number of elements as the 

vector Zjk, which mediate the effects of the 

intensity variables; and

Zjk is the vector of spatially varying intensity 

variables.

To improve the interpretability of the crop-group-specific 

application factors, the intensity variables are specified as 

deviations from their mean values. If the intensity variables are 

approximately normally distributed, then their transformation 

as deviations from a mean imply the application factor can be 

interpreted as the approximate median mass per unit area of 

elemental fertilizer applied to the given crop group.

Let Fi

u represent the elemental fertilizer use for spatial 

reporting unit i, either a state, county, or multicounty reporting 

unit. Let Ji represent the set of catchment/counties, j, that 

comprise reporting unit i, and let there be K identified crop 

groups, the collection of which comprises all agricultural area. 

The formulated model for reporting unit i is 

ln F A ln exp ' u i

u

k jk k jk k

K

j J i i

          α β Z 1 (2)

where 

Ji is the set of catchment/counties, j, that 

comprise reporting unit i;

K is the number of crop groups that comprise all 

agricultural area;

αk is a crop-group-specific application factor;

Ajk is the area of the crop group;

βk is a crop-group-specific vector of coefficients, 

having the same number of elements as the 

vector Zjk, which mediate the effects of the 

intensity variables;

Zjk is the vector of spatially varying intensity 

variables; and

ui is a model residual, possibly spatially 

correlated, having a mean of zero and a 

heteroscedastic variance σ2

ui

.

The model described by equation 2 adheres to massbalance constraints in that total fertilizer application for a 

reporting unit is simply a summation of the application at 

each fundamental spatial unit, j, and crop group, k. Moreover, 

fertilizer application scales directly with agricultural area in 

that a doubling of all Ajk causes a doubling of Fi

u

. The model 

formulation can be characterized as spatially referenced 

because the interaction of the intensity variables with the crop 

areas occurs at the smallest spatial scale of the model, the 

catchment/county scale.

The elemental composition of fertilizer use is unknown; 

therefore, the model in equation 2, is impossible to estimate 

directly. The available data for model estimation relate to the 

elemental composition of fertilizer sales, Fi

s

, with the relation 

between sales and use described by

ln F F ln x i

s

i

u   i (3)

where

Fi

u is the logarithm of elemental fertilizer use 

for spatial reporting unit t, either a state, 

county, or multicounty reporting unit; and

xi is the logarithm of a proportional net-export 

factor, assumed to have a mean of zero and 

heteroscedastic variance, σ2

xi, whereby if 

sales in reporting unit i exceed use then xi

is positive, and is negative otherwise. 

An assumption of the analysis is that the variance of xi goes 

to zero at the state level of aggregation. This assumption is 

consistent with the assumption employed in other studies 

(Ruddy and others, 2006; International Plant Nutrition 

Institute, 2012) that at the state scale, fertilizer sales 

approximate use. 

If Fi

u

is independent of xi, then estimation of equation 2

using Fi

s

instead of Fi

u does not lead to inconsistency in the 

estimated coefficients, although for smaller reporting units the 

residuals will tend to have a variance that exceeds σ2

i because 

of the additional variability induced by the variability of ηi. 

Because net exports are assumed to go to zero, the variance of 

residuals computed at the state level should better approximate 

σ2

i

. Similarly, if the CDL data contain classification error, then 

this error could also cause the variability of county-based 

residuals to overstate the variance of the fertilizer use residual. 

If the spatial correlation of this error is not too great, the error 

can be assumed to diminish with reporting-unit size, making 

the residual variance computed at larger scales essentially free 

of this bias. 
12 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

The residual obtained in model estimation, denoted ei, 

is a combination of the use residual, ui, and the net-export 

residual, xi, 

e u x i i   i (4)

where

ui is the use residual for reporting unit i; and

xi is the net-export residual. 

The analysis allows for spatial correlation of the residual 

terms, ui and xi, according to multicomponent processes. 

Without significant loss of generality, the fundamental spatial 

unit for correlation is assumed to be the county, indexed by 

c. The county use residual, uc, is assumed to consist of two 

components—a component that is common to all use residuals 

in the same state, denoted ηs(c), where s(c) refers to the state 

containing county c and a local component, εc. The state 

component is assumed to be independent across states, with 

a zero mean and variance of σ2

η. The local component has a 

zero mean and variance of σ2

ε and has a correlation of ρε

 with 

neighboring counties within the same state, but is independent 

across nonneighboring counties. 

A complication in empirically estimating the spatial 

correlation structure of the use residual is that the residual 

can only be observed at the state level; residuals at the county 

level are obscured by the net-export residual. However, it is 

possible to infer the correlation structure using only the statelevel residuals. Fertilizer use for state s, Fs

u

, is given by the 

sum of county use across all counties, C(s), in the state. In the 

logarithm space used for model estimation, 

ln F F ln e F ln w u s

u

c

u u

c C s c

u

c C s c c c C s

c                    (5)

where

C(s) is the set of all counties in state s;

Fc

u is the fertilizer use for county c;

euc is the estimated residual for county c;

wc is the county-c share of state-s elemental 

fertilizer use (wc = Fc

u

/Fs

u

); and

uc is the use residual for county c.

The approximation in equation 5 is based on a first-order 

Taylor series expansion of ln Fs

u

, with respect to uc evaluated 

at zero, for all c ε C(s). Thus, the state-level model residual, es, 

which is assumed to reflect the state-level residual for use, is 

given by

e w s cu w c s c c C s c c C s        η ε ( ) ( ) (6)

If the state residual is squared, the expectation is given by

E es c w wc d w c C s c C s d D c

2 2 2 2 2 

           σ σ η ε σ ρε ε ( ) ( ) ( ) (7)

where

C(s) is the set of all counties in state s;

wc is the county-c share of state-s elemental 

fertilizer use (wc = Fc

u

/Fs

u

);

D(c) is the set of all counties, indexed by d, that are 

in the same state and share a border with 

county c; and

wd is the share of state-elemental sales for a 

set of counties that share a border with 

county c.

In the empirical analysis, the set of county neighbors is 

based on the 1:2,000,000-scale county spatial coverage 

(Lanfear, 1994).

Equation 7 can be estimated for the parameters {σ2

η, σ2ε, 

ρε} using nonlinear least squares applied to the regression 

of the squared state residuals on state values of the two 

model-estimated sums, ∑cεC(s)w2

c and ∑cεC(s)wc ∑dεD(s)wd

, with 

an intercept (which serves as the estimate of σ2

η, where the 

nonlinear regression is constrained to have nonnegative values 

for the variance terms (σ2

η and σ2ε) and the correlation term, ρε

, 

is constrained to be between zero and one.

The county net-export residual, xc, has no state 

component because net exports are assumed to be zero at 

the state level of aggregation. The county component of 

the net-export residual has a variance σ2

x and is assumed to 

be correlated with the nearest neighbor, with correlation 

coefficient ρx.

The model described in equation 2 is estimated using 

nonlinear least squares to obtain estimates of the coefficients 

θ = (αk, βk), k = 1, ... , K. In estimation, the crop-group 

parameters, αk, are constrained to be nonnegative, causing 

the statistical significance of these parameters to be evaluated 

using one-sided significance tests. Although the βk can vary 

with crop group, for most intensive variables the coefficients 

are restricted to be equal across all groups, with one notable 

exception discussed in the "Results" section.

The existence of spatial correlation in the county 

residuals complicates the determination of the estimated 

coefficient covariance matrix. Generally, spatial correlation 

implies the standard estimates of coefficient variances are 

downward biased. To correct for this bias, an alternative 

estimator of the covariance matrix is adopted. The estimator 

accounts for covariance in the residuals and is asymptotically 

consistent (Amemiya, 1985). With observations ordered by 

state, the alternative covariance matrix, V[

^

θ], takes the form

V    

      G G   G G     G G  ' ' ' 1 1  (8)

where

Gθ is the matrix of gradients of the model (the 

partial derivatives of the right-hand side of 

equation 2 with respect to the coefficient 

vector, θ); and

Ω is the covariance matrix for reporting-unit 

residuals.
Methods 13

Because observations are assumed to be independent across 

state borders, Ω is a block diagonal matrix, with state 

blocks, Ωs, having dimension determined by the number of 

reporting units, ns, in each state s. For states with no county 

observations, Ωs is a single element having the value given by 

the right-hand side of equation 7; for states with county-level 

reporting units, the state block takes the form

  s n   σ σ η ε s s n x   σ ρ   n x s     ε εσ ρ σ x s  2 2 2 2 2 i i' (9)

where

ins is a ns-element vector of ones;

Ins is the ns × ns identity matrix; and

Ms is an ns × ns matrix with element Msc1c2 equal 

to one if counties c1 and c2 are neighbors, 

and equal to zero otherwise.

For states with no county observations, Ωs is evaluated 

by the estimation of equation 7. For states with county 

observations, note that the variance of the county residuals 

is [σ2

η] + σ2ε + σ2x and the variance of the difference, (ec1

 – ec2), 

where counties c1 and c2 are neighbors, is 2(σ2

ε + σ2x – (ρε

σ2

ε + ρx

σ2

x)). Therefore, equation 9 can be rewritten as follows:

  s n n n    s c  n s  c c s s s s s      V e      V e e  

 ση

2

1 2 i i 2 ' / (10)

where

ins is defined in equation 9;

Ins is defined in equation 9; 

Ms is defined in equation 9;

V[ec] is the variance of the county residuals; and

V[ec1 – ec2] is the variance of the difference in 

neighboring county (within the same state) 

residuals.

Because the grouped-county observations have no definitive 

neighbors, these observations are excluded from the 

calculation of V[ec1 – ec2]. Given the estimate of σ2

η from 

the regression in equation 7, all terms in equation 10 are 

estimated allowing for the corrected estimate of the coefficient 

covariance matrix through equation 8.

Prediction Methodology

The estimated models are used to generate two types of 

predictions—unconditional prediction, whereby the model 

is the sole basis for the predicted use, with a correction for 

retransformation bias and conditional prediction, whereby 

sales information at the state level is used to modify the 

unconditional predictions generated by the model. Both 

methods of prediction invoke the assumption that fertilizer 

sales and use are equivalent at the state level of aggregation. 

Predictions of elemental fertilizer use using the model are 

initially made at the finest spatial scale, the catchment/

county, which are then aggregated to the catchment scale. 

Because the estimation of the model uses statistical methods, 

to generate measures of prediction uncertainty is possible for 

all predictions.

Unconditional Prediction

In deriving the unconditional and conditional predictions 

a preliminary estimate is made based on equation 1, summed 

for all crop types, with coefficients evaluated at their estimated 

values (denoted by the accent “^”) and without an accounting 

for the model error estimated in equation 2. Accordingly, 

the preliminary prediction of fertilizer use for catchment/

county j, ^Wj, is given by 

W A j k jk k jk k

^ K ^ exp ^ ' 















   α β X 1 (11)

where

K identifies crop groups, the collection of which 

comprises all agricultural area;

αk is a crop-group-specific application factor;

Ajk is the area of the crop group;

βk is a crop-group-specific vector of coefficients; 

and

Xjk is the intensity variable for catchment/county 

unit j, associated with crop group k.

Because nonlinear least squares (under standard assumptions) 

results in consistent estimates of the coefficients (the standard 

error of the coefficients goes to zero as the sample size goes 

to infinity) and because of the continuity of equation 11 with 

respect to the coefficients, the preliminary prediction also is 

consistent (Amemiya, 1985). 

As indicated in equation 2, fertilizer use also depends 

on a residual term, u, which is additive in logarithm space 

and multiplicative in real space, with a distribution that 

is independent of scale, location, and the values of the 

explanatory variables. The expectation of the exponential 

transform of this residual is likely greater than one due to 

skew, implying the preliminary prediction given in equation 4

is biased downwards. 

To correct this bias, the preliminary prediction is scaled 

by a retransformation-bias correction factor, denoted ^eu, 

representing the expectation of the exponential transform 

of the fertilizer use residuals. If the use residual is normally 

distributed, the appropriate correction factor is given by 

exp((σ2

η + σ2ε)/2), the parameters of which can be consistently 

estimated from the state-level squared-residual, nonlinear 

least-squares regression described in equation 7. However, 

as explained in the results, to obtain a logically consistency 

estimate for σ2

ε is not possible.
14 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Therefore, σ2

ε is set to zero, making the state-level 

residuals a direct estimate of the state-level component of 

the use residual, η, without the confounding variation from a 

possibly spatially correlated, county-level component. Under 

these conditions, the state residuals are homoscedastic, and 

a smearing estimator approach can be used to estimate the 

retransformation-bias correction factor. This estimate, which 

does not require the residuals to have a known distribution, is 

given by

e S e u s s

 S

 

















 

1

1

exp (12)

where

S is the total number of states for which statelevel residuals, ^

es, are available.

Given the preliminary prediction for catchment/county j, ^

Wj, and the national estimate of the retransformation-bias 

correction factor, ^eε, the unconditional prediction of fertilizer 

use for catchment t, ^Fu

tu, is obtained by product according to 

F W e tU

u

j

u

j J t

^ ^ ^     

(13)

where

J(t) is the set of catchment/counties contained in 

catchment t; ^

Wj is the preliminary prediction for catchment/

county j; and ^

eu is the retransformation-bias correction factor.

Conditional Prediction

The conditional prediction of fertilizer use, for 

catchment/county j derives from a slightly alternative 

formulation of the fertilizer-use model,

F A j W m u

k jk k jk

k

K

   s j c j j j





 



       

α η exp exp ε θ ' β X ( ) ( )

1

(14)

where

Wj(θ) is the model component of the prediction, 

the estimation of which is the preliminary 

prediction indicated in equation 11; and

mj is a multiplicative-residual component 

composed of the exponentiated sum of the 

state- and county-component residuals, 

ηs(j) and εc(j), corresponding to the state, 

s(j), and county, c(j), in which catchment/

county j resides. 

The conditional prediction replaces Wj with its consistent 

estimate, ^Wj, and the multiplicative factor, mj, with its 

expectation given sample data, E[mj|F], where F is a vector of 

the observed elemental fertilizer sales data used to estimate the 

model. Given the assumption that residuals are independent 

across states, only sales within the same state as catchment/

county j are relevant, implying E[mj|F] = E[mj|Fs(j)], where Fs(j) 

is the vector of elemental fertilizer sales for catchment/county 

state, s(j).

In general, evaluating the expectation E[mj|Fs(j)] requires 

specifying explicit distributions for each of the residual terms, 

η, ε, and x, these distributions reflecting the assumed spatial 

correlation properties described previously. One way to 

simplify the analysis is to restrict the evaluation to functions 

that are linear in the dependent variable and select the precise 

linear relation to meet unbiased criteria and minimize the 

variance of the prediction error. Such an estimator is said to be 

a Best Linear Unbiased Estimator (BLUE) (Amemiya, 1985). 

Let Ws be an Ns × 1 vector of the preliminary predictions 

aggregated to the Ns reporting units in state s, and define m ̃ s as 

the Ns × 1 aggregate sales multiplicative factor, 

m W F ~

s  D s s

1

(15)

where

D(x) is a diagonal matrix with vector x along the 

diagonal;

Ws is an Ns × 1 vector of the preliminary 

predictions aggregated to the Ns reporting 

units in state s; and

Fs(j) is the vector of elemental fertilizer sales for 

catchment/county state, s.

For states with county-level reporting units, Ns is the number 

of counties (or counties and combined-counties), and m ̃ s 

is a vector of exponentiated sums of the county- and statelevel error components ηs

, εc, and xc. If state s has no county 

reporting units, then m ̃ s is a scaler equal to exp(es), where es is 

approximately equal to the relation indicated in equation 6.

In appendix 4, the BLUE estimator for mj, defined as mˆ j, 

takes the form

m D j js j s j s D D j s j s j s j s j

^ ~                    



        W W W F W 1 



 



 (16)

where

μ is the unconditional expectation of mj; ~μ is the unconditional expectation of the 

aggregate sales multiplicative factor, m ̃ s (if 

m ̃ s is a vector the assumptions imply it has 

the same unconditional expectation across 

all reporting units);
Methods 15

∑js(j) is 1 × Ns(j) a row vector of the covariance 

between mj and each of the Ns(j) reportingunit real-space multiplicative residuals, m ̃ s

;

D(Ws(j)) is a Ns(j) × Ns(j) diagonal matrix with diagonal 

elements given by the Ns(j) × 1 vector Ws(j)

representing the preliminary predictions 

aggregated to the Ns(j) reporting units in 

state s(j);

∑s(j) is the Ns(j) × Ns(j) covariance matrix between 

the state’s Ns(j) reporting unit real-space, 

zero-mean multiplicative residuals, m ̃ s; and

Fs(j) is the vector of elemental fertilizer sales for 

catchment/county state, s(j).

Appendix 4 describes the evaluation of the covariance 

matrices under the assumption that the error components in 

logarithm space are each normally distributed. As remarked 

with the unconditional predictions, an acceptable variance was 

possible to obtain only for the state component, the countylevel variance component being set to zero. This result greatly 

simplifies the evaluation of the BLUE estimator for mj. Under 

the assumption that the use residual consists only of a statelevel component and given the previous assumption that the 

state-level residual excludes the net-export component, then 

the state-level use residual is fully revealed by the state-level 

model residual. That is, from equation 6, if εc has no variance, 

then es = ηs and mj can be estimated perfectly by knowing the 

state value of fertilizer sales, Fs. In terms of equation 16, with 

σ2

ε = 0 and Fs(j)

 set to the state-level reporting unit for all states, 

then ~μ = μ and ∑js(j) = ∑(j), all vector terms are scalers, and the 

determination of mˆ j simplifies to

mj s j s j

^  /     F W (17)

where

Fs(j) is defined in equation 16; and

Ws(j) is defined in equation 16.

Because some catchments straddle state borders, the 

BLUE conditional estimate for elemental fertilizer use in 

catchment t, Fˆu

tC, is a combination of the normalized statelevel sales of the respective states comprising the catchment. 

Accordingly, the estimate is 

F W tC

u

j J t j s j s j

^ ^ / ^    ( ) F W ( ) ( ) (18)

where

J(t) is the set of catchment/counties contained in 

catchment t; ^

Wj is the preliminary prediction of fertilizer use, 

for catchment/county j;

Fs(j) is defined in equation 16; and

Ws(j) is defined in equation 16.

Effectively, the BLUE conditional estimates represent 

an allocation of the state-level fertilizer sales according to 

the share of model-predicted use in each catchment/county. 

In this regard, the conditional estimates are like the estimates 

generated by Brakebill and Gronberg (2017), the difference 

being the method used to derive the share allocations.

Prediction Uncertainty

An advantage of the statistical approach to estimating 

fertilizer use is the ability to assess the uncertainty of the 

estimate. For the unconditional predictions, the two sources 

of uncertainty are as follows: (1) uncertainty because of 

finite random sampling, which is manifest in the estimated 

model coefficients, these coefficients being based on a finite 

sample that is randomly drawn; and (2) model uncertainty, 

which is the uncertainty arising from the model residual. 

The consistency of the estimated coefficients implies that 

uncertainty because of random sampling goes to zero as the 

sample size goes to infinity; uncertainty because of model 

error does not go to zero in large samples.

Because of the nonlinear specification of the model 

coefficients, uncertainty is assessed using a parametric 

bootstrap method, as described in Schwarz and others (2006). 

The nonlinear least-squares estimates of the coefficients and 

their covariance matrix as defined by equation 8 are used to 

randomly generate 200 sets of coefficients assuming a normal 

multivariate distribution. The use of a normal distribution is 

justified because it is the large-sample limiting distribution for 

nonlinear least-squares estimation (Amemiya, 1985).

Let ^θ(r) denote the r-th bootstrap iteration randomly 

generated coefficient vector and let Ŵj

(r)

 be the r-th iteration 

realization of the preliminary prediction for catchment/county 

j. To obtain bootstrap evaluations of the retransformation-bias 

correction factor, the Ŵj

(r)

 are aggregated to the state level, 

log transformed, and subtracted from log-transformed state 

fertilizer sales to obtain bootstrap-iteration evaluations of 

state-level residuals. Using the smearing estimator defined 

in equation 12, the residuals are used to compute bootstrap 

iteration estimates of the retransformation bias factor, ^eu(r). 

Accordingly, the r-th bootstrap iteration unconditional 

estimate of fertilizer use for catchment t is 

F W e tU

u

j J t

r u r

j

^ ( ) ^ ^( ) r

( )

( )    (19)

where

J(t) is the set of catchment/counties contained in 

catchment t; ^

Wj

(r) is the r-th iteration realization of the 

preliminary prediction for catchment/

county j; and ^

eu(r) is the retransformation bias factor.
16 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

As described in Schwarz and others (2006), the 

coefficient of variation of unconditional estimated fertilizer 

use for catchment t, expressed as a percent, is given by

COV F

V e

e

V F

F

tU

u

u

u

tU

u r

tU

u

^

^

^

^

( )

















 

  

















100

2 2

(20)

where ^

Fu

tU is defined in equation 13;

V[eu] is the variance of exponentiated state-level 

residuals, exp(^es), from the original 

nonlinear least-squares model estimates; ^

eu is given by equation 12; and

V[

^

Fu

tU(r)

] is the variance over bootstrap iterations, r, of 

the bias-corrected unconditional estimate 

of fertilizer use for catchment t, ^Fu

tU(r)

, as 

given in equation 19.

Prediction uncertainty for the conditional estimates 

is simpler to evaluate because the model error for use, 

u, is assumed to be identified by the state-level fertilizer 

sales, which is known and not subject to uncertainty. From 

equation 20, the first term in the sum under the radical is zero. 

The only source of uncertainty, therefore, is due to random 

sampling, which affects the modeled shares used to allocate 

state-level sales to individual catchments. Therefore, the 

coefficient of variation for the conditional elemental fertilizer 

use estimate for catchment t, expressed as a percent, is 

COV F

V F

F

tC

u

tC

u r

tC

u

^

^

^

( )



































100

2

(21)

where ^

Fu

tC is given by equation 18; and

V[

^

Fu

tC

(r)

] is the variance over bootstrap iterations, r, 

of the catchment-aggregated, conditional 

estimates, j J t j

r

s j s j

r

   W F   W  

^ ( ) ( ) / ^ .

Results

Model Estimation

The specifications of the nitrogen and phosphorus 

fertilizer-use models include the same extensive variables 

and nominally similar intensive variables. The extensive 

variables consist of a delineation of the agricultural area 

within each catchment/county reporting unit into five distinct 

crop groups. These crop groups include corn, grass-pasturehay, miscellaneous, nitrogen-fixing, and small grains (fig. 4, 

table 2.1). These variables are not subject to any mathematical 

transformation in order to maintain a strict scaling of 

fertilizer use with agricultural cropping area. Because of 

this specification and the mean difference form specified for 

the intensive variables, the coefficients associated with the 

extensive variables can be interpreted as the elemental mass of 

fertilizer application per unit area of the given crop group.

The coefficients for the extensive variables are 

estimated with the constraint that the estimated value must 

be nonnegative. The constraint implies the significance of the 

coefficient is determined by applying a one-sided t-test, and 

the reported p-values are one-sided—meaning the p-values are 

one-half the value of a standard, two-sided p-value. 

The intensive factors included in the models are 

derived from a set of five variables—fertilizer expenditures 

per unit of cropland area; mean precipitation; mean actual 

evapotranspiration; the elemental mass of manure per 

unit of cropland area, which is the nitrogen mass for the 

nitrogen fertilizer-use model; and the phosphorus mass for 

the phosphorus fertilizer-use model. Each of the intensive 

variables is included in the model after transformation via the 

natural logarithm function and subsequent differencing from 

each of the intensive variables mean logarithm-transformed 

value. This transformation implies the estimated coefficients 

can be interpreted as the rate of change of fertilizer use, 

per rate of change of the intensive variable. A physical 

interpretation of the coefficients is complicated by the 

inclusion of the expenditure variable, which likely varies 

in response to variations in the other intensive variables. 

Thus, for example, the precipitation coefficient measures the 

effect on elemental fertilizer use holding total expenditure 

for fertilizer fixed, without allowing for the effect of varying 

precipitation on expenditures.

All intensive variables, except for manure, are specified 

in the model to interact with the extensive variables in an 

equivalent way, implying βk does not vary by source, k. The 

exception is the manure variable that likely has a differential 

effect on delivery depending on the extensive variable with 

which it interacts. Manure often serves as a direct substitute 

for commercial fertilizer for grass-pasture-hay crops and is 

less of a substitute for other crop groups. Consequently, the 

specification of the intensive variables has manure included 

twice—first as an interaction with all crop groups and 

second as an interaction with grass-pasture-hay only. This 

specification implies the coefficient associated with manure 

interacting with grass-pasture-hay represents a differential 

effect of manure on commercial fertilizer use, as compared 

to the full effect determined by the coefficient of manure 

interacting with all crop groups.
Results 17

The decision to use similar specifications for both 

the nitrogen and phosphorus fertilizer-use models is based 

on evaluations of preliminary models that consistently 

demonstrate the statistical significance of like variables 

in the two models. An effort was made to incorporate 

variables having a special influence on the phosphorus 

model, through variables reflecting natural concentrations 

of phosphorus in the soil, but such efforts failed to discern 

statistical significance either regionally or nationally. The 

nearly identical specification of the two models (identical 

except for the elemental content of the manure variable) 

implies spatial variation in the predicted ratio of nitrogen to 

phosphorus fertilizer content in use is largely a consequence of 

different valuations of the mediating coefficients estimated for 

the models. 

Nitrogen

The estimation results for the nitrogen fertilizer-use 

model are listed in table 2, with the standard errors of the 

coefficients reflecting the correction described by equation 8. 

All but one coefficient in the model is significant at the 

0.05 level (the significance of extensive variable coefficients 

is reported by a one-sided p-value; the significance of the 

intensive variables is a standard two-sided p-value). The two 

crop groups having the largest model coefficients are small 

grains (2.04) and corn (1.54). The smaller coefficient for corn 

may in part, be a result of crop rotation, whereby the nitrogen 

demands for growing corn are partially supplied by rotation 

with nitrogen-fixing crops in previous years, a factor that is 

not accounted for in the model. The coefficient for nitrogenfixing crops is unexpectedly larger than some of the other crop 

groups, possibly because of misclassification errors for this 

crop group, which is the second least accurate after grasspasture-hay (table 3.1). 

The grass-pasture-hay coefficient has the smallest 

standard error. Grass-pasture-hay accounts for more than 

50 percent of all cropland in the conterminous United States 

and occurs with the greatest variation of any crop group, 

although the rate of application of fertilizer to this crop group 

is probably the most variable, as indicated by relating the 

absolute magnitude of model residuals to crop group areas. 

This enhanced variability associated with grass-pasture-hay is 

likely a consequence of misclassification error, which is most 

prevalent for this crop group.

Three intensive variables—the logarithms of fertilizer 

expenditures per unit cropland, actual evapotranspiration, and 

nitrogen from manure per unit cropland as interacted with all 

crops—have a positive coefficient, indicating an enhancement 

of fertilizer use; and, two intensive variables—precipitation 

and manure interacting with grass-pasture-hay—have a 

negative coefficient, indicating suppressed use. The coefficient 

on manure interacting with all crops is the only coefficient that 

is not statistically significant at the 0.05 level. 

The overall effect of manure interacting with grasspasture-hay is given by the sum of the two manure 

coefficients, which at –0.072 (0.067 plus –0.139) is negative 

and statistically significant (the two-sided p-value equals 

0.026). The negative effect of manure interacting with grasspasture-hay is consistent with the interpretation that manure 

is a substitute for commercial fertilizer for application to that 

crop group. The positive sign of the coefficient for manure 

interacting with all crops is consistent with this interpretation, 

given the presence of expenditures in the model. With 

expenditures on fertilizer held fixed, a decrease in commercial 

fertilizer applied to grass-pasture-hay implies an increase 

in fertilizer applied to all other crops, which the model 

results confirm.

The nitrogen fertilizer-use model explains 74 percent 

of the variation in the logarithm of total nitrogen sold at the 

county level, and the root mean square residual (RMSE) 

for the log-transformed residuals is 1.20. The reporting-unit 

residuals are likely more variable than the residuals associated 

with use because of net exportation of fertilizer sales across 

county boundaries, and the fit of the model as it pertains to 

use is better represented by the residuals computed at the 

state level. This assumption is supported by the fit statistics 

computed from state-level residuals that indicate the model 

explains 93 percent of the logarithm of nitrogen fertilizer use. 

The RMSE for the use log-transformed residuals is 0.473, a 

much better fit than indicated by the reporting-unit residuals 

(note, the residual for Wyoming is excluded from this 

evaluation as the data for that State are deemed unreliable). 

The state-RMSE implies, as an approximation, that predictions 

of fertilizer use within any given catchment will be within 

47 percent of the true use for a one standard-deviation error 

(Schwarz and others, 2006).

Studentized residuals are the residuals divided (or 

scaled) by the reporting-unit RMSE and are mapped for the 

state- and county-level reporting units (fig. 5), with a negative 

residual indicating overprediction and a positive residual 

indicating underprediction. The largest residual for state-level 

reporting units is for Wyoming, where the positive residual 

indicates that actual sales greatly overstate likely use, and 

where fertilizer sales data are deemed unreliable—although 

Wyoming is mostly pasture, the grass-pasture-hay crop group 

is known to be poorly classified (table 3.1). For county-level 

reporting units, the largest residuals reside in Texas and 

Tennessee, where positive and negative residuals are observed, 

and in the isolated cases of Baltimore City, Maryland, and 

Elliott County, Kentucky. The export and import of fertilizer 

across county borders can partially explain these anomalies, 

although reporting errors are also a possible cause.
18 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Table 2. Model coefficient estimates and statistics for the 2012 nitrogen and phosphorus fertilizer-use models of the conterminous 

United States.

[Standard errors of model estimates are based on the coefficient covariance matrix with the correction for spatial correlation (see eq. 8). Reported p-values for 

crop-group variables are one sided probabilities from the t-distribution, all other p-values are two-sided. Fertilizer use R2 and RMSE are based on logarithm 

transforms of state-level aggregations of actual and predicted use. t/km2, metric ton per square kilometer; <, less than; km2, square kilometer; NA, not applicable; 

R2, coefficient of determination for model estimated in logarithm space; RMSE, root mean square error computed from residuals in logarithm space]

Variables or statistic

Model 

coefficient 

units

Nitrogen Phosphorus

Estimate 

of model 

coefficient

Standard 

error of model 

coefficient

Probability 

level 

(p-value)

Estimate 

of model 

coefficient

Standard 

error of model 

coefficient

Probability 

level 

(p-value)

Crop (extensive) variables

Corn crops t/km2 1.54 0.749 0.020 0.107 0.078 0.085

Grass-pasture-hay crops t/km2 1.07 0.161 <0.0001 0.205 0.029 <0.0001

Miscellaneous crops t/km2 1.05 0.460 0.011 0.044 0.028 0.062

Nitrogen-fixing crops t/km2 1.09 0.612 0.037 0.210 0.074 0.002

Small grain crops t/km2 2.04 0.926 0.014 0.117 0.069 0.045

Intensive variables

Log expenditures (times $1000) per km2

cropland (2012) in 1992 dollars

Unitless 0.961 0.061 <0.0001 1.07 0.058 0.000

Log precipitation (2012) Unitless –0.761 0.315 0.016 –0.610 0.313 0.052

Log actual evapotranspiration (2000–14 

average)

Unitless 1.35 0.513 0.008 1.01 0.514 0.050

Log nitrogen from manure (interacted 

with all crops) per km2 cropland

Unitless 0.067 0.047 0.153 NA NA NA

Log nitrogen from manure (interacted 

with pasture only) per km2 cropland

Unitless –0.139 0.063 0.026 NA NA NA

Log phosphorus from manure (interacted 

with all crops) per km2 cropland

Unitless NA NA NA 0.134 0.053 0.011

Log phosphorus from manure (interacted 

with pasture only) per km2 cropland

Unitless NA NA NA –0.260 0.069 <0.0001

Model diagnostics

Reporting unit (state/county)

Estimated model R2 NA 0.738 NA NA 0.725 NA NA

Estimated model RMSE NA 1.200 NA NA 1.221 NA NA

Number of sites (model) NA 2,097 NA NA 2,090 NA NA

State

Fertilizer use R2 NA 0.927 NA NA 0.929 NA NA

Fertilizer use RMSE NA 0.473 NA NA 0.473 NA NA

Number of sites (state level) NA 47 NA NA 46 NA NA
130° 120° 110° 100° 90° 80° 70° 

40°30°20°

Base from U.S. Census Bureau county boundaries, 1:500:000, 2014 0 200 400 800 MILES

0 200 400 800 KILOMETERS

EXPLANATION

StateState-level sales dataCounty-level sales data

Studentized residuals, nitrogen fertilizer-use model(negative value = overprediction)(positive value = underprediction)

-20 to -2Greater than -2 to -1Greater than -1 to 1Greater than 1 to 2Greater than 2 to 20

Figure 5. Studentized residuals for 2012 nitrogen fertilizer-use model.
20 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Scatter plots relating observed nitrogen fertilizer sales 

to predicted use at the county and state scales are shown 

in figures 6 and 7, respectively. State-level reporting units 

are also displayed (yellow dots) in the county-level plot in 

figure 6. County-level reporting units in Mississippi are also 

displayed (black dots) in figures 6 and 7 as part of the model 

evaluation, as described in the “Model Evaluation” section 

of this report. In both figures, because the observations are 

plotted on a logarithmic scale, predicted use is computed 

prior to applying the retransformation bias correction factor. 

Figure 6 shows that the relation between sales and predicted 

use is much more variable for lower values, with variability 

declining as predicted use increases—significantly so 

for observations reported at the state level. Figure 6 also 

shows a bias whereby predicted use tends to exceed sales 

for lower mass, a result that is likely due to counties with 

less agriculture tending to have a less-developed fertilizer 

distribution network, implying more use is met by imports 

of sales made in neighboring counties. The state-level 

relation between use and sales, displayed in figure 7, shows 

even variability for the full range of use, with considerably 

less variability than is evident at the county level. Again, 

state-level variability supports the assumption that sales 

approximate use at the state scale.

Phosphorus

Estimation results for the phosphorus fertilizer-use model 

are listed in table 2. All coefficients are statistically significant 

at the 0.10 level and most are significant at the 0.05 level. 

The two crop groups with the largest model coefficients are 

nitrogen-fixing crops (0.210) and grass-pasture-hay (0.205). 

Phosphorus enhances nitrogen fixation and nitrogen-fixing 

plants (alfalfa, soybeans, and other legumes) generally require 

more phosphorus than grasses for growth and development. 

Phosphorus is not nearly as important for the growth of corn, 

as reflected by the model coefficients.

The nitrogen fertilizer-use model crop coefficients are 

approximately 5 to 24 times higher than the phosphorus model 

coefficients. Presumably, this is a consequence of greater 

mass plant uptake of nitrogen as compared to phosphorus. 

Nitrogen plays a key role in photosynthesis and the formation 

of protein. Moreover, the greater solubility of nitrogen implies 

more nitrogen mass can be lost through infiltration through 

the soil, necessitating a need for more replacement after each 

growing season. Additionally, phosphorus in fertilizer is more 

expensive per unit mass than nitrogen (U.S. Department of 

Agriculture, Economic Research Service, 2016) and farmers 

rely on soil and tissue testing to economize on phosphate 

(P2O5) fertilizer application.

0 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000

0

1

10

100

1,000

10,000

100,000

1,000,000

10,000,000

EXPLANATION

County-level sales

State-level sales

State of Mississippi (county-level sales)

One-to-one line

Estimated nitrogen fertilizer use (metric tons), log scale

Nitrogen fertilizer sales (metric tons), log scale

Figure 6. Observed 2012 nitrogen fertilizer sales compared to estimated 2012 nitrogen fertilizer use summarized at the county level.
Results 21

0

1

10

100

1,000

10,000

100,000

1,000,000

10,000,000

0 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000

Estimated nitrogen fertilizer use (metric tons), log scale

Nitrogen fertilizer sales (metric tons), log scale

EXPLANATION

County-level sales

State-level sales

State of Mississippi (county-level sales)

One-to-one line

Figure 7. Observed 2012 nitrogen fertilizer sales compared to estimated 2012 nitrogen fertilizer use summarized at the state level.

The coefficients associated with the intensive variables 

have values that conform to those for the nitrogen model. 

The combined effect of phosphorus manure interacting 

with grass-pasture-hay is negative at –0.126 (coefficient for 

phosphorus from manure without interaction equals 0.134 and 

coefficient for phosphorus from manure interacting with grasspasture-hay equals –0.260; the two-sided p-value is 0.0004), 

supporting the finding from the nitrogen fertilizer-use model 

that manure is a substitute for commercial fertilizer for that 

crop group. In magnitude, the total effect is nearly twice that 

for nitrogen, suggesting that manure is a better substitute for 

commercial fertilizer for phosphorus than for nitrogen. This 

result may be due to large sources of phosphorus coming from 

poultry manure, a major source of manure in some regions of 

the country.

The model does not include any variable that accounts 

for natural sources of phosphorus occurring in the soil and 

attempts to include such a variable were unsuccessful. The 

lack of such a variable does not indicate that the model is 

blind to this factor. The phosphorus content of fertilizer has 

a significant effect on fertilizer price; therefore, variations 

in phosphorus fertilizer application possibly are reflected in 

fertilizer expenditures. The presumption, therefore, is that 

given the inclusion of fertilizer expenditures as a determinant 

of phosphorus fertilizer use, the inclusion of variables related 

to natural fertilizer content of the soil fail to improve model 

fit. This result could be a consequence of noise confounding 

the signal contained in the natural fertilizer variables. It could 

also be a consequence of past agricultural activity depleting 

soils of their natural phosphorus content or to the availability 

of phosphorus in the soil in a form for plant uptake, or both.

The phosphorus fertilizer-use model explains 72 percent 

of the variation in the logarithm of total elemental phosphorus 

sold, and the RMSE for the residuals in logarithm space is 

1.22, a fit that is similar to the nitrogen model. The fit statistics 

computed from state-level residuals (without application of 

the retransformation bias correction factor) have a coefficient 

of determination (R2) of 93 percent and an RMSE of 0.47, 

indicating that the method determines nearly identical fit 

statistics for both the nitrogen and phosphorus fertilizeruse models. As with nitrogen, the residual for Wyoming is 

excluded from the determination of state-level statistics. Also 

excluded is Nevada, which has no phosphorus sales.

Phosphorus model studentized residuals (residuals 

normalized by the reporting-unit RMSE) for the stateand county-level reporting units are mapped in figure 8. 

The location of extreme residuals, those having absolute 

studentized values exceeding one, is nearly the same as 

the nitrogen case (fig. 5), implying the errors in predicting 

fertilizer use relate more to total fertilizer application than 

to errors specific to elemental composition. This observation 

is consistent with the notion that errors in the models arise 

primarily from idiosyncrasies in the distribution system 

leading to a greater range in net exports and misclassification 

of cropping patterns inducing common errors in both models.

Patterns contained in the scatterplot of predicted nitrogen 

use relative to nitrogen sales at the state- and county-level 

reporting units (fig. 6) are apparent in the corresponding 

plot for phosphorus, shown as figure 9, although countylevel observations display a slightly greater variability for 

phosphorus. The scatterplot of phosphorus predicted use 

relative to sales exclusively at the state level (fig. 10) shows a 

better fit than for nitrogen (fig. 7), particularly in the higheruse states. Together, these results imply a slightly greater 

tendency for exporting and importing fertilizers with high 

phosphorus content.
EXPLANATION State State-level sales data County-level sales data Studentized residuals, phosphorus fertilizer-use model (negative value = overprediction) (positive value = underprediction) -20 to -2 Greater than -2 to -1 Greater than -1 to 1 Greater than 1 to 2 Greater than 2 to 20 130° 120° 110° 100° 90° 80° 70° 

40° 30° 20° 0 200 400 800 MILES 0 200 400 800 KILOMETERS Base from U.S. Census Bureau county boundaries, 1:500:000, 2014 Figure 8. Studentized residuals for 2012 phosphorus fertilizer-use model.
Results 23

0

1

10

100

1,000

10,000

100,000

1,000,000

0 1 10 100 1,000 10,000 100,000 1,000,000

Estimated phosphorus fertilizer use (metric tons), log scale

Phosphorus fertilizer sales (metric tons), log scale

EXPLANATION

County-level sales

State-level sales

State of Mississippi (county-level sales)

One-to-one line

Figure 9. Observed 2012 phosphorus fertilizer sales compared to estimated 2012 phosphorus fertilizer use summarized at the county 

level.

0

1

10

100

1,000

10,000

100,000

1,000,000

0 1 10 100 1,000 10,000 100,000 1,000,000

EXPLANATION

County-level sales

State-level sales

State of Mississippi (county-level sales)

One-to-one line

Estimated phosphorus fertilizer use (metric tons), log scale 

Phosphorus fertilizer sales (metric tons), log scale

Figure 10. Observed 2012 phosphorus fertilizer sales compared to estimated 2012 phosphorus fertilizer use summarized at the state 

level.
24 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Residual Analysis

The correction to the covariance matrix, described by 

equation 8, requires an estimate of the residual covariance 

matrix, Ω. The estimates of the residual variances and 

correlations used to evaluate this matrix are listed in table 3.

Attempts to estimate the fertilizer use error-components 

model based on equation 7 resulted in several inconsistencies. 

The nonlinear regression for squared, nitrogen model 

residuals yielded an insignificant state-component variance 

(σ2

η), a significant county-component variance (σ2ε), and a 

county-component spatial correlation (ρε) of 1; however, 

a spatial correlation of 1 implies the county component is 

perfectly correlated. The county component is observationally 

equivalent to a simpler covariance structure consisting of 

just a state-level component—the state-level component 

is inconsistent with the insignificant result for the statecomponent variance. The nonlinear regression for squared, 

phosphorus model residuals gave significant state- and 

county- component variances and a county-component spatial 

correlation of zero; however, the summed values of the stateand county- component variances greatly exceeds the variance 

of the county residuals from the phosphorus model, implying 

the net-export variance is negative—an impossible result.

Because of these inconsistencies, and given that county 

use is not observable, the decision was made to suppress the 

county component to the use residual and base all variance 

and covariance on the state component, the implication being 

that residual use throughout a state is perfectly correlated but 

uncorrelated across states. This assumption may overstate 

the degree to which use is correlated across substate units 

within the same state. Moreover, counter to the assumption, 

if a component of the state residual is independent across 

substate units, then the variability of the use residual at 

the substate level is understated by the variability of the 

state-level residual.

The first three rows of table 3 reflect the assumed use 

residual covariance structure, with a positive state-component 

variance (σ2

η) and the county-component variance (σ2ε) set 

to zero. With no county-component variance, the countycomponent spatial correlation (ρε

) indicated in the third row 

among neighboring counties becomes irrelevant. The fourth 

and fifth rows of table 3 give the estimates of the variance 

of the county-level residuals, V[ec], and the variance of the 

difference in the residuals among neighboring counties in the 

same state, V[ec

1

 – ec

2

], for those states having county-level 

reporting units. Counties that are merged because of no sales 

are included in the variance of county-level residuals but 

excluded from the variance of the difference in neighboring 

county residuals. 

The last two rows of table 3 give the values of the 

variance and spatial correlation among neighboring counties 

of the county-level net-export residual component, x, as 

derived from the estimates of σ2

η and the two variances, V[ec

] 

and V[ec

1

 – ec

2

]. The net-export residual component is more 

than five times the magnitude of the use component, implying 

the fertilizer sales data are noisy indicators of fertilizer use 

at the county level, as expected. The spatial correlation of 

net exports among neighboring counties is expected to be 

negative, because a county with significant positive net 

exports is likely to locate next to a county with large imports 

(that is, negative net exports). The finding of a positive spatial 

correlation implies exports must be balanced by imports across 

a larger spatial extent than neighboring counties. The fact that 

the state-level residuals have a much smaller variance than 

county-level residuals implies that a negative correlation must 

exist at some scale below the state level.

Table 3. Residual variance and covariance estimates for the nitrogen and phosphorus fertilizer models.

[Note that ση

2

 and σε

2

 are the variances of the state- and county-component use residuals; ρε is the correlation coefficient between the county-level components; 

the variances V[ec] and V[ec1 – ec2] are computed without centering; V[ec1 – ec2] excludes grouped-county observations; and σx

2

 and ρx are derived from other 

estimates in the table using equations described in the estimation methods section. <, less than; NA, not available]

Parameter Description

Nitrogen Phosphorus

Estimate Standard 

error

Probability 

level 

(p-value)

Estimate Standard 

error

Probability 

level 

(p-value)

ση

2 Variance of state-component use residual 0.223 0.049 <0.0001 0.224 0.039 <0.0001

σε

2 Variance of county-component use residual 0.000 NA NA 0.000 NA NA

ρε Correlation among neighboring county use 

residuals

NA NA NA NA NA NA

V[ec

] Variance of county residuals 1.434 0.298 <0.0001 1.488 0.286 <0.0001

V[ec1 – ec2]/2 Variance of difference of neighboring county 

residuals (halved)

1.060 0.093 <0.0001 1.118 0.090 <0.0001

σx

2 Variance of county net export residual 1.211 NA NA 1.264 NA NA

ρx Correlation among neighboring county net export 

residuals

0.309 NA NA 0.293 NA NA
Results 25

Model Predictions

The spatial patterns of predicted nitrogen and phosphorus 

fertilizer use in mass units, shown in figures 11 and 12, 

respectively, are based on the unconditional predictions. 

For reference, states with county-level reporting units have 

bolded black borders. The fertilizer-use patterns for nitrogen 

and phosphorus reflect known agricultural cropping patterns 

and practices. The maps show the pattern of high use largely 

mimics the pattern of cropland shown in figure 4, although 

high use is prevalent in the coastal region of the Southeast, 

Florida, and isolated areas in the Southwest despite the 

presence of extensive nonagricultural land—a possible 

consequence of double cropping. Nitrogen and phosphorus 

applications are lower relative to cropland occurrence in 

portions of states in the Southeast where poultry production 

is common (U.S. Department of Agriculture, National 

Agricultural Statistics Service, 2015), including eastern 

Arkansas, northern Louisiana, southeastern Mississippi, 

Alabama, and northern Georgia.

Nitrogen and phosphorus fertilizer use expressed per 

unit area of cropland, based on unconditional predictions, 

are shown in figures 13 and 14, respectively. Unlike the 

maps of fertilizer use in mass, both figures display a 

pronounced affinity for political boundaries, a consequence 

of normalization of use by catchment-level agricultural area 

advancing the less-resolved county fertilizer expenditures 

variable as the preeminent determinant of concentrated 

use. The most intensive use of nitrogen (fig. 13) occurs 

in areas dominated by crops that require high application 

rates (U.S. Department of Agriculture, Economic Research 

Service, 2013). High application rates are predicted for corn 

(Iowa, Illinois, Indiana, western Ohio, southern Minnesota, 

eastern North and South Dakota, and eastern Nebraska), 

rice (southeastern Missouri, eastern Arkansas, northwestern 

Mississippi, and the gulf coast of Louisiana and Texas), cotton 

(the coastal plain of North and South Carolina and southern 

Georgia), and vegetables (southern Florida and central 

California) (U.S. Department of Agriculture, Office of the 

Chief Economist, 2014). Lower nitrogen application rates are 

predicted in the semiarid, West particularly the Plains states 

where grassland and pasture predominate. 

The highest application rates of phosphorus fertilizer 

(fig. 14) closely follow the pattern for nitrogen, with 

some differences derived from the availability of natural 

phosphorus, such as the Delta region of northwestern 

Mississippi or in areas where poultry production is common 

and phosphorus-rich manure is made readily available 

(Oldham, 2012; U.S. Department of Agriculture, National 

Agricultural Statistics Service, 2015). The model may respond 

to the availability of natural phosphorus indirectly through an 

adjustment of fertilizer expenditures, despite the absence of 

natural phosphorus as an explicit model variable.

Previous studies combining fertilizer sales with Census 

expenditure data (Ruddy and others, 2006; Gronberg and 

Spahr, 2012) constrain the ratio of nitrogen to phosphorus to 

be constant for all areas within a state. Because the present 

study derives nitrogen and phosphorus use separately, with 

different model coefficients and slightly different data, the 

ratio of nitrogen to phosphorus use is not constrained. The 

ratio derived from unconditional predictions is displayed 

in figure 15 to better highlight the modeled determinants of 

variability. The ratio of nitrogen to phosphorus fertilizer use is 

generally highest in the Plains states, between the Mississippi 

River and Rocky Mountains, where nitrogen-fixing crops, 

such as soybeans, are uncommon (U.S. Department of 

Agriculture, Office of the Chief Economist, 2014). The ratio 

is also elevated in Florida and Southeastern states. Lower 

values of the ratio are observed in the corn-soybean cropgrowing region of the Midwest (Iowa, Illinois, Indiana, 

western Ohio, southern Minnesota, eastern North and South 

Dakota, and Nebraska) where crop rotation with nitrogenfixing crops is a common practice to supplement application of 

commercial fertilizer.

Prediction Uncertainty Estimates

The coefficient of variation, expressed as a percent of 

the fertilizer use prediction, is derived for each catchment 

according to equations 20 and 21 using a parametric bootstrap 

analysis based on 200 randomly generated sets of coefficients. 

A summary of the estimates for two-digit hydrologic regions 

(U.S. Geological Survey and others, 2017) is given in table 4. 

The average coefficient of variation for nitrogen use 

is greater than that for phosphorus use in all cases listed 

in table 4, although the difference is small. The average 

coefficient of variation for conditional predictions is 

significantly smaller than that for unconditional predictions. 

This result is expected because the conditional predictions, 

unlike the unconditional predictions, have no model error—a 

consequence of assuming the use residuals consist only 

of a state-level component, a component that is known 

with certainty because of the observation of state sales and 

the presumed equivalence of sales and use at that level of 

aggregation.

The average coefficient of variation for unconditional 

predictions in all catchments exhibits a considerable range 

in values across regions, with the maximum value among 

regions being approximately twice the minimum value. Much 

of this variation is due to the prevalence within a region of 

catchments with no agricultural land, which the model predicts 

with certainty to have no fertilizer use. As listed on the righthand side of table 4, the average coefficient of variation among 

catchments with positive use is significantly larger and less 

variable across regions.
Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012 130° 120° 110° 100° 90° 80° 

40° 30° EXPLANATION State with state-level sales data State County Predicted nitrogen fertilizer use, metric tons (quantile) 00.0001 to 0.0003 (20th) 0.0004 to 0.052 (40th) 0.0521 to 0.46 (60th) 0.4601 to 3.11 (80th) 3.1101 to 48,217.1 (100th) 0 200 400 800 MILES 0 200 400 800 KILOMETERS Figure 11. Predicted 2012 nitrogen fertilizer use at the catchment scale (metric tons), unconditional predictions.
Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012

130° 120° 110° 100° 90° 80° 

40°30°

EXPLANATION

State with state-level sales dataStateCountyPredicted phosphorus fertilizer use, metric tons (quantile)

00.00001 to 0.00003 (20th)0.00004 to 0.0083 (40th)0.0084 to 0.068 (60th)0.0681 to 0.417 (80th)0.4171 to 3,896 (100th)

0 200 400 800 MILES

0 200 400 800 KILOMETERS

Figure 12. Predicted 2012 phosphorus fertilizer use at the catchment scale (metric tons), unconditional predictions.
Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012 130° 120° 110° 100° 90° 80° 

40° 30° EXPLANATION State with state-level sales data State County Predicted nitrogen fertilizer use, metric tons per square kilometer of cropland (quantile) 00.0001 to 0.56 (20th) 0.5601 to 1.39 (40th) 1.3901 to 3.06 (60th) 3.0601 to 6.57 (80th) 6.5701 to 315.1 (100th) 0 200 400 800 MILES 0 200 400 800 KILOMETERS Figure 13. Predicted 2012 nitrogen fertilizer use at the catchment scale (metric tons per square kilometer of cropland), unconditional predictions.
130° 120° 110° 100° 90° 80° 

40°30°

Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012and County Boundaries, 1:100,000 (LaMotte, 2016)

EXPLANATION

State with state-level sales dataStateCountyPredicted phosphorus fertilizer use, metric tons per square kilometer cropland (quantile)

00.0001 to 0.08 (20th)0.0801 to 0.21 (40th)0.2101 to 0.46 (60th)0.4601 to 0.95 (80th)0.9501 to 101.2 (100th)

0 200 400 800 MILES0 200 400 800 KILOMETERS

Figure 14. Predicted 2012 phosphorus fertilizer use at the catchment scale (metric tons per square kilometer of cropland), unconditional predictions
EXPLANATION State with state-level sales data State County Predicted fertilizer use, nitrogen to phoshporus (NP) ratio (quantile) 00.0001 to 6.13 6.1301 to 7.04 7.0401 to 7.88 7.8801 to 9.39 9.3901 to 59.9 0 195 390 780 MILES 0 195 390 780 KILOMETERS Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012 130° 120° 110° 100° 90° 80° 

40° 30° Figure 15. Ratio of predicted 2012 nitrogen fertilizer use to predicted 2012 phosphorus fertilizer use, unconditional predictions.
Results 31

Table 4. Average of catchment predicted nitrogen and phosphorus fertilizer use coefficient of variation, by hydrologic region and 

conterminous United States.

[COV, coefficient of variation expressed as a percent of predicted use]

Hydrologic 

region 

(code)

All catchments Catchments with positive use

Nitrogen Phosphorus Nitrogen Phosphorus

Unconditional 

prediction 

COV 

(percent)

Conditional 

prediction 

COV 

(percent)

Unconditional 

prediction 

COV 

(percent)

Conditional 

prediction 

COV 

(percent)

Unconditional 

prediction 

COV 

(percent)

Conditional 

prediction 

COV 

(percent)

Unconditional 

prediction 

COV 

(percent)

Conditional 

prediction 

COV 

(percent)

New England 

(01)

33.9 12.0 33.8 10.8 53.3 18.8 53.2 16.9

Mid-Atlantic 

(02)

38.8 14.4 37.9 13.9 51.6 19.1 50.5 18.5

South 

AtlanticGulf (03)

43.9 18.3 42.0 15.0 51.4 21.5 49.3 17.6

Great Lakes 

(04)

42.5 15.9 41.0 15.2 50.9 19.1 49.1 18.2

Ohio (05) 46.7 16.3 44.6 14.8 50.6 17.6 48.3 16.0

Tennessee 

(06)

44.4 14.7 42.6 12.6 49.6 16.4 47.6 14.1

Upper 

Mississippi 

(07)

48.6 16.8 46.4 16.3 52.3 18.1 49.9 17.5

Lower 

Mississippi 

(08)

42.1 17.7 40.3 16.5 54.1 22.8 51.8 21.2

Souris-RedRainy (09)40.1 20.0 38.6 19.2 53.6 26.8 51.7 25.7

Missouri (10) 51.6 20.6 50.0 18.8 53.1 21.2 51.5 19.3

ArkansasWhite-Red 

(11)

49.5 23.4 47.9 21.5 52.6 24.9 50.8 22.8

Texas-Gulf 

(12)

46.7 24.4 45.5 23.3 51.0 26.7 49.7 25.4

Rio Grande 

(13)

42.6 18.2 41.9 17.5 56.1 24.0 55.2 23.0

Upper 

Colorado 

(14)

38.6 20.8 37.7 18.2 56.4 30.2 55.1 26.5

Lower 

Colorado 

(15)

26.2 14.4 26.1 16.1 59.9 33.0 59.5 36.4

Great Basin 

(16)

28.7 10.4 27.9 13.7 56.1 20.7 54.5 21.3

Pacific 

Northwest 

(17)

34.3 18.9 33.9 16.8 59.8 33.2 59.1 29.3

California 

(18)

46.7 22.9 46.5 21.1 62.9 30.9 62.6 28.2

Conterminous 

United 

States

43.4 18.4 42.1 17.0 53.6 22.7 52.0 20.8
32 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Among catchments with positive fertilizer use, the 

greatest unconditional prediction uncertainty, for nitrogen 

and phosphorus, is in the West (regions 13–18). This result is 

likely a consequence of the absence of county-level sales data 

in these regions, causing the cropping patterns of the West to 

be underrepresented in the regression models. The uncertainty 

for conditional predictions has a similar pattern, although the 

average uncertainty estimate for phosphorus use in the Great 

Basin region is suspect; Nevada has no phosphorus sales and 

consequently no conditional predictions are generated for 

Nevada’s catchments. The same result is true for Wyoming, 

which has no conditional predictions because of suspect sales 

data; however, Wyoming comprises a smaller portion of the 

Missouri Basin, its predominant region, than does Nevada of 

the Great Basin.

Model Evaluation

Model results are evaluated visually by comparing 

scatterplots of the predictions of elemental fertilizer use 

to estimates obtained from fertilizer sales data using other 

methods and to estimates based on direct survey data compiled 

for selected crop groups and subregions of the Nation in 

scatterplots. Model evaluation datasets are also compared 

statistically by calculating a measure of relative bias and 

correlation (table 5). For relative bias, the average of the 

differences between the natural logarithms of the paired 

evaluation and prediction values is computed. A negative 

relative bias indicates the model predictions from this study 

are high relative to the evaluation estimates. For small relative 

bias, the relative bias statistic is approximately equal to the 

difference between the two compared estimates, normalized 

by this study’s model prediction. Also reported is a p-value 

to evaluate the statistical significance of the relative bias. 

This statistic indicates little evidence of bias in any of the 

evaluation comparisons. Correlation is estimated using the 

Pearson correlation statistic and is applied to the paired natural 

logarithms of the evaluation and prediction values.

The first comparison relates the unconditional predictions 

aggregated at the county level to 2012 county predictions from 

Brakebill and Gronberg (2017) using the Gronberg and Spahr 

(2012) method. The plot for nitrogen (fig. 16) shows consistent 

agreement between the two estimates across the full range of 

predictions, with a Pearson correlation coefficient of 0.983 

(table 5) for the paired natural logarithms of the evaluation and 

prediction values. The comparison for phosphorus (fig. 17) is 

similar to nitrogen, although slightly less interrelated with a 

correlation coefficient of 0.979.

Table 5. Statistical summary of model evaluation comparisons for 2012 nitrogen and phosphorus fertilizer-use models.

[N is number of paired values; relative bias is the average of the natural logarithms of the evaluation data minus the average of the natural logarithms of the 

paired prediction values; p-value pertains to the statistical significance of the relative bias estimate; correlation is the Pearson correlation coefficient of the paired 

natural logarithms of the evaluation and prediction values]

Prediction type Evaluation dataset

Nitrogen Phosphorus

N

Relative bias

Correlation N

Relative bias

Correlation

Estimate p-value Estimate p-value

2012 unconditional 

prediction of 

fertilizer use

2012 estimates of fertilizer 

use, Gronberg and Spahr 

method1

2,100 –0.034 0.919 0.983 2,100 0.049 0.890 0.979

2012 conditional 

prediction of 

fertilizer use

2012 estimates of fertilizer 

use, Gronberg and Spahr 

method1

2,100 0.126 0.606 0.992 2,085 0.074 0.736 0.993

2012 unconditional 

prediction of 

fertilizer use

2012 estimates of fertilizer 

use by International Plant 

Nutrition Institute2

3,067 –0.002 0.998 0.894 3,067 –0.006 0.994 0.899

2012 unconditional 

prediction of 

fertilizer use

2010 survey data of fertilizer 

use on corn and soybeans3

14 0.173 0.540 0.944 14 0.129 0.734 0.896

2012 unconditional 

prediction of 

fertilizer use

2012 estimates of fertilizer 

use for Mississippi based 

on Mississippi Extension 

fertilizer recommendations4

82 –0.309 0.698 0.875 82 0.177 0.834 0.843

1

Gronberg and Spahr (2012).

2

International Plant Nutrition Institute (2012).

3

U.S. Department of Agriculture, Economic Research Service (2016a).

4

Oldham (2012); U.S. Department of Agriculture, Farm Service Agency (2013).
Model Evaluation 33

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Predicted nitrogen fertilizer use (metric tons), unconditional predictions, 

spatially-referenced method, log scale

Predicted nitrogen fertilizer use (metric tons), Gronberg and Spahr (2012) method, log scale

EXPLANATION

County-level sales

One-to-one line

Figure 16. Predicted 2012 nitrogen fertilizer use from the unconditional, spatially referenced method aggregated to the county level 

compared to 2012 county predictions from Brakebill and Gronberg (2017) using the Gronberg and Spahr (2012) method. Predicted phosphorus fertilizer use (metric tons), unconditional predictions, spatially-referenced method, log scale

Predicted phosphorus fertilizer use (metric tons), Gronberg and Spahr (2012) method, log scale

EXPLANATION

County-level sales

One-to-one line

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Figure 17. Predicted 2012 phosphorus fertilizer use from the unconditional, spatially referenced method aggregated to the county level 

compared to 2012 county predictions from Brakebill and Gronberg (2017) using the Gronberg and Spahr (2012) method.
34 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

The conditional predictions of elemental fertilizer use are 

compared to the Brakebill and Gronberg (2017) 2012 county 

predictions, at the county level, based on the Gronberg and 

Spahr (2012) method in figures 18 and 19. The figures show 

a closer relation with Brakebill and Gronberg than observed 

for the unconditional predictions in the respective figures 16

and 17, with a correlation coefficient of 0.992 for nitrogen and 

0.993 for phosphorus (table 5). The conditional adjustment is 

included in this report for comparison, given the adjustment 

is most similar to predictions generated by Brakebill and 

Gronberg (2017). The implication of these figures is that 

if the spatially referenced predictions of fertilizer use are 

to outperform the Brakebill and Gronberg predictions in a 

water-quality model, the unconditional predictions are more 

likely to do so because they are more flexible in allowing 

for net export of sales across state borders. Nevertheless, the 

spatially referenced models incorporate many more subcounty 

resolution variables than the Gronberg and Spahr method in 

determining fertilizer use at the catchment scale, making the 

comparisons in figures 18 and 19 inconclusive with regards to 

the relative performance of the predictions in a water-quality 

model. Future testing in SPAtially Referenced Regressions 

On Watershed attributes models know as SPARROW, could 

help further evaluate the unconditional compared to the 

conditional predictions.

The IPNI computes county estimates of elemental 

fertilizer use based on AAPFCO elemental fertilizer sales data 

(International Plant Nutrition Institute, 2012). The approach 

of the IPNI is to interpret fertilizer sales as fertilizer use for 

counties where sales estimates are deemed reliable. For states 

with no county-level sales data, the IPNI computes county 

estimates of use from state sales apportioned by the county 

share of 2012 USDA COA expenditures. The county sales per 

unit of cropland area are then located at the centroid of each 

county and statistically smoothed across the conterminous 

United States. A national map of fertilizer use is generated 

by assigning the smoothed fertilizer sales per unit area to 

classified cropland area using the CDL. The comparisons 

between the nitrogen and phosphorus unconditional 

predictions of 2012 county fertilizer use to the IPNI 2012 

county estimates are shown in figures 20 and 21, respectively. 

The comparisons indicate generally greater scatter than the 

Brakebill and Gronberg predictions (2017) with a correlation 

coefficient of 0.894 for nitrogen and 0.899 for phosphorus 

(table 5). This result may largely be due to the presence of 

significant imports and exports in county sales data that are 

not directly addressed by the IPNI method.

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Predicted nitrogen fertilizer use (metric tons), Gronberg and Spahr (2012) method, log scale

Predicted nitrogen fertilizer use (metric tons), conditional predictions, 

spatially-referenced method, log scale

EXPLANATION

County-level sales

One-to-one line

Figure 18. Predicted 2012 nitrogen fertilizer use from the conditional, spatially referenced method aggregated to the county level 

compared to 2012 county predictions from Brakebill and Gronberg (2017) using the Gronberg and Spahr (2012) method.
Model Evaluation 35

Predicted phosphorus fertilizer use (metric tons), conditional predictions, 

spatially-referenced method, log scale

Predicted phosphorus fertilizer use (metric tons), Gronberg and Spahr (2012) method, log scale

EXPLANATION

County-level sales

One-to-one line

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Figure 19. Predicted 2012 phosphorus fertilizer use from the conditional, spatially referenced method aggregated to the county level 

compared to 2012 county predictions from Brakebill and Gronberg (2017) using the Gronberg and Spahr (2012) method.

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Predicted nitrogen fertilizer use (metric tons),

unconditional predictions, log scale

Estimated nitrogen fertiliizer use (metric tons), log scale

(International Plant Nutrient Institute, 2012)

EXPLANATION

County-level sales

One-to-one line

Figure 20. Predicted 2012 nitrogen fertilizer use from the unconditional, spatially referenced method aggregated to the county level 

compared to 2012 county estimates produced by the International Plant Nutrition Institute (2012).
36 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

0

1

10

100

1,000

10,000

100,000

0 1 10 100 1,000 10,000 100,000

Predicted phosphorus fertilizer use (metric tons), 

unconditional predictions, log scale

Estimated phosphorus fertiliizer use (metric tons), log scale 

(International Plant Nutrient Institute, 2012)

EXPLANATION

County-level sales

One-to-one line

Figure 21. Predicted 2012 phosphorus fertilizer use from the unconditional, spatially referenced method aggregated to the county level 

compared to 2012 county estimates produced by the International Plant Nutrition Institute (2012).

The nitrogen unconditional predictions (fig. 22), with 

a Pearson correlation coefficient of 0.944, conform better to 

the USDA Economic Research Service (2016) estimates than 

the phosphorus predictions (fig. 23), with correlation of only 

0.896, but both comparisons indicate a reasonable relation 

(table 5). The stronger relation of the unconditional predictions 

for nitrogen use may partly be explained by the wide range 

of use among the USDA selected states; however, evidence 

indicates that the unconditional predictions overestimate 

nitrogen use, as compared to the USDA Economic Research 

Service estimates in the largest using states. Phosphorus does 

not display such a pattern in bias.

Lastly, model predictions are compared to 2012 estimates 

of total nitrogen and P2O5 fertilizer applied in Mississippi 

counties generated from USDA, Farm Service Agency 2012 

crop acreage data and Mississippi State University Extension 

Service fertilizer recommendations (U.S. Department of 

Agriculture, Farm Service Agency, 2013; Oldham, 2012). 

Planted crop acreage is determined for all crops by county 

using USDA, Farm Service Agency 2012 data. Fertilizer 

recommendations for nitrogen and P2O5 on 13 major crops 

are determined based on Mississippi State University 

Extension Service fertilizer recommendations (Oldham, 

2012; Larry Oldham, written commun., 2017). The 13 crops 

account for at least 93 percent of all crop acreage within 

a given county (U.S. Department of Agriculture, Farm 

Service Agency, 2013). Fertilizer recommendations for 

each crop are determined based on soil test-based fertilizer 

recommendations. The recommendations are based on the 

amount of P2

O5 measured in the soil compared to the amount 

needed by a specific crop. The P2O5 recommendations for soils 

testing medium for P2O5 are used in this comparison, as listed 

in Oldham (2012) with the exception for the Delta region 

of Mississippi. For the 18 counties that compose the Delta 

region of Mississippi, where plant-available phosphorus in 

the soil is high, the need for P2

O5 from commercial fertilizer 

sources is lower, so the Mississippi State University Extension 

Service P2O5 recommendation (based on medium soil testing 

values) is reduced to one-half of the recommendation used in 

non-Delta counties, for comparison with model predictions. 

(Oldham, 2012; Larry Oldham, written commun., 2017) 

(table 5.1). According to soil testing, soils of the Delta 

region are generally high in plant-available phosphorus, so 

P2O5 fertilization is likely less than the “medium soil test” 

recommendations in that region than other parts of the State 

(Oldham, 2012; Larry Oldham, written commun., 2017). The 

estimated nitrogen and P2O5 applied to each crop is determined 

by multiplying the recommended application rate by the 

individual crop acreage for each county. For comparison, 

phosphorus model unconditional predictions are converted to 

P2O5, and then nitrogen and P2O5 are summed at the county 

level. Both estimates are reported in metric units.

The comparison between the unconditional, spatially 

referenced model predictions and the recommendation-based 

Mississippi estimates for nitrogen and P2O5 fertilizer mass 

applied among Mississippi counties is shown in figures 24 and 

25, respectively. The Pearson correlation coefficient is 0.875 

for nitrogen and 0.843 for P2O5 (table 5). For phosphorus, 

evidence indicates an upward relative bias in the unconditional 

predictions, particularly for the higher use counties. Given the 

somewhat subjective adjustment made to P2

O5 requirements 

for the Delta counties, which also tend to be the larger use 

counties, some decline in agreement can be expected; although 

evidence also indicates a consistent upward relative bias 

among the non-Delta counties.
Model Evaluation 37

0

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 1,400,000

Predicted 2012 nitrogen fertilizer use on corn and soybeans, 

unconditional predictions, in metric tons

Estimated 2010 nitrogen fertilizer applied on corn and soybeans, in metric tons 

(U.S. Department of Agriculture Economic Research Service, 2016a)

Figure 22. U.S. Department of Agriculture, Economic Research Service survey data of 2010 nitrogen fertilizer use on corn and 

soybeans compared to unconditional predictions of 2012 nitrogen fertilizer use on corn and soybeans (metric tons) for selected states.

0

0 100,000 200,000 300,000

100,000

200,000

300,000

Predicted 2012 phosphorus fertilizer use on corn and soybeans, 

unconditional predictions, in metric tons

Estimated 2010 phosphorus fertilizer applied on corn and soybeans, in metric tons 

(U.S. Department of Agriculture Economic Research Service, 2016a)

Figure 23. U.S. Department of Agriculture, Economic Research Service survey data of 2010 phosphorus fertilizer use on corn and 

soybeans compared to unconditional predictions of 2012 phosphorus fertilizer use on corn and soybeans (metric tons) for selected 

states.
38 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

0 4,000 8,000 12,000

0

4,000

8,000

12,000

Predicted nitrogen fertilizer use, unconditional predictions, 

in metric tons

Recommended nitrogen fertilizer applied, 

in metric tons

EXPLANATION

One-to-one line

Non-Delta region counties

Delta region counties

Figure 24. Recommendation-based nitrogen fertilizer estimates applied to agricultural crops in Mississippi counties compared to 

unconditional predictions, 2012 nitrogen fertilizer use (metric tons). 

0

1,000

2,000

3,000

4,000

0 1,000 2,000 3,000 4,000

Predicted phosphate fertilizer use, unconditional predictions, 

in metric tons

Recommended phosphate fertilizer applied, 

in metric tons

EXPLANATION

One-to-one line

Non-Delta region counties

Delta region counties

Figure 25. Recommendation-based phosphate fertilizer estimates applied to agricultural crops in Mississippi counties compared to 

unconditional predictions, 2012 phosphate fertilizer use (metric tons).
Model Evaluation 39

Maps of the unconditional predictions of nitrogen and 

P2O5 application rates for catchments in Mississippi are 

shown in figures 26 and 27, respectively. The predicted 

nitrogen to phosphorus ratio in Mississippi catchments is 

shown in figure 28. The highest rates of application, for both 

elemental components, are in counties of the Delta (counties 

with widened borders) and southeastern Mississippi regions, 

which are both major agricultural crop-growing regions of the 

State. Most of the non-Delta counties have lower estimates 

of nitrogen and P2O5 use, but higher nitrogen to phosphorus 

ratios. This result is consistent with known agricultural 

practices in those regions, where poultry and hog industries 

are prevalent, making manure a significant alternative to 

commercial fertilizer.

92° 90° 88°

34°

32°

Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012 County Boundaries, 1:100,000 (LaMotte, 2016)

ARKANSAS

LOUISIANA

ALABAMA

GULF OF MEXICO

EXPLANATION

State

Delta region counties

County

Predicted nitrogen fertilizer use, metric tons

per square kilometer cropland (quantiles)

0

0.0001 to 0.94 (20th)

0.9401 to 1.48 (40th)

1.4801 to 2.48 (60th)

2.4801 to 4.52 (80th)

4.5201 to 13.17 (100th)

0 25 50 100 MILES

0 25 50 100 KILOMETERS

Figure 26. Predicted 2012 nitrogen fertilizer use at the catchment scale, metric tons per square kilometer of cropland in the State of 

Mississippi, based on unconditional predictions from the spatially referenced model.
40 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012

ARKANSAS

LOUISIANA

ALABAMA

GULF OF MEXICO

92° 90° 88°

34°

32°

0 25 50 100 MILES

0 25 50 100 KILOMETERS

EXPLANATION

State

Delta region counties

County

Predicted phosphate (P205) fertilizer use, 

metric tons per square kilometer 

cropland (quantile)

0

0.0001 to 0.32 (20th)

0.3201 to 0.53 (40th)

0.5301 to 0.85 (60th)

0.8501 to 1.38 (80th)

1.3801 to 8.77 (100th)

Figure 27. Predicted 2012 phosphate fertilizer use at the catchment scale, metric tons per square kilometer of cropland in the State of 

Mississippi, based on unconditional predictions from the spatially referenced model.
Model Evaluation 41

GULF OF MEXICO

92° 90° 88°

34°

32°

30° Base from National Hydrography Data Plus version 2 (NHDPlusV2), 1:100,000, 2012

ARKANSAS

LOUISIANA

ALABAMA

0 25 50 100 MILES

0 25 50 100 KILOMETERS

EXPLANATION

State

Delta region counties

County

Predicted 2012 fertilizer use, nitrogen to

phosphorus ratio (quantile) 

0

0.0001 to 6.43 (20th)

6.4301 to 6.94 (40th)

6.9401 to 7.52 (60th)

7.5201 to 8.41 (80th)

8.4101 to 62.22 (100th)

Figure 28. Predicted 2012 fertilizer use, nitrogen to phosphorus ratio for catchments in the State of Mississippi, based on unconditional 

predictions from the spatially referenced model.
42 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Summary and Conclusions

Elemental nitrogen and phosphorus fertilizer use on 

agricultural cropland is estimated for the conterminous United 

States for the year 2012 at the National Hydrography Dataset 

Plus version 2.0 (NHDPlusV2) catchment scale. An approach 

is developed that uses spatially referenced statistical modeling 

methods to relate the Association of American Plant Food 

Control Officials commercial fertilizer sales data to a set of 

explanatory variables to produce separate estimates of nitrogen 

and phosphorus fertilizer use. The spatially referenced 

method improves upon earlier techniques by allowing for 

varying nitrogen to phosphorus ratios at the catchment scale, 

rather than assuming the nitrogen to phosphorus ratio was 

constant across all counties in a state and by expanding the 

set of variables used to allocate county-level sales data to the 

catchment scale.

The nitrogen model explains 74 percent and the 

phosphorus model 72 percent of the variation in total nitrogen 

or phosphorus tonnage sold at the county level, and both 

models, based on state-level results, are estimated to explain 

more than 90 percent of the variation in fertilizer use. The 

planted areas of five major crop groups are determined to 

be statistically significant determinants of fertilizer use for 

nitrogen and phosphorus components. Similar explanatory 

variables were used in both models to determine the intensity 

with which elemental fertilizer is applied to the crops. The 

common factors, including climate variables and areanormalized fertilizer expenditures, had consistent results in 

both models. The respective area-normalized manure variables 

demonstrated the importance of manure as a substitute for 

commercial fertilizer, particularly for the grass-pasture-hay 

crop group as reflected by model coefficients. 

The spatial patterns of predicted nitrogen and phosphorus 

fertilizer use reflect known agricultural cropping patterns 

and practices across the United States. Total nitrogen and 

phosphorus tonnage use is predicted to be higher in states 

located in the eastern one-half of the United States and in 

major crop growing regions of the West. For nitrogen, the 

highest use (per unit area of cropland) is predicted to be in 

areas dominated by crops that require high application rates of 

nitrogen (such as corn, rice, cotton, and vegetables), and lower 

nitrogen use per unit area is predicted to be where grassland 

and pasture predominate. The highest rates of phosphorus 

fertilizer use, per unit area, closely follow the pattern of 

high nitrogen use, except for areas where soils are naturally 

rich in phosphorus, such as the Delta region of northwestern 

Mississippi or in areas where poultry production is common 

and phosphorus-rich manure available. Unlike previous 

studies, the nitrogen to phosphorus ratio in this study is not 

constant and reproduces recognized patterns in agricultural 

cropping practices. Highest values of the ratio are in the Plains 

states where nitrogen-fixing crops are not commonly grown. 

Lower ratios are in the corn-soybean crop-growing region of 

the Midwest where crop rotation with nitrogen-fixing crops is 

a common practice to supplement the nitrogen content in soil.

The estimates of this study compare favorably with 

other estimates of nitrogen and phosphorus use that are based 

on earlier U.S. Geological Survey modeling techniques 

and International Plant Nutrition Institute methods using 

2012 Association of American Plant Food Control Officials 

data. Reasonable comparisons are obtained relating model 

predictions to application rates derived from independent 

sources consisting of U.S. Department of Agriculture, 

Economic Research Service survey data for corn and soybeans 

and recommendations from the Mississippi State University 

Extension Service.

The primary products of this study are separate 

predictions of nitrogen and phosphorus fertilizer use at the 

NHDPlusV2 catchment scale. This information will prove 

useful for water-quality models that estimate total-nitrogen 

and total-phosphorus loads to streams and for other studies 

requiring fertilizer-use estimates related to agricultural 

cropping practices in the United States. The statistical 

methodology used to derive the estimates allows for the 

determination of uncertainty of the estimates, which has not 

previously been assessed by other methods. The unconditional 

predictions of nitrogen and phosphorus fertilizer use at the 

NHDPlusV2 catchment scale for the conterminous United 

States are available for download as a U.S. Geological Survey 

data release (Stewart and others, 2019b).

Determining fertilizer use for other years should be 

possible by extending this method, for years in which 

Association of American Plant Food Control Officials data 

are available. It should be possible to extend the method to 

determine fertilizer application rates for other years in which 

Association of American Plant Food Control Officials data 

are available. The main complication in doing so is finding 

reasonable substitutes for the Cropland Data Layer (CDL) 

cropping data. A shortcoming of the CDL is the classification 

accuracy for pasture/grassland, a principle crop determining 

the extent of fertilizer use. A revised CDL that reduces this 

error could yield significant improvements in fertilizer use 

estimates. Alternatively, the method could be generalized to 

include lands that are classified as close substitutes to pasture/

grassland according to the CDL error matrices. Inclusion of 

these lands as an additional source may improve model fit and 

correct possible biases in the estimated application factor for 

pasture/grassland. Another shortcoming of CDL is that the 

CDL only goes back consistently to 2009, limiting the ability 

to create a historical time series.
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Appendixes 45

Appendix 1. Description of the Method Used to Assign County Origin of Sales For 

Reported Fertilizer Sales With No County Identifier

The Association of American Plant Food Control 

Officials sales data include a locational category corresponding 

to an unknown county of origin. A signal extraction method 

is implemented to address this issue, allowing sales to be 

fully attributed to a county, or multicounty representing the 

grouping of counties with no sales information. The method 

makes use of the Census of Agriculture county fertilizer 

expenditure data, or county cropland data if expenditure data 

are not available. The methodology apportions to counties 

the total fertilizer sales having no assigned county, without 

regard to elemental composition. The elemental allocation of 

unassigned nitrogen and phosphorus sales is based on the total 

fertilizer sales apportionment.

For county c with reported sales, the true share of 

state total fertilizer sales, Sc, can be expressed in at least 

the following two ways: (1) in terms of the state share of 

expenditures, Sc, plus a random term representing net exports, 

uc, assumed to have zero mean and constant variance, s2

u

; or 

(2) in terms of the observed (but incomplete) share of state 

sales, S0c, plus a simple expenditure-based apportionment of 

the unassigned-county share of state sales, S*, plus a zeromean residual term having a variance that scales with S*

. Thus, 

the two equations describing the unknown county c true share 

of state sales are,

S s u c c   c and (1–1)

S S s S S e c c   c c  

0

* (1–2)

where

Sc is the unknown county c state share of 

expenditures;

uc is a random term representing net exports;

S0c is observed (but incomplete) share of state 

sales;

S* is a simple expenditure-based apportionment 

of the unassigned-county share of state 

sales; and

ec is the normalized residual for equation 1–2

having mean zero and variance σ2

ε. 

If a county has no fertilizer expenditure estimate, then sc is set 

equal to the county share of cropland.

Solving the two equations for true sales share, Sc, gives 

a relation among observable shares for county expenditures, 

sc

, reported sales, S0c, and the share of unassigned sales, S*, in 

terms of the unobserved residuals, ui and S ec

∗ ,

S s S u S e 0c c c c     1     (1–3)

where

uc is a random term representing net exports;

S* is a simple expenditure-based apportionment 

of the unassigned-county share of state 

sales; and

ec is the normalized residual for equation 1–2

having mean zero and variance σ2

e. 

The minimum variance unbiased linear estimate of uc, 

call it ^uc, is given by (Sargent, 1979) as follows:

u k S s S c c c

^       

0 1 (1–4)

where

k is   u u e S 2 2 2 /   ; 

S0c is observed (but incomplete) share of state 

sales;

sc is set equal to the county share of cropland, 

if a county has no fertilizer expenditure 

estimate; and

S* is a simple expenditure-based apportionment 

of the unassigned-county share of state 

sales.

To obtain an estimate of ^uc requires a state estimate 

of k, which requires estimates of σ2

u and σ2e. To obtain such 

estimates, the observable variance is compiled across counties 

among states with county-level sales data of the quantity 

S0c + sc

(S* – 1), a separate value for each state. A linear 

regression is conducted of these state-level variances on state 

values of the share of sales classified as having an unassigned 

county, S*. According to equation 1–3, the state-level variance 

of S0c + sc(S* – 1) equals σ2

u + S*

σ2

e, implying the intercept 

of the linear regression is an estimate σ2

u and the coefficient 

associated with S* is an estimate of σ2

e. 

The results of the regression are given in table 1.1, which 

indicate that both coefficients are marginally statistically 

significant, although overall model fit is poor. The results 

indicate that greater variability is associated with ec

 as 

compared to the net-export residual, uc.
46 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Given estimates of σ2

u and σ2e, which gives a state 

estimate of k, equation 1–4 can be evaluated for every county 

and subsequently substituted into equation 1–1 to obtain an 

estimate of county sales inclusive of unassigned sales, ^Sc. Let 

pc represent the proportion of state county-unassigned sales 

assigned to county (or multicounty) c. This proportion is 

computed according to 

p

S S

S

k s s ks S

S c

c c c c c          







0 0 1 ^

(1–5)

where

k is   u u e S 2 2 2 /   ;

sc is the county-c share of state expenditures, 

set equal to the county share of cropland, 

if a county has no fertilizer expenditure 

estimate;

S0c is observed (but incomplete) share of state 

sales; and

S* is a simple expenditure-based apportionment 

of the unassigned-county share of state 

sales.

Because this apportion factor can be either positive or 

negative, the factor is constrained to be nonnegative then 

factors are renormalized so that the sum of pc across all 

counties in the state is 1. Thus, the final proportion used to 

assign county-unassigned sales to specific counties is given by

p p p c c  ma   i c x ,0 0 / m  ax , (1–6)

where

pc is the proportion of state county-unassigned 

sales assigned to county (or multicounty) c.

This same proportion is applied to assign countyunassigned sales of nitrogen and phosphorus to specific 

counties.

Table 1.1. Results of the regression of state estimates of the variance across counties of the quantity defined by the reported county 

share of total state fertilizer sales plus the product of the county share of state fertilizer expenditures and the state share of county 

assigned sales (S0c

 + sc (S* – 1)) on the state share of sales unassigned to a county (S*) where * represents an individual state. The 

intercept of this regression is an estimate of the variance of net exports, σ2

u, and the slope coefficient associated with S*

 is an estimate 

of the variance σ2

e.

[t value, test statistic that is the ratio of the parameter estimate to its standard error; >, greater than; NA, not applicable]

Variable Parameter estimate Standard error t value Probability > absolute 

value of t value

Intercept 0.00024 0.00016 1.55 0.1359

Unassigned county sales share (S*) 0.00369 0.00232 1.59 0.1271

Model diagnostics

Root mean square error 0.0006072 NA NA NA

Coefficent of determination 0.1025 NA NA NA

Number of observations 24 NA NA NA
Appendixes 47

Appendix 2. Summary of Information For Cropland Data Layer 2012 Crops by 

Crop Group Used in the 2012 Nitrogen and Phosphorus Fertilizer-Use Models

(U.S. Department of Agriculture, National Agricultural Statistics Service, 2013)

Table 2.1. Summary of information for Cropland Data Layer 2012 crops by crop group used in the 2012 nitrogen and phosphorus 

fertilizer-use models (U.S. Department of Agriculture, National Agricultural Statistics Service, 2013).

Cropland Data Layer 

2012, crop(s)

Cropland Data Layer 

2012, crop code

Percent of 

total cropland

Corn crops 

Corn 1 13.241

Double crop—

barley and corn

237 0.005

Double crop—

corn and soybeans

241 0.002

Double crop—oats and corn 226 0.016

Double crop—

winter wheat and corn

225 0.058

Popcorn or ornamental corn 13 0.017

Sweet corn 12 0.042

Grass-pasture-hay crops

Grass and pasture 176 53.596

Other hay and 

nonalfalfa hay

37 3.329

Miscellaneous crops

Almonds 75 0.164

Apples 68 0.062

Apricots 223 0.001

Asparagus 207 0.003

Blueberries 242 0.013

Broccoli 214 0.002

Buckwheat 39 0.003

Cabbage 243 0.003

Camelina 38 0.001

Caneberries 55 0.002

Canola 31 0.237

Cantaloupes 209 0.003

Carrots 206 0.006

Cauliflower 244 0.000

Celery 245 0.000

Cherries 66 0.028

Chick peas 51 0.000

Christmas trees 70 0.009

Citrus 72 0.020

Cotton 2 1.835

Cranberries 250 0.005

Cropland Data Layer 

2012, crop(s)

Cropland Data Layer 

2012, crop code

Percent of 

total cropland

Miscellaneous crops—Continued

Cucumbers 50 0.005

Double crop—

lettuce and cantaloupe

231 0.001

Double crop—

lettuce and cotton

232 0.001

Double crop—

lettuce and barley

233 0.000

Double crop—

lettuce and durum wheat

230 0.006

Double crop—

soybeans and cotton

239 0.001

Double crop—

winter wheat and cotton

238 0.045

Dry beans 42 0.244

Eggplants 248 0.000

Flaxseed 32 0.040

Garlic 208 0.002

Gourds 249 0.000

Grapes 69 0.161

Greens 219 0.002

Herbs 57 0.015

Honeydew melons 213 0.001

Hops 56 0.004

Lentils 52 0.054

Lettuce 227 0.004

Mint 14 0.001

Miscellaneous vegetables 

and fruits

47 0.007

Mustard 35 0.005

Nectarines 218 0.000

Olives 211 0.006

Onions 49 0.020

Oranges 212 0.143

Other crops 44 0.024

Other tree crops 71 0.010

Peaches 67 0.007

Peanuts 10 0.231
48 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Table 2.1. Summary of information for Cropland Data Layer 2012 crops by crop group used in the 2012 nitrogen and phosphorus 

fertilizer-use models (U.S. Department of Agriculture, National Agricultural Statistics Service, 2013).—Continued

Cropland Data Layer 

2012, crop(s)

Cropland Data Layer 

2012, crop code

Percent of 

total cropland

Miscellaneous crops—Continued

Pears 77 0.004

Peas 53 0.108

Pecans 74 0.056

Peppers 216 0.003

Pistachios 204 0.029

Plums 220 0.008

Pomegranates 217 0.003

Potatoes 43 0.155

Prunes 210 0.000

Pumpkins 229 0.003

Radishes 246 0.001

Rape seed 34 0.000

Rice 3 0.371

Safflower 33 0.021

Sod and grass seed 59 0.111

Squash 222 0.003

Strawberries 221 0.006

Sugarbeets 41 0.173

Sugarcane 45 0.143

Sunflower 6 0.224

Sweet potatoes 46 0.012

Switchgrass 60 0.001

Tobacco 11 0.016

Tomatoes 54 0.050

Turnips 247 0.000

Walnuts 76 0.048

Watermelons 48 0.005

Cropland Data Layer 

2012, crop(s)

Cropland Data Layer 

2012, crop code

Percent of 

total cropland

Nitrogen-fixing crops

Alfalfa 36 2.273

Clover and wildflowers 58 0.020

Soybeans 5 9.720

Vetch 224 0.001

Small grain crops

Barley 21 0.403

Double crop—

barley and sorghum

235 0.002

Double crop—

barley and soybeans

254 0.014

Double crop—durum wheat 

and sorghum

234 0.001

Double crop—

soybeans and oats

240 0.002

Double crop—winter wheat 

and sorghum

236 0.054

Double crop—winter wheat 

and soybeans

26 0.739

Durum wheat 22 0.261

Millet 29 0.065

Oats 28 0.180

Other small grains 25 0.001

Rye 27 0.063

Speltz 30 0.000

Sorghum 4 0.875

Spring wheat 23 1.721

Triticale 205 0.022

Winter wheat 24 4.861
Appendixes 49

Appendix 3. Accuracy of Crop-Group Classification in the 2012 Cropland 

Data Layer (U.S. Department of Agriculture, National Agricultural Statistics 

Service, 2013)

Table 3.1. Accuracy of crop-group classification in the 2012 Cropland Data Layer (U.S. Department of Agriculture, National 

Agricultural Statistics Service, 2013).

[Accuracy is assessed by aggregation of Cropland Data Layer error matrices for individual states (or group of states), and aggregation of individual classes 

into the crop groups defined in appendix 2; the nonagriculture class represents all classes in the Cropland Data Layer not included in the classes appearing in 

appendix 2. Producer accuracy is given by the number of cases with correctly classified reference classes (the diagonal elements of the error matrix) divided by 

the sum of cases having the given reference class (the column sum); user accuracy is the number of cases with correctly classified reference classes divided by 

the sum of cases of the given classification (the row sum). Overall accuracy (reported in the row labeled “Sum”) is the sum of the diagonal elements divided by 

the sum of all cases. NA, not applicable]

Classified 

classes

Reference classes

Corn Grasspasture-hay

Nitrogenfixing Small grains Other Nonagriculture Sum

Corn 38,246,605 295,169 892,885 296,300 159,137 359,300 40,249,396

Grass-pasture-hay 555,077 24,974,287 1,541,971 1,149,371 379,012 70,302,329 98,902,047

Nitrogen-fixing 811,486 961,188 31,457,247 452,519 316,645 673,648 34,672,733

Small grains 217,325 307,184 356,315 35,457,949 454,375 709,349 37,502,497

Other 85,143 94,378 231,492 337,649 15,160,464 215,813 16,124,939

Nonagriculture 902,737 80,264,218 1,652,383 1,884,057 749,366 106,401,146 191,853,907

Sum 40,818,373 106,896,424 36,132,293 39,577,845 17,218,999 178,661,585 419,305,519

Classified 

classes

Accuracy 

(percent)

All classes Agriculture only

Producer User Producer User

Corn 93.7 95.0 95.8 95.9

Grass-pasture-hay 23.4 25.3 93.8 87.3

Nitrogen-fixing 87.1 90.7 91.2 92.5

Small grains 89.6 94.5 94.1 96.4

Other 88.0 94.0 92.1 95.3

Nonagriculture 59.6 55.5 NA NA

Sum 60.0 93.6
50 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Appendix 4. Derivation of the Best Linear Unbiased (Conditional) Estimates of 

Fertilizer Use

The Best Linear Unbiased Estimator (BLUE) prediction 

of fertilizer use, as described in equation 16, in the main 

section of this report, requires an estimate of the covariance 

matrices ∑js(j) and ∑s(j); the covariances between the catchment/

county j multiplier, mj; and the vector of multipliers associated 

with the reporting units of state s(j), ~ms(j). The following 

describes how these covariances are evaluated given estimates 

of the use and net-export residual components variances 

and covariances. Also, derived is a proof that the prediction 

equation 16 for these multipliers is BLUE.

The covariance matrix, ∑s(j), represents the covariances 

of the vector ~ms(j) – ~μs(j)i, where i is a vector of ones and ~μs(j) is 

a scaler equal to the unconditional expectation of the elements 

of ~ms(j), which under the assumptions of the analysis are 

all equal to the same value. If state s(j) consists of countylevel reporting units, then the county-c element of ~

ms(j) is 

exp(ηs(j) + εc + xc), where the three error components are jointly 

independent and have the following assumed properties: 

ηs(j) is a state-specific use error component, having variance 

σ2

η, which is common to all catchments and counties within 

state s(j) but independent across states; εc is a county-specific 

use error component, having variance σ2

e, which is common 

to all catchments within the same county, independent across 

nonneighboring counties or counties in different states, but 

has correlation ρε with counties and their catchments that 

neighbor county c; and xc is a county-specific net-export 

error component, having variance σ2

x, which is common to 

all catchments within the same county, independent across 

nonneighboring counties or counties in different states, but 

has ρx correlation with counties and their catchments that 

neighbor county c. Conversely, if state s(j) consists of only a 

state-level observation, then ~ms(j) is a scaler and approximately 

equals (see equation 6 in the main text of this report) 

exp(ηs(j) + ∑cεC(s(j))wcεc), where is the county-c share of state s(j) 

fertilizer use. This specification reflects the assumption that 

the net-export component is not present at the state level. 

Under the assumption of normality, the unconditional 

expectation of ~ms(j) in the case of county-level reporting 

units is 

μs j         σσσ η ε  x

~ exp / 222 2 (4–1)

which is the same for all states with county-level 

reporting units. 

For a state-level reporting unit, the assumption of 

normality implies the unconditional expectation of ~ms(j) is

μ σ s j η ε σ ρ c C s j c c ε c M c C s j w w w c ( )

~   exp             

2 2 2

1 1 1 2 1 2        / 2 (4–2)

where

M(c1) is the set of neighboring counties to county c1

in the same state. 

This evaluation varies across states with different shares and 

county neighbor configuration.

If state s(j) has county-level observations, then the (a, b) 

element of ∑s(j) takes the form

s j a b s j a b x c M c x x x b         ,         ~μ σ exp η ε δ σ σ δ ρ σε ρ σ α

2 2 2 2 2 2     1 (4–3)

where ~μs(j) is given by equation 4–1;

δα=b is 1 if the element is on the diagonal (both 

indices refer to the same county), and 0 

otherwise; and

δc∈M(cα) is 1 if the county referenced by index b, cb, is 

a neighbor of the county referenced by the 

index α, cα, and 0 otherwise. 

Conversely, if state s(j) has only a state-level observation, then 

Σs(j) consists of a single element and

s j   s j   s j     



 

    2 2 1 ~ ~ (4–4)

where ~μs(j) is given by equation 4–2.

The evaluation of the row vector Σjs(j) can similarly be 

characterized in terms of δ conditional factors, where the a-th 

element of the vector is given by

 js j s j c c j   c M   c j  C s   j   α η ε ρ σ μ μ σ δ σ δ α α ε ε

~ exp 2 2  2     1 (4–5)

where ~μs(j) is given by equation 4–1;

δcα=c(j) is equal to 1 if element α refers to the same 

county in which catchment/county j is 

located, and 0 otherwise; and

δcα∈M(c(j)) is 1 if element α refers to a neighboring 

county to that which contains catchment/

county j, and 0 otherwise.
Appendixes 51

The situation where state s(j) has only a state-level 

observation implies the covariance matrix is a scaler and is 

evaluated as





js j s j c j c C s j c c s j

js j

E w                       

 exp  ~ 2η μ  μ

        



 

 

       

 



E w s j c C s j c c s j

js j

exp

exp

~ 2

2

η μ μ

σ



 η σ ρ σ 2 2 2 2

1 1 2 1

       



      



         c C s j c c C s j c c M c C s j w w   

 

      

     

   

w s j

js j

c C s j c

c

w

2

1 1

2

2 1 2 2 2

/

exp

~ μ μ

σ σ η ρ



 w w c w c M c C s j c c j

c M c j C s j wc

1 2 1 2  2



        

        

   /  

ρ 

             






















    js j s j c j c M c j C s j μ μ σ σ η w ρ ~ exp 2 2             wc 1

(4–6)

where ~μs(j) is given by equation 4–2; and

w*

c is equal to wc

 + 1 if c = c(j) and equals wc

otherwise.

The proof that m^

j

 defined by equation 16 (in the main text 

of this report) is BLUE is as follows. Let the general linear 

unbiased estimator of mj be given by modifying the prediction 

equation 16,

m D j js j s j s D D j s j s j s j

^           



 

           



   B W   W W F 1

 



 

     ~

Ws j (4–7)

where

Β is an arbitrary row vector. 

To simplify notation, let Σj ≡ Σjs(s), Σs ≡ Σs(j), m~ s = m~ s(j), and 

Dw ≡ D(Ws(j)). The difference between the true multiplier, mj, 

and the predicted value is 

m m j j  m D j j     w w   D Ds w Dw s  



 



  ^ ~ ~ μ μ Β   1

m i (4–8)

where

D(Ws(j))

~

ms(j) is substituted for Fs(j). 

The prediction variance is given by

V mj j E m j w D Dw sD D w wE mj s

^ 















    

           

μ μ 2 12 Β   m

~ ~  



 



 



 



 

    

μ i

 B   j w D Dw s  D D w w

1

E D m m s s w

~ ~ ~ ~  ' 



 



  



 



 



 



 μ μ i i 

 B

B

    

















    





 

 

j w w s w

j m j w w s w

D D D

V m D D D

1

2 1

2

'

^ σ   

      

D

D D D D D D

w j

j w w s w w s w j w



   

'

 B B 1

D D

V m D D D D D D

w s w

j m j w w s w w j w s w



  

   

















    





1

2 1

'

^ ' σ  B B'

(4–9)

where

 σ2

m is the unconditional variance of the 

multiplier, mj. 

The last term in the last line of equation 4–9 is the positivedefinite matrix (in this case, a scaler), BDwΣs

DwB', and is 

strictly greater than zero if any element of B is nonzero. 

Therefore, because B is an arbitrary vector, to minimize 

prediction error variance B should be set to a vector of zeros, 

implying prediction equation 16 (in the main text of this 

report) results in the minimum prediction uncertainty, making 

it BLUE.
52 Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012

Appendix 5. Mississippi State University Extension Service Fertilizer 

Recommendations For Selected Crops Used to Estimate Total Nitrogen and 

Phosphate Fertilizer Applied to Crops in Mississippi Counties (Oldham, 2012)

Table 5.1. Mississippi State University Extension Service 

fertilizer recommendations for selected crops used to estimate 

total nitrogen and phosphate fertilizer applied to crops in 

Mississippi counties (Oldham, 2012).

Crops

Fertilizer recommendations 

(pounds per acre)

Nitrogen Phosphate 

All 

counties

Delta region 

counties

Non-Delta 

region counties

Corn—irrigated 112 80 40

Corn—Sorghum 180 50 25

Cotton 100 40 20

Grass 80 30 15

Mixed forage 80 30 15

Peanuts 20 60 30

Rice 170 30 15

Rye 90 45 22.5

Sorghum 100 45 22.5

Soybeans—small grains 100 80 40

Soybeans 20 30 15

Sweet potatoes 35 60 30

Wheat 100 45 22.5
Publishing support provided by the U.S. Geological Survey 

Science Publishing Network, Madison, Pembroke, and Sacramento 

Publishing Service Center

For more information concerning the research in this report, contact the

Director, Upper Midwest Water Science Center

U.S. Geological Survey 8505 Research Way 

Middleton, Wisconsin 53562 

(608) 828–9901 https://wi.water.usgs.gov ISSN 2328-0328 (online)

https://doi.org/10.3133/sir20185145
Stewart and others—Catchment-Level Estimates of Nitrogen and Phosphorus Agricultural Use from Commercial Fertilizer Sales for the United States, 2012—SIR 2018–5145