*#

*#

*#

*#

*#

#

*#

#

Potomac

River

Patuxent R

Rappahannock R

James

River

Pamunkey R

Mattaponi R

Chesapeake Bay

ATLANTIC OCEAN

Potomac

River

Choptank R

Chesapeake Bay

Watershed 

River

Appomattox

Altoona

Harrisburg

Baltimore

Annapolis

Richmond

Harrisonburg

York

Norfolk

PENNSYLVANIA

NEW

YORK

WEST

VIRGINIA

VIRGINIA

Washington DC

MARY LAND

DELAWARE

NEW

JERSEY

Susquehanna River

*#

*#

*#

*#

*#

*#

*#

*#

*# U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2012–5244

Prepared in cooperation with the Virginia Department of Environmental Quality, 

Maryland Department of Natural Resources, and the U.S. Environmental 

Protection Agency Chesapeake Bay Program

Comparison of Two Regression-Based Approaches 

for Determining Nutrient and Sediment Fluxes and 

Trends in the Chesapeake Bay Watershed

Comparison of Two Regression-Based Approaches 

for Determining Nutrient and Sediment Fluxes and 

Trends in the Chesapeake Bay Watershed

By Douglas L. Moyer, Robert M. Hirsch, and Kenneth E. Hyer

Prepared in cooperation with the Virginia Department of Environmental Quality, 

Maryland Department of Natural Resources, and the U.S. Environmental Protection 

Agency Chesapeake Bay Program 

Scientific Investigations Report 2012–5244

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, 

natural hazards, and the environment, visit http://www.usgs.gov or call 1-888-ASK-USGS

For an overview of USGS information products, including maps, imagery, and publications, 

visit http://www.usgs.gov/pubprod

To order this and other USGS information products, visit http://store.usgs.gov

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the 

U.S. Government.

Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation:

Moyer, D.L., Hirsch, R.M., and Hyer, K.E., 2012, Comparison of two regression-based approaches for determining 

nutrient and sediment fluxes and trends in the Chesapeake Bay watershed: U.S. Geological Survey Scientific 

Investigations Report 2012–5244, 118 p. (Available online at http://pubs.usgs.gov/sir/2012/5244/)

ISBN 978-1-4113-3525-7
iii

Contents

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

Introduction.....................................................................................................................................................2

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

Description of Nutrient and Sediment Data .....................................................................................4

Comparison of Regression Models ............................................................................................................6

Description of Model Forms................................................................................................................6

Parameter Estimation...........................................................................................................................7

Censored Data.......................................................................................................................................9

Determination of Trends in Water-Quality Conditions ..................................................................11

Reporting Error Associated with Flux and Trend Estimates ........................................................12

Comparison of Estimate Accuracy and Bias ..........................................................................................13

Observed and Estimated Flux Comparison Results.......................................................................13

Sources of Flux Bias Discrepancies................................................................................................15

Category I Combinations ..........................................................................................................16

Category II Combinations .........................................................................................................18

Category III Combinations........................................................................................................21

Comparison of ESTIMATOR and WRTDS Annual Fluxes.......................................................................24

Nitrogen................................................................................................................................................24

Phosphorus..........................................................................................................................................24

Suspended Sediment .........................................................................................................................33

Trends in Nutrient and Sediment Flux to the Chesapeake Bay............................................................35

Comparison of ESTIMATOR and WRTDS Trend Results .............................................................36

Changes in Flow-Normalized Flux at Individual Stations.............................................................41

Nitrogen................................................................................................................................................41

Phosphorus..........................................................................................................................................44

Suspended Sediment .........................................................................................................................48

Summary and Conclusions.........................................................................................................................51

Acknowledgments.......................................................................................................................................53

References Cited..........................................................................................................................................54

Appendix 1. Total nitrogen, nitrate, total phosphorus, orthophosphorus, and 

suspended sediment collected at the nine River Input Monitoring stations 

in the Chesapeake Bay watershed .............................................................................................57

Appendix 2. WRTDS-derived annual and flow-normalized annual fluxes for 

total nitrogen, nitrate, total phosphorus, orthophosphorus, and suspended 

sediment at the nine Chesapeake Bay River Input Monitoring stations.............................103

Appendix 3. Rates of change in WRTDS flow-normalized flux for total nitrogen, 

nitrate, total phosphorus, orthophosphorus, and suspended sediment at the

nine Chesapeake Bay River Input Monitoring stations .........................................................109

Appendix 4. Time series of WRTDS annual and flow-normalized annual fluxes of total 

nitrogen, nitrate, total phosphorus, orthophosphorus, and suspended sediment 

at the nine River Input Monitoring stations in the Chesapeake Bay watershed...............113
iv

Figures

1. Map showing location of the nine River Input Monitoring (RIM) stations 

in the Chesapeake Bay watershed............................................................................................5

2. Graph showing total nitrogen record, showing interval censored values as 

vertical lines, for Rappahannock River at Fredericksburg, Virginia...................................10

3–6. Graphs showing the observed concentration versus discharge relation, 

observed and ESTIMATOR-predicted concentration versus discharge relation, 

residual (observed minus predicted) plot for ESTIMATOR predictions, observed 

and WRTDS-predicted concentration versus discharge relation, and residual 

(observed minus predicted) plot for WRTDS predictions—

3. Nitrate at Patuxent River near Bowie, Maryland (USGS Station 01594440) ...........17

4. Suspended-sediment concentration at Rappahannock River near 

Fredericksburg, Virginia (USGS Station ID 01668000)..................................................19

5. Nitrate concentration at Rappahannock River near Fredericksburg, 

Virginia (USGS Station ID 01668000)...............................................................................20

6. Suspended-sediment concentration at Susquehanna River near 

Conowingo, Maryland (USGS Station ID 01578310).....................................................22

7–9. Graphs showing at the Susquehanna, Potomac, James, Rappahannock, Appomattox, 

Pamunkey, Mattaponi, Patuxent, and Choptank River Input Monitoring stations—

7. WRTDS-derived versus ESTIMATOR-derived annual total nitrogen 

fluxes and associated average percent difference.....................................................25

8. WRTDS-derived versus ESTIMATOR-derived annual nitrate fluxes 

and associated average percent difference.................................................................26

9. WRTDS-derived versus ESTIMATOR-derived annual total phosphorus 

fluxes and associated average difference....................................................................27

10–11. Graphs of total phosphorus concentration showing the observed concentration 

versus discharge relation, observed and ESTIMATOR-predicted concentration 

versus discharge relation, residual (observed minus predicted) plot for ESTIMATOR 

predictions, observed and WRTDS-predicted (concentration versus discharge 

relation, and residual (observed minus predicted) plot for WRTDS predictions—

10. At Susquehanna River near Conowingo, Maryland 

(USGS Station ID 01578310)..............................................................................................29

11. At Rappahannock River near Fredericksburg, Virginia 

(USGS Station ID 01668000)..............................................................................................30

12. Graphs showing WRTDS-derived versus ESTIMATOR-derived annual 

orthophosphorus fluxes and associated average difference at the Susquehanna, 

Potomac, James, Rappahannock, Appomattox, Pamunkey, Mattaponi, Patuxent, 

and Choptank River Input Monitoring stations ......................................................................31

13. Graphs of orthophosphorus concentration at Patuxent River near Bowie, 

Maryland (USGS Station ID 01594440), showing the observed concentration 

versus discharge relation, observed and ESTIMATOR-predicted concentration 

versus discharge relation, residual (observed minus predicted) plot for ESTIMATOR 

predictions, observed and WRTDS-predicted concentration versus discharge 

relation, and residual (observed minus predicted) plot for WRTDS predictions .............32

14. Graphs showing WRTDS-derived versus ESTIMATOR-derived annual suspendedsediment fluxes and associated average percent difference at the Susquehanna, 

Potomac, James, Rappahannock, Appomattox, Pamunkey, Mattaponi, Patuxent, 

and Choptank River Input Monitoring stations ......................................................................34
v

15–17. Graphs showing observed concentration collected during high (greater than 

the 90th percentile of discharge), intermediate (greater than the 60th and less 

than the 90th percentile of discharge), and low (less than or equal to the 

60th percentile of discharge) discharges—

15. Total nitrogen at the Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310)..............................................................................................37

16. Total phosphorus at the James River at Cartersville, Maryland 

(USGS Station ID 02035000)..............................................................................................38

17. Nitrate at the Pamunkey River near Hanover, Virginia 

(USGS Station ID 01673000)..............................................................................................40

18. Graphs showing total nitrogen yields and flow-normalized yields for the 

nine RIM stations using the WRTDS method.........................................................................42

19. Graphs showing nitrate yields and flow-normalized yields for the nine 

RIM stations using the WRTDS method..................................................................................43

20. Graphs showing total phosphorus yields and flow-normalized yields for 

the nine RIM stations using the WRTDS method..................................................................45

21. Graphs showing orthophosphorus yields and flow-normalized yields for 

the nine RIM stations using the WRTDS method..................................................................46

22. Graphs showing suspended sediment yields and flow-normalized yields 

for the nine RIM stations using the WRTDS method............................................................49

Appendix Figures

Appendix 1. Graphs showing the observed concentration versus discharge relation, 

observed and ESTIMATOR-predicted concentration versus discharge relation, 

residual (observed minus predicted) plot for ESTIMATOR predictions, observed 

and WRTDS-predicted concentration versus discharge relation, and residual 

(observed minus predicted) plot for WRTDS predictions—

 1–1. Total nitrogen at Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310) ...........................................................................................58

 1–2. Nitrate at Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310) ...........................................................................................59

 1–3. Total phosphorus at Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310) ..........................................................................................60

 1–4. Orthophosphorus at Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310) ...........................................................................................61

 1–5. Suspended sediment at Susquehanna River at Conowingo, Maryland 

(USGS Station ID 01578310) ...........................................................................................62

 1–6. Total nitrogen at Potomac River at Chain Bridge at Washington, D.C. 

(USGS Station ID 01646580) ...........................................................................................63

 1–7. Nitrate at Potomac River at Chain Bridge at Washington, D.C.

(USGS Station ID 01646580) ..........................................................................................64

 1–8. Total phosphorus at Potomac River at Chain Bridge at Washington, D.C. 

(USGS Station ID 01646580) ..........................................................................................65

 1–9. Orthophosphorus at Potomac River at Chain Bridge at Washington, D.C.

(USGS Station ID 01646580) ..........................................................................................66

 1–10. Suspended sediment at Potomac River at Chain Bridge at Washington, D.C.

(USGS Station ID 01646580) ..........................................................................................67
vi

Appendix Figures—Continued

Appendix 1. Graphs showing the observed concentration versus discharge relation, 

observed and ESTIMATOR-predicted concentration versus discharge relation, 

residual (observed minus predicted) plot for ESTIMATOR predictions, observed 

and WRTDS-predicted concentration versus discharge relation, and residual 

(observed minus predicted) plot for WRTDS predictions—

 1–11. Total nitrogen at James River at Cartersville, Virginia 

 (USGS Station ID 02035000) ........................................................................................68

 1–12. Nitrate at James River at Cartersville, Virginia

 USGS Station ID 02035000) .........................................................................................69

 1–13. Total phosphorus at James River at Cartersville, Virginia

 (USGS Station ID 02035000).........................................................................................70

 1–14. Orthophosphorus at James River at Cartersville, Virginia

(USGS Station ID 02035000) ........................................................................................71

 1–15. Suspended sediment at James River at Cartersville, Virginia

 (USGS Station ID 02035000).........................................................................................72

 1–16. Total nitrogen at Rappahannock River near Fredericksburg, Virginia

 (USGS Station ID 01668000).........................................................................................73

 1–17. Nitrate at Rappahannock River near Fredericksburg, Virginia

(USGS Station ID 01668000) ........................................................................................74

 1–18. Total phosphorus at Rappahannock River near Fredericksburg, Virginia

(USGS Station ID 01668000) ........................................................................................75

 1–19. Orthophosphorus at Rappahannock River near Fredericksburg, Virginia

 (USGS Station ID 01668000) ........................................................................................76

 1–20. Suspended sediment at Rappahannock River near Fredericksburg, Virginia

 (USGS Station ID 01668000) ........................................................................................77

 1–21. Total nitrogen at Appomattox River near Matoaca, Virginia

 (USGS Station ID 02041650) ........................................................................................78

 1–22. Nitrate at Appomattox River near Matoaca, Virginia

 (USGS Station ID 02041650).........................................................................................79

 1–23. Total phosphorus at Appomattox River near Matoaca, Virginia

 (USGS Station ID 02041650).........................................................................................80

 1–24. Orthophosphorus at Appomattox River near Matoaca, Virginia

 (USGS Station ID 02041650).........................................................................................81

 1–25. Suspended sediment at Appomattox River near Matoaca, Virginia

 (USGS Station ID 02041650) ........................................................................................82

 1–26. Total nitrogen at Pamunkey River near Hanover, Virginia

 (USGS Station ID 01673000) ........................................................................................83

 1–27. Nitrate at Pamunkey River near Hanover, Virginia

 (USGS Station ID 01673000).........................................................................................84

 1–28. Total phosphorus at Pamunkey River near Hanover, Virginia

 (USGS Station ID 01673000) ........................................................................................85

 1–29. Orthophosphorus at Pamunkey River near Hanover, Virginia

 (USGS Station ID 01673000) .......................................................................................86

 1–30. Suspended sediment at Pamunkey River near Hanover, Virginia

 (USGS Station ID 01673000) ........................................................................................87
vii

 1–31. Total nitrogen at Mattaponi River near Beulahville, Virginia

 (USGS Station ID 01674500) ........................................................................................88

 1–32. Nitrate at Mattaponi River near Beulahville, Virginia

 (USGS Station ID 01674500).........................................................................................89

 1–33. Total phosphorus at Mattaponi River near Beulahville, Virginia

 (USGS Station ID 01674500) ........................................................................................90

 1–34. Orthophosphorus at Mattaponi River near Beulahville, Virginia

 (USGS Station ID 01674500) ........................................................................................91

 1–35. Suspended sediment at Mattaponi River near Beulahville, Virginia

 (USGS Station ID 01674500).........................................................................................92

 1–36. Total nitrogen at Patuxent River near Bowie, Maryland

 (USGS Station ID 01594440) ........................................................................................93

 1–37. Nitrate at Patuxent River near Bowie, Maryland

 (USGS Station ID 01594440).........................................................................................94

 1–38. Total phosphorus at Patuxent River near Bowie, Maryland

 (USGS Station ID 01594440).95

 1–39. Orthophosphorus at Patuxent River near Bowie, Maryland

 (USGS Station ID 01594440) ........................................................................................96

 1– 40. Suspended sediment at Patuxent River near Bowie, Maryland

 (USGS Station ID 01594440).........................................................................................97

 1– 41. Total nitrogen at Choptank River near Greensboro, Maryland

 (USGS Station ID 01491000) ........................................................................................98

 1– 42. Nitrate at Choptank River near Greensboro, Maryland

 (USGS Station ID 01491000).........................................................................................99

 1– 43. Total phosphorus at Choptank River near Greensboro, Maryland

 (USGS Station ID 01491000) ......................................................................................100

 1– 44. Orthophosphorus at Choptank River near Greensboro, Maryland

 (USGS Station ID 01491000) ......................................................................................101

 1– 45. Suspended sediment at Choptank River near Greensboro, Maryland

 (USGS Station ID 01491000).......................................................................................102

Appendix 2. Graphs showing WRTDS-derived annual and flow-normalized annual fluxes 

at the nine Chesapeake Bay River Input Monitoring stations: Susquehanna, 

Potomac, James, Rappahannock, Appomattox, Pamunkey, Mattaponi, Patuxent, 

Choptank River Input Monitoring stations for—

 2–1. Total nitrogen.................................................................................................................104

 2–2. Nitrate.............................................................................................................................105

 2–3. Total phosphorus...........................................................................................................106

 2– 4. Orthophosphorus..........................................................................................................107

 2–5. Suspended sediment....................................................................................................108
viii

Tables

1. Chesapeake Bay River Input Monitoring Stations and associated 

watershed characteristics..........................................................................................................4

2. Measures of RMSE and flux bias relating WRTDS- and ESTIMATOR-derived 

nutrient and sediment fluxes compared to discrete flux observations at the 

nine River Input Monitoring (RIM) stations ............................................................................14

3. Total nitrogen trends for WRTDS flow-normalized yields and ESTIMATOR 

flow-adjusted concentration at the nine River Input Monitoring (RIM) stations 

for the time periods 1985 to 2010 and 2001 to 2010 ...............................................................35

4. Nitrate trends in WRTDS flow-normalized yields and ESTIMATOR flow-adjusted 

concentration at the nine River Input Monitoring (RIM) stations for the time 

periods 1985 to 2010 and 2001 to 2010 ....................................................................................39

5. Total phosphorus trends in WRTDS flow-normalized yields and ESTIMATOR flowadjusted concentration at the nine River Input Monitoring (RIM) stations for 

the time periods 1985 to 2010 and 2001 to 2010 .....................................................................47

6. Orthophosphorus trends in WRTDS flow-normalized yields and ESTIMATOR flowadjusted concentration at the nine River Input Monitoring (RIM) stations for 

the time periods 1985 to 2010 and 2001 to 2010 .....................................................................47

7. Suspended sediment trends in WRTDS flow-normalized yields and ESTIMATOR 

flow-adjusted concentration at the nine River Input Monitoring (RIM) stations 

for the time periods 1985 to 2010 and 2001 to 2010................................................................50

Appendix Tables

3–1. Total nitrogen trend slopes for WRTDS flow-normalized fluxes at the nine River Input 

Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010...............110

3–2. Nitrate trend slopes for WRTDS flow-normalized fluxes at the nine River Input 

Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010...............110

3–3. Total phosphorus trend slopes for WRTDS flow-normalized fluxes at the nine River 

Input Monitoring stations for the time periods 1985 to 2010 and 2001 to 2010 ...............111

3–4. Orthophosphorus trend slopes for WRTDS flow-normalized fluxes at the nine River

Input Monitoring stations for the time periods 1985 to 2010 and 2001 to 2010 ...............111

3–5. Suspended sediment trend slopes for flow-normalized at the nine River Input 

Monitoring stations for the time periods 1985 to 2010 and 2001 to 2010..........................112

4–1. Total nitrogen annual flux and flow-normalized annual flux, in tons per day, for the 

nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS...........114

4–2. Nitrate annual flux and flow-normalized annual flux, in tons per day, for the nine 

Chesapeake Bay River Input Monitoring Stations obtained using WRTDS....................115

4–3. Total phosphorus annual flux and flow-normalized annual flux, in tons per day, for 

the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS .......116

4–4. Orthophosphorus annual flux and flow-normalized annual flux, in tons per day, for 

the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS .......117

4–5. Suspended sediment annual flux and flow-normalized annual flux, in tons per

day, for the nine Chesapeake Bay River Input Monitoring Stations obtained 

using WRTDS.............................................................................................................................118
ix

Conversion Factors and Datums

Inch/Pound to SI

Multiply By To obtain

Length

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2) 

Volume

cubic foot (ft3) 0.02832 cubic meter (m3) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

cubic foot per second per square 

mile [(ft3/s)/mi2]

 0.01093 cubic meter per second per 

square kilometer [(m3/s)/km2]

Mass

ton, short (2,000 lb) 0.9072 megagram (Mg) 

ton per day (ton/d) 0.9072 metric ton per day

ton per day per square mile 

[(ton/d)/mi2]

 0.3503 megagram per day per square

kilometer [(Mg/d)/km2]

ton per year (ton/yr) 0.9072 megagram per year (Mg/yr)

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

(NAVD 88).

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

Abbreviations

AMLE Adjusted Maximum Likelihood Estimation 

BNR biological nitrate reduction 

CBP Chesapeake Bay Program

ESTIMATOR a multiple regression model

(lb/d)/mi2 pounds per day per square mile 

MDDNR Maryland Department of Natural Resources 

mg/L milligram per liter 

NO2 nitrite

NO3 nitrate

PO4

3–

 orthophosphorus

RIM River Input Monitoring 

SSC suspended-sediment concentration 

TMDL total maximum daily load

TSS total suspended solids 

USEPA U.S. Environmental Protection Agency

USGS U.S. Geological Survey 

VADEQ Virginia Department of Environmental Quality

WRTDS Weighted Regressions on Time, Discharge, and Season
Comparison of Two Regression-Based Approaches 

for Determining Nutrient and Sediment Fluxes and 

Trends in the Chesapeake Bay Watershed

By Douglas L. Moyer, Robert M. Hirsch, and Kenneth E. Hyer

Abstract 

Nutrient and sediment fluxes and changes in fluxes over 

time are key indicators that water resource managers can use 

to assess the progress being made in improving the structure 

and function of the Chesapeake Bay ecosystem. The U.S. 

Geological Survey collects annual nutrient (nitrogen and 

phosphorus) and sediment flux data and computes trends 

that describe the extent to which water-quality conditions are 

changing within the major Chesapeake Bay tributaries. Two 

regression-based approaches were compared for estimating 

annual nutrient and sediment fluxes and for characterizing 

how these annual fluxes are changing over time. The two 

regression models compared are the traditionally used 

ESTIMATOR and the newly developed Weighted Regression on Time, Discharge, and Season (WRTDS). The model 

comparison focused on answering three questions: (1) What 

are the differences between the functional form and construction of each model? (2) Which model produces estimates 

of flux with the greatest accuracy and least amount of bias? 

(3) How different would the historical estimates of annual 

flux be if WRTDS had been used instead of ESTIMATOR? 

One additional point of comparison between the two models 

is how each model determines trends in annual flux once the 

year-to-year variations in discharge have been determined. 

All comparisons were made using total nitrogen, nitrate, 

total phosphorus, orthophosphorus, and suspended-sediment 

concentration data collected at the nine U.S. Geological 

Survey River Input Monitoring stations located on the 

Susquehanna, Potomac, James, Rappahannock, Appomattox, 

Pamunkey, Mattaponi, Patuxent, and Choptank Rivers in the 

Chesapeake Bay watershed.

Two model characteristics that uniquely distinguish 

ESTIMATOR and WRTDS are the fundamental model form 

and the determination of model coefficients. ESTIMATOR and 

WRTDS both predict water-quality constituent concentration 

by developing a linear relation between the natural logarithm 

of observed constituent concentration and three explanatory 

variables—the natural log of discharge, time, and season. 

ESTIMATOR uses two additional explanatory variables—the 

square of the log of discharge and time-squared. Both models 

determine coefficients for variables for a series of estimation 

windows. ESTIMATOR establishes variable coefficients for 

a series of 9-year moving windows; all observed constituent 

concentration data within the 9-year window are used to 

establish each coefficient. Conversely, WRTDS establishes 

variable coefficients for each combination of discharge and 

time using only observed concentration data that are similar 

in time, season, and discharge to the day being estimated. As 

a result of these distinguishing characteristics, ESTIMATOR 

reproduces concentration-discharge relations that are closely 

approximated by a quadratic or linear function with respect 

to both the log of discharge and time. Conversely, the linear 

model form of WRTDS coupled with extensive model 

windowing for each combination of discharge and time allows 

WRTDS to reproduce observed concentration-discharge 

relations that are more sinuous in form. 

Another distinction between ESTIMATOR and WRTDS 

is the reporting of uncertainty associated with the model 

estimates of flux and trend. ESTIMATOR quantifies the 

standard error of prediction associated with the determination 

of flux and trends. The standard error of prediction enables the 

determination of the 95-percent confidence intervals for flux 

and trend as well as the ability to test whether the reported 

trend is significantly different from zero (where zero equals 

no trend). Conversely, WRTDS is unable to propagate error 

through the many (over 5,000) models for unique combinations of flow and time to determine a total standard error. As a 

result, WRTDS flux estimates are not reported with confidence 

intervals and a level of significance is not determined for 

flow-normalized fluxes.
2 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

The differences between ESTIMATOR and WRTDS, 

with regard to model form and determination of model 

coefficients, have an influence on the determination of nutrient 

and sediment fluxes and associated changes in flux over time 

as a result of management activities. The comparison between 

the model estimates of flux and trend was made for combinations of five water-quality constituents at nine River Input 

Monitoring stations. 

The major findings with regard to nutrient and sediment 

fluxes are as follows: (1) WRTDS produced estimates of 

flux for all combinations that were more accurate, based on 

reduction in root mean squared error, than flux estimates 

from ESTIMATOR; (2) for 67 percent of the combinations, 

WRTDS and ESTIMATOR both produced estimates of flux 

that were minimally biased compared to observed fluxes 

(flux bias=tendency to over or underpredict flux observations); however, for 33 percent of the combinations, WRTDS 

produced estimates of flux that were considerably less biased 

(by at least 10 percent) than flux estimates from ESTIMATOR; 

(3) the average percent difference in annual fluxes generated 

by ESTIMATOR and WRTDS was less than 10 percent at 

80 percent of the combinations; and (4) the greatest differences related to flux bias and annual fluxes all occurred for 

combinations where the pattern in observed concentrationdischarge relation was sinuous (two points of inflection) rather 

than linear or quadratic (zero or one point of inflection). 

The major findings with regard to trends are as follows: 

(1) both models produce water-quality trends that have 

factored in the year-to-year variations in flow; (2) trends in 

water-quality condition are represented by ESTIMATOR as a 

trend in flow-adjusted concentration and by WRTDS as a flownormalized flux; (3) for 67 percent of the combinations with 

trend estimates, the WRTDS trends in flow-normalized flux 

are in the same direction and magnitude to the ESTIMATOR 

trends in flow-adjusted concentration, and at the remaining 

33 percent the differences in trend magnitude and direction are 

related to fundamental differences between concentration and 

flux; and (4) the majority (85 percent) of the total nitrogen, 

nitrate, and orthophosphorus combinations exhibited longterm (1985 to 2010) trends in WRTDS flow-normalized flux 

that indicate improvement or reduction in associated flux and 

the majority (83 percent) of the total phosphorus (from 1985 to 

2010) and suspended sediment (from 2001 to 2010) combinations exhibited trends in WRTDS flow-normalized flux that 

indicate degradation or increases in the flux delivered.

Introduction

Excessive nutrient (nitrogen and phosphorus) and 

sediment transport to the Chesapeake Bay from the watershed 

is detrimental to the overall structure and function of the 

bay ecosystem and is a major concern for local, State, and 

Federal entities that benefit from and work to protect the 

living resources of the bay. The flux (also called load) of 

nutrients to the Chesapeake Bay is in part natural but has 

been accelerated as a result of anthropogenic inputs of these 

nutrients through sewage disposal, agricultural runoff, urban 

runoff, and acid rain (Officer and others, 1984; Nixon, 1987; 

Schlesinger, 1997). Accelerated eutrophication through excessive nutrient flux has been linked to the loss of critical habitat 

for living resources within the Chesapeake Bay estuary (U.S. 

Environmental Protection Agency, 1983). Cooper and Brush 

(1991) found that accelerated algal production resulting from 

elevated nutrient fluxes has led to an increased occurrence of 

anoxic conditions in bottom waters and associated sediment 

throughout the Chesapeake Bay estuary. Similarly, the flux 

of sediment to the Chesapeake Bay results from both natural 

processes associated with upland erosion, lateral movement 

of channels into streambanks, and downcutting of streambeds 

(Waters, 1995) as well as anthropogenic processes such as 

agriculture, logging, mining, and urbanization. Anthropogenically derived sediment can overwhelm the natural 

assimilative capacity of the aquatic system (Cairns, 1977) and 

may bury filter-feeding organisms, reduce habitat available for 

macroinvertebrates, contribute to decreased fish populations, 

and impair growth of aquatic vegetation by reducing available 

light (Lenat and others, 1981; Dennison and others, 1993; 

Box and Mossa, 1999; Madsen and others, 2001).

The Chesapeake Bay Program (CBP) was initiated in 

1983 to direct the restoration and protection of the Chesapeake Bay. The CBP is composed of various Federal, State, 

academic, and local watershed organizations. In 1987, the 

CBP established its first nutrient reduction goal, which was 

to reduce nitrogen and phosphorus fluxes to the Chesapeake 

Bay. In 2000, the CBP recommitted to achieve the nutrient and 

sediment reduction goals established in 1987and established 

criteria for dissolved oxygen, chlorophyll, and water clarity 

(Chesapeake Bay Program, 2000). Despite extensive restoration efforts made by the CBP, however, established waterquality goals were not being obtained for the Chesapeake 

Bay and associated tributaries (Chesapeake Bay Foundation, 

2010). As a result, in 2010, the U.S. Environmental Protection Agency (USEPA) established the Chesapeake Bay total 

maximum daily load (TMDL) for nitrogen, phosphorus, and 

sediment (U.S. Environmental Protection Agency, 2010). This 

TMDL assigns accountability for nutrient and sediment fluxes 

to New York, Pennsylvania, Maryland, Delaware, Virginia, 

West Virginia, and the District of Columbia and serves as a 

catalyst for rigorous implementation of management actions 

to mitigate the transport of excessive nutrients and sediment to 

the Chesapeake Bay and tidal estuaries. 

Since the early 1990s, the U.S. Geological Survey 

(USGS), in cooperation with the Virginia Department of 

Environmental Quality (VADEQ) and the Maryland Department of Natural Resources (MDDNR), has been responsible 

for monitoring nutrient and sediment conditions in the major 

rivers of the Chesapeake Bay watershed. Additionally, the 

USGS is responsible for quantifying annual nutrient and 
Introduction 3

sediment fluxes to the bay as well as determining long-term 

changes in water-quality conditions in the Chesapeake Bay 

watershed to facilitate continuing evaluation of the progress 

being made toward reducing nutrient and sediment inputs 

to the bay. The primary method for quantifying fluxes and 

determining trends in concentration at the River Input 

Monitoring (RIM) stations has been through the use of a 

multiple regression model (ESTIMATOR). ESTIMATOR 

had its origins in the work of Cohn and others (1992) who 

determined that the multiple regression approach generates 

valid flux estimates for nutrients (suspended sediment was 

not evaluated); the evaluation was based on data collected 

between 1980 and 1988 at the four RIM stations in Maryland. 

The model was modified to accommodate censored values 

(water-quality concentrations reported as less than a specified 

reporting or analytical limit) (Cohn, 2005). ESTIMATOR has 

been widely used for flux determination and identification of 

changing water-quality conditions over time (Langland and 

others, 2006; Aulenbach and others, 2007). As the USGS RIM 

program generated water-quality datasets for longer periods 

of time, however, it became apparent that improvements 

needed to be made to the ESTIMATOR approach. These 

improvements included (1) enhanced model flexibility to 

accommodate complex concentration-discharge relations 

(that is, concentration-discharge relations that are not linear or 

quadratic in form and that can change over a period of several 

decades) and (2) the ability to provide estimates of trends 

in flux, which may be different from trends in concentration 

(that is, an inherent property of ESTIMATOR is that, when 

expressed in terms of percentage change over time, estimates 

of concentration changes are constrained to be equal to 

percentage changes in flux). An evolving understanding of 

these apparent shortfalls in the ESTIMATOR approach has led 

to research and development of new methods for the determination of nutrient and sediment fluxes and associated trends 

at long-term (greater than 20 years) monitoring stations with 

large (greater than 300 observations) water-quality datasets. 

To address these needs, the USGS recently developed a 

new method for the determination of nutrient and sediment 

fluxes and trends using multiple weighted regressions (Hirsch 

and others, 2010). The Weighted Regressions on Time, 

Discharge, and Season (WRTDS) method was developed to 

provide a more robust tool for quantifying concentrations, 

fluxes, and descriptions of long-term changes in these 

quantities at monitoring stations with long-term datasets. 

In particular, it is designed to provide these descriptions of 

long-term changes in a manner that is not influenced by the 

particular year-to-year variations in river discharge, but rather 

provides a description of the evolving nature of the overall 

behavior of the watershed system in terms of nutrient and 

sediment concentrations and fluxes. This technique produces 

results that are directly relevant to the needs of the CBP by 

describing estimates of the yearly or seasonal nutrient and 

sediment inputs to the bay, as well as providing insight into 

the effects land-management actions have on water-quality 

conditions in the major tributaries to the bay. In addition to 

its use with Chesapeake Bay RIM data (Hirsch and others, 

2010), the WRTDS method also has been applied to studies 

of long-term changes in the Mississippi River Basin (Sprague 

and others, 2011) and in the Lake Champlain Basin (Medalie 

and others, 2012).

In 2011, the USGS began an investigation to compare 

the flux and trend estimates derived from ESTIMATOR and 

WRTDS. The overall objective of the investigation was to 

evaluate the nature and extent of the differences between 

nutrient and sediment flux estimates generated by each 

method and to determine which model provides the highest 

level of accuracy in annual flux estimates provided to the 

Chesapeake Bay Program. The model comparison focused 

on answering three questions: (1) What are the differences 

between the functional form and construction of the two 

models? (2) Which model produces discrete daily estimates of 

flux with the greatest accuracy and least amount of bias? and 

(3) How different would the historical estimates of annual flux 

be if WRTDS had been used instead of ESTIMATOR? One 

additional point of comparison between the two models was 

how each model determines the changes in annual flux once 

the year-to-year variations in discharge have been accounted 

for. All comparisons were made using total nitrogen, nitrate, 

total phosphorus, orthophosphorus, and suspended-sediment 

concentration data collected at the nine USGS RIM stations 

located on the Susquehanna, Potomac, James, Rappahannock, 

Appomattox, Pamunkey, Mattaponi, Patuxent, and Choptank 

Rivers in the Chesapeake Bay watershed. The investigation 

provided valuable information that extends beyond the 

boundaries of the Chesapeake Bay watershed with regard to 

determining the appropriate regression-based approach for 

quantifying nutrient and sediment fluxes and identifying how 

these fluxes are changing over time. 

Purpose and Scope

This report documents the comparison of two multiple 

regression approaches, ESTIMATOR and WRTDS, for 

the determination of nutrient and sediment fluxes in the 

Chesapeake Bay watershed. Discharge, nutrient, and sediment 

data collected from 1985 through 2010 at the RIM stations 

on the Susquehanna, Potomac, James, Rappahannock, 

Appomattox, Pamunkey, Mattaponi, Patuxent, and Choptank 

Rivers were used to construct ESTIMATOR and WRTDS 

models for predicting daily constituent concentrations and 

fluxes. ESTIMATOR and WRTDS discrete flux estimates were 

compared to direct observations of flux at each of the nine RIM 

stations to determine the accuracy and bias associated with 

each model. Annual estimates of nutrient and sediment flux 

were compared to determine the average difference in annual 

fluxes generated by ESTIMATOR and WRTDS. Finally, this 

report provides trends in WRTDS-derived annual nutrient and 

sediment fluxes, represented as the flow-normalized annual 

flux, for the periods 1985–2010 and 2001–2010.
4 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Description of Nutrient and Sediment Data

The USGS monitors nutrient and sediment conditions 

in the Chesapeake Bay watershed at nine long-term RIM 

stations (fig. 1; table 1). These RIM stations are situated at 

the furthest downstream point on the river, prior to where the 

river becomes tidally influenced. Because of historical data 

and logistical issues, however, the James River RIM station 

is located approximately 45 miles upstream from the tidal 

influence. The nine RIM stations, combined, account for 

streamflow from approximately 78 percent of the land area in 

the entire watershed (Langland and others, 1995). Monitoring 

water-quality conditions at these RIM stations allows for a 

nearly comprehensive representation of the total flux delivered 

to the tidal estuaries of the Chesapeake Bay from the nontidal 

portion of the watershed.

Since the early 1980s, the USGS has collected a 

minimum of 20 samples per year at each of the nine RIM 

stations. These samples are collected across the full range of 

the hydrologic conditions and are composed of 12 monthly 

samples and 8 targeted stormflow (that is, periods of elevated 

discharge) samples. These samples are analyzed for a variety 

of constituents that include dissolved and particulate phases of 

nitrogen and phosphorus as well as suspended sediment. 

For the model comparison, the following constituents 

were compiled into datasets for use in ESTIMATOR and 

WRTDS: total nitrogen, nitrate-nitrogen (analyzed as the mass 

of nitrogen in both nitrite+nitrate (NO2

–+NO3–

)) (referred 

to hereafter as nitrate), total phosphorus, orthophosphorus 

(PO4

3–), and suspended sediment. These water-quality data 

exist as either measured or calculated values. Measured 

values are those that are directly quantified through laboratory 

analysis; calculated values are those that are determined as 

the sum of a set of associated measured constituents. The 

priority, in constructing water-quality datasets as input for 

both ESTIMATOR and WRTDS, was to use a measured value 

whenever possible before using a calculated value. Nitrate, 

orthophosphorus, and suspended sediment are only available 

as measured values; however, historical values of total nitrogen 

and total phosphorus exist in some cases as both measured and 

calculated. Therefore, if there are concurrent data, the order 

of operation for constructing the time series datasets for total 

nitrogen, in order of decreasing priority, is as follows:

1. Total nitrogen=total nitrogen (measured),

2. Total nitrogen=total dissolved nitrogen

+total particulate nitrogen, or

3. Total nitrogen=total Kjeldahl nitrogen+nitrate.

Table 1. Chesapeake Bay River Input Monitoring Stations and associated watershed characteristics. 

[mi2, square mile; TN, total nitrogen; NO3, nitrate; TP, total phosphorus; OP, orthophosphorus; SSC, suspended sediment]

Station name Station 

number

Drainage 

area

(mi2)

Number of samples collected during 1985 to 2010 Site

abbreviation

Map

identifier TN NO3 TP OP SSC

Susquehanna River at 

Conowingo, Md.

01578310 27,100 904 909 906 906 901 SUS 1

Potomac River at 

Chain Bridge at 

Washington, D.C.

01646580 11,600 1,427 1,429 1,432 1,361 475 POT 2

James River at 

Cartersville, Va.

02035000 6,252 769 784 772 780 245 JAM 3

Rappahannock River near

Fredericksburg, Va.

01668000 1,595 695 711 702 705 225 RAP 4

Appomattox River at 

Matoaca, Va.

02041650 1,342 714 731 721 725 246 APP 5

Pamunkey River near 

Hanover, Va.

01673000 1,078 758 783 769 776 235 PAM 6

Mattaponi River near 

Beulahville, Va.

01674500 603 746 765 756 767 221 MAT 7

Patuxent River near 

Bowie, Md.

01594440 348 786 874 864 843 850 PAT 8

Choptank River near 

Greensboro, Md.

01491000 113 622 624 616 612 684 CHO 9
Introduction 5

Figure 1. Location of the nine River Input Monitoring (RIM) stations in the Chesapeake Bay watershed. Station numbers and 

names are provided in table 1.

CHESAPEAKE BAY

50 100 MILES

0 50 100 KILOMETERS

0

80°

75°

45°

40°

38°

42°

ATLANTIC

OCEAN

EXPLANATION

Physiographic province

Appalachian Plateau

Valley and Ridge

Blue Ridge

Piedmont

Coastal Plain

Chesapeake Bay

0 10 20 30 40 50 MILES

0 10 20 30 40 50 KILOMETERS

76°

42°

78°

40°

80°

38°

EXPLANATION

River input nontidal basin

River Input Monitoring (RIM)

 station and identifier

Figure 1. Location of the nine River Input Monitoring (RIM) stations in the Chesapeake Bay watershed. Station numbers and names are

provided in table 1.

Choptank River 

Susquehanna River 

Patuxent River 

Potomac River 

Rappahannock River 

Pamunkey River 

Mattaponi River 

James River 

Appomattox River 

Base modified from U.S. Geological Survey

1:2,000,000-scale digital line graphs

Albers Equal-Area Conic Projection

North American Datum of 1983 

Fall

Line

Chesapeake Bay

Watershed boundary

Chesapeake Bay

Watershed boundary

Fall Line

8

2

4

7

6

3

5

9

1

1

NEW YORK

WEST

VIRGINIA

VIRGINIA

NEW

DELAWARE

PENNSYLVANIA

Elmira

Scranton

Williamsport

Altoona

Harrisburg

York

Hagerstown

Baltimore

Annapolis

Wa shington

 DC

Charlottesville

Richmond

Harrisonburg

Norfolk

NEW YORK

PENNSYLVANIA

MARYLAND

DELAWARE

VIRGINIA

WEST

VIRGINIA

Front 

Royal

Chesapeake Bay

ATLANTIC OCEAN

Potomac

River

Potomac

River

Susquehanna

River

Patuxent River

James

James

Appomattox

Pamunkey River

River

Mattaponi

Rappahannock River

York River

River

River

River

Choptank R

Chester R

Cr et in gu odono C

South Fork

Shenandoah

River

North Fork

MAR YLAND

JERSEY
6 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Likewise, the order of operation for constructing the time 

series datasets for total phosphorus, in order of decreasing 

priority, is as follows:

1. Total phosphorus=total phosphorus (measured) or

2. Total phosphorus=total dissolved phosphorus

+total particulate phosphorus.

A distinction in the construction of these input datasets for 

ESTIMATOR and WRTDS is that the summation of total 

nitrogen and total phosphorus for ESTIMATOR input datasets 

is performed externally to the model and the summations for 

WRTDS are handled internally to the model; however, the 

same order of summation priority applies to both models. 

When a censored value (measured value that has a 

concentration below the analytical detection limit and denoted 

by a “less than” symbol) is used in the summation of total 

nitrogen and total phosphorus, censoring is propagated 

differently for datasets used in ESTIMATOR and WRTDS. 

For ESTIMATOR, 

1. if there are no censored constituents, then total nitrogen 

or total phosphorus is equal to the sum of the analytical 

constituents according to the order of operation listed 

above; however,

2. if any of the analytical constituents are censored, then the 

final summed value of total nitrogen or total phosphorus 

is censored .

An example of censoring propagation shown in (2) can be 

found in the summation of total nitrogen at the Rappahannock 

RIM station on October 10, 2007. The constituents available for the summation of total nitrogen are total particulate 

nitrogen (<0.03 milligram per liter (mg/L)) and total dissolved 

nitrogen (0.28 mg/L). Therefore, total nitrogen for use in 

ESTIMATOR is calculated as <0.03 mg/L+0.28 mg/L, which 

is equal to <0.31 mg/L. In WRTDS, the summation of total 

nitrogen and total phosphorus with censored data is handled 

within WRTDS and will be discussed in the following section. 

Comparison of Regression Models 

An important step in comparing model results obtained 

from ESTIMATOR and WRTDS is to compare how the 

models are used to estimate water-quality constituent flux 

at the nine RIM stations. This section of the report provides 

information that compares (1) the functional form of each 

model, (2) how each model’s coefficients are estimated, 

 data are handled, (4) how these models are used to determine 

temporal changes in the concentration and flux, and (5) the 

ability of each model to quantify uncertainty associated with 

flux estimates. 

Description of Model Forms

The primary goal for the USGS monitoring effort is 

to quantify the flux of nitrogen, phosphorus, and sediment 

delivered from each of the nine RIM basins into the receiving 

estuaries of the Chesapeake Bay. Flux is the mass of nutrients 

and sediment that passes the monitoring station before 

entering the Chesapeake Bay during a specified time period. 

A daily flux (mass per day) is calculated by multiplying the 

daily streamflow by the constituent concentration. The USGS 

continuously monitors streamflow conditions at each RIM 

station; however, water-quality conditions are only monitored 

approximately 20 days during a given year. Therefore, daily 

concentrations of nitrogen, phosphorus, and sediment are 

estimated using a multiple regression approach. Since the 

early 1990s, the USGS RIM program has used a log-linear 

multiple regression model (ESTIMATOR) developed 

by Cohn and others (1989) to estimate daily nutrient and 

sediment concentration and flux at each of the nine RIM 

stations. The ESTIMATOR approach produces a best fit 

relation between the logarithm of observed concentration and 

logarithm of discharge, time, and season as described in the 

following equation: 

c q q q q t t

t t t t

ln( ) ˆ ˆ ln( / ) ˆ ln( / ) ˆ ( )

ˆ ( ) ˆ sin(2 ) ˆ cos(2 ) ,

c c c

c

0 1 2

2

3

4

2

5 6

β β β β

β β π β π ε

= + + 

  + −

+ − + + + (1)

where

ln is the natural log function;

c is the measured concentration, in 

milligrams per liter;

q is measured daily-mean discharge, in 

cubic feet per second; 

t is time, in decimal years;

qc, t

c are centering variables for streamflow 

and time;

βˆ are coefficients estimated by ordinary least 

squares (non-censored observation) and 

Adjusted Maximum Likelihood Estimation 

(AMLE) (censored observations);

βˆ 0 is a constant;

βˆ 1

, βˆ 2 describe the relation between concentration 

and streamflow;

βˆ 3

, βˆ 4 describe the relation between concentration 

and time, independent of flow;

βˆ 5

, βˆ 6 describe seasonal variation in concentration; 

and 

	 ε is the unexplained variation.
Comparison of Regression Models 7

ESTIMATOR predicts constituent concentration in log 

space using seven explanatory variables. Cohn and others 

(1992) demonstrated that for several datasets considered 

(Chesapeake Bay RIM stations in Maryland, for nitrogen 

and phosphorus constituents only), this model accounts for a 

substantial portion of the variation in concentration and results 

in residuals that are approximately homoscedastic (unchanging 

variability) and not highly correlated with any of the predictor 

variables. Variation in concentration as a function of discharge 

is addressed by including both discharge and dischargesquared terms; these terms allow predicted concentration to 

change linearly or parabolically as a function of streamflow. 

Therefore, ESTIMATOR does well with modeling observed 

concentration-discharge relations that are linear or quadratic 

in form. Variation in concentration as a function of time is 

accounted for by including both time and time-squared; these 

variables are used to remove linear and (or) parabolic trends in 

the concentration residuals resulting from long-term increase/

decrease in concentration. Variations in concentration as a 

function of season is addressed by including sine and cosine 

functions with period 2πt; variable coefficients on these terms, 

determined as part of model estimation, allow the model to 

account for a single sinusoidal cycle with any magnitude 

or phase. In addition to the seven explanatory variables, 

ESTIMATOR uses centering variables for flow (qc ) and 

time (t

c ) to ensure that time and time-squared and discharge 

and discharge-squared are orthogonal (independent). These 

centering variables simplify calculations in ESTIMATOR 

and have no effect on flux estimates (Cohn and others, 1992). 

There is an implicit assumption in ESTIMATOR that the shape 

of the relation between concentration and flow is constant 

throughout the estimation period although the intercept is free 

to vary as a quadratic function of time. Similarly, there is an 

implicit assumption that the seasonality of this flow versus 

concentration relation can be described as a sine wave of fixed 

amplitude and phase throughout the estimation period.

The USGS is now considering adding the WRTDS 

approach, developed by Hirsch and others (2010), for the 

determination of nutrient and sediment fluxes and associated 

trends at each of the nine RIM stations to ensure the integrity 

of flux and trend estimates and to help overcome the potential 

limitations associated with ESTIMATOR. The WRTDS 

approach is described in detail by Hirsch and others (2010) 

and Sprague and others (2011). Like ESTIMATOR, WRTDS 

produces a best-fit relation defined between the logarithm of 

observed concentration and logarithm of discharge, time, and 

season as described in the following equation: 

 

ln( ) c t = + β β + + β β q t π β + + π ε t ˆ ˆ ˆ ln( ) ˆ sin(2 ) ˆ cos(2 ) 0 1 2 3 4 

 (2)

where

ln is the natural log function;

c is the measured concentration, in 

milligrams per liter;

q is measured daily-mean discharge, in 

cubic feet per second; 

t is time, in years;

βˆ are coefficients;

βˆ 0 is a constant;

βˆ 1 describes the relation between 

concentration and time;

βˆ 2 describes the relation between 

concentration and flow;

βˆ 3

, βˆ 4 describe seasonal variation 

in concentration; and 

ε is the unexplained variation.

WRTDS is similar in functional form to ESTIMATOR 

in that it uses discharge, time, and season as variables to 

explain the variation associated with observed water-quality 

constituents. WRTDS, however, explains this variation with 

five explanatory variables compared to the seven variables 

within ESTIMATOR. WRTDS does not include dischargesquared and time-squared variables and as a result does not 

need centering variables for time and discharge to ensure 

orthogonality. The key difference between the two methods 

is the way that the coefficients vary as a function of time, 

discharge, and season.

Parameter Estimation

Parameter estimation refers to the process by which model 

parameters/coefficients are estimated to determine constituent 

concentration and flux. Two key steps in this process that 

distinguishes ESTIMATOR from WRTDS is the width of the 

model estimation window and the means by which observed 

water-quality data are incorporated in the estimation window. 

The model construction processes used in this investigation 

will be described for ESTIMATOR and WRTDS.

The parameter estimation approach that the USGS 

currently (2012) uses for the determination of constituent 

concentration and flux, using ESTIMATOR, begins with 
8 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

the establishment of the model window. Until 2000, a 

single model window was used that encompassed the entire 

period of the water-quality monitoring record (for example, 

1979 to 1999). This approach employed two assumptions: 

(1) time invariance in the relation between concentration 

and discharge, time, and season and (2) equal uncertainty 

associated with all annual estimates of flux within the full estimation window. These assumptions held true when the model 

estimation window was relatively short (<10 years); however, 

as the monitoring program matured and the datasets expanded 

to include 15 or more years of data, these assumptions became 

less valid (Yochum, 2000; Milly, 2008). As the dataset grew, 

the model fit became overly sensitive to data from the beginning and ending years. Particularly with the use of the timesquared variable, errors could become substantial, especially 

if the trends in the data did not conform to the quadratic or 

linear shape (Yochum, 2000). Yochum (2000) recommended 

the use of a series of 9-year estimation windows when using 

ESTIMATOR to quantify monthly and annual fluxes. This 

approach minimizes estimation error primarily by allowing the 

relation between concentration and discharge, time, and season 

flexibility to change over time. Yochum (2000) showed that to 

produce fluxes with the greatest accuracy, a 9-year estimation 

window should be used to estimate fluxes for the centered 

(5th) year only and to estimate fluxes for the entire period a 

9-year moving window approach should be used. 

Since 2000, the USGS has used a 9-year moving window 

approach to estimate monthly and annual fluxes at the nine 

RIM stations. All available observations for the constituent 

being modeled, for each 9-year window, are used to estimate 

the seven model coefficients in equation 1, including those for 

flow, time, and season through best-fit ordinary least squares. 

Once the seven model coefficients are defined, daily concentrations are calculated from daily discharge, time, and season. 

ESTIMATOR uses the AMLE developed by Cohn (2005) to 

(1) estimate daily concentrations when censored constituent 

data are present and (2) account for retransformation bias that 

occurs when the natural log of estimated concentrations are 

retransformed from log space. Finally, the constituent flux is 

determined for each day by multiplying the daily concentration by the daily discharge. Monthly and annual fluxes, for 

the centered year in the 9-year window (beginning with 1985 

which is centered in the 1981 to 1989 9-year window), are the 

summation of the daily fluxes for the associated month or year, 

respectively. Fluxes for the most recent 4 years are obtained 

from the last 4 years of the final 9-year estimation window 

and labeled as provisional (Yochum, 2000). As an example, 

the current investigation obtained annual fluxes for 2007, 

2008, 2009, and 2010 from the ESTIMATOR 9-year window 

spanning 2002 to 2010.

In contrast, WRTDS uses a considerably different 

approach than ESTIMATOR for the determination of nutrient 

and sediment fluxes. The first major distinction is that instead 

of the 9-year estimation window used by ESTIMATOR, 

unique WRTDS models (eq. 2) are constructed for a large 

number of combinations of discharge (Q) and time (T) defined 

over a grid. The first dimension of the grid is time. The 

number of values of T is set to (years×16)+1, where years is 

the length of the period for which estimates are being made. 

Therefore, the grid spacing is set to 1/16 of a year (approximately every 23 days). The second dimension in the estimation grid is discharge, where discharge has 14 equally spaced 

(in log of discharge) levels that range from slightly higher than 

the highest daily discharge to slightly lower than the lowest 

daily discharge. At each unique grid intersection between 

discharge and time, unique WRTDS models are constructed, 

and estimated values of concentration are stored. The total 

number of unique WRTDS models is 14×((years×16)+1); 

therefore, for each water constituent modeled, as part of 

this report, there are 14×((26×16)+1) WRTDS models or 

5,838 unique WRTDS models. WRTDS uses bilinear interpolation to estimate concentration for combinations of Q and T 

that do not coincide with the estimation grid nodes. 

Another factor that distinguishes WRTDS from 

ESTIMATOR is the method by which water-quality observations are chosen for the estimation of model coefficients. 

WRTDS identifies which water-quality observations are 

included in each unique model by weighting each observation 

in the entire dataset on the basis of similarities/distance from 

the target condition in three dimensions: time, season, and 

discharge (Hirsch and others, 2010). The first distance assessed 

is the time distance where greater weight is given to observations that were collected closer in time to the target time. For 

this investigation, the half-window width is set to 10 years; 

therefore, observations that are approximately 6 years from the 

center of the window are assigned weights that are less than 

half of the weights at the center of the window. Observations 

10 years and greater from the center of the window are assigned 

weights equal to zero. The second distance is the seasonal 

distance where greater weight is given to observations that were 

collected during the same time of year. The half-window width, 

for this investigation, is set to 0.5 meaning, for example, if 

July 1, 2010, is the unique time where a WRTDS model is being 

constructed, then observations collected during the summer 

would have the greatest weights followed by observations 

collected during spring and fall; observations collected a half a 

year from the time being estimated (in this example January 1) 

would be assigned weights equal to zero. The third distance is 

the discharge distance where greater weight is given to observations collected during similar discharge conditions/magnitude. 

The half-window width for discharge (log discharge), for this 

investigation, is set to 2; therefore, water-quality observations 

that were collected during discharge conditions that are within 

two natural log cycles of the target discharge condition will be 

weighted greater than zero and included in the model coefficient 

estimation process. The “tri-cubed weight function” (Tukey, 

1977) is defined in each of these three dimensions, and then 

these three weights are combined by multiplying these together 
Comparison of Regression Models 9

to determine the overall weight of the observation. The greater 

the overall weight of an observation, the greater the influence 

it has in establishing the parameters in equation 2. A minimum 

of 100 observations with weights greater than zero is required 

in each model window. Hirsch and others (2010) provide 

extensive details pertaining to the how weights are assigned 

to observations. The coefficients in equation 2 are fitted, using 

the weighted observations, and used to estimate constituent 

concentration that best represents the targeted discharge and 

time. Retransformation bias associated with transforming 

concentration back from log space was addressed in the 

version of WRTDS used by Hirsch and others (2010) by using 

the smearing factor developed by Duan (1983); however, the 

version of WRTDS used for this investigation (version 4) uses a 

method that addresses retransformation bias and fitting of model 

coefficients in the presence of censored data simultaneously and 

is discussed in detail in the section “Censored Data.” Following 

this approach, WRTDS produces a concentration derived from 

unique models of individual flow and time for every day in the 

monitoring record. Daily flux is then determined by multiplying 

the daily concentration by the daily discharge. Daily fluxes are 

summed to obtain monthly and annual fluxes. 

The estimation of model coefficients for ESTIMATOR 

and WRTDS is similar in that they both use the concept of 

a moving window so that the regression model is based on 

observations that are from a period of years “near” to the year 

for which the estimates are being computed. One difference 

is that the windowed ESTIMATOR approach treats observations as either “in” or “out” of the regression, so that at each 

annual increment of time individual observations are added to 

the regression dataset or deleted. In contrast, in WRTDS the 

window moves continuously through time rather than moving 

forward a year at a time. The portion of the total weight 

attributed to time on each observation changes gradually from 

zero to one (versus an abrupt shift from zero to one and back 

to zero again). As a consequence, in WRTDS the influence 

of each observation on the regression gradually changes with 

the passage of time. The other difference is that ESTIMATOR 

creates a window only in the time dimension, whereas 

WRTDS uses additional windows in discharge and season. 

Thus, in ESTIMATOR data from all discharge values and 

seasons of the year are given equal weight in the regression as 

long as they are in the time window. In WRTDS, the weighting 

scheme provides high weights for data with similar time, 

discharge, and season when compared to the target condition, 

and low weights are provided for data with dissimilar time, 

discharge, and season when compared to the target condition. 

Because of this fundamental difference, WRTDS theoretically 

replicates the behavior of the constituents at higher discharges 

where data tend to be relatively sparse, but which are highly 

important in the overall annual flux. In a sense, the estimates 

for the high discharges are decoupled from the behavior 

observed at much lower discharges, which may depend on 

very different factors and processes. 

Censored Data

The ability of ESTIMATOR and WRTDS to determine 

daily concentrations and fluxes for constituents that have 

censored data observations is essential. ESTIMATOR uses the 

AMLE methodology to handle the estimation of constituent 

concentration and flux in the presence of censored data; details 

are given in Cohn (2005). The version of WRTDS used in this 

investigation (version 4) is different from the version reported 

in Hirsch and others (2010) because version 4 allows for 

computation of estimated concentration values in the presence 

of censored values. The technique is an adaptation of “survival 

analysis,” originally developed for medical or industrial 

applications, which is also known as “censored regression 

analysis.” WRTDS (version 4) allows for left censoring 

and interval censoring of the observed concentration data. 

Left censoring is a common characteristic of water-quality 

data (Cohn, 2005; Helsel, 2012). Interval censoring occurs 

when the concentration of interest is the sum of two or more 

concentration values and at least one is reported as a “less 

than” value and at least one is reported without censoring; this 

situation occurs frequently in the RIM data. For example, for 

the Rappahannock River RIM station, 41 of 695 samples are 

reported for total nitrogen for which there is interval censoring 

(and none with left censoring). One of these instances is the 

sample taken on September 11, 2003, where total dissolved 

nitrogen was reported as 0.64 mg/L and total particulate 

nitrogen was reported was <0.01 mg/L. Because the analysis 

for which equation 2 is being fit is for total nitrogen, these 

two results must be added together to constitute an estimate 

of total nitrogen concentration. Given this information, the 

value of total nitrogen lies between 0.64 mg/L and 0.65 mg/L. 

Note that if the analysis was restricted to only using left 

censoring representations of results, as with ESTIMATOR, a 

value of <0.65 mg/L would be assigned to this observation, 

which conveys very different information. Figure 2 illustrates 

the representation of all 695 values in this dataset, using the 

vertical lines to indicate the range of each interval estimate. 

The general rule, in WRTDS, for computing interval 

estimates between the concentration of analyte 1 (c1 ) and 

the concentration of analyte 2 (c 2 ) is accomplished by 

implementing 1 of 3 possible cases. The reporting limits 

for the two analytes are r1 and r2, respectively. The method 

computes two values: IL, which is the lower limit of the range 

of possible values for the sum, and IU , which is the upper limit 

of the range. If c1 and c 2 are not censored, then I L and I U are 

identical. For illustrative purposes, it is assumed that there 

are never more than two analytes present for the summation 

of total nitrogen and total phosphorus; however, this intervalcensored approach is generalized in the WRTDS software to 

apply to any number of analytes. The three interval censoring 

cases are
10 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

• Case 1: the concentration of analytes 1 and 2 are 

greater than or equal to their reporting limits, that 

is c1≥ r

1

 and c2 ≥ r

2

. In this case, the concentration 

assigned to the lower (I L ) and upper (I U ) limits of 

the concentration interval is defined as I L =c1+ c2 and 

IU =c1+c 2. This means that the interval estimate collapses to a single value, c1+c 2.

• Case 2: the concentration of analyte 1 is reported as 

less than the reporting limit, but the concentration 

of analyte 2 is reported as greater than or equal to its 

reporting limit, that is c1< r

1

 and c 2 ≥ r2. In this case, 

I L = c 2 and I U = r1+c 2. This case was illustrated above 

with the example using total nitrogen data from the 

Rappahannock River. 

• Case 3: the concentration of both analytes are reported 

as less than their reporting limits, that is c1< r1 and 

c2< r2. In this case, I L =0 and IU =r1+ r2, which is 

identical to a left censoring cas e, where the reporting 

limit is r1+ r2. 

The estimation of equation 2 now can be accomplished 

by assigning an interval value of concentration for every 

observation. Because the estimation is for the logarithm of 

concentration, for those cases where IL is equal to zero the 

lower bound on the interval in log space becomes negative 

infinity. The weights for the estimation are computed exactly 

as described in Hirsch and others (2010). The method of 

estimation is an extension of Tobit analysis defined by Tobin 

(1958) and described more fully in Judge and others (1985). 

The Tobit analysis requires that the user specify a distributional assumption for the residuals. The Gaussian (normal) 

distribution was used for all Chesapeake Bay data analyzed 

in this report. This estimation method returns values for the 

individual coefficients in equation 2, along with a “scale” 

parameter, which is the analog of a standard error of the 

residuals in the usual weighted least squares implementation. 

The method of handling the retransformation bias differs 

from that in the previous versions of WRTDS. In those 

versions, the smearing estimate (Duan, 1983) was used, but 

that estimate requires computed values for all of the residuals. 

In the censored case, the residuals are not known for all of 

the samples. Therefore, the approach was to use the scale to 

estimate the residual variance and thus

σˆε = scale 2 2 (3)

Concentration, in milligrams per liter 

1975 1980 1985 1990 1995

Year

2000 2005 2010 2015

0.1

0.2

0.5

1.0

2.0

5.0

10

Figure 2. Plot of Rappahannock River at Fredericksburg, Va., total nitrogen record 

showing interval censored values as vertical lines.

Figure 2. Total nitrogen record, showing interval censored values as vertical 

lines, for Rappahannock River at Fredericksburg, Virginia.
Comparison of Regression Models 11

Note that in WRTDS (versions 3 and 4) the variance estimate 

is specific to the particular values of discharge and time for 

which concentration is being estimated. This is very different from the approach used in ESTIMATOR (Cohn, 2005) 

in which the variance is assumed constant over all values of 

discharge and time. This difference in the approach can have 

substantial effects on the retransformation method and resulting biases. The datasets analyzed in this investigation show 

wide variations in scale, and these variations typically are 

very different at high discharges versus low discharges. The 

retransformation bias correction term α is:

α

σ = 











ε exp ˆ

2

2

(4)

and the estimate of concentration for each model is obtained 

by modifying equation 2 to the following:

c Q ˆ exp( ln( ) t t sin(2 ) cos(2 )t ) 1 2 3 4 5 = + α β × β β + +β π +β π 

 (5)

Note that all of the parameters in equation 5 (α β, ,β β, ,β β, 1 2 3 4 5) 

are estimates that are the result of each unique weighted 

survival regression analysis where the weights are based on 

“distances” between the point of estimation and the sample 

values of time, discharge, and season. Thus, one important 

difference between WRTDS and ESTIMATOR is that this 

retransformation bias correction term in WRTDS can vary by 

a substantial amount across the range of time, discharge, and 

season but in ESTIMATOR it is constant. 

The survival regression method produces estimates that 

are virtually identical to those computed by the linear regression and smearing estimator method used in Hirsch and others 

(2010) even when there is no censoring. For consistency of 

implementation, this survival regression approach is used, in 

WRTDS, for all datasets considered, regardless of whether 

they contained censored data.

Determination of Trends in Water-Quality Conditions

The next point of comparison between the two models 

is how ESTIMATOR and WRTDS address the question of 

whether water-quality conditions in the Chesapeake Bay 

watershed are changing. Two considerations should be 

addressed when choosing an analytical approach for this 

purpose. First, natural variation in discharge confounds the 

determination of water-quality trends. Because water-quality 

concentrations are highly correlated with discharge, a large 

portion of the variation in water quality is simply a reflection 

of the year-to-year variations in discharge. These year-to-year 

variations in discharge confound the attribution of changes in 

water quality to land-management practices. ESTIMATOR can 

generate information about changes in water-quality constituent 

concentrations that are independent of the random variations in 

discharge through a process called “flow adjustment;” while, 

WRTDS can generate information about changes in both 

water-quality concentration and flux that are independent of 

the random variations in discharge through a process called 

“flow normalization.” Both the flow-adjustment and flownormalization processes are described below in detail. 

The second consideration is how water-quality conditions 

will be represented. Will they be characterized by changes 

in concentration or by changes in flux or both, and how are 

the changes calculated? This is important because trends 

in concentration and flux can tell very different stories and 

are best suited to different purposes. The computation of an 

average concentration over a period, such as a year, treats 

the values for each day equally. The average of 365 daily 

concentrations is the sum of these values divided by 365. In 

contrast, the computation of an average (or total) flux over a 

period, such as a year, gives much more weight to the concentrations on high-discharge days than it does to concentrations 

on low-discharge days, because the average flux for the year 

is the sum of the fluxes for each day, which is the product of 

concentration and discharge for that day, divided by 365. Thus, 

large changes in concentration on low-discharge days can have 

a big influence on average concentration but minimal effect on 

average flux. Conversely, large changes in concentration on 

high-discharge days can have a big influence on average flux 

but minimal effect on average concentration. Information on 

trends in concentration is useful for identifying whether water 

quality is improving or degrading at a monitoring location and 

is particularly useful for assessing progress toward attainment 

of water-quality standards. Information about trends in flux 

is much more relevant to assessing conditions in a large 

downstream water body such as the Chesapeake Bay. Trends 

in flux focus on the total inputs of nutrient and sediment, 

which can be crucial to ecological conditions in the bay. For 

example, eutrophication is likely to be most responsive to the 

total input of nutrients over periods of time such as a year or a 

season. Accurate information about the trends in concentration 

and the trends in flux are both vital to assessing the changing 

influences on the estuary.

The USGS RIM program currently (2012) uses 

ESTIMATOR to determine trends in flow-adjusted concentration. The flow-adjusted concentration is determined using 

ESTIMATOR (eq. 1) for a single model window, typically 

1985 to present for long-term trends and the most recent 

10 years for shorter-term trends. The variability in waterquality concentration that is directly related to flow is removed 

by the log discharge and log discharge-squared terms. 

Similarly, variability associated with time of year is removed, 

using the seasonal terms (eq. 1). The trend in flow-adjusted 

concentration is determined based on the magnitude, direction, 

and significance of the time and time-squared coefficients. The 

flow-adjusted concentration trend is presented as a percentage 

change that occurred over the entire time period defined by 

the model window. A full description of the ESTIMATOR 

flow-adjustment methodology can be found in Langland and 

others (2006). Note that flow-adjusted concentration and 
12 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

flow-adjusted flux trends presented as a percentage change 

per year are forced by ESTIMATOR’s fixed mathematical 

form to be identical, which can be an unrealistic model 

constraint. As a result, changes in water-quality conditions, 

using ESTIMATOR, are represented using the flow-adjusted 

concentration only. The CBP has been using flow-adjusted 

concentration trends, calculated using ESTIMATOR, as an 

indicator of observed changes in water-quality conditions 

across the Chesapeake Bay watershed, resulting from human 

activities. With the recent establishment of the total maximum 

daily load for the entire Chesapeake Bay watershed, however, 

the CBP has shifted focus from concentrations to flux of 

nutrients and sediment to the bay. Accordingly, the CBP has 

requested that the USGS develop a more appropriate tool to 

use as an indicator of changing water-quality conditions that 

more accurately characterizes temporal changes in the flux of 

nutrients and sediment being delivered to the bay.

WRTDS uses a different approach to characterize 

temporal changes in water-quality conditions. Instead of a 

flow-adjustment approach, WRTDS uses “flow normalization” 

to remove the variability in water-quality conditions that 

is directly related to the random variations in discharge. 

The theory behind flow normalization is that the flow that 

occurred on a given day is one realization from a probability 

distribution of flows that can be expected for that particular 

time of year. For computational purposes, that probability 

distribution is the set of all flows that occurred on that same 

date throughout the monitoring record, with each considered 

equally likely to occur. For example, the flow that occurred at 

the James River RIM station on April 6, 2004, is one realization of 31 possible realizations during the 31-year period 1980 

to 2010. WRTDS determines the flow-normalized concentration for April 6, 2004, by estimating 31 possible concentrations by running the model 31 times, using equation 2, 

each centered in time to April 6, 2004, but with the value 

of flow equal to each of the 31 observed values. The flownormalized concentration for April 6, 2004, is the mean of 

the 31 estimated concentrations. Similarly, WRTDS estimates 

flow-normalized flux for April 6, 2004, as the mean of the 

31 estimated flux values for that date. This process is repeated 

for every day in the record (31×365) to obtain a daily time 

series of flow-normalized concentration and flow-normalized 

flux. Daily flow-normalized flux values are aggregated to 

obtain monthly and annual total flux. A complete description 

of the WRTDS flow-normalization method can be found in 

Hirsch and others (2010).

To summarize, WRTDS produces two complete daily 

time series of concentration and two daily time series of flux 

for every modeled water-quality constituent. The first is a time 

series for both concentration and flux derived directly from the 

estimation grid of unique time and discharge combinations. 

The second is a record of both flow-normalized concentration 

and flux calculated as described in the previous section.

Changes in water-quality conditions, represented in this 

report as a total annual flux, during any given time period 

are determined using two different approaches. These two 

approaches present the change in flux as a slope over a given 

time period. The first approach is to define the flux change as a 

slope change per year as follows:

Slope Percent f f f / 100 / Years t t t ( ) 2 1 1

( )= − ( ) ( ) × (6)

where, 

f

t

2

is the total annual flow-normalized flux 

in year t

2

 , 

f

t

1

is the total annual flow-normalized flux in 

year t

1

, and 

Years is the total number of years over which the 

slope is defined (Years = t

2 – t1 ).

The second approach is to define the flux change as a change 

in mass (tons per day) per year. To facilitate the comparison of 

changes in mass between the nine RIM stations, the mass from 

each watershed was normalized on the basis of the watershed 

drainage area, thus producing a slope in yield with units of tons 

per day per square mile per year and is calculated as follows: 

Slope ( ) Yield f = − ( ) ( ) ( ) / / DA ( ) f DA f / / ( ) DA / Years t t t 2 1 1 

 (7)

where, DA is the watershed drainage area (in square miles) for 

a given RIM station. 

Reporting Error Associated with Flux and 

Trend Estimates

The last point of comparison between each model’s 

form and function is to evaluate the ability of each model 

to assign uncertainty to estimates of flux and associated 

trends in water-quality conditions. The standard error of 

prediction is the primary statistic used to assign uncertainty to 

estimates derived from regression models. From the standard 

error, and statistics used to compute the standard error, the 

95-percent confidence interval for flux and trend estimates 

can be determined. Additionally, using the standard error 

will enable hypothesis testing to determine if a measured 

trend in concentration or flux is statistically different from 

zero (zero indicating no trend). ESTIMATOR does compute 

a standard error associated with the estimates of flux and 

flow-adjusted concentration trends. The standard errors are 

determined following the approach in Gilroy and others 

(1990) and Cohn and others (1992). As a result, all flux and 

trend results estimated by ESTIMATOR are presented with 

associated 95-percent confidence intervals, and the trend 

results are subjected to hypothesis testing to determine if 

the trend is statistically different from zero, following the 

procedure outlined in Langland and others (2006). The 

reliability of the estimates and associated standard error 

depends on three assumptions: (1) the model form represents 
Comparison of Estimate Accuracy and Bias 13

the actual behavior of the system, (2) the error variance of 

the model is equal across all of the days (regardless of season 

or discharge), and (3) the errors are uncorrelated in time. 

Conversely, the estimates of flux and trend in flow-normalized 

flux, derived from WRTDS, are not associated with a measure 

of uncertainty. Consequently, all flux and trend estimates 

derived from WRTDS are not reported within the context of a 

95-percent confidence interval; as a consequence, the reported 

flow-normalized flux trends do not support hypothesis testing 

to determine if the slope of the trend is significantly different 

from zero. The lack of an estimate of uncertainty in WRTDS is 

a function of the complexity of propagating error through each 

of the 5,838 models created for each water-quality constituent. 

Research is underway to develop an approach to assign 

uncertainty to WRTDS-derived estimates.

Comparison of Estimate Accuracy 

and Bias 

The next point of comparison between ESTIMATOR and 

WRTDS is to identify the accuracy of each model’s predictions compared to observations of flux. It should be noted that 

no model of environmental systems will yield 100 percent 

accuracy; however, the goal in developing these models is 

to produce estimates of flux with maximum accuracy and 

minimal bias. This section evaluates the differences between 

ESTIMATOR- and WRTDS-derived estimates of nutrient and 

sediment flux when compared to the sampled data on which 

they were based. To answer this question, model accuracy and 

bias associated with ESTIMATOR and WRTDS flux estimates 

for total nitrogen, nitrate, total phosphorus, orthophosphorus, 

and suspended sediment at each of the nine RIM stations were 

compared. Model accuracy is represented by the root mean 

square error (RMSE) and is defined by the following equation:

∑ =

− 











−

=

RMSE

y y

n 2

i i i

n ^

1

2

(8) 

where

yi is observed yield, in pounds per day per 

square mile;

yi

^

is estimated yield, in pounds per day per 

square mile; and

n is the total number of observations.

As the value of RMSE approaches zero, the model predictions more closely represent actual observations. The values 

of estimated and observed flux are normalized by drainage 

area (presented as yield) to facilitate the comparison of RMSE 

across all nine RIM stations. Model bias (reported here as flux 

bias) is a measure of the model’s tendency to over or underpredict observed fluxes and is defined as:

∑

∑ =

















=

=

Flux Bias

L

L

p i i

n

o i i

n

1 ,

1 ,

(9)

where

Lp,i is estimated yield for day I, in pounds

 per day per square mile;

Lo,i is observed yield for day i, in pounds 

per day per square mile; and

n is the total number of observations.

As the value of flux bias approaches 1.0, the model prediction 

of flux more closely represents the observed flux. Flux bias 

values greater than 1.0 indicate that the model estimates of 

flux tend to be greater than observed fluxes, and values less 

than 1.0 are indicative of the model’s tendency to underpredict 

observed fluxes. 

RMSE and flux bias are valuable measures for comparing 

how well model estimates reflect environmental observations; 

however, both of these measures are sensitive to the presence 

of extreme values, as well as the particular pattern of sampled 

days and would be expected to change if the sampled days 

included more high-discharge days or more low-discharge days. 

Therefore, for this investigation, only major differences in RMSE 

and flux bias are used to signify noteworthy differences between 

ESTIMATOR and WRTDS. For RMSE, differences between 

ESTIMATOR and WRTDS are considered significant when there 

is at least a 20-percent difference (percent difference=(((WRTDS 

RMSE–ESTIMATOR RMSE)/ESTIMATOR RMSE)×100)). 

Similarly for flux bias, differences between ESTIMATOR and 

WRTDS are considered significant when the magnitude of the 

difference in the two flux-bias ratios is greater than or equal 

to 0.10. Note that the use of the word “significant” here is not 

to be interpreted in the statistical sense (that is, a probability of 

falsely rejecting a null hypothesis). Rather, it is used to refer 

to an indication of practical significance. Ideally, one would 

like to test the quality of the two methods using datasets that 

contain a complete record of daily flux, but such records do 

not exist within the Chesapeake Bay watershed.

Throughout this section, results are compared for 

observed versus predicted (either ESTIMATOR or WRTDS) 

total nitrogen, nitrate, total phosphorus, orthophosphorus, 

and suspended sediment at each of the nine RIM stations. To 

simplify the discussion, the term “combinations” refers to the 

45 different combinations made up of nine RIM stations and 

five constituents. 

Observed and Estimated Flux Comparison Results

For all 45 possible combinations, RMSE for WRTDS 

is smaller (more accurate) than the RMSE for ESTIMATOR 

(table 2). Of the 45 possible combinations, 22 combinations 

have values of RMSE that show WRTDS is considerably 

more accurate (20 percent or greater reduction in RMSE) 

than ESTIMATOR. Of these 22 combinations, 7 show 
14 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Table 2. Measures of RMSE and flux bias relating WRTDS- and ESTIMATOR-derived nutrient and sediment fluxes compared to 

discrete flux observations at the nine River Input Monitoring (RIM) stations.—Continued

[(lbs/day)/mi2, pounds per day per square mile; bold text, percent difference between 20 and 40; bold text, percent difference between 41 and 60; 

bold text, percent difference greater than 61; text, WRTDS flux bias ratio closer to 1.0 by at least 0.10; bold text, difference between WRTDS and 

ESTIMATOR flux bias ratio 0.10 or greater; Category, variable used to group station and constituent combinations based on differences between 

WRTDS and ESTIMATOR flux-bias results] 

RIM station

Root mean squared error ((lbs/day)/mi2) Flux bias ratio

Category WRTDS ESTIMATOR Percent

difference WRTDS ESTIMATOR Difference

Total nitrogen

Susquehanna 25.22 64.44 –61 0.99 0.95 0.04 I

Potomac 29.06 38.42 –24 0.99 0.95 0.04 I

James 12.31 12.94 –5 1.01 1.00 0.01 I

Rappahannock 12.55 21.50 –42 1.00 1.12 – 0.12 II

Appomattox 2.32 2.57 –10 1.01 1.02 – 0.01 I

Pamunkey 3.59 4.27 –16 0.98 1.03 – 0.05 I

Mattaponi 1.62 1.70 –4 1.00 1.03 – 0.03 I

Patuxent 7.02 7.32 –4 1.00 1.01 – 0.01 I

Choptank 11.85 13.31 –11 0.99 1.02 – 0.03 I

Nitrate

Susquehanna 6.41 6.65 –4 0.99 1.00 – 0.01 I

Potomac 5.81 7.01 –17 0.98 1.03 – 0.05 I

James 1.46 2.03 –28 0.99 1.12 – 0.13 II

Rappahannock 2.94 6.70 –56 0.99 1.36 – 0.37 II

Appomattox 0.89 1.16 –23 1.03 1.09 – 0.06 I

Pamunkey 0.90 1.12 –20 0.98 1.04 – 0.06 I

Mattaponi 0.61 0.70 –13 1.00 1.02 – 0.02 I

Patuxent 7.02 7.32 –4 1.00 1.01 – 0.01 I

Choptank 8.01 8.52 –6 0.97 1.01 – 0.04 I

Total phosphorus

Susquehanna 1.57 2.82 –44 1.06 0.96 0.10 III

Potomac 8.54 11.41 –25 1.03 1.12 – 0.09 I

James 4.91 7.90 –38 1.03 1.07 – 0.04 I

Rappahannock 6.36 23.88 –73 1.02 1.53 – 0.51 II

Appomattox 0.43 0.45 –4 1.02 1.03 – 0.01 I

Pamunkey 0.79 0.95 –17 0.98 1.03 – 0.05 I

Mattaponi 0.25 0.27 –7 1.00 1.03 – 0.03 I

Patuxent 1.93 2.41 –20 1.04 1.01 0.03 I

Choptank 1.42 4.07 –65 1.04 1.17 – 0.13 II

Orthophosphorus

Susquehanna 0.20 0.22 –9 1.05 1.16 – 0.11 II

Potomac 0.45 1.21 –63 1.02 1.34 – 0.32 II

James 0.39 0.40 –3 0.99 1.01 – 0.02 I

Rappahannock 0.25 0.50 –50 1.02 1.14 – 0.12 II

Appomattox 0.08 0.09 –11 1.03 1.04 – 0.01 I

Pamunkey 0.13 0.14 –7 1.00 1.00 0.00 I

Mattaponi 0.06 0.07 –1 0.98 1.01 – 0.03 I

Patuxent 0.23 0.25 –8 1.09 1.04 0.05 I

Choptank 0.44 1.03 –57 1.06 1.15 – 0.09 I
Comparison of Estimate Accuracy and Bias 15

a 20 to 40 percent reduction, 7 show a 41 to 60 percent 

reduction, and 8 show a 61 percent or greater reduction/

improvement in model accuracy when WRTDS is used to 

estimate flux. The remaining 23 combinations show that the 

RMSE for WRTDS is marginally smaller (0 to 19 percent) 

when compared to the RMSE for ESTIMATOR. These RMSE 

results indicate that WRTDS generates flux estimates that 

are universally more accurate than flux estimates generated 

by ESTIMATOR and in nearly half of the combinations the 

improvement in model accuracy is considerable. The RMSE 

also can be viewed here as a surrogate for uncertainty in the 

model estimates where WRTDS would show a reduction in 

estimate uncertainty (that is, reduction in the width of the 

confidence interval) compared to ESTIMATOR predictions. 

For 36 of the possible 45 (80 percent) combinations, 

WRTDS generated fluxes that were less biased (that is, 

flux-bias ratio closer to 1.0) than fluxes generated by 

ESTIMATOR (table 2). Of these 36 combinations, there 

were 16 combinations where WRTDS exhibited significantly 

less bias than ESTIMATOR (closer to 1.0 by at least 0.1). 

There were 7 of the 45 (16 percent) combinations where 

ESTIMATOR-generated fluxes were less biased than fluxes 

generated by WRTDS; however, the biases associated 

with ESTIMATOR and WRTDS-derived fluxes, for these 

7 combinations, were both within 0.10 of 1.0. The remaining 

2 of the 45 combinations, WRTDS and ESTIMATOR fluxes 

were equally biased and within 0.10 of 1.0. These results 

show that for the majority of the combinations (67 percent, 

30 of 45 combinations) the differences in flux bias from 

1.0 were marginal. For 33 percent of the combinations 

(15 of 45 combinations), however, WRTDS-derived fluxes 

showed a marked improvement in flux bias (flux bias closer to 

1.0) compared to fluxes generated using ESTIMATOR. 

Sources of Flux Bias Discrepancies

The results of the flux bias analyses provide valuable 

information on the overall tendency of WRTDS and 

ESTIMATOR to over or underpredict fluxes. To better 

understand how to interpret these flux bias results, however, 

the 45 combinations were categorized by presumed source of 

discrepancy between WRTDS and ESTIMATOR. The first 

two categories are established on the basis of the comparison 

of flux bias between WRTDS and ESTIMATOR (that is, the 

total difference between WRTDS and ESTIMATOR flux 

biases) (table 2) and are (1) marginal difference, less than 0.10 

(Category I) and (2) considerable improvements (greater than 

or equal to 0.10) of WRTDS over ESTIMATOR flux-bias ratio 

(Category II). The third category contains two combinations 

of improvements of flux-bias ratio (by more than 0.10) of 

WRTDS over ESTIMATOR; however, for both of these 

combinations, ESTIMATOR has a tendency to underpredict 

flux and WRTDS tends to overpredict flux (Category III). 

Within each category, the differences between ESTIMATOR 

and WRTDS predictions of flux are investigated. In addition to 

flux bias, patterns in the concentration versus discharge (CQ) 

plots and concentration residual versus discharge (residual) 

plots were used to illustrate model fit. CQ and residual plots 

for each RIM combination are provided in appendix 1. 

Table 2. Measures of RMSE and flux bias relating WRTDS- and ESTIMATOR-derived nutrient and sediment fluxes compared to 

discrete flux observations at the nine River Input Monitoring (RIM) stations.—Continued

[(lbs/day)/mi2, pounds per day per square mile; bold text, percent difference between 20 and 40; bold text, percent difference between 41 and 60; 

bold text, percent difference greater than 61; text, WRTDS flux bias ratio closer to 1.0 by at least 0.10; bold text, difference between WRTDS and 

ESTIMATOR flux bias ratio 0.10 or greater; Category, variable used to group station and constituent combinations based on differences between 

WRTDS and ESTIMATOR flux-bias results] 

RIM station

Root mean squared error ((lbs/day)/mi2) Flux bias ratio

Category WRTDS ESTIMATOR Percent

difference WRTDS ESTIMATOR Difference

Suspended sediment

Susquehanna 2,758 9,632 –71 1.06 0.80 0.26 III

Potomac 6,883 26,536 –74 1.15 1.79 –0.64 II

James 3,271 12,153 –73 1.21 1.80 –0.59 II

Rappahannock 8,632 41,288 –79 1.32 2.57 –1.25 II

Appomattox 145 174 –17 1.02 1.13 –0.11 II

Pamunkey 1,229 1,370 –10 0.98 1.07 –0.09 I

Mattaponi 155 158 – 2 0.94 1.13 –0.19 II

Patuxent 1,396 2,379 – 41 0.99 1.17 –0.18 II

Choptank 609 1,122 – 46 1.02 1.21 –0.19 II
16 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Category I Combinations

Category I combinations are those where the analysis 

of model bias yielded only marginal differences (that is, 

less than 0.10 in absolute magnitude) between WRTDS 

and ESTIMATOR-derived fluxes compared to discrete 

observations. Twenty-eight of 45 RIM combinations exhibit 

Category I type results (table 2). A representative combination 

for all Category I combinations is nitrate at Patuxent River 

near Bowie, Maryland. This combination was chosen because 

of the complex concentration-discharge relation exhibited 

for nitrate at this location (fig. 3A). The Patuxent River is 

a point-source dominated system receiving discharge of 

effluent from multiple sewage treatment plants (Sprague and 

others, 2000). The concentration versus discharge relation 

for nitrate has been altered as a result of various sewage 

treatment plant upgrades that specifically reduce nitrate from 

effluent (for example, biological nitrate reduction (BNR)). The 

implications of these management actions are evident based 

on the reduction in nitrate concentration associated with low 

discharges that range between 50 and 400 ft3/s (cubic feet 

per second; between 4 and 6 log units; fig. 3A). Prior to these 

management actions (1981–89), the concentration of nitrate in 

the Patuxent River ranged from approximately 2.0 to 6.0 mg/L 

(approximately 0.7 to 2.0, log units) at low discharges; this is 

evident as the upper left lobe of the concentration-discharge 

relation in figure 3(A). Following the most recent upgrade of 

sewage treatment plants to BNR during the early 1990s, nitrate 

concentrations associated with low discharges range from 

approximately 1.0 to 2.0 mg/L (approximately 0.0 to 0.70, log 

units); this is evident as the lower left lobe of the concentration-discharge relation in figure 3(A). Figure 3B shows this 

same observed concentration-discharge relation for nitrate 

in the Patuxent (red dots) with the ESTIMATOR-predicted 

concentrations overlain (black dots). Figure 3D shows the 

observed concentration-discharge relation (red dots) overlain 

by WRTDS-predicted concentrations (black dots). Both of 

these plots show that ESTIMATOR and WRTDS accurately 

reproduce the complex nature of the concentration-discharge 

relation for nitrate at the Patuxent. The residual (observed 

minus predicted concentration) plots for ESTIMATOR and 

WRTDS (figures 3C and 3E) illustrate changes in model error 

as a function of changing discharge. The desired pattern in 

residual plots is a homoscedastic distribution around zero 

across the full range of discharge. The residual plots shown 

in figure 3C and 3E indicate that ESTIMATOR and WRTDS, 

respectively, produced estimates of nitrate concentrations 

that were similar in accuracy. This example for nitrate at 

the Patuxent RIM station is representative of all Category I 

combinations in that both ESTIMATOR and WRTDS produce 

estimates of concentration and flux that are similar and the 

differences between flux-bias ratios are marginal. 
Comparison of Estimate Accuracy and Bias 17

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

–1.5

–1.0

– 0.5

0

0.5

1.0

1.5

2.0

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

3 4 5 6 7 8 9 10

–1.5

–1.0

– 0.5

0.0

0.5

1.0

1.5

2.0

3 4 5 6 7 8 9 10

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

Figure 3. Plots for nitrate at Patuxent River near Bowie, Md. (USGS Station ID 01594440) showing the (A) observed concentration (red dots) versus 

discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual (observed minus 

predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions .

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 3. Nitrate at Patuxent River near Bowie, Maryland (USGS Station 01594440), showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) 

concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, 

(D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and (E) residual 

(observed minus predicted) plot for WRTDS predictions.
18 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Category II Combinations

Category II combinations are those where WRTDSderived estimates of flux are significantly less biased 

compared to ESTIMATOR-derived fluxes. A total of 15 RIM 

combinations exhibit Category II type discrepancies (table 2). 

Two examples illustrate situations that lead to this type of 

discrepancy and provide insight for potential causes. The first 

situation, which occurs in 13 of 15 cases of Category II, is 

when ESTIMATOR considerably overpredicts concentrations 

associated with high-discharge conditions. A representative 

example is suspended sediment at the Rappahannock River 

RIM station. The concentration-discharge relation (fig. 4A) 

shows sediment concentration (red dots) increases as discharge 

increases; however, the slope of the relation between log of 

sediment concentration and the log of discharge decreases 

when the log of discharge is above about 9 (equivalent to 

about 8,000 ft3/s). ESTIMATOR reasonably reproduces the 

concentration-discharge relation for discharge conditions less 

than about 8,000 ft3/s; however, ESTIMATOR considerably 

overestimates sediment concentration for discharge conditions 

greater than about 8,000 ft3/s (fig. 4B). The residual plot shown 

in figure 4C also clearly shows the overprediction (negative 

residuals) of suspended-sediment concentration for discharge 

values greater than about 9 log units. A discharge value of 

about 8,000 ft3/s occurs frequently during storm-runoff periods. 

For context, a discharge value of 25,000 ft3/s (10.1 log units) at 

the Rappahannock River station has a 2-year flood-recurrence 

interval. The peak observed sediment concentration of 

1,204 mg/L was predicted by ESTIMATOR to be 4,105 mg/L. 

This overprediction of sediment concentration, during highdischarge conditions, causes an overprediction of flux, which 

explains the high flux-bias ratio of 2.57 (table 2). Conversely, 

suspended-sediment concentrations, estimated by WRTDS, 

more accurately reflect the shape of the observed concentration-discharge relation throughout the full range of flows from 

the Rappahannock River (fig. 4D). The WRTDS prediction of 

the highest observed concentration was 1,423 mg/L. Although 

WRTDS has a tendency also to overpredict suspendedsediment concentration at discharge values greater than about 

8,000 ft3/s (fig. 4E), which is the primary reason for the 

flux-bias ratio of 1.32 (table 2), this overprediction of flux by 

WRTDS is considerably less than ESTIMATOR flux estimates 

during these same high-discharge conditions. 

The second situation of Category II combination occurs 

when ESTIMATOR underpredicts concentrations that occur 

during extreme low-discharge and high-discharge periods and 

overpredicts concentrations during data rich intermediatedischarge conditions. Of the 15 Category II combinations, 

only 2 combinations exhibit this type of discrepancy; these 

2 combinations are nitrate at the James River RIM station 

and nitrate at the Rappahannock River RIM station. Nitrate 

concentration at the Rappahannock River RIM station will 

be used as an example to define this situation and to discuss 

the root causes. The observed concentration-discharge 

relation for nitrate at the Rappahannock River RIM station 

(fig. 5A) has three distinguishing features. The first and 

most important feature is that over a very broad range of 

discharge values, from about 400 to 22,000 ft3/s (about 

6 to 10 in log units), nitrate concentrations appear to be 

essentially unrelated to discharge and cluster in a range 

between about 0.2 and 1 mg/L (about –1.5 to 0 log units). The 

second feature is that during low discharges that range from 

about 7 to 400 ft3/s (2 to 6 log units), nitrate concentrations 

increase with discharge from about the reporting limit of 

0.004 mg/L (–6.2 log units) to about 0.2 mg/L (–1.5 log units). 

These very low concentrations are probably due to some 

combination of the following factors: (1) the water is derived 

from deeper groundwater, which may not be subjected to 

anthropogenic increases in nitrate to the same extent as 

shallow groundwater, and has had more opportunity for 

subsurface denitrification on its way to the stream; (2) the 

river has a low ratio of volume to streambed surface area 

providing more opportunity for denitrification, which typically 

occurs at the interface between the riverbed and the water 

column (hyporheic zone); and (3) typically at times of high 

temperature, nitrate uptake by aquatic biota may greatly 

reduce the amount of nitrate remaining in the river. The third 

feature is that at discharge values above about 20,000 ft3/s 

(about 10 log units), the slope of the relation between log 

concentration and log discharge is negative. This negative 

slope most likely arises from one or both of two causes: 

(1) at these high discharges, the water in the river reflects the 

chemistry of the rain water, rather than the more nitrate-rich 

soil water and groundwater that dominate streamflow in the 

lower to middle range of discharges; and (2) the steeper, more 

forested portions of the watershed become more dominant 

contributors to runoff at these high discharges and thus 

waters tend to be lower in nitrate than the waters derived 

from the more gently sloping parts of the landscape where 

urban and agricultural activities are more dominant. Because 

ESTIMATOR is constrained to use the quadratic function to 

fit this relation, the data have very few observations in this 

high discharge range, and the anomalous observations cannot 

be accounted for by time or season. Thus the fit in this range 

is largely based on the fit in the middle to low discharge 

range. As shown in figure 5B, ESTIMATOR overpredicts 

nitrate concentrations for discharges between 1,000 ft3/s 

(7 log units) and 22,000 ft3/s (10 log units) and underpredicts 

concentrations for discharges above about 30,000 ft3/s (about 

10.3 log units). The residual plot shown in figure 5C also 

clearly shows that ESTIMATOR underpredicts (positive 

residuals) nitrate at low and high discharges and overpredicts 

nitrate concentrations during intermediate discharges. This 

overprediction of nitrate concentration for discharges between 

about 1,000 and 22,000 ft3/s (about 7 to 10 log units) directly 

contributes to the flux-bias ratio of 1.36 (table 2). WRTDS 

produces nitrate concentrations that more closely follow 

the patterns exhibited in the observed nitrate concentrationdischarge relation (fig. 5D), because it is not constrained to 

follow the quadratic functional form. Residuals are produced 

that are much more symmetrical around zero over most of 
Comparison of Estimate Accuracy and Bias 19

–1

0

1

2

3

4

5

6

7

8

9

–1

0

1

2

3

4

5

6

7

8

9

– 3

–2

–1

0

1

2

3

2 4 6 8 10 12

–1

0

1

2

3

4

5

6

7

8

9

2 4 6 8 10 12

–3

–2

–1

0

1

2

3

Figure 4. Plots for suspended sediment at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (green dots) and WRTDS-predicted (black dots) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 4. Suspended-sediment concentration at Rappahannock River near Fredericksburg, Virginia (USGS Station 

ID 01668000), showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and 

ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot 

for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.
20 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

–14

–12

–10

– 8

– 6

– 4

–2

0

2

–14

–12

–10

– 8

– 6

– 4

–2

0

2

–6

– 4

–2

0

2

4

6

2 4 6 8 10 12

–14

–12

–10

– 8

– 6

– 4

–2

0

2

2 4 6 8 10 12

– 6

– 4

–2

0

2

4

6

Figure 5. Plots for nitrate at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed concentration

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, 

(C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (green dots) and WRTDS-predicted (black dots) concentration

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 5. Nitrate concentration at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.
Comparison of Estimate Accuracy and Bias 21

the range of discharges (certainly those above about 7 log 

units or 1,100 ft3/s). The residuals for both methods appear 

to be highly heteroscedastic. In either method the residual 

variance is fairly small and constant above a discharge of 

about 1,000 ft3/s (about 7 log units), but it is much larger for 

the lower discharge values (less than 7 log units) (fig. 5C and 

5E). Note that ESTIMATOR assumes that the residuals have 

constant variance and uses that overall variance in making the 

bias adjustment (see Cohn, 2005). This assumption is likely to 

result in too large a bias adjustment at high discharges and too 

small a bias adjustment for concentrations at low discharges. 

In addition to the general problem of lack of fit, this assumption also contributes to the positive flux bias associated with 

ESTIMATOR. In contrast, WRTDS makes no assumption of 

constant variance, and the bias adjustment (eq. 4) tends to 

be small at high discharges and large at low discharges. As a 

result of the improved accuracy associated with the WRTDS 

nitrate predictions, the flux-bias ratio improved from 1.36 

(ESTIMATOR predictions) to 0.99 (table 2). 

Both ESTIMATOR (fig. 5C) and WRTDS (fig. 5E) show 

relatively poor ability to predict log nitrate concentrations 

at low discharge values. Although the graphs in figure 5 

indicate a high degree of variability at low discharge, it 

should be noted that these are graphs of log concentration. 

Concentrations at these low discharges are generally between 

0 and 0.1 mg/L and most concentrations at middle to high 

discharges range from 0.2 to 1.0 mg/L. This relatively poor fit 

(using either method) is of little consequence in the calculation 

of annual or long-term average fluxes, because the concentrations are so close to zero that they contribute little to the total 

flux. What is important from the standpoint of the performance 

of the two methods is that the inability of both methods to 

accurately predict concentrations in this low discharge range 

influences the predictions by ESTIMATOR at the much more 

important high discharges but does not affect predictions by 

WRTDS at high discharges. 

Both examples of Category II discrepancies highlight 

that although ESTIMATOR does an exceptional job accurately 

estimating concentration-discharge relations that are linear or 

quadratic in nature (Category I combinations), ESTIMATOR 

is unable to fully reproduce concentration-discharge relations that have multiple points of inflection (that is, more 

sinuous in form). The primary reason that ESTIMATOR 

cannot fully reproduce concentration-discharge relations for 

Category II combinations is because these relations are not 

linear or quadratic, and the deviation in the relation cannot 

be accounted for by the remaining model variables, time 

and season. WRTDS, conversely, more closely reproduces 

the observed shape of the concentration-discharge relation 

because WRTDS is not constrained to linear or quadratic 

forms, but rather uses smoothing techniques to more closely 

approximate the sinuous shape of the relation. Additionally, 

data from low discharge days have limited influence on the 

model coefficients for days of high discharge because the 

weights assigned to these observed data collected during 

low-discharge conditions are at or near zero (and the reverse 

also is true). This is not the case with ESTIMATOR where all 

water-quality data are used to determine model coefficients for 

each 9-year model window.

Category III Combinations

Category III contains two combinations, Susquehanna 

River total phosphorus and Susquehanna River suspended 

sediment, that exhibit some features consistent with the 

sources of flux bias discrepancies discussed in Category II; 

however, other unique characteristics warrant labeling these 

two combinations as Category III. The general characteristics 

that define Category III combinations are (1) ESTIMATOR 

underpredicts flux associated with the highest discharges; 

(2) WRTDS overpredicts flux associated with moderate 

discharges; (3) the magnitude of the difference between 

flux-bias ratio associated with WRTDS and the flux-bias ratio 

associated with ESTIMATOR is at least 0.10. The net effect 

of these three characteristics is that WRTDS results will show 

that larger fluxes of suspended sediment and total phosphorus 

are being delivered from the Susquehanna River RIM station 

when compared to ESTIMATOR. 

Susquehanna River suspended-sediment concentrations 

will be used to represent both Category III combinations; the 

patterns and discrepancies associated with total phosphorus 

are similar but less pronounced as those associated with 

Susquehanna River suspended sediment. The flux-bias ratio 

for Susquehanna River suspended sediment increases from 

0.80 to 1.06 (net change of 0.26) when ESTIMATOR flux 

predictions are replaced by those derived from WRTDS 

(table 2). This flux-bias ratio indicates that WRTDS produces 

flux estimates that are positively biased (1.06) but are closer 

to 1.0 than the flux-bias ratio associated with ESTIMATOR, 

which is negatively biased (0.80). To better understand the 

reasons associated with the ESTIMATOR and WRTDSassociated flux biases, one needs to take a close look at the 

observed relation between log of suspended-sediment concentration and log of discharge (fig. 6A). The concentrationdischarge relation (fig. 6A) shows that suspended-sediment 

concentration (red dots) increases as discharge increases; 

however, the slope of the relation between log of suspendedsediment concentration and the log of discharge steepens at 

two separate locations. The first increased slope is represented 

for log of discharges between 11 and 13 (equivalent to 

discharges between about 60,000 and 450,000 ft3/s); and an 

additional increase occurs at log of discharges greater than 

about 13 (albeit only four observations are defining this 

increased slope). The Susquehanna River RIM station is 

located at the outfall of Conowingo reservoir. The changing 

shape of the concentration-discharge relation is related to an 

abrupt shift from net depositional at lower discharges to net 

scour of the reservoir sediments at higher discharges. Langland (2009) estimated that a minimum discharge threshold of 

at least 390,000 ft3/s must be exceeded to mobilize the stored 

reservoir sediments. ESTIMATOR accurately reproduces the 

concentration-discharge pattern for all discharge values up 
22 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

–1

0

1

2

3

4

5

6

7

8

9

–1

0

1

2

3

4

5

6

7

8

9

–3

–2

–1

0

1

2

3

7 8 9 10 11 12 13 14

–1

0

1

2

3

4

5

6

7

8

9

7 8 9 10 11 12 13 14

–3

–2

–1

0

1

2

3

Figure 6. Plots for suspended sediment at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 6. Suspended-sediment concentration at Susquehanna River near Conowingo, Maryland (USGS Station 

ID 01578310), showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and 

ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) 

plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus 

discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.
Comparison of Estimate Accuracy and Bias 23

to about 450,000 ft3/s (13 log units); however, ESTIMATOR 

tends to underestimate suspended-sediment flux for discharges 

greater than about 450,000 ft3/s (fig. 6B). It is this underprediction of suspended-sediment concentration associated with 

the highest discharges (that is, greater than 450,000 ft3/s) 

that primarily produces the flux-bias ratio of 0.80 (table 2). 

The residuals for the ESTIMATOR predictions are evenly 

distributed around zero for discharges up to 450,000 ft3/s 

(fig. 6C); conversely; the residuals skew to greater than zero 

(indicative of underestimation of flux by ESTIMATOR) for 

discharges greater than 450,000 ft3/s. The primary reason that 

ESTIMATOR underpredicts suspended-sediment concentrations associated with discharges greater than 450,000 ft3

/s 

is that the observed concentration-discharge relation is not 

truly quadratic in form at high discharges (greater than 

approximately 400,000 ft3/s, or 13 log units). WRTDS is 

more successful than ESTIMATOR at reproducing the 

concentration-discharge relation for discharges greater than 

about 450,000 ft3/s; however, WRTDS is not as successful 

as ESTIMATOR at reproducing log suspended-sediment 

concentrations associated with log discharges between 

approximately 11 and 12.5 (equivalent to discharges between 

about 60,000 and 270,000 ft3/s) (fig. 6D). It is this overprediction of suspended-sediment concentration for discharges 

between 60,000 and 270,000 ft3/s that produces the flux-bias 

ratio of 1.06 (table 2). The reason for the overprediction of 

suspended-sediment concentration for discharges between 

60,000 and 270,000 ft3/s by WRTDS may be related to lack of 

a log discharge squared term. 

What can be concluded from this comparison of 

WRTDS and ESTIMATOR model estimates of flux to 

the fluxes observed at the nine RIM stations? For 28 of 

the 45 RIM combinations (Category I), there is little to 

no difference between the ESTIMATOR and WRTDS 

estimates of flux with regard to flux-bias ratio; however, for 

all 28 combinations, WRTDS produced flux estimates that 

were more accurate (reduced RMSE) than those derived 

from ESTIMATOR with 8 of the 28 combinations having 

greater than 20 percent reduction in RMSE. For 15 of the 

45 combinations, WRTDS fluxes were less biased (fluxbias ratio closer to 1.0 by at least 0.10) and had a marked 

improvement in accuracy (12 of the 15 combinations 

showed a 20-percent reduction in RMSE) compared to flux 

estimates obtained from ESTIMATOR (Category II). These 

results show that ESTIMATOR’s strength is reproducing 

concentration-discharge relations (in log space) that are 

either linear or quadratic in form; the majority (62 percent 

or 28 of 45 combinations) of the combinations exhibits 

this relation. The problem arises, for ESTIMATOR, when 

the concentration-discharge relation is more sinuous (more 

than one point of inflection) in form than a simple linear or 

quadratic pattern, and the differences between the observed 

concentration-discharge relation and a linear or quadratic form 

cannot be accounted for by the additional model variables, 

time and season. WRTDS, conversely, is able to represent 

these complex concentration-discharge relations because of 

the model flexibility brought about by the combination of its 

functional form and how the model coefficients are determined 

for every unique combination of time and discharge. 

Only 2 combinations of the 45 cannot be categorized 

as Category I or Category II. What makes these Category III 

combinations unique is that they are either particulate or 

particulate-dominated constituents (suspended sediment and 

total phosphorus) collected from the Susquehanna River 

downstream from the Conowingo reservoir. ESTIMATOR 

tends to outperform WRTDS in reproducing the concentration 

and flux associated with the majority of the range in discharge; 

WRTDS outperforms ESTIMATOR in reproducing concentrations and fluxes associated with the highest discharges. 

The difficulty that both models have with reproducing 

concentrations and fluxes across the full range of discharges 

is directly related to transport processes that change (shift 

from net depositional to net scour) depending on conditions 

in the Conowingo reservoir. In this situation, WRTDS seems 

to be the better model to address questions associated with 

flux and changes in flux, and ESTIMATOR seems to be better 

at reproducing fluxes and concentrations associated with 

non-extreme hydrologic events. 
24 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Comparison of ESTIMATOR and 

WRTDS Annual Fluxes

The results of the analysis thus far have shown that flux 

estimates derived from WRTDS are in all cases except one 

more accurate and for the majority of RIM combinations have 

an improved flux-bias ratio when compared to flux estimates 

derived from ESTIMATOR. The focus of the analysis shifts 

now to address how different estimates of annual fluxes would 

have been for all RIM combinations had WRTDS been used 

instead of ESTIMATOR for estimates for 1980 to 2010. To 

answer this question the average percent difference in annual 

flux estimates was quantified between WRTDS-derived and 

ESTIMATOR-derived annual fluxes of total nitrogen, nitrate, 

total phosphorus, orthophosphorus, and suspended sediment 

at each of the nine RIM stations. The average difference in 

estimated annual fluxes (expressed in percent) is defined as 

∑ ∑

∑ = − 















= =

=

AF AF

AF

Average Difference 100 W i E i i

n

i

n

E i i

n

1 , , 1

1 ,

×

 

 (10)

where 

AFW,i is WRTDS estimated annual flux 

for year i; in tons per day;

AFE,i is ESTIMATOR estimated annual flux 

for year i, in tons per day; and

n is the total number of years.

Positive values for the average percent difference in annual 

flux indicate that annual fluxes would have been, on average, 

greater coming from WRTDS; and, negative values indicate 

that, on average, annual fluxes derived from WRTDS would 

have been smaller than those derived from ESTIMATOR. 

An average percent difference less than 10 percent is 

considered minimal.

Nitrogen

Total nitrogen annual fluxes generated by WRTDS for the 

nine RIM stations are consistent with those generated using 

ESTIMATOR (fig. 7). All stations show a close correspondence for annual fluxes of total nitrogen derived from WRTDS 

and ESTIMATOR. Average percent differences for annual 

fluxes of total nitrogen range from –5.13 to 3.04 percent 

(fig. 7). These differences are all less than 10 percent and 

are considered minimal. The greatest average difference 

(–5.13 percent) occurs at the Rappahannock RIM station 

(fig. 7D). The Rappahannock is the only station that exhibits 

Category II discrepancies that are related to overprediction 

of flux, by ESTIMATOR, during the highest discharge 

conditions. Recall that flux-bias ratios for total nitrogen at 

the Rappahannock RIM station are 1.00 (no bias) and 1.12 

for WRTDS and ESTIMATOR-generated fluxes, respectively 

(table 2). Figure 7D shows that the greatest discrepancy 

between ESTIMATOR and WRTDS annual fluxes occurs 

during years with the greatest flux. The highest total nitrogen 

annual flux, between 1985 and 2010, occurred during 2003 

when the annual flux determined by ESTIMATOR was 

15.5 tons per day (tons/d) and the annual flux determined by 

WRTDS was 13.6 tons/d (fig. 7D). The annual discharge for 

2003 is the greatest annual discharge for this same period. 

Thus, the effect of ESTIMATOR’s tendency to overpredict 

concentrations associated with the highest discharges becomes 

most apparent during the years with the greatest discharges. 

The average difference for total nitrogen at the Rappahannock 

RIM station is interpreted as follows: historical annual fluxes 

would have been, on average, 5.13 percent lower coming 

from WRTDS compared to annual fluxes generated using 

ESTIMATOR, and much of this difference comes from the 

estimates in the years for which ESTIMATOR produced its 

two highest annual values of the entire period of record. 

Nitrate annual fluxes generated by WRTDS for the nine 

RIM stations are consistently less than those generated using 

ESTIMATOR (fig. 8). Average differences for annual fluxes 

of nitrate range from –28.13 to –1.61 percent (fig. 8). The 

average differences are all negative; this means the annual 

nitrate fluxes derived from WRTDS are typically smaller 

than those derived from ESTIMATOR. The two stations with 

the greatest average percent difference are the Rappahannock (–28.13 percent) (fig. 8D) and James (–13.55 percent) 

(fig. 8C). Both of these stations, being identified as having 

Category II discrepancies for the flux-bias ratio, have large 

negative average differences because ESTIMATOR overpredicts nitrate flux during intermediate- to high-discharge 

conditions, and WRTDS-derived fluxes more accurately 

represent observed nitrate concentrations/fluxes for this 

discharge interval. The remaining seven RIM stations, listed 

as having Category I-type discrepancies, have average percent 

differences that range from –8.38 to –1.61 percent, which are 

considered to be minimal differences because they are all less 

than 10 percent different. 

Phosphorus

Total phosphorus annual fluxes generated by WRTDS 

for the nine RIM stations are generally consistent with 

those generated using ESTIMATOR (fig. 9). Average 

differences for annual fluxes of total phosphorus range from 

–17.68 to 12.38 percent (fig. 9). For seven of the nine RIM 

stations, differences between annual total phosphorus fluxes 

generated by ESTIMATOR and WRTDS are minimal, ranging 

from –2.10 to 8.17 percent (all less than 10 percent. The 

average difference for total phosphorus at the Susquehanna 

(12.38 percent) (fig. 9A) and Rappahannock (–17.68 percent) 

(fig. 9D) RIM stations, however, exceeds the 10 percent 

threshold, which indicates that the annual fluxes generated 

by WRTDS and ESTIMATOR are different. It has been 
Comparison of ESTIMATOR and WRTDS Annual Fluxes 25

0 100 200 300 400

0

400300200100

0 50 100 150 200

0

50

200150100

0 10 20 30 40

0

40302010

0 5 10 15 20

50

201510

0 1 2 3 4 5 6

6543210

0 1 2 3 4 5

543210

0 0.5 1.0 1.5 2.0

0

0.5

1

1.5

2

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

3.53.02.52.01.51.00.5

0 0.5 1.0 1.5

0

1.51.00.5

= 3.04%

2.37%

 Susquehanna, 

–

0.21%

verage difference

= verage difference

= –

(A)

 James

verage difference

C.  A

 Pamunkey

 A

I. Choptank

F.

 A

TOR-derived annual total nitrogen fluxes and associated average percent difference at the 

(I) Choptank River Input Monitoring stations. (H) Patuxent, and 

1.32%–

0.33%

=

= 

= 0.64%

verage difference

verage difference

verage difference

Estimator flux, in tons per day

 Mattaponi, (G)

 Potomac

 Pamunkey, 

 Patuxent

B. 

 A

E. Appomattox A

H.  A

(F) Appomattox, (E) Rappahannock, 

5.13%

0.01%

(D)

1.02% 

 James, 

 Susquehanna

verage difference=

= –

= –

verage difference

WRTDS-derived versus ESTIMA

(C)

A. 

1:1 line

 Rappahannock

 Mattaponi

verage difference

 A

D.  A

G.  A

 Potomac, 

Figure 7.(B)

WRTDS flux, in tons per day

Figure 7. Plot showing WRTDS-derived versus ESTIMATOR-derived annual total nitrogen fluxes and associated average difference at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Black line represents the 1:1 line. 
26 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

0 50 100 150 200 250

0

50

250200150100

0 20 40 60 80 100 120

0

8060

4020

120

100

0 2 4 6 8 10 12

86

420

1210

0 2 4 6 8 10

8

6420

10

0 0.5 1.0 1.5

0

1.51.0

0.5

0 0.5 1.0 1.5

0

1.51.0

0.5

0 0.1 0.2 0.3 0.4

0

0.40.30.20.1

0 0.5 1.0 1.5 2.0 2.5

0

2.52.01.51.00.5

0 0.2 0.4 0.6 0.8 1.0

0

1.00.80.6

0.40.2

 Potomac, (B)

= –13.55%

4.99%

.66%2

–=

e

 Susquehanna, 

 James

verage differenc

C. 

 Pamunkey

verage difference

= –

 A

 Choptank

verage difference

(A)

. F

 A

I.  A

TOR-derived annual nitrate fluxes and associated average percent difference at the 

(I) Choptank River Input Monitoring stations.

y da er s p on

8.38%

7.63%

1.61%

 t n

–

–

x, i

= –

 Patuxent, and 

=

u

= 

l

e

e

e

f

verage differenc

 Patuxent

verage differenc

(H)

verage differenc

ro at

 Potomac

stim E

 Mattaponi, 

B. 

 A

E. Appomattox

 A

H.  A

(G) Pamunkey, (F) Appomattox, (E)

3.39% 

28.13%

6.09%

–

 Susquehanna

 Rappahannock

= –

= –

 Rappahannock, 

verage difference=

verage difference

 Mattaponi

verage difference

WRTDS-derived versus ESTIMA

(D)

A.  A

1:1 line

D.  A

G.  A

 James, 

Figure 8.

(C)

ya d per s on t n x, i ulf DS TRW

Figure 8. Plot showing WRTDS-derived versus ESTIMATOR-derived annual nitrate fluxes and associated average difference at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Black line represents the 1:1 line. 
Comparison of ESTIMATOR and WRTDS Annual Fluxes 27

0 5 10 15 20 25

50

25201510

0 5 10 15 20

50

201510

0 2 4 6 8 10

86420

10

0 1 2 3 4

43210

0 0.2 0.4 0.6 0.8

0

0.80.60.40.2

0 0.2 0.4 0.6 0.8

0

0.80.60.40.2

0 0.05 0.10 0.15 0.20

0

0.200.150.100.05

0 0.1 0.2 0.3 0.4

0

0.40.30.20.1

0 0.02 0.04 0.06 0.08 0.10 0.12

0

0.120.100.080.060.040.02

 Potomac, (B)

= 8.17%

.10%

0.33%

2

=

(A) Susquehanna, 

verage difference

= –

 James

verage difference

verage difference

C.  A

 Pamunkey F.

 A

I. Choptank A

4.12%

2.84%

=

= 

= 5.52%

 Potomac

verage difference

verage difference

 Patuxent

verage difference

Estimator flux, in tons per day

B. 

 A

 Appomattox E. 

 A

H.  A

1:1 line

TOR-derived annual total phosphorus fluxes and associated average difference at the 

(H) Patuxent, and (I) Choptank River Input Monitoring stations. (G) Mattaponi, (F) Pamunkey, Appomattox, (E)

12.38% 

17.68%= –

= 0.80%

 Susquehanna

verage difference=

 Rappahannock

verage difference

WRTDS-derived versus ESTIMA

(D) Rappahannock, 

verage difference

 Mattaponi

A. 

 A

D.  A

G.  A

 James, 

Figure 9.(C)

WRTDS flux, in tons per day

Figure 9. Plot showing WRTDS-derived versus ESTIMATOR-derived annual total phosphorus fluxes and associated average difference at the (A) Susquehanna, (B) Potomac, (C) James,(D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Black line represents the 1:1 line. 
28 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

determined that total phosphorus at the Susquehanna station 

is a Category III combination, which means that WRTDS has 

a tendency to overestimate concentration and flux at intermediate discharges, and ESTIMATOR tends to underestimate 

concentration and flux during the highest discharge conditions. 

The flux-bias ratios (table 2) reveal the cumulative effect 

of the overestimation by WRTDS and underestimation by 

ESTIMATOR with ratios of 1.06 and 0.96, respectively. The 

flux-bias ratios for both models are within 0.10 of 1.0, which 

indicates both models are minimally biased; however, the 

total difference between WRTDS and ESTIMATOR flux-bias 

ratios is 10 percent, which is comparable to the annual flux 

average difference of 12.38 percent (fig. 9A). The greatest 

discrepancy in estimated annual fluxes occurred during 2004 

when ESTIMATOR predicted an annual flux of 15.9 tons/d 

and WRTDS predicted an annual flux of 21.7 tons/d. An 

explanation for this discrepancy can be found in the observed 

concentration-discharge relations for total phosphorus at 

the Susquehanna RIM station and how well both WRTDS 

and ESTIMATOR reproduce this relation (fig. 10). First, 

ESTIMATOR (fig. 10B) and WRTDS (fig. 10D) appear 

to reproduce total phosphorus concentration equally well; 

however, a closer inspection shows that between discharges 

of approximately 11 and 12.5 log units, WRTDS (black dots) 

overestimates a greater portion of the observations (red dots) 

(fig. 10D) compared to ESTIMATOR (fig. 10B). The residuals 

(observed minus predicted concentration) for ESTIMATOR 

(fig. 10C) and WRTDS (fig. 10E) predicted concentrations 

show that WRTDS has a greater density of negative residuals 

(negative residual means estimated concentration is greater 

than observed concentration) for discharges between about 

11 and 12.5 log units than does ESTIMATOR. The average 

residual for concentrations in this discharge range is –0.16 

(fig. 10C) and – 0.22 log units as determined by ESTIMATOR 

(fig. 10C) and WRTDS (fig. 10D), respectively. Second, 

ESTIMATOR has a greater tendency to underpredict total 

phosphorus concentrations/fluxes for extreme high-discharge 

conditions (greater than 440,000 ft3/s, 13 log units); whereas, 

WRTDS concentration predictions more closely approximate 

observed concentrations during these high-discharge conditions (fig. 10B–10E). Annual discharge for the Susquehanna 

River at the RIM station for 2004 was the highest during the 

1985 to 2010 period when the peak daily discharge reached 

622,000 ft3/s (13.3 log units). This example shows that the 

cumulative effect of the over and underprediction of concentration and flux by WRTDS and ESTIMATOR, respectively, 

is more pronounced in years with greater annual discharge 

(fig. 9A). 

At the Rappahannock station, the average total phosphorus difference of –17.68 percent (fig. 9D) occurs because 

ESTIMATOR considerably overpredicts total phosphorus 

concentration associated with discharge conditions greater 

than 22,000 ft3/s (10 log units) (fig. 11B and 11C); whereas, 

WRTDS produces estimates of total phosphorus concentration 

that more closely reproduce observed concentrations during 

these high-discharge conditions (fig. 11D and 11E). Observed 

total phosphorus concentration at the highest discharge (approximately 55,000 ft3

/s or 11 log units) is 0.46 mg/L. ESTIMATOR 

predicted a concentration of 2.72 mg/L and WRTDS predicted 

a concentration of 0.77 mg/L at this same discharge. This 

overprediction of concentration by ESTIMATOR during 

the highest discharges is directly related to Category II 

discrepancies, as previously discussed, where the observed 

concentration-discharge relation exhibits a more sinuous 

pattern with multiple points of inflection. The functional form 

of the ESTIMATOR model can only represent concentrationdischarge relations that are linear or quadratic in form, and 

the differences between the observed concentration-discharge 

pattern and a linear or quadratic form cannot be accounted for 

by the remaining ESTIMATOR variables, time and season. 

Orthophosphorus annual fluxes generated by WRTDS for 

the nine RIM stations are in close agreement with those generated using ESTIMATOR (fig. 12). For eight of the nine RIM 

stations, the average difference for annual fluxes ranged from 

–6.85 to 0.17, all of which are considered minimal differences 

(less than 10 percent); however, the average difference for 

orthophosphorus at the Patuxent RIM station (10.30 percent) 

(fig. 12H) just exceeds the 10 percent threshold. The 

flux-bias ratio results show a very similar pattern in that the 

flux-bias ratio for WRTDS is 1.09 and the flux-bias ratio for 

ESTIMATOR is 1.04 (table 2). Both models produce estimates 

of flux that are positively biased with WRTDS more biased 

than ESTIMATOR; however, both models are considered 

minimally biased because both are within 0.10 of 1.0 (no 

bias). Inspection of each model’s performance in reproducing 

the observed concentration-discharge relation reveals that both 

models produce similar patterns in orthophosphorus concentration across the full range of flow (fig. 13). Figure13E shows 

a slight increase in negative residuals (WRTDS predicted 

concentrations greater than observed concentrations) for low 

to intermediate discharges that range from approximately 

5 to 7 log units (approximately 150 to 1,100 ft3/s); conversely, 

the ESTIMATOR-derived concentration residuals are more 

symmetrically distributed around zero for these intermediatedischarge conditions (fig. 13C). This positive bias associated 

with WRTDS-derived concentrations, however, is diminished 

(more symmetrical distribution around zero) for discharges 

greater than approximately 8 log units (approximately 

3,000 ft3/s) (fig.13E) compared to ESTIMATOR-derived 

concentrations residuals (fig. 13C), which are shifted negative 

within this region of discharge. Therefore, the minor positive 

shift from the 1:1 line (fig. 12H) may be attributed to the 

tendency for WRTDS to slightly overpredict orthophosphorus 

concentrations during intermediate-discharge conditions. 

This shift is diminished for the highest annual flux, which 

occurred in 1985, when WRTDS predicted that annual flux 

was 0.095 tons/d and ESTIMATOR predicted annual flux was 

0.088 tons/d (fig.12H).
Comparison of ESTIMATOR and WRTDS Annual Fluxes 29

– 6

–5

– 4

–3

–2

–1

0

1

– 6

– 5

– 4

–3

–2

–1

0

1

– 3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

7 8 9 10 11 12 13 14

– 6

–5

– 4

–3

–2

–1

0

1

7 8 9 10 11 12 13 14

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

Figure 10. Plots for total phosphorus at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310)showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

WRTDS over 

estimation

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 10. Total phosphorus concentration at Susquehanna River near Conowingo, Maryland (USGS Station ID 01578310), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions.
30 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

–6

–5

–4

–3

–2

–1

0

1

2

–6

–5

–4

–3

–2

–1

0

1

2

–3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

2 4 6 8 10 12 –6

–5

–4

–3

–2

–1

0

1

2

2 4 6 8 10 12 –3.0

–2.5

–2.0

–1.5

–1.0

–0.5

0

0.5

1.0

1.5

2.0

Figure 11. Plot for total phosphorus at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 

A

B C

D E

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 11. Total phosphorus concentration at Rappahannock River near Fredericksburg, Virginia (USGS Station 

ID 01668000), showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and 

ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot 

for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.
Comparison of ESTIMATOR and WRTDS Annual Fluxes 31

0 0.5 1.0 1.5 2.0 2.5 3.0

0

3.02.52.01.51.00.5

0 0.5 1.0 1.5 2.0 2.5 3.0

0

3.02.52.01.51.00.5

0 0.5 1.0 1.5 2.0 2.5

0

2.52.01.51.00.5

0 0.05 0.10 0.15 0.20 0.25

0

0.250.200.150.100.05

0 0.02 0.04 0.06 0.08 0.10

0

0.100.080.060.040.02

0 0.02 0.04 0.06 0.08 0.10

0

0.100.080.060.040.02

0 0.01 0.02 0.03 0.04

0

0.040.030.020.01

0 0.02 0.04 0.06 0.08 0.10

0

0.100.080.060.040.02

0 0.005 0.010 0.015 0.020 0.025 0.030

0

0.0300.0250.0200.0150.0100.005

 Potomac, (B)

6.85%

3.71%

.04%2

= –

–=

 Susquehanna, 

verage difference

= – verage difference

(A)

 James

verage difference

C.  A

 Pamunkey

 A

I. Choptank

F.

 A

TOR-derived annual orthophosphorus fluxes and associated average difference at the 

 Choptank River Input Monitoring stations. (I)

4.70%–

0.17%

=

= 

= 10.30%

 Potomac

verage difference

Estimator flux, in tons per day

(H) Patuxent, and 

verage difference

 Patuxent

verage difference

B. 

 A

 Appomattox

 Mattaponi, 

E.  A

H.  A

(G) (F) Pamunkey, 

1:1 line

 Appomattox, (E)

3.17% 

4.27%

3.53%

– verage difference=

 Rappahannock

= –

= –

 Susquehanna

verage difference

verage difference

WRTDS-derived versus ESTIMA

(D) Rappahannock, 

 Mattaponi

A.  A

D.  A

G.  A

Figure 12.

 James, (C)

WRTDS flux, in tons per day

Figure 12. Plot showing WRTDS-derived versus ESTIMATOR-derived annual orthophosphorus fluxes and associated average difference at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Black line represents the 1:1 line. 
32 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

–6

–5

–4

–3

–2

–1

0

1

–6

–5

–4

–3

–2

–1

0

1

–5

–4

–3

–2

–1

0

1

3 4 5 6 7 8 9 10

–6

–5

–4

–3

–2

–1

0

1

3 4 5 6 7 8 9 10

–5

–4

–3

–2

–1

0

1

2

A

B C

D E

Figure 13. Plot for orthophosphorus at Patuxent River near Bowie, Md. (USGS Station ID 01594440) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, 

(C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 

Log concentration, in milligrams per liter

Residual log concentration, in milligrams per liter

Log discharge, in cubic feet per second Log discharge, in cubic feet per second

Figure 13. Orthophosphorus concentration at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.
Comparison of ESTIMATOR and WRTDS Annual Fluxes 33

Suspended Sediment

Suspended sediment annual fluxes generated by WRTDS 

for the nine RIM stations are, in general, considerably 

different from those generated using ESTIMATOR (fig. 14). 

Average differences for annual fluxes of suspended sediment 

range from –39.50 to 38.33 percent (fig. 14). The average 

difference for suspended sediment at the nine RIM stations 

can be split into two groups: (1) stations (eight of nine 

stations) with negative average percent differences (that 

is, ESTIMATOR tends to produce higher annual estimates 

than WRTDS: fig. 14B–14I) and (2) the Susquehanna RIM 

station, which has a positive average percent difference (that 

is, WRTDS tends to produce higher annual estimates than 

ESTIMATOR: fig. 14A). For the first group, average percent 

difference between WRTDS and ESTIMATOR annual fluxes 

ranges from –39.50 percent to –2.35 percent. Of the eight 

stations in this first group, only three have average percent 

differences greater than 10 percent; these stations are the 

Potomac (–19.34 percent) (fig. 14B), James (–19.20 percent) 

(fig. 14C), and Rappahannock (–39.50 percent) (fig. 14D) 

RIM stations. These three stations are Category II combinations and as a result each has the same root cause for these 

large negative average percent differences—considerable 

overprediction of flux by ESTIMATOR during high-discharge 

conditions. The concentration-discharge relation for suspended 

sediment at the Rappahannock River RIM station will once 

again be used to highlight this discrepancy (fig. 4). Figure 4B

shows that ESTIMATOR (black circles) does a good job 

reproducing the observed concentration-discharge relation for 

discharge conditions up to 8,100 ft3/s (9 log units); however, 

ESTIMATOR considerably overpredicts observed suspendedsediment concentration associated with discharges greater 

than 8,100 ft3/s. This overprediction, by ESTIMATOR, at the 

Rappahannock, James, and Potomac RIM stations is directly 

is related to Category II discrepancies, previously discussed, 

where the observed concentration-discharge relation exhibits 

a more sinuous pattern with multiple points of inflection. 

The functional form of the ESTIMATOR model can only 

represent concentration-discharge relations that are linear or 

quadratic in form, and the differences between the observed 

concentration-discharge pattern and a linear or quadratic 

form cannot be accounted for by the remaining ESTIMATOR 

variables, time and season. These three stations exhibit a 

sigmoidal pattern in the observed concentration-discharge 

relation that can only be partially reproduced by ESTIMATOR 

(fig. 4B); conversely, WRTDS, which has a more flexible 

design, is able to reproduce the full sigmoidal pattern of 

the observed concentration-discharge relation (fig. 4D). 

At the Susquehanna station, two issues are contributing 

to the elevated average percent difference in suspended 

sediment annual fluxes (38.33 percent). First, WRTDS 

has a tendency to overpredict suspended sediment flux for 

intermediate- to high-discharge conditions ranging from 

22,000 to 163,000 ft3/s (10 to 12 log units) (fig. 6D and 6E); 

whereas, ESTIMATOR produces more accurate estimates in 

this range of flow conditions (fig. 6B and 6C). This discharge 

condition typically dominates the annual hydrographs given 

that over 95 percent of the daily discharges, occurring between 

1985 and 2010, are less than 163,000 ft3/s (approximately 

12 log units). Therefore, for the majority of years, the annual 

flux is generated from discharge conditions that extend up to 

163,000 ft3/s, and the positive shift from the 1:1 line (fig. 14A) 

for the range extending from 5,000 to 10,000 tons/d can 

primarily be attributed to the overestimation by WRTDS 

during these intermediate-high discharge conditions. Second, 

ESTIMATOR has a greater tendency to considerably underpredict suspended-sediment concentrations/fluxes for extreme 

high-discharge conditions (that is, greater than 440,000 ft3/s, 

13 log units) (fig. 6B and 6C); whereas, WRTDS flux predictions are more accurate during these discharge conditions 

(fig. 6D–6E). The greatest Susquehanna total phosphorus 

annual flux estimated by both WRTDS (34,721 tons/d) and 

ESTIMATOR (13,007 tons/d) occurred in 2004, which was 

the wettest year for the period 1985 to 2010. The deviation 

from the 1:1 line (fig. 7A) is more a result of ESTIMATOR 

tending to underpredict total phosphorus concentrations/fluxes 

associated with extreme high-flow conditions (greater than 

440,000, ft3/s, 13 log units) than the positive bias of WRTDS 

estimates during intermediate-discharge conditions. 
34 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

 Susquehanna, TOR-derived annual suspended-sediment fluxes and associated average percent difference at the (A)

 Choptank River Input Monitoring stations. (I) Patuxent, and (H) Mattaponi, (G) , Pamunkey (F) Appomattox, 

WRTDS-derived versus ESTIMA

(E) Rappahannock, (D) James, (C) Potomac, 

0

Figure 14.(B)

10,000 20,000 30,000 40,000

0

40,00030,00020,00010,000

0 5,000 10,000 15,000 20,000 25,000

0

5,000

25,00020,00015,00010,000

0 2,000 4,000 6,000 8,000 10,000 12,000

0

8,0006,0004,0002,000

12,00010,000

0 1,000 2,000 3,000 4,000 5,000

0

5,0004,0003,0002,0001,000

WRTDS flux, in tons per day

0 50 100 150 200 250

0

50

250200150100

0 100 200 300 400

0

400300200100

0 10 20 30 40 50

0

5040302010

0 50 100 150 200

0

50

200150100

Estimator flux, in tons per day

0 5 10 15 20 25

50

25201510

= –19.20%

7.74%–

2.35%

verage difference

= verage difference

= –

 James

verage difference

C.  A

 Pamunkey

 A

I. Choptank

F.

 A

–19.34%

3.37%

9.04%

–

=

= –

verage difference

= 

 Potomac

 Appomattox

verage difference

 Patuxent

verage difference

B.  A

E.  A

H.  A

1:1 line

38.33% 

39.50%

9.12%

=

= –

= –

 Susquehanna

verage difference

 Rappahannock

verage difference

 Mattaponi

verage difference

A.  A

D.  A

G.  A

Figure 14. Plot showing WRTDS-derived versus ESTIMATOR-derived annual suspended sediment fluxes and associated average percent difference at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. 
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 35

Trends in Nutrient and Sediment Flux 

to the Chesapeake Bay

For the presentation of changes in nutrient and sediment flux, two perspectives are provided. First, the WRTDS 

flow-normalized changes in flux (table 3) will be compared to 

the historical trends in flow-adjusted concentration (table 3) 

obtained from ESTIMATOR. Second, the way in which nutrient 

and sediment fluxes are changing in each of the nine major 

Chesapeake Bay tributaries will be discussed. Changes in flownormalized nutrient and sediment flux are reported for two time 

periods: 1985 to 2010 and 2001 to 2010. These time periods are 

used because 1985 is viewed by the CBP as the base year that 

all subsequent years are compared in order to assess changes 

in nutrient and sediment concentrations and now fluxes, and 

2001 serves as the beginning year for the most recent 10-year 

period. The CBP requests that results be provided for both 

long-term (since 1985) and short-term (last 10 years) trends to 

assess the influence of management activities on nutrient and 

sediment transport throughout the Chesapeake Bay watershed. 

Historically, analysis of 10 years of data was the shortest 

period for which ESTIMATOR could determine flow-adjusted 

concentration trends and minimizing associated uncertainty. 

WRTDS has the ability to determine trends for any specified 

time period for records of at least 20 years. The slope defining 

flux changes at each of the nine RIM stations can be reported 

as (1) a percent change per year (eq. 6) and (2) a mass change 

per year (eq. 7). Because the range in drainage areas for the 

nine RIM stations extends from 113 square miles (mi2) for 

the Choptank River to 27,100 mi2 for the Susquehanna River, 

yield (flux per unit of drainage area and expressed as pounds 

per day per square mile) is used instead of flux so that yield 

and changes in yield comparison can be made among all nine 

RIM stations. Plots of annual and flow-normalized flux, tables 

of changes in annual flux, and time series of annual and flownormalized flux for each of the nine RIM stations are provided 

in appendixes 2, 3, and 4, respectively.

Table 3. Total nitrogen trends for WRTDS flow-normalized yields and ESTIMATOR flow-adjusted concentration at the nine River Input 

Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010. 

[%, percent; %/yr, percent per year; WRTDS flow-normalized yield results are presented as both the total percent change in yield and average slope (percent 

change in yield per year); ESTIMATOR changes in flow-adjusted concentration are reported as a 95-percent confidence interval (CI) for the average change 

in concentration per year; in the two ESTIMATOR columns of the table, shaded cells are those where the ESTIMATOR flow-adjusted concentration trend is 

significant, those with no shading are not significant. Pink shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS flownormalized yield trend have the same sign; black text indicates that the WRTDS trend in flow-normalized yield falls within the 95-percent confidence interval 

for ESTIMATOR flow-adjusted concentration; red text indicates that the WRTDS trend in flow-normalized yield falls outside the 95-percent confidence interval for ESTIMATOR flow-adjusted concentration; all values are rounded to the nearest tenth]

RIM station

WRTDS flow-normalized yield ESTIMATOR flow-adjusted concentration

1985 to 2010 2001 to 2010 1985 to 2010 2001 to 2010

Total change

(%)

Slope 

(%/yr)

Total change

(%)

Slope

(%/yr)

Slope

(Range of 95% CI,

%/yr)

Slope

(Range of 95% CI,

%/yr)

Susquehanna –20.8 –0.8 –5.8 –0.6 –1.2 to – 0.8 –1.0 to 1.4

Potomac –14.3 –0.6 –3.6 –0.4 –1.1 to – 0.8 –2.0 to – 0.2

James –8.2 –0.3 6.5 0.7 –1.1 to – 0.3 –1.8 to 1.2

Rappahannock –8.5 –0.3 – 4.3 –0.5 –0.8 to 0.1 –3.0 to 0.9

Appomattox –3.5 –0.1 3.3 0.4 –0.1 to 0.6 –1.1 to 1.4

Pamunkey 2.8 0.1 4.9 0.5 0.4 to 1.2 –1.4 to 1.0

Mattaponi –4.2 –0.2 4.8 0.5 –0.3 to 0.3 –1.2 to 0.6

Patuxent –49.3 –2.0 –10.6 –1.2 –2.4 to – 2.2 –3.0 to – 1.4

Choptank 7.4 0.3 7.4 0.8 0.0 to 0.5 –0.5 to 1.6
36 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Comparison of ESTIMATOR and WRTDS 

Trend Results 

To effectively present the new WRTDS results, information is provided on how similar or different the WRTDS trends 

in flow-normalized flux are compared to the historical trends 

in flow-adjusted concentration, derived from ESTIMATOR, 

and explanation for major differences in trend (as determined 

by changes in trend direction) is also provided. The problem 

with comparing changes in flow-normalized flux (WRTDS) 

to changes in flow-adjusted concentration (ESTIMATOR) is 

related to their respective limitations. ESTIMATOR’s assumption of time-invariance in the relation between constituent 

concentration and discharge, time, and season inherent in 

the single model window approach is not appropriate for 

all constituents at the nine RIM stations (as documented 

earlier in the comparison of methods for the determination 

of flux). Also, the flow-adjusted concentration trend from 

ESTIMATOR may not be representative of the actual trends 

in flux. Two limitations with the WRTDS approach are the 

lack of measurements of uncertainty associated with trend (for 

example, 95-percent confidence interval) and the inability to 

assign significance to the trend (that is, determine if the trend 

is significantly different from zero).

ESTIMATOR flow-adjusted concentration and WRTDS 

flow-normalized flux trends were compared to answer two 

questions: (1) Does the WRTDS flow-normalized flux trend 

fall within the 95-percent confidence interval associated 

with the ESTIMATOR flow-adjusted concentration trend? 

and (2) Is the direction of change (that is, improving or 

degrading conditions) consistent between flow-normalized 

flux and flow-adjusted concentration for stations and 

constituents that have a significant (p-value < 0.05) flowadjusted concentration trend? For this comparison, there are 

85 constituent/station combinations that have trend information for the two time periods (the reason for 85 instead of 

90 combinations is that the five RIM stations in Virginia do 

not have trend information for suspended sediment for the 

period 1985 to 2010). For 55 of 85 possible combinations 

(64 percent), the WRTDS flow-normalized flux (in percent 

change per year) resides within the 95-percent confidence 

interval for the ESTIMATOR flow-adjusted concentration 

trend (in percent change per year) (bold black text, 

tables 3 –7). For 48 of the 85 combinations, the ESTIMATOR 

flow-adjusted concentration trend is significantly different 

from zero (shaded cells, tables 3–7) (historically, the magnitude 

of flow-adjusted concentration trends was reported only for 

combinations with significant trends; whereas, the magnitude 

was not reported for combinations with non-significant trends 

(Langland and others, 2006)). There is agreement between 

flow-normalized flux and flow-adjusted concentration, 

with respect to the direction of change (that is, improving 

or degrading conditions), at 38 of the 48 combinations 

(79 percent) with significant flow-adjusted concentration trends 

(pink shaded cells, tables 3–7). Ten of the 48 combinations 

(21 percent) have trend results where WRTDS flow-normalized 

flux trends are in an opposite direction when compared to 

ESTIMATOR flow-adjusted concentration trend (blue shaded 

cells, tables 3–7). So, the question becomes why are there 

cases of differences in the trend directions between WRTDS 

flow-normalized flux and ESTIMATOR flow-adjusted 

concentration? To answer this question, an investigation was 

conducted to determine how observed concentrations are 

changing (presumably as a result of management activities) 

within different portions of the complete range of hydrologic 

conditions. Three examples are provided to illustrate when 

(1) the direction of the flow-normalized flux and flow-adjusted 

concentration trends agree; (2) flow-normalized flux trends are 

positive (degrading conditions) and flow-adjusted concentration trends are negative (improving conditions); and (3) flownormalized flux trends are negative (improving conditions) 

and flow-adjusted concentration trends are positive (degrading 

conditions). For each of these examples, observed concentrations were separated into three discharge intervals: discharges 

less than or equal to the 60th percentile (base-flow conditions); 

discharges that are greater than the 60th percentile but less than 

the 90th percentile (intermediate discharges); and discharges 

greater than the 90th percentile (high discharges). The first 

example is for combinations where the trend in flow-adjusted 

concentration is in the same direction as the trend in flownormalized flux, which accounts for 38 of the 48 combinations 

that have significant ESTIMATOR flow-adjusted concentration 

trends. Figure 15 shows total nitrogen collected at the Susquehanna River RIM station and is an example of this category. 

The figure shows that total nitrogen concentrations collected 

during high (discharge greater 11.32 log units or 82,700 ft3/s) 

(fig. 15A), intermediate (discharge between 9.96 and 11.32 log 

units or 21,200 to 82,700 ft3/s) (fig. 15B), and low (less than 

9.96 log units or 21,200 ft3/s) (fig. 15C) discharges all exhibit 

patterns of decreasing concentrations from 1985 to 2010 (the 

red line is a loess smooth fit line to serve as a visual aid for 

detecting direction of change and not a function of either 

WRTDS or ESTIMATOR). The WRTDS flow-normalized 

flux for the period 1985 to 2010 is – 0.8 percent per year, 

and the 95-percent confidence interval for the flow-adjusted 

concentration trend is –1.2 to – 0.8 percent (table 3). Because 

the slopes in total nitrogen concentration are similar for all 

three discharge categories, the flow-normalized flux and 

flow-adjusted concentration trends are statistically indistinguishable. The second example shows total phosphorus 

collected at the James River RIM station and is representative 

of 7 of 48 combinations where the trend in WRTDS flownormalized flux is positive (degrading condition) and the 

trend in ESTIMATOR flow-adjusted concentration is negative 

(improving condition) (fig. 16). Figure 16C shows that total 

phosphorus concentrations associated with low-flow/base-flow 

conditions have decreased sharply during the period from 

1985 to 2010. This pattern also is evident but less pronounced 

for concentrations associated with intermediate discharges 

(fig. 16B); however, total phosphorus concentrations associated 

with high discharges (fig. 16A) show a gradual increase starting 

about 2000. The effect of these patterns is evident in the flux 
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 37

0

1

2

3

0

1

2

3

0

1

2

3

4

Figure 15. Plot for total nitrogen at the Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing observed 

concentration collected during (A) high (greater than the 90th percentile of discharge), (B) intermediate (greater than the 60th

and less than the 90th percentile of discharge), and (C) low (less than or equal to the 60th percentile of discharge) discharges. 

Red line represents the Loess smooth fit line used as a visual aid for changing concentrations and is not associated with either

WRTDS or ESTIMATOR. 

1985 1990 1995 2000 2005 2010

Year

Concentration, in milligrams per liter

A

B

C

Figure 15. Total nitrogen at the Susquehanna River at Conowingo, Maryland (USGS Station 

ID 01578310), showing observed concentration collected during (A) high (greater than the 

90th percentile of discharge), (B) intermediate (greater than the 60th and less than the 

90th percentile of discharge), and (C) low (less than or equal to the 60th percentile of discharge) 

discharges. Red line represents the Loess smooth fit line used as a visual aid for changing 

concentrations and not associated with either WRTDS or ESTIMATOR.
38 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

0

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 16. Plot for total phosphorus at the James River at Cartersville, Va. (USGS Station ID 02035000) showing observed 

concentration collected during (A) high (greater than the 90th percentile of discharge), (B) intermediate (greater than the 

60th and less than the 90th percentile of discharge), and (C) low (less than or equal to the 60th percentile of discharge) 

discharges. Red line represents the Loess smooth fit line used as a visual aid for changing concentrations and is not 

associated with either WRTDS or ESTIMATOR. 

1985 1990 1995 2000 2005 2010

Year

Concentration, in milligrams per liter

A

B

C

Figure 16. Total phosphorus at the James River at Cartersville, Maryland (USGS Station ID 

02035000), showing observed concentration collected during (A) high (greater than the 

90th percentile of discharge), (B) intermediate (greater than the 60th and less than the 

90th percentile of discharge), and (C) low (less than or equal to the 60th percentile of discharge) 

discharges. Red line represents the Loess smooth fit line used as a visual aid for changing 

concentrations and not associated with either WRTDS or ESTIMATOR.
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 39

and concentration trends where WRTDS flow-normalized 

flux trend shows an increase of 0.4 percent per year whereas 

the ESTIMATOR flow-adjusted concentration trend has a 

95-percent confidence interval showing a decrease of –2.7 to 

–2.2 percent per year (table 5). These trends and patterns in 

the observed concentration suggest that management efforts 

(primarily upgrades to municipal wastewater-treatment 

plants and the phosphorus detergent ban) to reduce total 

phosphorus in the James River watershed are most evident in 

concentrations associated with low discharges. Conversely, 

the increasing trend in flow-normalized flux and observed 

concentration associated with high discharges indicates that 

management activities have not had a net effect in reducing 

total phosphorus loading from nonpoint sources. The last 

example shows nitrate collected at the Pamunkey RIM station 

and is representative of 3 of 48 combinations where the trend 

in WRTDS flow-normalized flux is negative (improving 

condition) and the trend in ESTIMATOR flow-adjusted 

concentration is positive (degrading condition) (fig. 17). 

Figure 17C shows that nitrate concentrations associated with 

base-flow conditions increased in the overall period 1985 

to 2010 with the greatest rate of increase occurring between 

1985 and 1999. For intermediate flows, nitrate concentrations 

increased between 1985 and 2002 and decreased from 2003 to 

2010 (fig. 17B). Conversely, nitrate concentrations associated 

with high discharges show a pattern of gradually decreasing 

concentrations between 2002 and 2010 (fig. 17A). The effect 

of these patterns is evident in the flux and concentration trends 

where WRTDS flow-normalized flux trend shows a gradual 

decrease of –0.3 percent per year whereas the ESTIMATOR 

flow-adjusted concentration trend has a 95-percent 

confidence interval showing a more pronounced increase of 

0.8 to 2.1 percent per year (table 4). These three examples 

show that (1) ESTIMATOR flow-adjusted concentration 

trends are heavily influenced by the patterns in constituent 

concentration during low and intermediate discharges, whereas 

WRTDS flow normalized fluxes are heavily influenced by the 

patterns in constituent concentrations associated with high 

discharges; and (2) the WRTDS flow-normalized flux trends 

and the ESTIMATOR flow-adjusted concentration trends are 

correctly characterizing changes in water-quality conditions 

specific to changes in flux and concentration, respectively 

and there are times that trends in flux will be in the opposite 

direction of trends in concentration.

Table 4. Nitrate trends in WRTDS flow-normalized yields and ESTIMATOR flow-adjusted concentration at the nine River Input 

Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010. 

[%, percent; %/yr, percent per year; WRTDS flow-normalized yield results are presented as both the total percent change in yield and average slope (percent 

change in yield per year); ESTIMATOR changes in flow-adjusted concentration are reported as a 95-percent confidence interval (CI) for the average change 

in concentration per year; in the two ESTIMATOR columns of the table, shaded cells are those where the ESTIMATOR flow-adjusted concentration trend 

is significant, those with no shading are not significant. Pink shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS 

flow-normalized yield trend have the same sign; blue shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS flownormalized yield trend have opposite signs. Black text indicates that the WRTDS trend in flow-normalized yield falls within the 95-percent confidence interval 

for ESTIMATOR flow-adjusted concentration; red text indicates that the WRTDS trend in flow-normalized yield falls outside the 95-percent confidence 

interval for ESTIMATOR flow-adjusted concentration; using this coding scheme, the most substantial apparent contradictions are those in the blue shaded cells 

(significant ESTIMATOR trend and opposite signs to the trend directions); all values are rounded to the nearest tenth]

RIM station

WRTDS flow-normalized yield ESTIMATOR flow-adjusted concentration

1985 to 2010 2001 to 2010 1985 to 2010 2001 to 2010

Total change

(%)

Slope

(%/yr)

Total change

(%)

Slope

(%/yr)

Slope

(Range of 95% CI,

%/yr)

Slope

(Range of 95% CI,

%/yr)

Susquehanna –14.9 – 0.6 –11.5 –1.3 –0.9 to – 0.5 –1.2 to 1.6

Potomac –20.9 – 0.8 –18.9 –2.1 –1.5 to –1.0 –3.2 to –0.4

James –20.8 – 0.8 –7.3 – 0.8 –1.7 to – 0.7 –1.2 to 8.3

Rappahannock –18.6 – 0.7 –14.8 –1.6 –1.5 to 0.1 –2.6 to 10.0

Appomattox –13.2 – 0.5 –8.3 – 0.9 –1.2 to 0.3 –1.5 to 7.3

Pamunkey – 6.5 – 0.3 –16.5 –1.8 0.8 to 2.1 –2.0 to 1.5

Mattaponi – 6.3 – 0.3 –4.9 – 0.5 –0.3 to 1.2 –2.4 to 4.1

Patuxent – 49.1 –2.0 –13.3 –1.5 –2.4 to –2.2 –3.1 to –1.2

Choptank 33.0 1.3 9.8 1.1 1.0 to 1.9 –0.7 to 2.3
40 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

1985 1990 1995 2000 2005 2010

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 17. Plot for nitrate at the Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing observed 

concentration collected during (A) high (greater than the 90th percentile of discharge), (B) intermediate (greater than the 

60th and less than the 90th percentile of discharge), and (C) low (less than or equal to the 60th percentile of discharge) 

discharges. Red line represents the Loess smooth fit line used as a visual aid for changing concentrations and is not 

associated with either WRTDS or ESTIMATOR. 

Year

Concentration, in milligrams per liter

A

B

C

Figure 17. Nitrate at the Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), 

showing observed concentration collected during (A) high (greater than the 90th percentile of 

discharge), (B) intermediate (greater than the 60th and less than the 90th percentile of discharge), 

and (C) low (less than or equal to the 60th percentile of discharge) discharges. Red line represents 

the Loess smooth fit line used as a visual aid for changing concentrations and not associated with 

either WRTDS or ESTIMATOR.
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 41

Changes in Flow-Normalized Flux at 

Individual Stations

The focus for the remainder of this section will be to 

report changes in WRTDS flow-normalized flux for total 

nitrogen, nitrate, total phosphorus, orthophosphorus, and 

suspended sediment at the nine RIM stations for the periods 

1985 to 2010 and 2001 to 2010. 

Nitrogen

Annual average (black circles) and flow-normalized 

annual average (red line) total nitrogen yield for the nine RIM 

stations are presented in figure 18 for the period 1985 to 2010. 

Total nitrogen yield at the majority of the nine RIM stations 

commonly ranges between 5 and 20 pounds per day per square 

mile [(lb/d)/mi2). However, the Appomattox, Pamunkey, and 

Mattaponi RIM stations exhibit yields that are the lowest of 

the nine RIM stations and are routinely less than 5 (lb/day)/mi2

(fig. 18E, F, and G). The Pamunkey and Mattaponi Rivers are 

low-gradient and primarily forested watersheds that reside 

almost entirely in the Coastal Plain Physiographic Province 

(fig. 1). These basin characteristics function to limit total 

nitrogen transport by limiting contributing nitrogen sources 

and increasing residence time, which promotes greater 

rates of biological processing and uptake. Not only is the 

Appomattox River a primarily forested watershed, but more 

importantly, it is located directly downstream from the Lake 

Chesdin reservoir, which limits the downstream transport of 

total nitrogen because of impounded sediment. The extent to 

which annual yields have changed outside the year-to-year 

variations in discharge is provided in table 3 for two time 

periods, 1985 to 2010 and 2001 to 2010. This change in 

total nitrogen flux is reported, in table 3 (in the WRTDS 

columns), as a total change in percent and as a percentage 

change per year. The rate of change in total nitrogen flux 

(and all subsequent constituents) for each RIM station 

reported as tons per day per year is provided in Appendixes 

3–1 through 3–5. For the period 1985 to 2010, seven of the 

nine RIM stations had negative slopes, indicating a decrease 

in the annual delivery of total nitrogen to the Chesapeake 

Bay. The three stations with the greatest improvement in 

total nitrogen are the Patuxent (–2.0 percent per year, total 

reduction of –49.3 percent), Susquehanna (–0.8 percent 

per year, total reduction of –20.8 percent), and the Potomac 

(– 0.6 percent per year, total reduction of –14.3 percent). Only 

two stations had increasing yields over the 1985 to 2010 

period: the Choptank (0.3 percent per year; total increase of 

7.4 percent) and the Pamunkey (0.1 percent per year, total 

increase of 2.8 percent). For the period 2001 to 2010, only the 

Susquehanna (–0.6 percent per year), Potomac (–0.4 percent 

per year), Rappahannock (–0.5 percent per year) and Patuxent 

(–1.2 percent per year) stations showed a continued reduction 

(negative slopes); however, the rate of improvement has 

decreased at the Susquehanna, Potomac, and Patuxent stations 

compared to the rates during 1985 to 2010. The Rappahannock is the only station where the rate of improvement has 

increased. The James, Appomattox, and Mattaponi stations 

had negative slopes during 1985 to 2010 that have become 

positive for the period 2001 to 2010. The positive slope in 

total nitrogen yields at the Choptank (0.3 percent per year) 

and Pamunkey (0.1 percent per year) stations during 1985 to 

2010 steepened during the more recent period of 2001 to 2010 

to 0.8 and 0.5 percent per year, respectively (table 3). These 

results for total nitrogen yield tell two very different stories. 

Flow-normalized trends in total nitrogen yields indicate that 

for the long-term period all but two of the RIM stations have 

total nitrogen yields that are improving. Conversely, flownormalized trends in total nitrogen yield at eight of the nine 

RIM stations show that the rate of improvement has either 

slowed (three of eight RIM stations), changed from improving 

to degrading (three of eight RIM stations), or the rate of 

degradation has accelerated (two of the eight RIM stations) 

during the more recent 2001 to 2010 period.

Annual average and flow-normalized annual average 

nitrate yields for the nine RIM stations are presented in 

figure 19 for the period 1981 to 2010. The James, Rappahannock, Appomattox, Pamunkey, and Mattaponi stations have 

nitrate yields that are noticeably lower than the remaining four 

RIM stations and routinely are below 5 (lb/d)/mi2. Conversely, 

the Susquehanna, Potomac, Patuxent, and Choptank stations 

have the highest yields that commonly range between 

5 and 15 (lb/d)/mi2. Data in table 4 indicate that nitrate yields 

are decreasing (negative slopes) at all RIM stations except 

for the Choptank station for the period 1985 to 2010. These 

negative slopes range from –2.0 to –0.3 percent per year. The 

greatest reductions in nitrate yield occurred at the Patuxent 

(–2.0 per year, total reduction of 49.1 percent), Potomac 

(– 0.8 percent per year, total reduction of 20.9 percent), James 

(– 0.8 percent per year, total reduction of 20.8 percent), and 

Rappahannock (–0.7 percent per year, total reduction of 

18.6 percent). At the Susquehanna, Potomac, Rappahannock, 

Appomattox, Pamunkey, and Mattaponi stations, the negative 

slopes in nitrate yield observed during 1985 to 2010 steepened 

considerably (greater rates of improvement) during the period 

of 2001 to 2010 (table 4). The Choptank is the only RIM 

station that had a positive slope for annual-average yield; the 

positive slope for 1985 to 2010 was 1.3 percent per year (total 

increase of 33.0 percent), which is greater than the 1.0 percent 

per year slope observed during the 2001 to 2010. These results 

show that nitrate yields are improving at eight of the nine 

RIM stations and that during the 2001 to 2010 time period the 

rate of improvement has increased at seven of the nine RIM 

stations. The flow-normalized trend results for nitrate yield are 

considerably different than the trends for total nitrogen during 

the period 2001 to 2010 in that nitrate is showing patterns of 

increased improvement; whereas, trends in total nitrogen yield 

indicate that the rate of improvement is slowing or becoming 

increasingly degraded.
42 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Figure 18. WRTDS-derived total nitrogen annual yields (black circles) and flow-normalized yields (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. 

0

201510

5

3025

0

201510

5

3025

0

201510

5

3025

A. Susquehanna B. Potomac C. James

D. Rappahannock

Yield, in pounds per day per square mile E. Appomattox F. Pamunkey

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

G. Mattaponi H. Patuxent I. Choptank

Figure 18. Total nitrogen yields (black circles) and flow-normalized yields (red line) for the nine RIM stations using the WRTDS method.
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 43 Figure 19. WRTDS-derived nitrate annual yields (black circles) and flow-normalized yields (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. 

50

201510

50

201510

50

201510

A. Susquehanna B. Potomac C. James

D. Rappahannock

Yield, in pounds per day per square mile E. Appomattox F. Pamunkey

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

G. Mattaponi H. Patuxent I. Choptank

Figure 19. Nitrate yields (black circles) and flow-normalized yields (red line) for the nine RIM stations using the WRTDS method.
44 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Phosphorus

Annual average and flow-normalized annual average total 

phosphorus yields for the nine RIM stations are presented in 

figure 20 for the period 1981 to 2010. Similar to the patterns 

in total nitrogen yields, total phosphorus yields from the 

Appomattox, Pamunkey, and Mattaponi Rivers stand out as 

having the smallest yields, generally less than 0.5 (lb/d)/mi2. The 

Susquehanna River also stands out as having relatively low yields 

(fig. 20A) commonly less than 0.8 (lb/d)/mi2 that are probably 

a direct result of total phosphorus being trapped in the reservoir 

just upstream from the Susquehanna RIM station. Unlike the 

predominance of RIM stations with improving conditions for 

total nitrogen and nitrate, total phosphorus at seven of the RIM 

stations exhibits positive (degrading) slopes during 1985 to 2010, 

which all steepen during 2001 to 2010 (fig. 20; table 5). The two 

RIM stations with the steepest positive slopes are the Rappahannock (4.0 percent per year, total increase of 99.8 percent) and the 

Pamunkey (2.4 percent per year; total increase of 60.5 percent) 

(table 5). Five RIM stations (Susquehanna, James, Appomattox, 

Mattaponi, and Choptank) have positive total phosphorus slopes 

that range from 0.0 to 0.6 percent per year (total increase of 

1.3 and 16.1 percent, respectively) for the period 1985 to 

2010. Two of the nine RIM stations that have negative slopes 

for 1985 to 2010 are the Patuxent (–2.4 percent per year, total 

reduction of –59.7 percent) and the Potomac (–0.5 percent per 

year, total reduction of –12.4) (table 5). The marked improvement at the point-source dominated Patuxent RIM station can 

be attributed, in part, to the phosphorus-based detergent ban 

that occurred around 1980 and improvements in the municipal 

wastewater system. The rate of total phosphorus delivery to the 

Chesapeake Bay increased dramatically at seven of the nine RIM 

stations for the period 2001 to 2010 (table 5). The greatest rates 

of change of total phosphorus yield for the period 2001 to 2010 

occurred at the Rappahannock (6.9 percent per year), James 

(5.1 percent per year), Pamunkey (3.5 percent per year), Susquehanna (2.0 percent per year), Choptank (1.3 percent per year), and 

Appomattox (1.3 percent per year) Rivers (table 5). The Patuxent 

RIM station exhibits no measurable trend in total phosphorus 

yield for the period 2001 to 2010 compared to the considerable 

negative slope exhibited during 1985 to 2010. The Potomac 

RIM station (–0.6 percent per year) is the only one that has a 

negative slope for phosphorus yield for the period 2001 to 2010 

(table 5). The long-term (1985 to 2010) trends in flow normalized 

total phosphorus yield indicate that the rate of total phosphorus 

delivery is increasing (degrading conditions) and this rate of total 

phosphorus delivery is further increasing at eight of the nine RIM 

stations for the more recent 2001 to 2010 period. 

Annual average and flow-normalized annual average 

orthophosphorus yields for the nine RIM stations are presented 

in figure 21 for the period 1981 to 2010. Orthophosphorus 

is important because it is the form of phosphorus that is the 

most bioavailable; therefore, the mass of orthophosphorus 

delivered to the Chesapeake Bay is immediately available for 

biotic uptake and assimilation. From figure 21, four of the nine 

RIM stations (Potomac, James, Patuxent, and Choptank) stand 

out as having higher orthophosphorus yields routinely above 

0.1 (lb/d)/mi2. Eight of nine RIM stations have negative slopes 

for orthophosphorus for the period 1985 to 2010 (fig. 18) 

(table 6). These negative slopes range from –0.6 percent per 

year (total reduction of –14.2 percent) to –3.4 percent per year 

(total reduction of 86.2 percent). The Patuxent (–3.4 percent 

per year, total reduction of 84.6 percent) and the James 

(–3.4 percent per year, total reduction of 86.2 percent) exhibit 

the greatest reductions in orthophosphorus; these reductions 

can be attributed, in part, to the implementation of the 

phosphorus detergent ban in the early- to mid-1980s as well as 

improvements to municipal sewage treatment systems. 

Three additional RIM stations that have sizable reductions in orthophosphorus are the Mattaponi (–2.0 percent per 

year, total reduction of 51.2 percent), Potomac (–1.9 percent 

per year, total reduction of 48.0 percent), and the Pamunkey 

(–1.5 percent per year; total reduction of 37.7 percent). The 

Choptank is the only RIM station that has a positive slope 

(1.8 percent per year, total increase of 44.2 percent) associated 

with orthophosphorus yield for the period 1985 to 2010. For 

the period 2001 to 2010, five of the nine RIM stations show 

an increased rate of orthophosphorus reduction compared 

to rates of reduction associated with the period 1985 to 

2010; these stations are the James (–7.6 percent per year), 

Pamunkey (–5.1 percent per year), Mattaponi (–4.1 percent 

per year), Potomac (–3.9 percent per year) and Appomattox 

(–1.7 percent per year). The rates of orthophosphorus reduction have slowed at the Susquehanna, Rappahannock, and 

Patuxent RIM stations for the periods 2001 to 2010 compared 

to rates associated with 1985 to 2010 (table 6). The Choptank 

RIM station has greater rates of increasing orthophosphorus 

yields (3.6 percent per year) for the period 2001 to 2010 

compared to the rate (1.8 percent per year) associated with 

1985 to 2010. The results that have been presented related to 

changes in orthophosphorus yields over time show that efforts 

to control this constituent are working in eight of the nine 

watersheds. However, the results from the Choptank River, 

typically considered as a watershed with nitrogen-related 

and not phosphorus-related water-quality issues, show that 

the trends in total phosphorus and orthophosphorus yields 

are steadily increasing. This could be indicative of trends in 

other Eastern Shore tributaries, which have similar geology 

and land-use history. The results for flow-normalized trends 

in orthophosphorus are encouraging in that eight of nine RIM 

stations show orthophosphorus yields are improving over the 

long-term and short-term periods; however, these trend results 

for orthophosphorus are directly opposite of the trend results 

for total phosphorus flow-normalized yields where eight of 

the nine RIM stations show that the rate of total phosphorus 

delivery is increasing (becoming more degraded) during 2001 

to 2010 compared to 1985 to 2010. One hypothesis for the 

dichotomy between orthophosphorus and total phosphorus is 

that the improvements in flow-normalized orthophosphorus 

yields are related to phosphorus controls associated with point 

sources in many of the watersheds (industrial and municipal 

wastewater discharges); whereas, increases in total phosphorus 

flow-normalized yields are related to increased loading from 

nonpoint sources (for example, phosphorus-saturated soils). 
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 45 Figure 20. WRTDS-derived total phosphorus annual yields (black circles) and flow-normalized yields (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. 

432104321043210

A. Susquehanna B. Potomac C. James

D. Rappahannock

Yield, in pounds per day per square mile E. Appomattox F. Pamunkey

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

G. Mattaponi H. Patuxent I. Choptank

Figure 20. Total phosphorus yields (black circles) and flow-normalized yields (red line) for the nine RIM stations using the WRTDS method.
46 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends NOTE: Y-scale truncated 3-years missing 1982 = 1.72 1983 = 1.24 1984 = 1.03

0

0.70.60.50.40.30.20.1

0

0.70.60.50.40.30.20.1

0

0.70.60.50.40.30.20.1

A. Susquehanna B. Potomac C. James

D. Rappahannock

Yield, in pounds per day per square mile E. Appomattox F. Pamunkey

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

G. Mattaponi H. Patuxent I. Choptank

Figure 21. WRTDS-derived orthophosphorus annual yields (black circles) and flow-normalized yields (red line) at the (A) Susquehanna, (B) Potomac, (C) James, Figure 21. (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Orthophosphorus yields (black circles) and flow-normalized yields (red line) for the nine RIM stations using the WRTDS method.
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 47

Table 5. Total phosphorus trends in WRTDS flow-normalized yields and ESTIMATOR flow-adjusted concentration at the nine River 

Input Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010. 

[%, percent; %/yr, percent per year; WRTDS flow-normalized yield results are presented as both the total percent change in yield and average slope (percent 

change in yield per year); ESTIMATOR changes in flow-adjusted concentration are reported as a 95-percent confidence interval (CI) for the average change 

in concentration per year; in the two ESTIMATOR columns of the table, shaded cells are those where the ESTIMATOR flow-adjusted concentration trend 

is significant, those with no shading are not significant. Pink shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS 

flow-normalized yield trend have the same sign; blue shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS flow-normalized yield trend have opposite signs. Black text indicates that the WRTDS trend in flow-normalized yield falls within the 95-percent confidence interval for 

ESTIMATOR flow-adjusted concentration; red text indicates that the WRTDS trend in flow-normalized yield falls outside the 95-percent confidence interval 

for ESTIMATOR flow-adjusted concentration; using this coding scheme, the most substantial apparent contradictions are those in the blue shaded cells (significant ESTIMATOR trend and opposite signs to the trend directions); all values are rounded to the nearest tenth]

RIM station

WRTDS flow-normalized yield ESTIMATOR flow-adjusted concentration

1985 to 2010 2001 to 2010 1985 to 2010 2001 to 2010

Total change

(%)

Slope 

(%/yr)

Total change

(%)

Slope

 (%/yr)

Slope

(Range of 95% CI,

%/yr)

Slope

(Range of 95% CI,

%/yr)

Susquehanna 1.9 0.1 18.4 2.0 –0.8 to 0.2 –1.7 to 2.0

Potomac –12.4 – 0.5 –5.0 –0.6 –1.0 to 0.1 –6.0 to –3.4

James 9.4 0.4 46.0 5.1 –2.7 to –2.2 –7.6 to –5.1

Rappahannock 99.8 4.0 62.0 6.9 –1.1 to 0.3 –1.1 to 5.8

Appomattox 16.1 0.6 11.5 1.3 0.4 to 1.9 0.2 to 5.1

Pamunkey 60.5 2.4 31.2 3.5 2.8 to 5.3 –4.5 to –1.2

Mattaponi 1.2 0.0 3.6 0.4 –0.9 to 0.0 –2.1 to 0.7

Patuxent –59.7 –2.4 0.2 0.0 –2.6 to –2.1 –4.3 to –0.8

Choptank 14.7 0.6 11.6 1.3 1.1 to 2.8 –2.1 to 1.2

Table 6. Orthophosphorus trends in WRTDS flow-normalized yields and ESTIMATOR flow-adjusted concentration at the nine River 

Input Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010. 

[%, percent; %/yr, percent per year; WRTDS flow-normalized yield results are presented as both the total percent change in yield and average slope (percent 

change in yield per year); ESTIMATOR changes in flow-adjusted concentration are reported as a 95-percent confidence interval (CI) for the average change 

in concentration per year; in the two ESTIMATOR columns of the table, shaded cells are those where the ESTIMATOR flow-adjusted concentration trend 

is significant, those with no shading are not significant. Pink shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS 

flow-normalized yield trend have the same sign; blue shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS flownormalized yield trend have opposite signs. Black text indicates that the WRTDS trend in flow-normalized yield falls within the 95-percent confidence interval 

for ESTIMATOR flow-adjusted concentration; red text indicates that the WRTDS trend in flow-normalized yield falls outside the 95-percent confidence 

interval for ESTIMATOR flow-adjusted concentration; using this coding scheme, the most substantial apparent contradictions are those in the blue shaded cells 

(significant ESTIMATOR trend and opposite signs to the trend directions); all values are rounded to the nearest tenth]

RIM station

WRTDS flow-normalized yield ESTIMATOR flow-adjusted concentration

1985 to 2010 2001 to 2010 1985 to 2010 2001 to 2010

Total change

(%)

Slope

(%/yr)

Total change

(%)

Slope

(%/yr)

Slope

(Range of 95% CI, 

%/yr)

Slope

(Range of 95% CI, 

%/yr)

Susquehanna –17.1 – 0.7 –2.2 –0.2 0.6 to 3.6 –2.3 to 4.9

Potomac –48.0 –1.9 –34.8 –3.9 –2.2 to –1.4 –7.4 to –4.1

James –86.2 –3.4 – 68.5 –7.6 –3.6 to –3.4 –10.0 to –9.2

Rappahannock –14.2 – 0.6 –0.4 0.0 –1.0 to 0.4 –2.9 to 2.4

Appomattox –19.0 – 0.8 –15.1 –1.7 –1.4 to – 0.2 –1.5 to 3.6

Pamunkey –37.7 –1.5 – 46.3 –5.1 1.1 to 3.1 –8.3 to –7.0

Mattaponi –51.2 –2.0 –36.9 –4.1 –2.1 to –1.4 –5.7 to –3.2

Patuxent –84.6 –3.4 –27.6 –3.1 –3.2 to –3.0 –3.9 to –0.3

Choptank 44.2 1.8 32.5 3.6 0.5 to 2.6 3.3 to 9.8
48 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Suspended Sediment

Annual average and flow-normalized annual average 

suspended sediment yields for the nine RIM stations are presented 

in figure 22 for the period 1981 to 2010 or for 2001 to 2010. 

The five RIM stations in Virginia (Rappahannock, Mattaponi, 

Pamunkey, James, and Appomattox) have suspended sediment 

monitoring records that span 2001 to 2010. Before 2001, water 

samples collected at these sites were analyzed for total suspended 

solids (TSS). Analysis for TSS (which uses an aliquot of the 

original sample) results in suspended-sediment concentrations 

that are biased low; the more reliable laboratory analysis is 

suspended-sediment concentration (SSC), which analyzes the 

entire water sample (Gray and others, 2000). In 2000, the SSC 

analysis was added to the analytical suite at the five RIM stations 

in Virginia. Based on 10 years of paired SSC and TSS results, the 

USGS RIM team decided to only use SSC derived suspendedsediment data for the determination of flux at these five RIM 

stations. At the Susquehanna, Patuxent, Potomac, and Choptank 

RIM stations, SSC has always been the primary analysis by 

which suspended-sediment concentration is determined. 

Another feature that stands out in figure 22 is that four 

of the RIM stations have suspended sediment yields that 

are greater than 500 (lb/d)/mi2 for about 50 percent of the 

years, (Potomac, James, Rappahannock, and Patuxent); 

four other RIM stations have suspended sediment yields 

below 500 (lb/d)/mi2 virtually every year (Appomattox, 

Pamunkey, Mattaponi, and Choptank). The first group 

includes watersheds that drain extensive parts of the Piedmont 

Physiographic Province (fig. 1) and others areas of high 

topographic relief and (or) are areas of significant amounts of 

urban and agriculture land uses (Sprague and others, 2000). 

The latter group, except the Appomattox, almost entirely 

drains low-relief coastal plain areas and has very limited 

urban land use. The Appomattox River watershed is situated 

almost entirely within the Piedmont; however, Lake Chesdin, 

which is situated upstream from the Appomattox RIM station, 

serves as a sink for suspended sediment. The Susquehanna 

(fig. 22A) is unique because it exhibits two different sediment 

transport processes such that (1) during average discharge 

years, such as 2010, the reservoir upstream from the RIM 

station acts as a sink for suspended sediment, thus reducing 

the average annual yield, and in most of those years the yield 

is below 500 (lb/d)/mi2 and (2) during years with extreme 

high-flow events, such as 2004, the reservoir upstream from 

the RIM station acts as a source of suspended sediment and 

can greatly exceed this level (reaching a maximum of over 

2,500 mg/L in 2004 (Langland, 2009). From the standpoint 

of landscape features, the Susquehanna is much more like the 

Potomac and James Rivers, for example, but the presence of 

the reservoir gives it a very different pattern of variability. 

The extent to which annual yields have changed outside the 

year-to year variations in discharge is provided in table 7 for 

two time periods, 1985 to 2010 and 2001 to 2010. For the 

period 1985 to 2010, the two largest watersheds, Susquehanna 

and Potomac, have positive slopes associated with the 

flow-normalized yields of 3.5 percent per year (total increase 

of 86.5 percent) and 0.5 percent per year (total increase of 

12.2 percent), respectively (table 7). At the Patuxent and 

Choptank, suspended sediment yields have been decreasing at 

a rate of –0.8 and –0.9 percent per year, respectively (table 7). 

For the period 2001 to 2010, eight of the nine RIM stations 

exhibit patterns of increasing yields; although there is considerable variation, sediment yield is generally on the rise at 

these stations. Suspended sediment yield for the period 2001 

to 2010 has the greatest rate of change at the Potomac and 

Susquehanna RIM stations where sediment yield is increasing 

at a rate of 9.9 and 7.9 percent per year, respectively (table 7). 

It is reasonable to think that this increase in the Potomac 

may be related to land-use changes in the watershed, while 

in the case of the Susquehanna River, the increase may be 

related to the decrease in the ability of the reservoir to trap 

sediment, and increased propensity for reservoir scour may be 

the dominant factor.
Trends in Nutrient and Sediment Flux to the Chesapeake Bay 49

0

500

3,0002,5002,0001,5001,000

0

500

3,0002,5002,0001,5001,000

0

500

3,0002,5002,0001,5001,000

A. Susquehanna B. Potomac C. James

D. Rappahannock

Yield, in pounds per day per square mile E. Appomattox F. Pamunkey

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

G. Mattaponi H. Patuxent I. Choptank

Figure 22. WRTDS-derived suspended sediment annual yields (black circles) and flow-normalized yields (red line) at the (A) Susquehanna, (B) Potomac, (C) James, Figure 22. (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Suspended sediment yields (black circles) and flow-normalized yields (red line) for the nine RIM stations using the WRTDS method.
50 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Table 7. Suspended sediment trends in WRTDS flow-normalized yields and ESTIMATOR flow-adjusted concentration at the nine River 

Input Monitoring (RIM) stations for the time periods 1985 to 2010 and 2001 to 2010.

[%, percent; %/yr, percent per year; WRTDS flow-normalized yield results are presented as both the total percent change in yield and average slope (percent 

change in yield per year); ESTIMATOR changes in flow-adjusted concentration are reported as a 95-percent confidence interval (CI) for the average change 

in concentration per year; in the two ESTIMATOR columns of the table, shaded cells are those where the ESTIMATOR flow-adjusted concentration trend 

is significant, those with no shading are not significant. Pink shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS 

flow-normalized yield trend have the same sign; blue shaded cells indicate that the ESTIMATOR flow-adjusted concentration trend and the WRTDS flownormalized yield trend have opposite signs. Black text indicates that the WRTDS trend in flow-normalized yield falls within the 95-percent confidence interval 

for ESTIMATOR flow-adjusted concentration; red text indicates that the WRTDS trend in flow-normalized yield falls outside the 95-percent confidence 

interval for ESTIMATOR flow-adjusted concentration; using this coding scheme, the most substantial apparent contradictions are those in the blue shaded cells 

(significant ESTIMATOR trend and opposite signs to the trend directions); all values are rounded to the nearest tenth]

RIM station

WRTDS flow-normalized yield ESTIMATOR flow-adjusted concentration

1985 to 2010 2001 to 2010 1985 to 2010 2001 to 2010

Total change

(%)

Slope

(%/yr)

Total change

(%)

Slope

(%/yr)

Slope

(Range of 95% CI, 

%/yr)

Slope

(Range of 95% CI, 

%/yr)

Susquehanna 86.5 3.5 71.1 7.9 –1.1 to – 0.3 1.5 to 9.2

Potomac 12.2 0.5 89.1 9.9 –2.8 to –2.0 –4.4 to 4.0

James NA NA 14.3 1.6 NA –3.0 to 5.4

Rappahannock NA NA 2.4 0.3 NA –3.3 to 6.3

Appomattox NA NA 10.1 1.1 NA –1.4 to 5.0

Pamunkey NA NA 23.1 2.6 NA –0.6 to 12.6

Mattaponi NA NA –28.7 –3.2 NA –6.0 to –1.3

Patuxent –20.8 – 0.8 16.1 1.8 –2.1 to –1.4 –2.1 to 7.1

Choptank –22.5 – 0.9 26.0 2.9 –1.8 to – 0.8 2.0 to 12.7
Summary and Conclusions 51

Summary and Conclusions

The accurate determination of nutrient (nitrogen and 

phosphorus) and sediment fluxes and changes in fluxes over 

time are essential components that can be used by waterresource managers to manage and improve the structure and 

function of the Chesapeake Bay ecosystem. The U.S. Geological Survey (USGS) has been responsible for providing annual 

nutrient and sediment flux data and trends that describe the 

extent to which water-quality conditions are changing within 

the major Chesapeake Bay tributaries. For the investigation 

summarized herein, the USGS compared two multiple linear 

regression approaches for the determination of annual fluxes 

and changes in annual fluxes over time. The two regression 

approaches compared are the traditionally used ESTIMATOR 

and the newly developed Weighted Regression on Time, 

Discharge, and Season (WRTDS). The overall objective for 

this investigation was to evaluate each model’s performance 

in reproducing observed nutrient and sediment concentrations 

and fluxes and to determine which model ultimately ensures 

the highest level of accuracy in annual flux estimates provided 

to the Chesapeake Bay Program. The model comparison 

focused on three questions: (1) What are the differences 

between each model’s functional form and how each model is 

constructed? (2) Which model produces discrete estimates of 

flux with the greatest accuracy and least amount of bias? and 

(3) How different would the historical estimates of annual flux 

have been had WRTDS been used instead of ESTIMATOR? 

One additional point of comparison was an evaluation of each 

model’s ability to determine the changes in annual flux once 

the year-to-year variations in discharge have been accounted 

for. These results are displayed for total nitrogen, nitrate, 

total phosphorus, orthophosphorus, and suspended-sediment 

concentration data collected at the nine USGS operated River 

Input Monitoring (RIM) stations located on the Susquehanna, 

Potomac, James, Rappahannock, Appomattox, Pamunkey, 

Mattaponi, Patuxent, and Choptank Rivers.

The first comparison addressed the differences between 

each model’s functional form and construction. Both 

ESTIMATOR and WRTDS predict water-quality constituent 

concentrations on the basis of river discharge, time, and 

season. ESTIMATOR is composed of seven parameters 

(discharge, discharge-squared, time, time-squared, two 

seasonal terms, and a model constant); whereas WRTDS is 

composed of five parameters (discharge, time, two seasonal 

terms, and a model constant). The functional form associated 

with ESTIMATOR allows constituent concentration to vary 

linearly and quadratically over the full range of discharge and 

time. The WRTDS model is much more flexible in terms of 

functional form even though the fitting process for any given 

set of explanatory variables constrains these relations to be 

(locally) linear. The greatest control governing the flexibility 

of each model, however, is directly related to how each model 

is constructed. ESTIMATOR models are constructed for a 

series of 9-year moving windows that estimate concentration 

and flux for the middle (5th) year and step forward in time 

1 year at a time. New ESTIMATOR model coefficients are 

established for every 9-year window using all observed data 

for that window. Conversely, WRTDS models are constructed 

for each unique combination of discharge and time, and only 

observed data collected during similar discharges and times 

(determined by weighting water-quality observations based 

on distance of observation from modeled condition with 

respect to time, season, and discharge) are used to define the 

model coefficients. The use of model windows enable both 

models to account for changing relations between constituent 

concentration and flow and time better than if a single model 

was used for the entire period. The 9-year windows used for 

determining ESTIMATOR model coefficients is shown to have 

less flexibility relative to WRTDS. For example, ESTIMATOR 

accurately reproduces observed constituent concentration when 

the concentration-discharge relation is expressed as a linear 

or quadratic function (that is, zero or one point of inflection) 

or when deviations from a linear/quadratic form are well 

accounted for by other terms in the model such as time or 

season; otherwise, ESTIMATOR is unable to fully reproduce 

concentration-discharge relations that are more sinuous in form 

(that is, exhibit two points of inflection). WRTDS, on the other 

hand, has much greater flexibility because it only reproduces 

a single concentration for a given unique combination of 

discharge and time and only uses similar observations to define 

the model coefficients. Therefore, WRTDS has a greater ability, 

compared to ESTIMATOR, to reproduce complex observed 

concentration-discharge and (or) concentration-time relations.

 The second model comparison addressed which model 

produced discrete estimates of concentration and flux with the 

greatest accuracy and least amount of bias. This comparison 

was accomplished by comparing discrete water-quality observations to associated estimates derived from ESTIMATOR 

and WRTDS. There are 45 combinations (nine RIM stations 

times five water-quality constituents) for comparison. With 

regard to model accuracy, WRTDS produced model estimates 

of flux for all 45 possible combinations that were more 

accurate than the estimates of flux that were derived from 

ESTIMATOR. The analysis of flux bias associated with 

WRTDS and ESTIMATOR-derived estimates resulted in the 

development of three categories of combinations. Category I 

contains combinations where no distinguishable differences 

existed between the modeled results with regard to flux bias. 

Of the 45 combinations, 28 combinations (62 percent) exhibited no or minor differences in flux bias between WRTDS 

and ESTIMATOR-derived estimates. Category II contains 

combinations where major differences with regard to flux bias 

occurred between the modeled results. Fifteen (33 percent) of 

the 45 combinations had at least a 10 percent difference in the 

value of flux bias associated with WRTDS and ESTIMATOR 

estimates. In all 15 combinations, WRTDS produced estimates 

of flux that were considerably less biased than ESTIMATORderived fluxes. The source of the elevated flux bias associated 

with ESTIMATOR resides in the complexity of the observed 

concentration-discharge relation. For all 15 combinations, 

the observed concentration-discharge relation exhibited a 
52 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

more sinuous pattern with two points of inflection at which 

the slope of the relation changed. ESTIMATOR can only 

reproduce concentration-discharge relations with either no 

points or one point of inflection (that is, the relation is either 

linear or quadratic in form); whereas, WRTDS is structured 

such that any number of shifts in the observed concentrationdischarge relation can be reproduced. Category III contains two 

combinations (Susquehanna total phosphorus and Susquehanna 

suspended sediment) that cannot be categorized as either 

Category I or Category II. What makes these two combinations 

unique is that they are either particulate or particulatedominated constituents collected just downstream from the 

Conowingo reservoir. For these two cases, ESTIMATOR 

outperformed WRTDS in reproducing concentrations and 

fluxes associated with the majority of the range of discharge; 

WRTDS outperforms ESTIMATOR in reproducing concentrations and fluxes associated with the highest discharges 

(greater than the 95 percentile). For Susquehanna suspended 

sediment, WRTDS produces flux estimates that are positively 

biased (flux-bias ratio equals 1.06) but are closer to 1.0 than 

the flux-bias ratio associated with ESTIMATOR, which is 

negatively biased (0.80). For Susquehanna total phosphorus, 

ESTIMATOR produces flux estimates that are negatively 

biased (flux-bias ratio equals 0.96) but are closer to 1.0 than 

flux estimates associated with WRTDS, which are positively 

biased (1.06). The difficulty WRTDS and ESTIMATOR have 

in reproducing the full range of the concentration-discharge 

relation, for Susquehanna suspended sediment and total 

phosphorus, is directly related to each model’s functional 

form being inadequate in fully reproducing the transport 

processes (that is, shift from net depositional to net scour 

during extreme high discharges) governed by the Conowingo 

reservoir. Therefore, for the vast majority of combinations 

(44 of 45, 98 percent) WRTDS produces estimates of flux that 

are at worst equal and at best considerably less biased than 

estimates of flux derived from ESTIMATOR. 

The third model comparison addressed how different 

the historical estimates of annual flux would be had WRTDS 

been used instead of ESTIMATOR. This comparison was 

made using the annual 1985 to 2010 flux estimates derived 

from ESTIMATOR and WRTDS. For total nitrogen, WRTDS 

would have generated annual fluxes that were, on average, 

–5.13 to 1.02 percent different than the annual fluxes generated 

by ESTIMATOR. The greatest difference (–5.13 percent) 

occurred at the Rappahannock RIM station. For nitrate, WRTDS 

would have generated annual fluxes that were, on average, 

–28.13 to –1.61 percent different. The James (–13.55 percent) 

and Rappahannock (–28.13 percent) are the only RIM stations 

to yield percent differences greater than 10 percent; these 

differences are related to the inability of ESTIMATOR to 

reproduce the complex concentration-discharge relation. For 

total phosphorus, WRTDS would have generated annual fluxes 

that were on average –17.68 to 12.38 percent different than the 

annual fluxes generated by ESTIMATOR. The Susquehanna 

(12.38 percent) and Rappahannock (–17.68 percent) are the 

only RIM stations to yield percent differences greater than 

10 percent. The difference observed at the Susquehanna station 

is related to Category III discrepancies; while the difference 

observed at the Rappahannock is related to Category II discrepancies. For orthophosphorus, WRTDS would have generated 

annual fluxes that were, on average, –10.30 to 0.17 percent 

different than the annual fluxes generated by ESTIMATOR. 

The Patuxent (10.30 percent) is the only RIM station to yield 

a percent difference greater than 10 percent. This difference is 

related to Category II type discrepancies. For suspended sediment, WRTDS would have generated annual fluxes that were, on 

average, –39.50 to 38.33 percent different than the annual fluxes 

generated by ESTIMATOR. The Susquehanna (38.33 percent), 

Potomac (–19.34 percent), James (–19.20 percent), and Rappahannock (–39.50 percent) are the only RIM stations to yield 

percent differences greater than 10 percent. The difference 

in annual fluxes observed at the Susquehanna are directly 

related to Category III-type discrepancies; while differences at 

the Potomac, James, and Rappahannock RIM stations are all 

related to Category II-type discrepancies. 

The last point of comparison is each model’s ability to 

quantify the changes in annual flux once the year-to-year 

variations in discharge have been accounted for. ESTIMATOR 

is not able to quantify the changes in annual flux (as being 

unique from changes in concentration) once the year-to-year 

variations in flow are removed. ESTIMATOR has historically 

defined the changes in water-quality conditions as a trend 

in flow-adjusted concentration; however, this information is 

not completely indicative of how flux is changing over time 

because the functional form of ESTIMATOR assumes that the 

percentage changes in concentration for a given discharge and 

time of year are the same across all discharges and seasons. 

WRTDS allows a wide range of relations and thus is able 

to estimate flux changes that more closely match the kinds 

of changes actually occurring in the watershed. Conversely, 

WRTDS allows for the direct determination of changes in 

annual flux once year-to-year variations in discharge have 

been addressed (flow normalization). Temporal changes in 

flow-normalized flux can be attributed to changes in nutrient/

sediment sources and (or) land uses and (or) implementation 

of water-quality improvement strategies. Temporal changes 

in flow-normalized flux (presented as changes in yield) 

were presented for two time periods, 1985 to 2010 and 2001 

to 2010. Before the changes in WRTDS flow-normalized 

flux were presented, the WRTDS flow-normalized fluxes 

were compared to the flow-adjusted concentration trends 

produced by ESTIMATOR. There are 55 of 85 combinations 

(64 percent) where the estimated WRTDS flow-normalized 

trend falls within the 95-percent confidence interval associated with the ESTIMATOR flow-adjusted concentration 

trends (significant and non-significant trends). A total of 

48 of the 85 combinations have ESTIMATOR flow-adjusted 

concentration trends that are significantly different from 

zero. Of these 48 combinations, there are 38 combinations 

(79 percent) where the trend in WRTDS flow-normalized 

flux is in complete agreement, with respect to the direction 

of change (that is, improving or degrading conditions), with 
Acknowledgments 53

ESTIMATOR flow-adjusted concentration trends. There are 

only 10 of the 48 combinations (21 percent) where the direction of change differs between the trends in WRTDS flownormalized flux and ESTIMATOR flow-adjusted concentration, 

and these differences for the 10 combinations are related to real 

differences between changes in concentrations and fluxes. Once 

the trends in WRTDS flow-normalized flux were compared to 

ESTIMATOR flow-adjusted concentration trends, the trends 

in WRTDS flow-normalized fluxes were further explored for 

total nitrogen, nitrate, total phosphorus, orthophosphorus, and 

suspended sediment at the nine RIM stations. 

For total nitrogen, seven of nine RIM stations exhibited 

negative slopes (improving conditions, decreasing fluxes) in 

flow-normalized fluxes for the period 1985 to 2010; whereas

two RIM stations, Choptank and Pamunkey, exhibited positive 

slopes (degrading condition, increasing fluxes). For the period 

2001 to 2010, only four of the nine RIM stations exhibited 

negative slopes in flow-normalized fluxes, and of these four 

RIM stations, only the Rappahannock showed an increase in 

the rate of improvement compared to the rate during 1985 to 

2010. The Patuxent exhibited the greatest rates of improvement 

with a rate of –2.0 percent per year (total improvement of 

49.3 percent) during 1985 to 2010 and a rate of –1.2 percent 

per year during 2001 to 2010. Conversely, the Choptank exhibited the greatest rates of increase in flux with a rate of 0.3 percent 

per year (total degradation of 7.8 percent) during 1985 to 2010 

and a rate of 0.8 percent per year during 2001 to 2010. 

 For nitrate, eight of nine RIM stations exhibited negative 

slopes in flow-normalized fluxes for the period 1985 to 2010; 

whereas only the Choptank exhibited a positive slope. For the 

period 2001 to 2010, eight of the nine RIM stations exhibited 

negative slopes in flow-normalized fluxes, and of these eight 

RIM stations, six stations exhibited increases in the rate of 

improvement. The Pamunkey RIM station exhibited the greatest 

rate of increase from –0.3 percent per year for 1985 to 2010 to 

–1.8 percent per year for 2001 to 2010. The Choptank is the only 

RIM station that exhibited positive slopes for nitrate flownormalized fluxes with a rate of 1.3 percent per year during 1985 

to 2010 and a rate of 1.1 percent per year during 2001 to 2010. 

For total phosphorus, only two of nine RIM stations 

(Potomac and Patuxent) exhibited negative slopes in flownormalized fluxes for the period 1985 to 2010. For the period 

2001 to 2010, only one of the nine RIM stations (Potomac) 

exhibited negative slopes in flow-normalized fluxes. The rates 

of improvement in total phosphorus flow-normalized flux are 

–0.5 percent per year (total improvement of 13.0 percent) 

during 1985 to 2010 and –0.6 percent during 2001 to 2010. 

The Patuxent showed a considerable rate of improvement 

(–2.4 percent per year or total reduction of 59.7 percent) 

during 1985 to 2010; conversely, during 2001 to 2010 the 

Patuxent exhibited no detectable slope (0.0 percent per year). 

The Rappahannock had the greatest positive slope for both 

time periods with rates equal to 4.0 percent per year (total 

increase of 99.8 percent) during 1985 to 2001 and 6.9 percent 

per year during 2001 to 2010. 

For orthophosphorus, eight of nine RIM stations exhibited negative slopes in flow-normalized fluxes for the period 

1985 to 2010; whereas only the Choptank station exhibited a 

positive slope. For the period 2001 to 2010, eight of the nine 

RIM stations exhibited negative slopes in flow-normalized 

fluxes and of these eight RIM stations five exhibited increases 

in the rate of improvement. The James River exhibited the 

greatest change in the rate of improvement from –3.4 percent 

per year (total reduction of 86.2 percent) for 1985 to 2010 to 

–7.6 percent per year for 2001 to 2010. The Choptank is the 

only RIM station that exhibited positive slopes for orthophosphorus flow-normalized fluxes with a rate of 1.8 percent 

per year during 1985 to 2010 and a rate of 3.6 percent per year 

during 2001 to 2010. 

For suspended sediment, eight of nine RIM stations exhibit 

positive slopes in flow-normalized fluxes for the period 2001 

to 2010 with slopes ranging from 0.3 to 9.9 percent per year. 

The Potomac River exhibited the greatest rate of change from 

0.5 percent per year during 1985 to 2010 to 9.9 percent per year 

during 2001 to 2010. The Mattaponi River is the only RIM 

station that had a negative slope for flow normalized flux during 

2001 to 2010; the rate of decrease is –3.2 percent per year. 

The results of this investigation have shown that WRTDS 

is a viable alternative to ESTIMATOR for the determination 

of annual nutrient and sediment fluxes at the nine Chesapeake 

Bay watershed RIM stations. WRTDS produces discrete 

concentrations and fluxes that are generally more accurate 

than those generated using ESTIMATOR, especially when the 

observed concentration-discharge relation is more sinuous in 

form with two points of inflection. The single most beneficial 

aspect associated with using WRTDS, however, is the added 

ability to report changes in flow-normalized annual fluxes that 

have occurred as a result of human activities in the watershed. 

Information on the changes in flow-normalized flux improves 

the relevancy of the USGS RIM data to the Chesapeake Bay 

Program and allows water-resource managers to better understand the flux of nutrients and sediment to the Chesapeake Bay 

and to directly measure the efficacy of their efforts to limit the 

delivery of nutrients and sediment to the bay. 

Acknowledgments

The authors wish to thank the Virginia Department of 

Environmental Quality, Maryland Department of Natural 

Resources, and U.S. Environmental Protection Agency 

Chesapeake Bay Program whose continued support and 

collaboration made this report possible. The authors also thank 

Laura Medalie and Jeff Chanat of the USGS for providing 

thorough technical reviews. Many other individuals within the 

USGS and Chesapeake Bay Program contributed to the report 

through various discussions and input; their contributions are 

sincerely appreciated. 
54 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

References Cited

Aulenbach, B.T., Buxton, H.T., Battaglin, W.A., and 

Coupe, R.H., 2007, Streamflow and nutrient fluxes of the 

Mississippi-Atchafalaya River basin and subbasins for the 

period of record through 2005: U.S. Geological Survey 

Open-File Report 2007–1080, accessed September 17, 2012, 

at http://toxics.usgs.gov/pubs/of-2007-1080/index.html. 

Box, J.B., and Mossa, JoAnn, 1999, Sediment, land use, and 

freshwater mussels—Prospects and problems: Journal of 

the North American Benthological Society, v. 18, no. 1, 

p. 99–117.

Cairns, J., Jr., 1977, Aquatic ecosystem assimilative capacity: 

Fisheries, v. 2, no. 2, p. 5–7.

Chesapeake Bay Foundation, 2010, State of the Bay 2010: 

Accessed June 25, 2012, at http://www.cbf.org/document.

doc?id=596.

Chesapeake Bay Program, 2000, CHESAPEAKE 2000: 

Accessed May 30, 2012, at http://www.chesapeakebay.net/

documents/cbp_12081.pdf.

Cohn, T.A., 2005, Estimating contaminant loads in rivers: 

An application of adjusted maximum Likelihood to type 1 

censored data: Water Resources Research, v. 41, W07003, 

doi:10.1029/2004WR003833.

Cohn, T.A., Caulder, D.L., Gilroy, E.J., Zynjuk, L.D., and 

Summers, R.M., 1992, The validity of a simple log-linear 

model for estimating fluvial constituent loads—An empirical study involving nutrient loads entering Chesapeake Bay: 

Water Resources Research, v. 28, no. 9, p. 2353–2364.

Cohn, T.A., DeLong, L.L., Gilroy, E.J., Hirsch, R.M., and 

Wells, R.M., 1989, Estimating constituent loads: Water 

Resources Research, v. 25, no. 5, p. 937–942.

Cooper, S.R., and Brush, G.S., 1991, Long-term history of 

Chesapeake Bay anoxia: Science, v. 254, p. 992–996.

Dennison, W.C., Orth, R.J., Moore, K.A., Stevenson, J.C., 

Carter, Virginia, Kollar, Stan, Bergstrom, P.W., and Batiuk, 

R.A., 1993, Assessing water quality with submersed aquatic 

vegetation: BioScience, v. 43, no. 2, p. 86–94.

Duan, Naihua, 1983, Smearing estimate—A nonparametric retransformation method: Journal of the American 

Statistical Association, v. 78, p. 605–610.

Gilroy, E.J., Hirsch, R.M., Kirby, W.H., and Cohn, T.A., 

1990, Mean square error of regression-based constituent 

transport estimates: Water Resources Research, v. 26, no. 9, 

p. 2069–2077.

Gray, J.R., Glysson, G.D., Turcios, L.M., Schwarz, G.E., 

2000, Comparability of suspended-sediment concentration 

and total suspended solids data: U.S. Geological Survey 

Water-Resources Investigations Report 00–4191, 20 p.

Helsel, D.R., 2012, Statistics for censored environmental data 

using Minitab and R, (2d ed.): Hoboken, N.J., John Wiley & 

Sons, 324 p.

Hirsch, R.M., Moyer, D.L., and Archfield, S.A., 2010, 

Weighted regression on time, discharge, and season 

(WRTDS), with an application to Chesapeake Bay river 

inputs: Journal of American Water Resources Association, 

v. 46, no. 5, p. 857–1064. 

Judge, G.G., Griffiths, W.E., Hill, R.C., Lutkepohl, K., and 

Lee, T.C., 1985, Qualitative and limited dependent variable 

models: Chapter 18, in The Theory and Practice of Econometrics: New York, John Wiley & Sons, 1,019 p.

Langland, M.J., 2009, Bathymetry and sediment-storage 

capacity change in three reservoirs on the Lower Susquehanna River, 1996–2008: U.S. Geological Survey Scientific 

Investigations Report 2009–5110, 21 p.

Langland, M.J., Lietman, P.L., and Hoffman, S.A., 1995, Synthesis of nutrient and sediment data for watersheds within 

the Chesapeake Bay drainage basin: U.S. Geological Survey 

Water-Resources Investigations Report 95–4233, 121 p.

Langland, M.J., Raffensperger, J.P., Moyer, D.L., Landwehr, J.M.,

and Schwarz, G.E., 2006, Changes in streamflow and water 

quality in selected nontidal basins in the Chesapeake Bay 

watershed, 1985–2004: U.S. Geological Survey Scientific 

Investigations Report 2006–5178, 75 p.

Lenat, D.R., Penrose, D.L., and Eagleson, K.W., 1981, 

Variable effects of sediment addition on stream benthos: 

Hydrobiologia, v. 79, no. 2, p. 187–194.

Madsen, J.D., Chambers, P.A., James, W.F., Koch, E.W., and 

Westlake, D.F., 2001, The interaction between water movement, sediment dynamics, and submersed macrophytes: 

Hydrobiologia, v. 444, nos. 1–3, p. 61–84.

Medalie, L., Hirsch, R.M., and Archfield, S.A., 2012, Use of 

flow-normalization to evaluate nutrient concentration and 

flux changes in Lake Champlain tributaries, 1990–2009: 

Journal of Great Lakes Research, v. 38, supplement 1, 

p. 58–67.

Milly, P.C.D., Betancourt, J., Falkenmark, M., Hirsch, R.M., 

Kundzewicz, Z.W., Lettenmaier, D.P., and Stouffer, R.J., 

2008, Stationarity is dead: Whither water management?: 

Science, v. 319, no. 5863, p. 573–574.
References Cited 55

Nixon, S.W., 1987, Chesapeake Bay nutrient budgets—A 

reassessment: Biogeochemistry, v. 4, p. 77–90.

Officer, C.B., Biggs, R.B., Talf, J.L., Cronin, L.E., Tyler, 

M.A., and Boynton, W.R., 1984 Chesapeake Bay anoxia—

Origin, development, and significance: Science, v. 223, 

p. 22–27.

Schlesinger, W.H., 1997, Biogeochemistry—An analysis of 

global change: New York, Academic Press, 588 p.

Sprague, L.A., Hirsch, R.M., and Aulenbach, B.T., 2011, 

Nitrate in the Mississippi River and its tributaries, 1980 to 

2008: Are we making progress?: Environmental Science & 

Technology, v. 45, no. 17, p. 7209–7216.

Sprague, L.A., Langland, M.J., Yochum, S.E., Edwards, R.E., 

Blomquist, J.D., Phillips, S.W., Shenk, G.W., and Preston, S.D.,

2000, Factors affecting nutrient trends in major rivers of the 

Chesapeake Bay watershed: U.S. Geological Survey WaterResources Investigations Report 00– 4218, 98 p.

Tobin, James, 1958, Estimation relationships for limited 

dependent variables: Econometrica, v. 26, no. 1, p. 24 –36.

Tukey, J.W., 1977, Exploratory data analysis: Reading, Mass., 

Addison-Wesley, 688 p.

U.S. Environmental Protection Agency, 1983, Chesapeake 

Bay—A framework for action: Philadelphia, Pa., U.S. 

Environmental Protection Agency, Region 3, 186 p.

U.S. Environmental Protection Agency, 2010, Chesapeake Bay 

TMDL: Accessed June 25, 2012, at http://www.epa.gov/

chesapeakebaytmdl/.

Waters, T.F., 1995, Sediment in streams: Sources, biological 

effects, and control: Bethesda, Md., American Fisheries 

Society, 251 p. 

Yochum, S.E., 2000, A revised load estimation procedure for 

the Susquehanna, Potomac, Patuxent, and Choptank Rivers: 

U.S. Geological Survey Water-Resources Investigations 

Report 00–4156, 49 p.

Appendix 1. Total nitrogen, nitrate, total phosphorus, 

orthophosphorus, and suspended sediment collected at the 

nine River Input Monitoring stations in the Chesapeake Bay 

watershed showing (A) observed concentration (red dots) versus 

discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, 

(C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black 

dots) concentration versus discharge relation, and (E) residual 

(observed minus predicted) plot for WRTDS predictions.
58 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Appendix 1

7 8 9 10 11 12 13 14

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-1.5

-1

-0.5

0

0.5

1

1.5

7 8 9 10 11 12 13 14

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-1. Plots for total nitrogen at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–1. Total nitrogen at Susquehanna River at Conowingo, Maryland (USGS Station ID 01578310), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 59

7 8 9 10 11 12 13 14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-2. Plots for nitrate at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 2. Nitrate at Susquehanna River at Conowingo, Maryland (USGS Station ID 01578310), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
60 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

7 8 9 10 11 12 13 14

-6

-5

-4

-3

-2

-1

0

1

7 8 9 10 11 12 13 14

-6

-5

-4

-3

-2

-1

0

1

7 8 9 10 11 12 13 14

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

7 8 9 10 11 12 13 14

-6

-5

-4

-3

-2

-1

0

1

7 8 9 10 11 12 13 14

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-3. Plots for total phosphorus at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–3. Total phosphorus at Susquehanna River at Conowingo, Maryland (USGS Station ID 01578310), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 61

7 8 9 10 11 12 13 14

-8

-7

-6

-5

-4

-3

-2

-1

7 8 9 10 11 12 13 14

-8

-7

-6

-5

-4

-3

-2

-1

7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

7 8 9 10 11 12 13 14

-8

-7

-6

-5

-4

-3

-2

-1

7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-4. Plots for orthophosphorus at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 4. Orthophosphorus at Susquehanna River at Conowingo, Maryland (USGS Station ID 01578310), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
62 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

7 8 9 10 11 12 13 14

-1

0

1

2

3

4

5

6

7

8

9

7 8 9 10 11 12 13 14

-1

0

1

2

3

4

5

6

7

8

9

7 8 9 10 11 12 13 14

-3

-2

-1

0

1

2

3

7 8 9 10 11 12 13 14

-1

0

1

2

3

4

5

6

7

8

9

7 8 9 10 11 12 13 14

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-5. Plots for suspended sediment at Susquehanna River at Conowingo, Md. (USGS Station ID 01578310) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 5. Suspended sediment at Susquehanna River at Conowingo, Maryland (USGS Station ID 01578310), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 63

4 6 8 10 12 14

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

4 6 8 10 12 14

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-6. Plots for total nitrogen at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 6. Total nitrogen at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
64 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

4 6 8 10 12 14

-4

-3

-2

-1

0

1

2

4 6 8 10 12 14

-4

-3

-2

-1

0

1

2

4 6 8 10 12 14

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

4 6 8 10 12 14

-4

-3

-2

-1

0

1

2

4 6 8 10 12 14

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-7. Plots for nitrate at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 7. Nitrate at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 65

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

1

2

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

1

2

LOG CONCENTRATION, IN MG/L

4 6 8 10 12 14

-4

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

1

2

4 6 8 10 12 14

-4

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-8. Plots for total phosphorus at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 8. Total phosphorus at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
66 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

4 6 8 10 12 14

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

-6

-5

-4

-3

-2

-1

0

4 6 8 10 12 14

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-9. Plots for orthophosphorus at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 9. Orthophosphorus at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 67

4 6 8 10 12 14

-2

0

2

4

6

8

10

4 6 8 10 12 14

-2

0

2

4

6

8

10

4 6 8 10 12 14

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

4 6 8 10 12 14

-2

0

2

4

6

8

10

4 6 8 10 12 14

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

A.

B. C.

D. E.

Figure 1-10. Plots for suspended sediment at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580) showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration 

versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted 

(black circles) concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–10. Suspended sediment at Potomac River at Chain Bridge at Washington, D.C. (USGS Station ID 01646580), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
68 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-11. Plots for total nitrogen at James River at Cartersville, Va. (USGS Station ID 02035000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 11. Total nitrogen at James River at Cartersville, Virginia (USGS Station ID 02035000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 69

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-6

-5

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-6

-5

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-12. Plots for nitrate at James River at Cartersville, Va. (USGS Station ID 02035000) showing the (A) observed concentration (red dots) 

versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual 

(observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration versus 

discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–12. Nitrate at James River at Cartersville, Virginia (USGS Station ID 02035000), showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) 

concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, 

(D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and (E) residual 

(observed minus predicted) plot for WRTDS predictions. 
70 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

6 7 8 9 10 11 12 13

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-6

-5

-4

-3

-2

-1

0

1

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-13. Plots for total phosphorus at James River at Cartersville, Va. (USGS Station ID 02035000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–13. Total phosphorus at James River at Cartersville, Virginia (USGS Station ID 02035000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 71

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

6 7 8 9 10 11 12 13

-7

-6

-5

-4

-3

-2

-1

0

6 7 8 9 10 11 12 13

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-14. Plots for orthophosphorus at James River at Cartersville, Va. (USGS Station ID 02035000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–14. Orthophosphorus at James River at Cartersville, Virginia (USGS Station ID 02035000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
72 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

6 7 8 9 10 11 12 13

-2

-1

0

1

2

3

4

5

6

7

8

6 7 8 9 10 11 12 13

-2

-1

0

1

2

3

4

5

6

7

8

6 7 8 9 10 11 12 13

-3

-2

-1

0

1

2

3

4

6 7 8 9 10 11 12 13

-2

-1

0

1

2

3

4

5

6

7

8

6 7 8 9 10 11 12 13

-3

-2

-1

0

1

2

3

4

A.

B. C.

D. E.

Figure 1-15. Plots for suspended sediment at James River at Cartersville, Va. (USGS Station ID 02035000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–15. Suspended sediment at James River at Cartersville, Virginia (USGS Station ID 02035000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 73

2 4 6 8 10 12

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10 12

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10 12

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 4 6 8 10 12

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10 12

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-16. Plots for total nitrogen at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–16. Total nitrogen at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
74 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

2 4 6 8 10 12

-14

-12

-10

-8

-6

-4

-2

0

2

2 4 6 8 10 12

-14

-12

-10

-8

-6

-4

-2

0

2

2 4 6 8 10 12

-6

-4

-2

0

2

4

6

2 4 6 8 10 12

-14

-12

-10

-8

-6

-4

-2

0

2

2 4 6 8 10 12

-6

-4

-2

0

2

4

6

A.

B. C.

D. E.

Figure 1-17. Plots for nitrate at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–17. Nitrate at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 75

2 4 6 8 10 12

-6

-5

-4

-3

-2

-1

0

1

2

2 4 6 8 10 12

-6

-5

-4

-3

-2

-1

0

1

2

2 4 6 8 10 12

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10 12

-6

-5

-4

-3

-2

-1

0

1

2

2 4 6 8 10 12

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-18. Plots for total phosphorus at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–18. Total phosphorus at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
76 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

2 4 6 8 10 12

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10 12

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10 12

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10 12

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10 12

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-19. Plots for orthophosphorus at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–19. Orthophosphorus at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 77

2 4 6 8 10 12

-1

0

1

2

3

4

5

6

7

8

9

2 4 6 8 10 12

-1

0

1

2

3

4

5

6

7

8

9

2 4 6 8 10 12

-3

-2

-1

0

1

2

3

2 4 6 8 10 12

-1

0

1

2

3

4

5

6

7

8

9

2 4 6 8 10 12

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-20. Plots for suspended sediment at Rappahannock River near Fredericksburg, Va. (USGS Station ID 01668000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–20. Suspended sediment at Rappahannock River near Fredericksburg, Virginia (USGS Station ID 01668000), 

showing the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATORpredicted (black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for 

ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge 

relation, and (E) residual (observed minus predicted) plot for WRTDS predictions. 
78 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-21. Plots for total nitrogen at Appomattox River near Matoaca, Va. (USGS Station ID 02041650) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–21. Total nitrogen at Appomattox River near Matoaca, Virginia (USGS Station ID 02041650), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 79

2 4 6 8 10

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

2 4 6 8 10

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

2 4 6 8 10

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 4 6 8 10

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

2 4 6 8 10

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-22. Plots for nitrate at Appomattox River near Matoaca, Va. (USGS Station ID 02041650) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–22. Nitrate at Appomattox River near Matoaca, Virginia (USGS Station ID 02041650), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
80 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-23. Plots for total phosphorus at Appomattox River near Matoaca, Va. (USGS Station ID 02041650) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 23. Total phosphorus at Appomattox River near Matoaca, Virginia (USGS Station ID 02041650), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 81

2 4 6 8 10

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 4 6 8 10

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-24. Plots for orthophosphorus at Appomattox River near Matoaca, Va. (USGS Station ID 02041650) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–24. Orthophosphorus at Appomattox River near Matoaca, Virginia (USGS Station ID 02041650), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
82 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

2 4 6 8 10

-1

0

1

2

3

4

5

2 4 6 8 10

-1

0

1

2

3

4

5

2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 4 6 8 10

-1

0

1

2

3

4

5

2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-25. Plots for suspended sediment at Appomattox River near Matoaca, Va. (USGS Station ID 02041650) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 25. Suspended sediment at Appomattox River near Matoaca, Virginia (USGS Station ID 02041650), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 83

3 4 5 6 7 8 9 10 11

-1.5

-1

-0.5

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11

-1.5

-1

-0.5

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11

-1.5

-1

-0.5

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11

-1.5

-1

-0.5

0

0.5

1

1.5

3 4 5 6 7 8 9 10 11

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-26. Plots for total nitrogen at Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–26. Total nitrogen at Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
84 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

3 4 5 6 7 8 9 10 11

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

3 4 5 6 7 8 9 10 11

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

3 4 5 6 7 8 9 10 11

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

3 4 5 6 7 8 9 10 11

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

3 4 5 6 7 8 9 10 11

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

A.

B. C.

D. E.

Figure 1-27. Plots for nitrate at Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing the (A) observed concentration (red dots) 

versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual 

(observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration versus 

discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 27. Nitrate at Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) 

concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, 

(D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and (E) residual 

(observed minus predicted) plot for WRTDS predictions. 
Appendix 1 85

3 4 5 6 7 8 9 10 11

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

3 4 5 6 7 8 9 10 11

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

3 4 5 6 7 8 9 10 11

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10 11

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

3 4 5 6 7 8 9 10 11

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-28. Plots for total phosphorus at Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–28. Total phosphorus at Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
86 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

3 4 5 6 7 8 9 10 11

-6

-5

-4

-3

-2

-1

0

3 4 5 6 7 8 9 10 11

-6

-5

-4

-3

-2

-1

0

3 4 5 6 7 8 9 10 11

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3 4 5 6 7 8 9 10 11

-6

-5

-4

-3

-2

-1

0

3 4 5 6 7 8 9 10 11

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

A.

B. C.

D. E.

Figure 1-29. Plots for orthophosphorus at Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 29. Orthophosphorus at Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 87

3 4 5 6 7 8 9 10 11

-1

0

1

2

3

4

5

6

7

3 4 5 6 7 8 9 10 11

-1

0

1

2

3

4

5

6

7

3 4 5 6 7 8 9 10 11

-3

-2

-1

0

1

2

3

3 4 5 6 7 8 9 10 11

-1

0

1

2

3

4

5

6

7

3 4 5 6 7 8 9 10 11

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-30. Plots for suspended sediment at Pamunkey River near Hanover, Va. (USGS Station ID 01673000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–30. Suspended sediment at Pamunkey River near Hanover, Virginia (USGS Station ID 01673000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
88 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

-2 0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 0 2 4 6 8 10

-1.5

-1

-0.5

0

0.5

1

1.5

-2 0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 0 2 4 6 8 10

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-31. Plots for total nitrogen at Mattaponi River near Beulahville, Va. (USGS Station ID 01674500) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 31. Total nitrogen at Mattaponi River near Beulahville, Virginia (USGS Station ID 01674500), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 89

-2 0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

-2 0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

-2 0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

-2 0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

-2 0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-32. Plots for nitrate at Mattaponi River near Beulahville, Va. (USGS Station ID 01674500) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–32. Nitrate at Mattaponi River near Beulahville, Virginia (USGS Station ID 01674500), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
90 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

-2 0 2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-2 0 2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-2 0 2 4 6 8 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 0 2 4 6 8 10

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-2 0 2 4 6 8 10

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-33. Plots for total phosphorus at Mattaponi River near Beulahville, Va. (USGS Station ID 01674500) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 33. Total phosphorus at Mattaponi River near Beulahville, Virginia (USGS Station ID 01674500), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 91

-2 0 2 4 6 8 10

-7

-6

-5

-4

-3

-2

-1

-2 0 2 4 6 8 10

-7

-6

-5

-4

-3

-2

-1

-2 0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 0 2 4 6 8 10

-7

-6

-5

-4

-3

-2

-1

-2 0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

A.

B. C.

D. E.

Figure 1-34. Plots for orthophosphorus at Mattaponi River near Beulahville, Va. (USGS Station ID 01674500) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–34. Orthophosphorus at Mattaponi River near Beulahville, Virginia (USGS Station ID 01674500), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
92 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

-2 0 2 4 6 8 10

-2

-1

0

1

2

3

4

5

6

-2 0 2 4 6 8 10

-2

-1

0

1

2

3

4

5

6

-2 0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

-2 0 2 4 6 8 10

-2

-1

0

1

2

3

4

5

6

-2 0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-35. Plots for suspended sediment at Mattaponi River near Beulahville, Va. (USGS Station ID 01674500) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 35. Suspended sediment at Mattaponi River near Beulahville, Virginia (USGS Station ID 01674500), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 93

3 4 5 6 7 8 9 10

-0.5

0

0.5

1

1.5

2

2.5

3 4 5 6 7 8 9 10

-0.5

0

0.5

1

1.5

2

2.5

3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0

0.5

1

1.5

3 4 5 6 7 8 9 10

-0.5

0

0.5

1

1.5

2

2.5

3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0

0.5

1

1.5

A.

B. C.

D. E.

Figure 1-36. Plots for total nitrogen at Patuxent River near Bowie, Md.. (USGS Station ID 01594440) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–36. Total nitrogen at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
94 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10

-1.5

-1

-0.5

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

A.

B. C.

D. E.

Figure 1-37. Plots for nitrate at Patuxent River near Bowie, Md.. (USGS Station ID 01594440) showing the (A) observed concentration (red dots) 

versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) residual 

(observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration versus 

discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 37. Nitrate at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) 

concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, 

(D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and (E) residual 

(observed minus predicted) plot for WRTDS predictions. 
Appendix 1 95

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-4

-3

-2

-1

0

1

2

3

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-4

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-38. Plots for total phosphorus at Patuxent River near Bowie, Md.. (USGS Station ID 01594440) showing the (A) observed concentration

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1–38. Total phosphorus at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
96 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3 4 5 6 7 8 9 10

-6

-5

-4

-3

-2

-1

0

1

3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-39. Plots for orthophosphorus at Patuxent River near Bowie, Md.. (USGS Station ID 01594440) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 39. Orthophosphorus at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 97

3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

-3

-2

-1

0

1

2

3

4

5

3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8 9 10

-3

-2

-1

0

1

2

3

4

5

A.

B. C.

D. E.

Figure 1-40. Plots for suspended sediment at Patuxent River near Bowie, Md.. (USGS Station ID 01594440) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 40. Suspended sediment at Patuxent River near Bowie, Maryland (USGS Station ID 01594440), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
98 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

A.

B. C.

D. E.

Figure 1-41. Plots for total nitrogen at Choptank River near Greensboro, Md.. (USGS Station ID 01491000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 41. Total nitrogen at Choptank River near Greensboro, Maryland (USGS Station ID 01491000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 99

0 2 4 6 8 10

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

-5

-4

-3

-2

-1

0

1

2

0 2 4 6 8 10

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8 10

-5

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-42. Plots for nitrate at Choptank River near Greensboro, Md.. (USGS Station ID 01491000) showing the (A) observed concentration 

(red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge relation, (C) 

residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) concentration 

versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 42. Nitrate at Choptank River near Greensboro, Maryland (USGS Station ID 01491000), showing the 

(A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
100 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10

-4

-3

-2

-1

0

1

2

A.

B. C.

D. E.

Figure 1-43. Plots for total phosphorus at Choptank River near Greensboro, Md.. (USGS Station ID 01491000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 43. Total phosphorus at Choptank River near Greensboro, Maryland (USGS Station ID 01491000), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 1 101

0 2 4 6 8 10

-8

-7

-6

-5

-4

-3

-2

-1

0 2 4 6 8 10

-8

-7

-6

-5

-4

-3

-2

-1

0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

1

2

3

0 2 4 6 8 10

-8

-7

-6

-5

-4

-3

-2

-1

0 2 4 6 8 10

-6

-5

-4

-3

-2

-1

0

1

2

3

A.

B. C.

D. E.

Figure 1-44. Plots for orthophosphorus at Choptank River near Greensboro, Md.. (USGS Station ID 01491000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 44. Orthophosphorus at Choptank River near Greensboro, Maryland (USGS Station ID 01491000), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and 

(E) residual (observed minus predicted) plot for WRTDS predictions. 
102 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

0 2 4 6 8 10

-1

0

1

2

3

4

5

6

0 2 4 6 8 10

-1

0

1

2

3

4

5

6

0 2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

-1

0

1

2

3

4

5

6

0 2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

A.

B. C.

D. E.

Figure 1-45. Plots for suspended sediment at Choptank River near Greensboro, Md.. (USGS Station ID 01491000) showing the (A) observed 

concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted (black dots) concentration versus discharge 

relation, (C) residual (observed minus predicted) plot for ESTIMATOR predictions, (D) observed (red dots) and WRTDS-predicted (black circles) 

concentration versus discharge relation, and (E) residual (observed minus predicted) plot for WRTDS predictions.

LOG DISCHARGE, IN CUBIC FEET PER SECOND

LOG CONCENTRATION, IN MILLIGRAMS PER LITER 

LOG CONCENTRATION, IN MILLIGRAMS PER LITER

RESIDUAL LOG CONCENTRATION, IN MILLIGRAMS PER LITER

LOG DISCHARGE, IN CUBIC FEET PER SECOND LOG DISCHARGE, IN CUBIC FEET PER SECOND

Figure 1– 45. Suspended sediment at Choptank River near Greensboro, Maryland (USGS Station ID 01491000), showing 

the (A) observed concentration (red dots) versus discharge relation, (B) observed (red dots) and ESTIMATOR-predicted 

(black dots) concentration versus discharge relation, (C) residual (observed minus predicted) plot for ESTIMATOR 

predictions, (D) observed (red dots) and WRTDS-predicted (black dots) concentration versus discharge relation, and (E)

residual (observed minus predicted) plot for WRTDS predictions. 
Appendix 2. WRTDS-derived annual and flow-normalized annual 

fluxes for total nitrogen, nitrate, total phosphorus, orthophosphorus, 

and suspended sediment at the nine Chesapeake Bay River Input 

Monitoring stations
104 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Appendix 21985 1990 1995 2000 2005 2010 50

400350300250200150100

A. SUS

1985 1990 1995 2000 2005 2010 0

50

200150100

B. POT

1985 1990 1995 2000 2005 2010 0

40302010

C. JAM

1985 1990 1995 2000 2005 2010

50

201510

D. RAP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

6543210

E. APP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

543210

F. PAM

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

0.5

1

1.5 G. MAT

1985 1990 1995 2000 2005 2010

543210

H. PAT

1985 1990 1995 2000 2005 2010 0

0.5

1

1.5 I. CHO

Figure 2-1. WRTDS-derived total nitrogen annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Figure 2–1. WRTDS-derived total nitrogen annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River Input Monitoring stations.
Appendix 2 105

1985 1990 1995 2000 2005 2010

50

250200150100

A. SUS

1985 1990 1995 2000 2005 2010

0

80604020

120100

B. POT

1985 1990 1995 2000 2005 2010

86420

1210

C. JAM

1985 1990 1995 2000 2005 2010

86420

10 D. RAP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0

0.80.60.40.2

1

1.41.2

E. APP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0

0.80.60.40.2

1

1.41.2

F. PAM

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0

0.40.30.20.1

G. MAT

1985 1990 1995 2000 2005 2010

0.5

1

1.5

2

2.5

H. PAT

1985 1990 1995 2000 2005 2010

0

0.80.60.40.2

1 I. CHO

Figure 2-2. WRTDS-derived nitrate annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Figure 2–2. WRTDS-derived nitrate annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River Input Monitoring stations.
106 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends 1985 1990 1995 2000 2005 2010

50

25201510

A. SUS

1985 1990 1995 2000 2005 2010

50

201510

B. POT

1985 1990 1995 2000 2005 2010

86420

C. JAM

1985 1990 1995 2000 2005 2010 0

0.5

1

1.5

2

2.5 D. RAP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

0.60.50.40.30.20.1

E. APP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

0.60.50.40.30.20.1

F. PAM

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

0.080.060.040.02

0.1

0.140.12

G. MAT

1985 1990 1995 2000 2005 2010 0

0.60.50.40.30.20.1

H. PAT

1985 1990 1995 2000 2005 2010 0

0.080.060.040.02

0.1

0.12 I. CHO

Figure 2-3. WRTDS-derived total phosphorus annual fluxess (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Figure 2–3. WRTDS-derived total phosphorus annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River Input Monitoring stations.
Appendix 2 107

1985 1990 1995 2000 2005 2010

0.5

1

1.5

2

2.5

A. SUS

1985 1990 1995 2000 2005 2010

0

0.5

1

1.5

2

2.5

3 B. POT

1985 1990 1995 2000 2005 2010

0

2.01.51.00.5

C. JAM

1985 1990 1995 2000 2005 2010

0

0.05

0.1

0.15

0.2

0.25 D. RAP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0

0.080.060.040.02

0.1 E. APP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0.090.080.070.060.050.040.030.02

F. PAM

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010

0

0.050.040.030.020.01

G. MAT

1985 1990 1995 2000 2005 2010

0

0.080.060.040.02

0.1

0.12

H. PAT

1985 1990 1995 2000 2005 2010

0

0.005

0.01

0.015

0.02

0.025

0.03 I. CHO

Figure 2-4. WRTDS-derived orthophosphorus annual fluxess (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Figure 2–4. WRTDS-derived orthophosphorus annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River Input Monitoring stations.
108 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends 1985 1990 1995 2000 2005 2010 0 5,000 20,000 15,000 10,000 A. SUS 1985 1990 1995 2000 2005 2010 0 5,000 20,000 15,000 10,000 B. POT 1985 1990 1995 2000 2005 2010 0 8,000 6,000 4,000 2,000 C. JAM 1985 1990 1995 2000 2005 2010 0 500 3,000 2,500 2,000 1,500 1,000 D. RAP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

50

200150100

E. APP

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

50

200150100

F. PAM

FLUX, IN TONS/DAY

1985 1990 1995 2000 2005 2010 0

40302010

G. MAT

1985 1990 1995 2000 2005 2010 0

50

200150100

H. PAT

1985 1990 1995 2000 2005 2010

50

25201510

I. CHO

Figure 2-5. WRTDS-derived suspended sediment annual fluxess (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, (D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River-Input Monitoring stations. Figure 2–5. WRTDS-derived suspended sediment annual fluxes (black circles) and flow-normalized fluxes (red line) at the (A) Susquehanna, (B) Potomac, (C) James, 

(D) Rappahannock, (E) Appomattox, (F) Pamunkey, (G) Mattaponi, (H) Patuxent, (I) Choptank River Input Monitoring stations.
Appendix 3. Rates of change in WRTDS flow-normalized flux for 

total nitrogen, nitrate, total phosphorus, orthophosphorus, and 

suspended sediment at the nine Chesapeake Bay River Input 

Monitoring stations
110 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Appendix 3

Table 3–1. Total nitrogen trend slopes for WRTDS flow-normalized fluxes at the nine River Input Monitoring 

(RIM) stations for the time periods 1985 to 2010 and 2001 to 2010. 

[ton/d/yr, tons per day per year; %/yr, percent per year] 

RIM station 

WRTDS flow-normalized flux

1985 to 2010 2001 to 2010

Slope×10 –2

(ton/d/yr)

Slope 

(%/yr)

Slope×10 –2

(ton/d/yr)

Slope

(%/yr)

Susquehanna –171.79 –0.8 –111.98 –0.6

Potomac –44.90 –0.6 –27.95 –0.4

James –6.03 –0.3 11.44 0.7

Rappahannock –2.19 –0.3 –2.95 –0.5

Appomattox –0.32 –0.1 0.79 0.4

Pamunkey 0.24 0.1 1.14 0.5

Mattaponi –0.16 –0.2 0.46 0.5

Patuxent –6.62 –2.0 –2.24 –1.2

Choptank 0.20 0.3 0.55 0.8

Table 3–2. Nitrate trend slopes for WRTDS flow-normalized fluxes at the nine River Input Monitoring (RIM) 

stations for the time periods 1985 to 2010 and 2001 to 2010. 

[ton/d/yr, tons per day per year; %/yr, percent per year] 

RIM station

WRTDS flow-normalized flux

1985 to 2010 2001 to 2010

Slope×10 –2

(ton/d/yr)

Slope 

(%/yr)

Slope×10 –2

(ton/d/yr)

Slope

 (%/yr)

Susquehanna –79.13 – 0.6 –162.80 –1.3

Potomac –37.47 – 0.8 –91.98 –2.1

James –4.71 – 0.8 –3.96 –0.8

Rappahannock –2.04 – 0.7 – 4.31 –1.6

Appomattox –0.31 – 0.5 – 0.52 –0.9

Pamunkey –0.16 – 0.3 –1.28 –1.8

Mattaponi –0.05 – 0.3 – 0.11 –0.5

Patuxent –3.98 –2.0 –1.75 –1.5

Choptank 0.46 1.3 0.46 1.1
Appendix 3 111

Table 3–3. Total phosphorus trend slopes for WRTDS flow-normalized fluxes at the nine River Input Monitoring 

stations for the time periods 1985 to 2010 and 2001 to 2010. 

[ton/d/yr, tons per day per year; %/yr, percent per year] 

RIM station

WRTDS flow-normalized flux

1985 to 2010 2001 to 2010

Slope×10 –3

(ton/d/yr)

Slope 

(%/yr)

Slope×10 –3

(ton/d/yr)

Slope

 (%/yr)

Susquehanna 6.10 0.1 143.33 2.0

Potomac –34.71 – 0.5 –36.02 –0.6

James 15.43 0.4 157.02 5.1

Rappahannock 30.30 4.0 64.50 6.9

Appomattox 1.30 0.6 2.68 1.3

Pamunkey 4.92 2.4 8.62 3.5

Mattaponi 0.04 0.0 0.32 0.4

Patuxent –8.75 –2.4 0.04 0.0

Choptank 0.24 0.6 0.54 1.3

Table 3–4. Orthophosphorus trend slopes for WRTDS flow-normalized fluxes at the nine River Input Monitoring 

stations for the time periods 1985 to 2010 and 2001 to 2010. 

[ton/y/yr, tons per day per year; %/yr, percent per year] 

RIM station

WRTDS flow-normalized flux

1985 to 2010 2001 to 2010

Slope×10 – 3

(ton/d/yr)

Slope 

(%/yr)

Slope×10 –3

(ton/d/yr)

Slope

 (%/yr)

Susquehanna –9.00 – 0.7 –2.72 –0.2

Potomac –25.76 –1.9 –41.47 –3.9

James – 43.44 –3.4 –42.03 –7.6

Rappahannock – 0.58 – 0.6 –0.04 0.0

Appomattox – 0.29 – 0.8 –0.61 –1.7

Pamunkey – 0.77 –1.5 –3.04 –5.1

Mattaponi – 0.44 –2.0 –0.68 –4.1

Patuxent –3.82 –3.4 –0.74 –3.1

Choptank 0.17 1.8 0.38 3.6
112 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends

Table 3–5. Suspended sediment trend slopes for flow-normalized at the nine River Input Monitoring stations for 

the time periods 1985 to 2010 and 2001 to 2010.

[ton/d/yr, tons per day per year; %/yr, percent per year; NA, not available] 

RIM station

WRTDS flow-normalized yield

1985 to 2010 2001 to 2010

Slope

(ton/d/yr)

Slope 

(%/yr)

Slope

(ton/d/yr)

Slope

 (%/yr)

Susquehanna 143.11 3.3 356.21 7.9

Potomac 33.16 0.5 400.12 9.9

James NA NA 32.39 1.6

Rappahannock NA NA 2.48 0.3

Appomattox NA NA 0.59 1.1

Pamunkey NA NA 3.68 2.6

Mattaponi NA NA –0.81 –3.2

Patuxent – 0.82 –0.8 1.20 1.8

Choptank – 0.09 –0.9 0.18 2.9
Appendix 4. Time series of WRTDS annual and flow-normalized 

annual fluxes of total nitrogen, nitrate, total phosphorus, 

orthophosphorus, and suspended sediment at the nine River 

Input Monitoring stations in the Chesapeake Bay watershed
114 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends Table 4–1. Total nitrogen annual flux and flow-normalized annual flux, in tons per day, for the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS. [SUS, Susquehanna River at Conowingo, Md.; POT, Potomac River at Chain Bridge at Washington, D.C.; JAM, James River at Cartersville, Va.; RAP, Rappahannock River near Fredericksburg, Va.; APP, Appomattox River at Matoaca, Va.; PAM, Pamunkey River near Hanover, Va.; MAT, Mattaponi River near Beulahville, Va.; PAT, Patuxent River near Bowie, Md.; CHO, Choptank River near Greensboro, Md.; FN flux, flow-normalized flux] Year SUS POT JAM RAP APP PAM MAT PAT CHO Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux 1981 131.77 199.25 30.98 74.27 5.72 19.23 1.39 6.42 0.78 2.30 0.47 2.18 0.32 0.92 1.98 3.10 0.34 0.62 1982 190.46 202.72 62.34 76.50 17.15 18.89 4.75 6.50 1.97 2.31 1.63 2.16 0.64 0.94 2.34 3.18 0.49 0.63 1983 177.20 204.76 73.40 77.96 19.53 18.63 7.00 6.47 2.75 2.31 2.14 2.14 1.16 0.95 3.25 3.26 0.77 0.64 1984 279.39 205.62 124.82 78.58 28.30 18.48 11.65 6.47 3.52 2.32 3.75 2.14 1.67 0.95 4.15 3.32 0.93 0.65 1985 140.84 206.92 45.70 78.26 11.44 18.31 3.01 6.45 1.37 2.33 1.59 2.15 0.69 0.95 2.37 3.36 0.24 0.67 1986 212.21 206.66 88.96 77.51 19.78 18.12 5.39 6.49 2.14 2.34 2.00 2.15 0.86 0.95 2.24 3.37 0.45 0.68 1987 188.83 205.86 65.89 76.20 24.72 17.83 5.12 6.55 2.52 2.35 2.25 2.15 1.01 0.94 2.75 3.32 0.59 0.69 1988 154.34 203.79 55.04 75.57 8.18 17.46 4.02 6.59 1.47 2.35 1.66 2.12 0.75 0.94 2.87 3.19 0.31 0.69 1989 187.56 201.66 60.42 75.14 17.64 16.98 5.19 6.47 2.31 2.33 2.09 2.09 0.91 0.93 3.39 3.02 0.93 0.68 1990 186.31 197.76 47.36 74.29 18.64 16.60 5.15 6.34 2.11 2.30 2.41 2.05 1.13 0.92 2.75 2.84 0.67 0.68 1991 216.27 193.80 73.42 73.32 18.78 16.27 5.38 6.19 1.37 2.26 1.26 2.02 0.46 0.90 2.31 2.67 0.45 0.67 1992 137.13 189.70 39.85 72.28 12.70 15.98 3.77 6.05 1.07 2.22 0.89 1.99 0.44 0.88 1.74 2.51 0.35 0.66 1993 248.73 186.64 102.62 71.41 20.65 15.69 9.77 5.89 2.83 2.19 2.67 1.99 1.13 0.87 2.69 2.38 0.61 0.65 1994 251.60 182.64 111.48 70.81 20.06 15.56 8.71 5.84 2.93 2.17 3.39 1.98 1.27 0.85 2.77 2.27 0.85 0.65 1995 128.79 179.38 42.98 70.48 13.67 15.50 5.35 5.84 1.48 2.16 1.39 1.99 0.49 0.85 1.47 2.18 0.39 0.65 1996 238.82 176.82 153.66 70.41 25.26 15.53 11.54 5.89 2.55 2.16 2.94 2.01 1.14 0.84 2.99 2.10 1.00 0.65 1997 196.60 175.97 87.41 70.58 16.29 15.53 6.30 5.90 2.45 2.16 2.76 2.04 1.14 0.85 2.80 2.04 0.92 0.65 1998 207.46 174.71 125.21 70.58 26.86 15.61 10.39 5.98 3.23 2.17 3.58 2.06 1.35 0.85 2.35 1.99 0.78 0.65 1999 99.87 174.26 24.01 70.41 5.68 15.65 1.61 6.03 1.10 2.17 0.76 2.08 0.32 0.86 1.29 1.95 0.44 0.65 2000 149.39 173.78 41.51 69.96 9.24 15.74 3.14 6.12 1.49 2.17 1.33 2.09 0.57 0.86 1.75 1.92 0.76 0.66 2001 101.94 174.05 37.14 69.56 5.89 15.77 3.20 6.17 1.16 2.18 1.03 2.11 0.44 0.87 1.64 1.90 0.80 0.67 2002 114.10 173.61 16.76 69.17 2.58 15.96 0.76 6.29 0.37 2.19 0.27 2.12 0.11 0.88 0.92 1.88 0.20 0.67 2003 214.19 173.68 138.63 68.92 32.72 16.14 13.58 6.38 5.00 2.21 3.72 2.13 1.35 0.89 2.95 1.87 1.41 0.68 2004 382.52 173.70 104.48 68.79 19.87 16.35 9.58 6.47 3.29 2.23 3.11 2.15 1.41 0.90 2.80 1.85 0.88 0.69 2005 235.74 174.23 67.18 68.99 13.82 16.43 6.60 6.47 2.05 2.25 2.03 2.16 0.95 0.92 1.93 1.84 0.66 0.69 2006 211.69 173.45 43.92 69.23 10.25 16.52 4.69 6.42 1.06 2.25 1.29 2.16 0.73 0.93 1.86 1.82 0.63 0.70 2007 174.47 172.74 56.36 68.98 16.33 16.47 5.13 6.27 2.65 2.25 2.31 2.16 1.15 0.93 1.68 1.79 0.75 0.71

2008 175.29 170.65 52.91 68.07 5.32 16.41 3.63 6.11 1.10 2.24 0.94 2.15 0.60 0.92 1.47 1.76 0.45 0.712009 126.04 168.80 42.60 67.27 7.36 16.43 3.17 5.96 1.23 2.24 1.03 2.16 0.55 0.92 1.45 1.73 0.64 0.712010 140.50 163.97 78.96 67.04 20.81 16.80 6.89 5.90 2.91 2.25 2.44 2.21 1.27 0.91 2.20 1.70 0.34 0.62

Appendix 4
Appendix 4 115

Table 4–2. Nitrate annual flux and flow-normalized annual flux, in tons per day, for the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS.[SUS, Susquehanna River at Conowingo, Md.; POT, Potomac River at Chain Bridge at Washington, D.C.; JAM, James River at Cartersville, Va.; RAP, Rappahannock River near Fredericksburg, Va.; APP, Appomattox River at Matoaca, Va.; PAM, Pamunkey River near Hanover, Va.; MAT, Mattaponi River near Beulahville, Va.; PAT, Patuxent River near Bowie, Md.; CHO, Choptank River near Greensboro, Md.; FN flux, flow-normalized flux] 

Year

SUS POT JAM RAP APP PAM MAT PAT CHO

Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux

1981 81.60 124.29 19.95 43.01 2.22 5.93 0.79 2.84 0.22 0.54 0.18 0.75 0.08 0.19 1.22 1.85 0.20 0.331982 118.51 126.10 37.81 43.37 5.52 5.85 2.26 2.83 0.50 0.56 0.56 0.71 0.15 0.19 1.46 1.91 0.30 0.331983 110.35 128.05 42.56 43.85 6.04 5.80 2.89 2.80 0.75 0.57 0.69 0.67 0.24 0.20 1.99 1.96 0.40 0.331984 173.42 129.88 67.96 44.29 8.47 5.73 4.82 2.77 0.94 0.58 1.09 0.64 0.33 0.21 2.47 2.00 0.48 0.341985 91.78 132.56 30.19 44.85 4.07 5.67 1.55 2.75 0.39 0.59 0.46 0.62 0.16 0.21 1.54 2.03 0.15 0.351986 138.47 134.62 43.28 45.31 4.71 5.60 2.36 2.73 0.53 0.60 0.55 0.62 0.19 0.21 1.49 2.04 0.27 0.361987 127.18 136.69 42.26 45.82 6.77 5.52 2.16 2.72 0.73 0.60 0.66 0.61 0.24 0.21 1.76 2.01 0.34 0.371988 106.68 137.97 35.09 46.46 2.95 5.44 1.93 2.71 0.40 0.59 0.50 0.61 0.19 0.21 1.82 1.94 0.20 0.381989 127.99 139.27 40.16 47.18 5.72 5.37 2.09 2.68 0.58 0.58 0.61 0.61 0.20 0.21 2.07 1.85 0.49 0.391990 133.44 139.31 34.82 47.66 6.64 5.31 2.53 2.64 0.57 0.57 0.74 0.61 0.26 0.21 1.76 1.75 0.41 0.391991 157.59 139.24 49.84 48.08 6.36 5.24 2.58 2.59 0.40 0.57 0.42 0.62 0.12 0.20 1.50 1.66 0.30 0.391992 101.73 138.48 29.18 48.40 4.21 5.16 1.80 2.53 0.31 0.56 0.33 0.63 0.11 0.20 1.19 1.58 0.24 0.401993 179.03 137.73 66.76 48.82 6.86 5.05 4.22 2.49 0.79 0.55 0.87 0.64 0.27 0.20 1.69 1.50 0.39 0.401994 183.11 135.54 74.33 49.03 6.50 4.93 3.60 2.46 0.81 0.54 1.05 0.65 0.29 0.19 1.72 1.43 0.49 0.401995 98.43 133.65 31.79 48.89 3.88 4.84 1.94 2.45 0.38 0.54 0.49 0.66 0.13 0.19 1.00 1.37 0.28 0.401996 175.09 132.11 93.85 48.26 6.87 4.78 4.54 2.46 0.65 0.54 0.92 0.66 0.25 0.19 1.82 1.31 0.60 0.411997 147.62 131.66 60.89 47.52 5.52 4.76 3.17 2.49 0.66 0.54 0.88 0.68 0.26 0.19 1.72 1.28 0.57 0.411998 157.37 130.48 78.55 46.59 7.96 4.76 4.17 2.53 0.93 0.55 1.14 0.68 0.32 0.19 1.48 1.25 0.48 0.411999 77.33 129.53 17.52 45.63 1.79 4.78 0.86 2.56 0.29 0.56 0.28 0.69 0.08 0.20 0.86 1.23 0.28 0.422000 113.36 128.29 29.73 44.63 3.18 4.81 1.67 2.59 0.41 0.56 0.52 0.70 0.15 0.20 1.16 1.21 0.49 0.422001 78.05 127.42 26.16 43.76 1.91 4.85 1.58 2.63 0.32 0.56 0.39 0.70 0.12 0.21 1.09 1.19 0.50 0.422002 84.92 125.75 11.24 42.85 0.76 4.90 0.41 2.66 0.11 0.56 0.11 0.69 0.03 0.21 0.65 1.17 0.15 0.432003 155.79 124.40 77.43 42.04 9.13 4.95 5.32 2.69 1.01 0.57 1.06 0.69 0.29 0.21 1.67 1.15 0.80 0.432004 202.50 123.17 63.27 41.29 6.77 5.00 4.31 2.72 0.77 0.57 0.94 0.68 0.32 0.22 1.66 1.14 0.56 0.442005 166.95 122.93 42.07 40.77 5.09 5.07 3.21 2.74 0.59 0.57 0.68 0.68 0.26 0.23 1.21 1.13 0.44 0.442006 146.10 122.20 29.26 40.30 3.42 5.10 2.43 2.71 0.32 0.58 0.47 0.67 0.20 0.23 1.13 1.11 0.43 0.452007 127.92 121.52 35.07 39.74 5.31 5.07 2.52 2.63 0.68 0.57 0.68 0.66 0.29 0.23 1.07 1.10 0.49 0.462008 128.97 119.90 31.80 38.72 1.92 4.93 1.37 2.51 0.26 0.55 0.29 0.64 0.14 0.23 0.92 1.08 0.31 0.462009 94.86 117.60 25.89 37.26 2.60 4.72 1.46 2.38 0.35 0.53 0.33 0.61 0.14 0.21 0.89 1.06 0.41 0.462010 102.52 112.77 39.84 35.48 5.64 4.50 2.72 2.24 0.74 0.52 0.66 0.58 0.28 0.20 1.35 1.03 0.70 0.47
116 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends Table 4–3. Total phosphorus annual flux and flow-normalized annual flux, in tons per day, for the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS. [SUS, Susquehanna River at Conowingo, Md.; POT, Potomac River at Chain Bridge at Washington, D.C.; JAM, James River at Cartersville, Va.; RAP, Rappahannock River near Fredericksburg, Va.; APP, Appomattox River at Matoaca, Va.; PAM, Pamunkey River near Hanover, Va.; MAT, Mattaponi River near Beulahville, Va.; PAT, Patuxent River near Bowie, Md.; CHO, Choptank River near Greensboro, Md.; FN Flux, flow-normalized flux] Year SUS POT JAM RAP APP PAM MAT PAT CHO Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux 1981 6.217 9.534 2.030 7.207 1.087 3.649 0.087 0.673 0.041 0.210 0.046 0.204 0.026 0.076 0.322 0.507 0.016 0.044 1982 8.093 9.028 4.877 7.370 3.256 3.764 0.514 0.697 0.150 0.205 0.145 0.205 0.051 0.079 0.319 0.471 0.022 0.043 1983 7.613 8.673 6.154 7.407 3.757 3.870 1.000 0.708 0.249 0.205 0.200 0.204 0.099 0.080 0.401 0.436 0.052 0.042 1984 13.398 8.361 12.344 7.321 5.763 3.990 1.540 0.732 0.346 0.203 0.366 0.203 0.146 0.080 0.525 0.401 0.057 0.041 1985 4.378 8.135 2.674 6.984 2.426 4.097 0.256 0.759 0.098 0.201 0.147 0.203 0.056 0.081 0.218 0.367 0.010 0.041 1986 8.104 7.894 11.685 6.566 6.891 4.212 0.542 0.790 0.187 0.198 0.191 0.204 0.075 0.082 0.165 0.335 0.018 0.040 1987 6.250 7.631 4.337 6.204 6.485 4.279 0.709 0.841 0.220 0.198 0.212 0.206 0.089 0.084 0.222 0.305 0.029 0.040 1988 4.500 7.340 3.942 6.031 1.888 4.295 0.453 0.903 0.104 0.204 0.157 0.209 0.066 0.086 0.214 0.276 0.011 0.040 1989 6.730 7.067 4.016 5.854 4.267 4.244 0.882 0.933 0.196 0.214 0.207 0.215 0.085 0.090 0.280 0.247 0.055 0.039 1990 5.390 6.783 2.231 5.659 4.260 4.142 0.605 0.945 0.182 0.226 0.253 0.219 0.111 0.091 0.199 0.222 0.032 0.038 1991 7.008 6.537 4.715 5.477 4.422 3.970 0.695 0.933 0.124 0.234 0.133 0.221 0.042 0.090 0.151 0.199 0.019 0.038 1992 3.516 6.339 1.882 5.308 2.960 3.771 0.478 0.916 0.093 0.238 0.086 0.221 0.040 0.089 0.097 0.181 0.012 0.037 1993 11.003 6.254 7.707 5.092 4.637 3.567 1.460 0.879 0.318 0.239 0.297 0.221 0.110 0.087 0.183 0.167 0.027 0.037 1994 10.307 6.244 8.281 4.952 4.328 3.431 1.385 0.868 0.346 0.237 0.385 0.221 0.127 0.086 0.200 0.157 0.054 0.037 1995 3.454 6.292 2.233 4.931 3.066 3.328 1.147 0.860 0.129 0.234 0.145 0.223 0.046 0.085 0.077 0.152 0.015 0.038 1996 10.651 6.354 18.224 5.147 5.354 3.256 1.948 0.866 0.261 0.230 0.330 0.227 0.113 0.084 0.247 0.149 0.064 0.038 1997 7.564 6.439 6.133 5.488 3.215 3.182 0.736 0.862 0.253 0.225 0.328 0.232 0.115 0.083 0.221 0.147 0.059 0.039 1998 7.652 6.535 11.438 5.875 5.599 3.142 1.735 0.879 0.357 0.219 0.399 0.236 0.126 0.082 0.167 0.146 0.045 0.039 1999 2.817 6.690 1.157 6.208 1.060 3.104 0.161 0.893 0.100 0.215 0.091 0.239 0.029 0.081 0.088 0.146 0.032 0.040 2000 4.724 6.836 2.339 6.393 1.631 3.086 0.313 0.919 0.125 0.211 0.153 0.243 0.052 0.079 0.120 0.146 0.045 0.041 2001 2.865 6.997 2.097 6.441 1.060 3.070 0.364 0.936 0.102 0.209 0.122 0.249 0.039 0.079 0.111 0.148 0.053 0.042 2002 3.846 7.158 0.948 6.436 0.489 3.120 0.050 0.982 0.028 0.209 0.041 0.255 0.009 0.079 0.047 0.150 0.008 0.043 2003 8.937 7.353 15.831 6.382 7.001 3.198 2.317 1.030 0.499 0.210 0.472 0.262 0.124 0.079 0.308 0.153 0.114 0.043 2004 21.724 7.519 9.735 6.310 3.767 3.316 1.497 1.090 0.304 0.212 0.407 0.269 0.129 0.079 0.274 0.156 0.058 0.044 2005 11.863 7.652 5.091 6.205 2.365 3.418 1.021 1.147 0.185 0.214 0.251 0.276 0.081 0.079 0.159 0.158 0.037 0.045 2006 9.920 7.709 2.675 6.100 2.026 3.550 0.619 1.204 0.088 0.216 0.164 0.279 0.063 0.079 0.177 0.156 0.039 0.045 2007 6.952 7.771 3.739 5.920 3.361 3.668 0.784 1.271 0.269 0.219 0.304 0.283 0.098 0.079 0.134 0.153 0.045 0.045

2008 7.149 7.854 3.722 5.774 0.778 3.827 0.864 1.334 0.099 0.222 0.123 0.290 0.053 0.079 0.122 0.149 0.024 0.0452009 3.778 8.126 2.790 5.803 1.101 4.100 0.483 1.399 0.114 0.227 0.134 0.305 0.048 0.080 0.123 0.147 0.046 0.0462010 5.549 8.287 7.078 6.117 5.542 4.483 1.970 1.516 0.331 0.233 0.346 0.326 0.111 0.082 0.194 0.148 0.098 0.047
Appendix 4 117

Table 4–4. Orthophosphorus annual flux and flow-normalized annual flux, in tons per day, for the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS.[SUS, Susquehanna River at Conowingo, Md.; POT, Potomac River at Chain Bridge at Washington, D.C.; JAM, James River at Cartersville, Va.; RAP, Rappahannock River near Fredericksburg, Va.; APP, Appomattox River at Matoaca, Va.; PAM, Pamunkey River near Hanover, Va.; MAT, Mattaponi River near Beulahville, Va.; PAT, Patuxent River near Bowie, Md.; CHO, Choptank River near Greensboro, Md.; FN flux, flow-normalized flux] 

Year

SUS POT JAM RAP APP PAM MAT PAT CHO

Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux

1982 1.627 1.803 1.003 1.281 0.916 0.946 0.063 0.097 0.032 0.039 0.035 0.046 0.017 0.022 0.299 0.340 0.008 0.0131983 1.224 1.639 1.215 1.312 1.087 1.057 0.083 0.100 0.054 0.039 0.047 0.049 0.029 0.022 0.215 0.228 0.015 0.0111984 1.834 1.468 2.229 1.335 1.464 1.174 0.171 0.103 0.066 0.038 0.087 0.050 0.037 0.022 0.180 0.158 0.015 0.0101985 0.821 1.314 0.751 1.343 1.149 1.260 0.051 0.103 0.024 0.038 0.038 0.051 0.016 0.021 0.095 0.113 0.002 0.0101986 1.189 1.187 1.275 1.332 1.171 1.306 0.112 0.101 0.037 0.037 0.048 0.050 0.019 0.020 0.067 0.085 0.005 0.0091987 1.046 1.098 1.116 1.316 1.435 1.327 0.075 0.096 0.041 0.037 0.048 0.048 0.022 0.019 0.062 0.069 0.006 0.0091988 0.754 1.044 0.936 1.297 1.069 1.327 0.059 0.091 0.027 0.037 0.038 0.046 0.016 0.019 0.055 0.060 0.003 0.0101989 0.907 1.019 1.173 1.262 1.347 1.295 0.068 0.087 0.039 0.038 0.045 0.045 0.019 0.018 0.058 0.054 0.016 0.0101990 0.949 1.012 0.797 1.216 1.411 1.228 0.076 0.085 0.039 0.038 0.052 0.045 0.023 0.018 0.049 0.049 0.010 0.0101991 1.286 1.024 1.140 1.176 1.265 1.140 0.082 0.084 0.026 0.040 0.032 0.045 0.010 0.018 0.040 0.045 0.006 0.0111992 0.690 1.040 0.583 1.148 0.951 1.041 0.054 0.083 0.022 0.041 0.026 0.046 0.011 0.018 0.031 0.042 0.005 0.0111993 1.405 1.062 1.516 1.124 1.088 0.946 0.138 0.082 0.054 0.041 0.057 0.047 0.023 0.019 0.041 0.039 0.009 0.0111994 1.511 1.077 1.725 1.109 0.971 0.867 0.125 0.082 0.058 0.041 0.074 0.049 0.026 0.019 0.043 0.037 0.016 0.0111995 0.766 1.098 0.678 1.104 0.743 0.805 0.072 0.083 0.029 0.041 0.041 0.051 0.013 0.019 0.024 0.035 0.005 0.0111996 1.470 1.120 2.921 1.106 0.944 0.756 0.171 0.084 0.051 0.040 0.073 0.053 0.027 0.019 0.052 0.034 0.019 0.0111997 1.364 1.144 1.493 1.115 0.784 0.718 0.096 0.084 0.049 0.040 0.071 0.056 0.026 0.019 0.046 0.032 0.016 0.0111998 1.384 1.151 1.915 1.125 0.880 0.684 0.150 0.086 0.062 0.039 0.083 0.058 0.027 0.019 0.033 0.030 0.012 0.0111999 0.621 1.144 0.348 1.123 0.417 0.649 0.023 0.087 0.020 0.038 0.033 0.059 0.007 0.018 0.021 0.028 0.012 0.0112000 0.924 1.129 0.673 1.104 0.522 0.603 0.049 0.088 0.027 0.037 0.050 0.060 0.013 0.017 0.024 0.026 0.012 0.0112001 0.592 1.114 0.530 1.072 0.366 0.552 0.047 0.088 0.020 0.036 0.041 0.059 0.009 0.016 0.021 0.024 0.013 0.0112002 0.639 1.089 0.227 1.033 0.243 0.500 0.011 0.090 0.006 0.035 0.024 0.057 0.003 0.016 0.013 0.023 0.002 0.0102003 1.436 1.073 2.234 0.986 0.750 0.452 0.209 0.091 0.080 0.035 0.079 0.055 0.023 0.015 0.036 0.021 0.028 0.0102004 2.067 1.073 1.637 0.932 0.508 0.409 0.152 0.092 0.052 0.034 0.071 0.052 0.022 0.014 0.031 0.020 0.013 0.0102005 1.603 1.087 0.864 0.870 0.362 0.368 0.091 0.093 0.032 0.033 0.049 0.049 0.014 0.013 0.020 0.019 0.009 0.0112006 1.423 1.088 0.542 0.804 0.248 0.325 0.074 0.093 0.018 0.033 0.036 0.045 0.010 0.012 0.022 0.018 0.011 0.0112007 1.179 1.084 0.586 0.744 0.275 0.278 0.072 0.092 0.041 0.033 0.044 0.042 0.014 0.012 0.017 0.018 0.011 0.0122008 1.105 1.080 0.506 0.701 0.124 0.233 0.054 0.090 0.017 0.032 0.022 0.038 0.008 0.011 0.015 0.017 0.007 0.0122009 0.784 1.084 0.438 0.690 0.135 0.197 0.048 0.088 0.019 0.032 0.021 0.034 0.007 0.011 0.016 0.017 0.017 0.0132010 0.931 1.089 0.613 0.699 0.183 0.174 0.063 0.097 0.041 0.031 0.033 0.032 0.013 0.010 0.021 0.017 0.024 0.014
118 Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends Table 4–5. Suspended sediment annual flux and flow-normalized annual flux, in tons per day, for the nine Chesapeake Bay River Input Monitoring Stations obtained using WRTDS. [SUS, Susquehanna River at Conowingo, Md.; POT, Potomac River at Chain Bridge at Washington, D.C.; JAM, James River at Cartersville, Va.; RAP, Rappahannock River near Fredericksburg, Va.; APP, Appomattox River at Matoaca, Va.; PAM, Pamunkey River near Hanover, Va.; MAT, Mattaponi River near Beulahville, Va.; PAT, Patuxent River near Bowie, Md.; CHO, Choptank River near Greensboro, Md.; FN flux, flow-normalized flux; NA, not available] Year SUS POT JAM RAP APP PAM MAT PAT CHO Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux Flux FN flux 1981 3,621.21 5,525.57 1,081.69 6,423.90 NA NA NA NA NA NA NA NA NA NA 19.24 76.94 2.72 12.50 1982 4,106.13 5,142.80 4,206.75 6,838.32 NA NA NA NA NA NA NA NA NA NA 34.61 82.01 3.85 11.40 1983 4,093.29 4,773.36 6,678.15 7,194.09 NA NA NA NA NA NA NA NA NA NA 92.91 87.73 14.87 10.69 1984 9,891.60 4,418.14 13,712.10 7,296.24 NA NA NA NA NA NA NA NA NA NA 139.12 93.44 14.85 10.25 1985 1,377.63 4,137.09 1,714.13 6,813.71 NA NA NA NA NA NA NA NA NA NA 34.67 98.16 1.71 10.05 1986 3,979.10 3,903.02 11,264.56 6,142.33 NA NA NA NA NA NA NA NA NA NA 23.31 101.13 3.53 9.96 1987 2,315.77 3,738.22 3,942.48 5,596.20 NA NA NA NA NA NA NA NA NA NA 58.83 101.18 6.82 9.93 1988 1,481.33 3,647.11 3,524.21 5,306.95 NA NA NA NA NA NA NA NA NA NA 65.24 98.05 2.00 9.83 1989 2,956.75 3,620.06 3,584.16 5,142.96 NA NA NA NA NA NA NA NA NA NA 129.07 92.60 12.46 9.66 1990 1,860.09 3,608.91 1,297.55 4,710.71 NA NA NA NA NA NA NA NA NA NA 71.78 86.42 6.50 9.21 1991 3,181.82 3,616.11 3,008.85 4,271.05 NA NA NA NA NA NA NA NA NA NA 47.41 80.39 3.52 8.74 1992 1,224.68 3,617.88 1,107.94 3,894.95 NA NA NA NA NA NA NA NA NA NA 28.18 75.13 1.86 8.27 1993 9,014.61 3,626.38 6,204.80 3,594.03 NA NA NA NA NA NA NA NA NA NA 81.92 71.08 6.08 7.88 1994 6,913.26 3,617.25 5,641.09 3,357.52 NA NA NA NA NA NA NA NA NA NA 96.16 68.62 14.08 7.55 1995 1,276.31 3,609.81 1,087.07 3,243.40 NA NA NA NA NA NA NA NA NA NA 22.38 67.23 2.35 7.26 1996 10,244.86 3,631.89 15,535.52 3,215.35 NA NA NA NA NA NA NA NA NA NA 125.46 66.66 10.69 7.02 1997 4,305.70 3,715.54 2,691.54 3,224.47 NA NA NA NA NA NA NA NA NA NA 103.29 66.35 11.69 6.77 1998 4,206.36 3,832.61 6,737.45 3,350.07 NA NA NA NA NA NA NA NA NA NA 82.53 66.36 9.72 6.49 1999 1,099.71 3,992.14 313.81 3,545.30 NA NA NA NA NA NA NA NA NA NA 34.82 66.29 3.86 6.30 2000 1,785.41 4,205.54 665.94 3,792.24 978.05 2,261.91 NA NA 31.00 51.34 79.23 164.15 13.31 30.65 46.44 66.47 6.22 6.18 2001 909.92 4,509.06 685.86 4,041.54 696.45 2,042.28 379.04 936.95 28.58 52.72 66.83 143.54 9.83 25.40 43.07 66.99 7.51 6.18 2002 1,451.45 4,866.30 304.76 4,278.02 157.93 2,039.98 27.03 905.06 5.03 53.67 5.49 131.63 1.33 22.42 10.25 68.06 0.67 6.23 2003 4,069.15 5,202.07 12,073.42 4,503.57 7,561.96 2,025.57 2,630.80 892.40 175.58 54.52 347.24 126.09 35.46 20.78 168.16 69.35 16.16 6.28 2004 34,721.41 5,412.79 6,291.05 4,745.19 2,752.18 2,078.46 1,389.33 889.37 89.96 55.19 237.88 124.46 35.49 19.68 129.76 70.75 8.73 6.37 2005 8,339.01 5,598.06 3,207.44 5,014.75 1,450.07 1,972.78 892.37 883.43 47.05 55.92 129.19 124.91 21.30 18.93 71.55 71.71 5.19 6.48 2006 7,567.06 5,670.85 1,394.72 5,374.35 1,702.64 2,041.59 448.00 888.72 18.81 56.32 65.00 127.14 14.25 18.22 81.97 71.83 4.75 6.63 2007 3,139.77 6,082.02 2,652.55 5,783.14 2,548.40 2,064.14 554.62 894.00 79.53 56.66 167.79 131.36 24.88 17.70 59.36 71.51 7.41 6.79

2008 4,017.29 6,396.86 3,562.38 6,337.82 398.59 2,172.14 799.26 906.69 22.35 56.85 69.88 139.16 12.29 17.36 60.48 71.81 3.49 7.052009 1,198.85 7,349.21 2,589.70 6,929.49 621.42 2,134.88 307.63 922.76 25.33 57.34 73.44 153.08 10.84 17.48 64.05 73.80 5.61 7.422010 3,125.31 7,714.94 9,802.96 7,642.59 4,359.82 2,333.74 1,737.02 959.29 93.40 58.05 232.88 176.64 27.41 18.10 100.54 77.77 19.58 7.79
Manuscript was approved October 25, 2012.

Edited by Kay (Hedrick) Naugle

Layout by Caryl J. Wipperfurth

Science Publishing Network

Raleigh Publishing Service Center

For additional information regarding this 

publication contact:

Douglas L. Moyer

Virginia Water Science Center

1730 East Parham Road

Richmond, VA 23228

dlmoyer@usgs.gov

http://va.water.usgs.gov
Moyer and ohters—Comparison of Two Regression-Based Approaches for Determining Nutrient and Sediment Fluxes and Trends—Scientific Investigations Report 2012– 5244

Printed on recycled paper