Prepared in cooperation with the St. Croix Watershed Research Station

Development of Regression Equations to Revise Estimates 

of Historical Streamflows for the St. Croix River at Stillwater, 

Minnesota (Water Years 1910–2011), and Prescott, Wisconsin 

(Water Years 1910–2007)

Scientific Investigations Report 2014–5239

U.S. Department of the Interior

U.S. Geological Survey
Cover photograph: The lift bridge at Stillwater, Minnesota is raised up, and a riverboat 

is passing underneath it. The solar panel, wire-weight gage, antenna, and box housing 

the Data Collection Platform are shown in the left-center of the photo, attached to the 

bridge. Photograph taken on June 2, 2014, by Brett Savage, U.S. Geological Survey, 

Minnesota Water Science Center.
Development of Regression Equations to 

Revise Estimates of Historical Streamflows 

for the St. Croix River at Stillwater, 

Minnesota (Water Years 1910–2011), and 

Prescott, Wisconsin (Water Years 1910–

2007)

By Jeffrey R. Ziegeweid and Suzanne Magdalene

Prepared in cooperation with the St. Croix Watershed Research Station

Scientific Investigations Report 2014–5239

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015

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Suggested citation:

Ziegeweid, J.R., and Magdalene, Suzanne, 2015, Development of regression equations to revise estimates of historical streamflows for the St. Croix River at Stillwater, Minnesota (water years 1910–2011), and Prescott, Wisconsin 

(water years 1910–2007): U.S. Geological Survey Scientific Investigations Report 2014–5239, 23 p., http://dx.doi.

org/10.3133/sir20145239.

ISSN: 2328-0328 (online)
iii

Acknowledgments

This report was produced in cooperation with the St. Croix Watershed Research Station. Funding 

support for the index-velocity streamgages that provided data used to develop regression equations was provided by Metropolitan Council Environmental Services, the St. Croix River Association, the Wisconsin Department of Natural Resources Stillwater Bridge Mitigation Fund, and the 

U.S. Geological Survey (USGS)-National Park Service Water Quality Partnership Program. Kent 

Johnson of Metropolitan Council Environmental Services (MCES) provided information about the 

St. Croix River and the discharge estimates used in nutrient-load monitoring. Scott Schellhaas of 

MCES provided discrete discharge measurement data used in evaluating revised estimates. 

Installation, operation, and maintenance of field equipment and field data collection and record 

working, checking, and reviewing were completed by staff in the Data Sections of the USGS 

Minnesota and Wisconsin Water Science Centers. From the USGS Minnesota Water Science 

Center Investigations Section, Dave Lorenz provided invaluable assistance with statistical analyses and R outputs, and Scott Bennett created the study area map. 
iv

Contents

Acknowledgments........................................................................................................................................iii

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

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

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

Physical Setting.....................................................................................................................................3

Methods...........................................................................................................................................................4

Previous Equations for Estimating Historical Streamflows ...........................................................4

Measurement of Continuous Streamflow at Index-Velocity Streamgages ................................5

Methods for Development of New Equations to Estimate Streamflows ....................................5

Methods for Evaluation of Equations Used to Estimate Historical Streamflows ......................6

Methods for Calculation of Revised Estimates of Historical Streamflows .................................6

Methods for Evaluation of Revised Estimates of Historical Streamflows...................................6

Development of Regression Equations to Estimate Historical Streamflows .......................................7

Regression Equations for St. Croix River at Stillwater, Minnesota...............................................7

Regression Equations for St. Croix River at Prescott, Wisconsin ..............................................10

Computation and Evaluation of Revised Estimates of Historical Streamflows .................................12

Computation of Revised Historical Streamflows for the St. Croix River at Stillwater, 

Minnesota...............................................................................................................................12

Computation of Revised Historical Streamflows for the St. Croix River at Prescott, 

Wisconsin...............................................................................................................................12

Evaluation of Revised Estimates of Historical Streamflows at Stillwater, Minnesota ...........13

Evaluation of Revised Estimates of Historical Streamflows at Prescott, Wisconsin ..............14

Limitations of the Study...............................................................................................................................18

Summary........................................................................................................................................................18

References Cited..........................................................................................................................................19

Appendixes 1–3............................................................................................................................................21

Appendix 1. Summary of Equations Listed in Text of the Report.................................................22

Appendix 2. Previous and Revised Estimates of Historical Daily Streamflows for the 

St. Croix River at Stillwater, Minnesota.............................................................................23

Appendix 3. Previous and Revised Estimates of Historical Daily Streamflows for the 

St. Croix River at Prescott, Wisconsin...............................................................................23

Figures

1. Map showing study area for the St. Croix River in Minnesota and Wisconsin .................2

2. Time-series plots used to visually compare measured streamflows (water years 

2012–13) for the St. Croix River at Stillwater, Minnesota (U.S. Geological Survey 

streamgage 05341550) to streamflows estimated using selected regression 

equations from table 1 .................................................................................................................8

3. Time-series plots of measured and estimated streamflow for the St. Croix River at 

Stillwater, Minnesota, using regression equations based on streamflow data from 

the St. Croix River at St. Croix Falls (U.S. Geological Survey streamgage 05340500) 

for 2 water years.........................................................................................................................11
v

4. Time-series plots used to visually compare measured streamflows (water years 

2008–13) for the St. Croix River at Prescott, Wisconsin (U.S. Geological Survey 

streamgage 05344490) to streamflows estimated using selected regression 

equations from table 3 ...............................................................................................................15

Tables

1. Streamgaging sites and corresponding date ranges for data used in developing 

regression equations and comparing the abilities of previous equations and new 

regression equations to estimate measured streamflows at Stillwater, Minnesota 

(U.S. Geological Survey streamgage 05341550) and Prescott, Wisconsin (U.S. 

Geological Survey streamgage 05344490)................................................................................4

2. Nash-Sutcliffe Efficiencies that evaluate the abilities of selected regression 

equations to fit measured streamflows (water years 2012–13) of the St. Croix 

River at Stillwater, Minnesota (U.S. Geological Survey streamgage 05341550) ................7

3. Nash-Sutcliffe Efficiencies for regression equations developed using measured 

streamflows of the St. Croix River at St. Croix Falls (U.S. Geological Survey 

streamgage 05340500) from one water year to fit measured streamflows for 

another water year at the St. Croix River at Stillwater, Minnesota (U.S. Geological 

Survey streamgage 05341550) to validate the regression method used to develop 

equation 9 for estimating historical streamflows at the St. Croix River at Stillwater, 

Minnesota ....................................................................................................................................10

4. Nash-Sutcliffe Efficiencies that evaluate the abilities of selected regression 

equations equations to fit measured streamflows (water years 2008–13) of the 

St. Croix River at Prescott, Wisconsin (U.S. Geological Survey streamgage 

05344490 for water years 2008–13 records) ...........................................................................12

5. Selected statistics for measured streamflows of the St. Croix River at St. Croix 

Falls, Wisconsin (U.S. Geological Survey streamgage 05340500) used to compute 

revised estimates of historical streamflows for the St. Croix River at Stillwater, 

Minnesota and Prescott, Wisconsin, and minimum and maximum estimated 

streamflows at Stillwater and Prescott using new regression equations........................13

6. Comparisons between estimated historical daily mean streamflows and discrete 

streamflow measurements made before the installation of an index-velocity 

streamgage at the St. Croix River at Stillwater, Minnesota.................................................14

7. Comparisons between estimated historical daily mean streamflows and discrete 

streamflow measurements made before the installation of an index-velocity 

streamgage at the St. Croix River at Prescott, Wisconsin...................................................17

1–1. Summary of all the equations listed in the text of the report ..............................................22
vi

Conversion Factors

Inch/Pound to International System of Units

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3

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

Mass

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

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 

83).

A water year is the 12-month period, October 1 through September 30, and is designated by 

the calendar year in which it ends.

Abbreviations

ADVM acoustic Doppler velocity meter

ANOVA analysis of variance

MCES Metropolitan Council Environmental Services

NSE Nash-Sutcliffe Efficiency

NWIS National Water Information System [U.S. Geological Survey database]

R2 coefficient of determination

TMDL total maximum daily load

USACE U.S. Army Corps of Engineers

USGS U.S. Geological Survey
Development of Regression Equations to Revise Estimates 

of Historical Streamflows for the St. Croix River at 

Stillwater, Minnesota (Water Years 1910–2011), and 

Prescott, Wisconsin (Water Years 1910–2007)

By Jeffrey R. Ziegeweid and Suzanne Magdalene

Abstract

A natural dam of glacial-era sediments at the confluence 

of the St. Croix and Mississippi Rivers forms Lake St. Croix, 

a riverine lake that comprises the lowest 25 miles of the St. 

Croix River. Historically, backwater effects from the Mississippi River prevented the use of traditional streamgages for 

collecting continuous streamflow data needed to calculate 

nutrient loads at the inlet to and outlet from Lake St. Croix at 

Stillwater, Minnesota and Prescott, Wisconsin, respectively. 

The development of index-velocity streamgages has enabled 

the measurement of continuous streamflow in backwater 

conditions using continuously measured velocities at the 

streamgage. Index-velocity streamgages were installed at 

Prescott, Wisconsin, and Stillwater, Minnesota, in 2007 and 

2011, respectively. 

Continuous daily mean streamflow data from the new 

index-velocity streamgages, long-term upstream streamgages, 

and tributary streamgages were used to (1) develop regression 

equations that improve estimates of historical streamflow at 

Stillwater and Prescott, (2) evaluate the accuracies of new and 

previous equations used to estimate historical streamflows, 

and (3) compute and evaluate revised estimates of historical 

streamflows for Stillwater for water years 1910‒2011 and for 

Prescott for water years 1910‒2007. The abilities of previous 

and newly developed regression equations to accurately estimate streamflows were evaluated using Nash-Sutcliffe Efficiency (NSE) values. The NSE values at Stillwater improved 

from 0.90 to 0.98, and the NSE values at Prescott improved 

from 0.77 to 0.94. 

The new regression equations were used to calculate 

revised estimates of historical streamflows for Stillwater and 

Prescott starting in 1910 and ending when index-velocity 

streamgages were installed. Monthly, annual, 30-year, and 

period of record statistics were examined between previous 

and revised estimates of historical streamflows. The abilities of the new regression equations to estimate historical 

streamflows were evaluated by using percent differences to 

compare new estimates of historical daily streamflows to 

discrete streamflow measurements made at Stillwater and 

Prescott before the installation of index-velocity streamgages. 

Although less variability was observed between estimated 

and measured streamflows at Stillwater compared to Prescott, 

the percent difference data indicated that the new estimates 

closely approximated measured streamflows at both locations. 

Introduction

A natural dam of glacial-era sediments at the confluence of the St. Croix and Mississippi Rivers creates Lake St. 

Croix, a riverine lake that comprises the lowest 25 miles (mi) 

of the St. Croix River along the border between Minnesota 

and Wisconsin (fig. 1; Triplett and others, 2009). Increased 

eutrophication in Lake St. Croix throughout recent decades 

is linked to increased agriculture and urban development in 

the St. Croix Basin (Minnesota Pollution Control Agency and 

Wisconsin Department of Natural Resources, 2012). Concerns 

about future water quality in the St. Croix River (including 

Lake St. Croix) prompted several agencies and organizations 

to form the St. Croix Basin Water Resources Planning Team 

in 1993 to coordinate research and monitoring efforts in the 

St. Croix Basin and address the issue of increased eutrophication (Magdalene and others, 2013). Monitoring efforts have 

included nutrient monitoring at the inlet to and outlet from 

Lake St. Croix by Metropolitan Council Environmental Services (MCES) since 1976. However, subsequent nutrient-loading analyses required streamflow data at the inlet to and outlet 

from Lake St. Croix at Stillwater, Minnesota, and Prescott, 

Wisconsin, respectively (hereafter referred to as Stillwater and 

Prescott, respectively; fig. 1).

Traditional streamgages could not be used to measure 

continuous streamflow at the inlet to and outlet from Lake 

St. Croix because backwater effects caused by the confluence 

with the larger Mississippi River confounded the relation 

between stage and discharge typically used to generate continuous streamflow records. In order to estimate nutrient loads 

entering and leaving Lake St. Croix, streamflows for the inlet 
2 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

Figure 1. Study area for the St. Croix River in Minnesota and Wisconsin.

Map area

MINNESOTA

WISCONSIN

"

!

!

!

!

!

!

!

!

#

#

#

#

#

##

#

#

St. Croix River

Minnesota River

Apple River

Willow River

Lake

St. Croix

05340500

05341500

05341550

05341752

05342000

05344490

05344500

05331000

05288500

Stillwater

St. Paul

St. Croix Falls

Hudson

Prescott

River Falls

Burkhardt

Somerset

MWTF

POLK

CHISAGO

ISANTI

BARRON

ANOKA

DUNN

HENNEPIN ST CROIX

RAMSEY

DAKOTA

PIERCE

SCOTT

GOODHUE

WASHINGTON

Mississippi River

Kinnickinnic River

-93°20' -93° -92°40' -92°20'

45°20'

45°

44°40'

Data from U.S. Geological Survey and U.S. Census Bureau

Digital Data 1:750,000 and 1:2,000,000

North American Datum of 1983 (NAD 83)

0

0

5 10 15 20 MILES

5 10 15 20 KILOMETERS

# U.S. Geological Survey streamgage with station number

" Metropolitan Wastewater Treatment Facility (MWTF)

Basins

Minnesota River

Mississippi River headwaters

St. Croix River

WISCONSIN

WISCONSIN

MINNESOTA

EXPLANATION

Mississippi River

MINNESOTA
Introduction 3

to and outlet from Lake St. Croix previously were estimated 

by combining measured daily streamflows from upstream 

and tributary streamgages not affected by backwater conditions (LaFrancois and others, 2009; Magdalene and others, 

2013). However, the accuracies of these estimates of historical 

streamflows for the St. Croix River at Stillwater and Prescott 

were not determined. 

In the late 1990s, advances in velocity meters enabled 

researchers with the U.S. Geological Survey (USGS) to 

develop index-velocity streamgages to measure continuous streamflow in backwater conditions using continuously 

measured velocities at the streamgages (Levesque and Oberg, 

2012). To obtain more accurate streamflow data for calculating 

nutrient loads for Lake St. Croix, index-velocity streamgages 

were installed at Prescott (USGS streamgage 05344490; St. 

Croix River at Prescott, Wisc.) in 2007 and Stillwater (USGS 

streamgage 05341550; St. Croix River at Stillwater, Minn.) in 

2011 (fig. 1). The new index-velocity streamgages account for 

backwater effects and provide accurate continuous streamflow data that has improved loading estimates calculated from 

nutrient data collected by MCES. 

The St. Croix Basin Water Resources Planning Team has 

developed a total maximum daily load (TMDL) for phosphorus in Lake St. Croix, with the goal of eliminating 100 tons 

in annual phosphorus loads by the year 2020 (Minnesota Pollution Control Agency and Wisconsin Department of Natural 

Resources, 2012). Improved historical streamflow estimates 

are needed to evaluate achievement of the TMDL-based phosphorus-reduction goal for Lake St. Croix. Improved estimates 

of daily streamflow from the index-velocity streamgages can 

be used in conjunction with streamflow values from upstream 

and tributary streamgages to develop new regression equations 

to improve historical streamflow estimates for the St. Croix 

River at Stillwater and Prescott and to determine the accuracy 

of historical nutrient-loading estimates at the inlet to and outlet 

from Lake St. Croix. To address these needs, the U.S. Geological Survey, in cooperation with the St. Croix Watershed 

Research Station, (1) developed new regression equations 

to improve historical streamflow estimates at Stillwater and 

Prescott, (2) evaluated the accuracies of new and previous 

equations used to estimate historical streamflows, (3) computed revised estimates of historical streamflows for Stillwater 

and Prescott, and (4) compared previous and revised estimates of historical streamflows for water years 1910‒2011 at 

Stillwater and for water years 1910‒2007 at Prescott; a water 

year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends. This 

effort built upon a previous study completed by the St. Croix 

Watershed Research Station, USGS, and MCES (Magdalene 

and others, 2013), which was funded by the St. Croix River 

Association. 

Purpose and Scope

The purposes of this report are to (1) describe the 

development of regression equations to estimate historical 

streamflows of the St. Croix River at Stillwater and Prescott, 

(2) describe the evaluation of the accuracies of new and previous equations used to estimate historical streamflows, and 

(3) present revised estimates and evaluations of estimates of 

historical streamflows for the St. Croix River at Stillwater, 

Minn. (water years 1910‒2011), and at Prescott, Wisc. (water 

years 1910‒2007). 

The study area is limited to the lower St. Croix River 

from St. Croix Falls, Wisconsin to the confluence with the 

Mississippi River at Prescott, Wisconsin (fig. 1). However, 

similar techniques could be used to improve discharge estimates in other basins with natural reservoirs of similar slopes, 

underlying geologies, and hydrologic conditions. Because of 

issues with serial correlation, the regression analyses used to 

develop new equations for estimating historical daily mean 

streamflows should not be used to make inferences about sample variances. However, the developed regression equations 

are valid for estimating streamflows at Stillwater and Prescott 

from measured streamflow at St. Croix Falls. In addition, the 

equations developed to estimate streamflows at Stillwater and 

Prescott from measured streamflow at St. Croix Falls should 

only be considered valid for the range of flows observed in 

the records used to generate these equations. For the rest of 

this report, “streamflows” will refer to daily mean streamflow 

unless specified otherwise, and “historical streamflows” will 

refer to streamflow estimated for the period before installation 

of the index-velocity streamgages at Stillwater (water years 

1910‒2011) and Prescott (water years 1910‒2007).

Physical Setting

The St. Croix Basin encompasses 7,730 square miles 

(mi2) in Minnesota and Wisconsin (Lenz and others, 2001) 

(fig. 1). Before European settlement, the St. Croix Basin 

was dominated by forests, peatlands, and prairie grasslands 

(Niemela and Feist, 2000; Payne and others, 2002). Starting in 

the mid-1800s, logging, agricultural development, and urbanization altered the landscape, increased stormwater runoff, and 

created the need for more effective wastewater treatment in 

the basin. Today (2014), the St. Croix Basin is dominated by 

forest, pastures, and croplands, with most of the urban lands 

concentrated in the areas around the lowest 25 mi of the St. 

Croix River (Heiskary and Vavricka 1993, Larson and others, 

2002). In addition, the St. Croix National Scenic Riverway 

provides 252 river miles for recreational use. Recreational use 

in the St. Croix National Scenic Riverway doubled from 1973 

to 1995, reaching nearly one million visitors annually (Robertson and Lenz, 2002). 

The lower St. Croix River comprises the 51.9 river 

miles from St. Croix Falls, Wisc. to the confluence with the 

Mississippi River at Prescott, Wisc. The final 25 mi of the 

St. Croix River, from Stillwater, Minn., to Prescott, Wisc., 

are referred to as Lake St. Croix because of near-zero slope, 

maximum water depths of more than 70 feet (ft), and a mean 

width greater than 0.3 mi (Robertson and Lenz, 2002). Several 

tributaries enter the lower St. Croix River, including the Apple 
4 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

River upstream from Stillwater, and the Willow and Kinnickinnic Rivers downstream from Stillwater and upstream from 

Prescott (fig. 1). 

Before installation of the index-velocity streamgages, 

streamflows for the inlet to and outlet from Lake St. Croix 

were estimated using streamflow-routing equations involving 

continuous streamflow records from upstream and tributary 

streamgages not affected by backwater conditions (LaFrancois 

and others, 2009; Magdalene and others, 2013). Streamflows 

of the St. Croix River at Stillwater were estimated using continuous streamflow data from an upstream streamgage at the 

St. Croix River at St. Croix Falls, Wisc. (USGS streamgage 

05340500; hereafter referred to as St. Croix Falls) and a 

streamgage on the Apple River near Somerset, Wisc. (USGS 

streamgage 05341500; hereafter referred to as Apple River) 

(fig. 1; LaFrancois and others, 2009; Magdalene and others, 2013). Streamflows of the St. Croix River at Prescott 

were estimated using continuous streamflow data from the 

records at St. Croix Falls and two tributary streamgages: Willow River at Willow River State Park near Burkhardt, Wisc. 

(USGS streamgage 05341752; hereafter referred to as Willow 

River) and Kinnickinnic River near River Falls, Wisc. (USGS 

streamgage 05342000; hereafter referred to as Kinnickinnic 

River) (fig. 1, LaFrancois and others, 2009; Magdalene and 

others, 2013). 

Methods

This section of the report describes various methods 

used in the development of regression equations to estimate 

historical streamflows for the St. Croix River at Stillwater 

and Prescott. Previous equations used for estimating historical streamflows are presented first, followed by methods for 

measurement of continuous streamflow at the index-velocity 

streamgages at Stillwater and Prescott. Methods for development of new regression equations and methods used to 

evaluate previous and new equations are then described. All 

equations used to estimate historical streamflows at Stillwater 

and Prescott are presented in table 1‒1 in the appendix. All 

streamflow variables used in this report are in units of cubic 

feet per second.

Previous Equations for Estimating Historical 

Streamflows

Streamflow data from several USGS streamgages were 

used in previous equations for estimating historical streamflows of the St. Croix River at Stillwater and Prescott (table 1). 

All USGS streamflow data are available from the 

National Water Information System database (NWIS; 

U.S. Geological Survey, 2014). Streamflow measurements 

by the USGS follow methods of Rantz and others (1982).

Historical streamflows at Stillwater (QStillwater) were estimated in previous studies (Kloiber, 2004; Triplett and others, 

2009; LaFrancois and others, 2009) using the equation:

QStillwater = QSCF + QApple (1)

when streamflow data were available for both St. Croix Falls 

(QSCF) and Apple River (QApple). However, the streamflow 

record at St. Croix Falls is more extensive than the streamflow 

Table 1. Streamgaging sites and corresponding date ranges for data used in developing regression 

equations and comparing the abilities of previous equations and new regression equations to estimate 

measured streamflows at Stillwater, Minnesota (U.S. Geological Survey streamgage 05341550) and Prescott, 

Wisconsin (U.S. Geological Survey streamgage 05344490).

[Station number, the station number used to identify streamgages in the National Water Information System (http://waterdata.

usgs.gov/nwis/); Station name, the descriptive name associated with the corresponding station number; Start date, the start date 

of data from the corresponding station used in presented analyses; End date, the end date of data from the corresponding station 

used in presented analyses]

Station 

number (fig. 1) Station name Start date End date

05340500 St. Croix River at St. Croix Falls, Wisconsin 01/01/1910 09/30/2013

05341500 Apple River near Somerset, Wisconsin 09/30/2011 09/30/2013

05341550 St. Croix River at Stillwater, Minnesota 10/01/2011 09/30/2013

05341752 Willow River at Willow R State Park nr Burkhardt, Wisconsin 10/01/2012 09/30/2013

05342000 Kinnickinnic River near River Falls, Wisconsin 10/01/2012 09/30/2013

05344490 St. Croix River at Prescott, Wisconsin 10/01/2007 09/30/2013

05331000 Mississippi River at St. Paul, Minnesota 09/30/2007 09/30/2013

1

NA Discharge from Metropolitan Wastewater Treatment Facility 09/30/2007 09/30/2013

05344500 Mississippi River at Prescott, Wisconsin 10/01/2007 09/30/2013

05331580 Mississippi River Below L & D #2 at Hastings, Minnesota 10/01/2007 09/30/2013

1

Site number assigned, data available as a data descriptor within 05331000 (internal only).
Methods 5

record for Apple River. Therefore, when Apple River data 

were unavailable, the mean ratio between Apple River streamflow and St. Croix Falls streamflow was used to estimate 

historical streamflows at Stillwater using only measured 

streamflow data from St. Croix Falls. On average, streamflows 

at Apple River were 11 percent of the streamflows at St. Croix 

Falls, resulting in the following equation (Kloiber, 2004; 

Triplett and others, 2009; LaFrancois and others, 2009):

QStillwater = 1.11 * QSCF (2)

For other USGS streamflow-routing equations used occasionally in project-specific situations, measured streamflows from 

St. Croix Falls and Apple River were lagged 1 day (QSCF_

lagged_1_day and QApple_lagged_1_day, respectively) and summed (Greg 

Mitton, U.S. Geological Survey, written commun., 2014), 

resulting in the following equation:

QStillwater = QSCF_lagged_1_day + QApple_lagged_1_day (3)

When streamflow data were available for Willow River 

(QWillow) and Kinnickinnic River (QKinnickinnic), previous studies 

used estimated streamflows at Stillwater and measured streamflows at Willow River and Kinnickinnic River to estimate 

historical streamflows at Prescott (QPrescott) using equation 4 

(Kloiber, 2004; Triplett and others, 2009; LaFrancois and others, 2009).

QPrescott =QStillwater + QWillow + QKinnickinnic (4)

However, the overlapping period of record for Willow River 

and Kinnickinnic River was brief, so the mean streamflow 

ratios of Willow River to St. Croix Falls (0.037) and Kinnickinnic River to St. Croix Falls (0.032) were added to the 

estimated multiplier of equation 2 for Stillwater to produce 

the following equation for estimating historical streamflows at 

Prescott (Kloiber, 2004; Triplett and others, 2009; LaFrancois 

and others, 2009):

QPrescott =1.18 * QSCF (5)

During certain past flood situations, USGS hydrographers 

calculated the Prescott streamflow using data from USGS 

streamgages on the Mississippi River and measured discharges from the Metropolitan Wastewater Treatment Facility 

(QMWTF), which were collected by MCES (Jinger Pulkrabek, 

MCES, written commun., 2014; data not publicly available). Streamflow at Prescott was determined by summing 

the streamflow from the Mississippi River at St. Paul, Minn. 

(USGS streamgage 05331000; QSt.Paul) and QMWTF, lagging that 

sum by 1 day [(QSt.Paul + QMWTF)lagged_1_day], and multiplying by 

a constant to account for unmeasured portions of the flow in 

the Mississippi River. The resulting total then was subtracted 

from the streamflow of the Mississippi River at Prescott, Wisc. 

(USGS streamgage 05344500; QMiss_Prescott) (Greg Mitton, U.S. 

Geological Survey, written commun., 2014), resulting in the 

following equation to calculate streamflow of the St. Croix 

River at Prescott (QPrescott):

QPrescott = QMiss_Prescott – {1.034[(QSt.Paul + QMWTF )lagged_1_day]} 

 (6)

Measurement of Continuous Streamflow at 

Index-Velocity Streamgages

The index-velocity streamgages at Prescott (USGS 

streamgage 05344490) and Stillwater (USGS streamgage 

05341550) have collected continuous streamflow data since 

2007 and 2011, respectively (U.S. Geological Survey, 2014). 

The index-velocity streamgage at Prescott (USGS streamgage 

05344490) was installed by USGS personnel on August 20, 

2007. At Stillwater, an existing U.S. Army Corps of Engineers 

(USACE) stage-only streamgage was converted to an indexvelocity streamgage on September 9, 2011, when USGS personnel added an acoustic Doppler velocity meter (ADVM) to 

collect continuous velocity data. Continuous stage and velocity data were collected at Prescott and Stillwater using methods and instrumentation in accordance with USGS protocols 

(Rantz and others, 1982; Kennedy, 1983; Fallon and others, 

2002; Mueller and others, 2013; Turnipseed and Sauer, 2010). 

Continuous streamflow data were calculated using stage-discharge and index-velocity rating curves developed according 

to methods specified in Levesque and Oberg (2012). Continuous streamflow records were worked, checked, and reviewed 

according to protocols described in Fallon and others (2002) 

and Levesque and Oberg (2012). Interruptions in continuous 

data collection because of ice cover or equipment malfunction 

caused a few gaps in the continuous streamflow record, and 

daily streamflows during these gaps were estimated by USGS 

hydrographers (U.S. Geological Survey, 2014). Estimated 

streamflow values were not used for the regression analyses 

presented in this report. Daily streamflow data (U.S. Geological Survey, 2014) from water years 2012 and 2013 for the 

Stillwater index-velocity streamgage and from water years 

2008 through 2013 for the Prescott index-velocity streamgage 

were used in this study.

Methods for Development of New Equations to 

Estimate Streamflows 

Simple and multiple linear regression analyses were used 

to evaluate the streamflow relations between the upstream and 

tributary streamgages and the new index-velocity streamgages 

at Stillwater and Prescott. Several regression equations with 

various explanatory variables were compared for Stillwater 
6 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

and Prescott to account for differing periods of operation 

for upstream and tributary streamgages (table 1). However, 

St. Croix Falls provides the longest and most consistent 

streamflow record for predicting downstream streamflows at 

Stillwater and Prescott. Model goodness-of-fit was evaluated 

using the Nash-Sutcliffe Efficiency (NSE; Nash and Sutcliffe, 

1970) combined with graphical analyses as described in the 

following section. The best regression equations for estimating 

streamflows at Stillwater and at Prescott were selected based 

on NSE values and were used to calculate revised estimates of 

historical streamflows at Stillwater and Prescott.

Methods for Evaluation of Equations Used to 

Estimate Historical Streamflows 

The goodness-of-fit of previous and new equations for 

estimating daily streamflows at Stillwater and Prescott were 

compared using the NSE (Nash and Sutcliffe, 1970), which is 

defined in equation 7;

 NSE = 1 – (Qm,i – Qe,i)

2

(Qm,i – Qm)2

∑n

i=1

∑n

i=1

— (7)

where Qm,i is the measured daily mean streamflow at the 

index-velocity streamgage for each day i, n is the upper bound 

of the summation, Qe,i is the estimated daily mean streamflow 

from the model for each day i, and is the mean of all measured daily streamflows for the examined period of record. 

The NSE is a normalized statistic that determines the relative 

magnitude of the residual variance compared to measured data 

variance (Nash and Sutcliffe, 1970). Furthermore, the NSE 

indicates how well the plot of the relation between observed 

(measured) and simulated (modeled) data fits a 1:1 line. When 

NSE is equal to 1, the simulated data match the observed data 

perfectly. When NSE is equal to 0, the simulated data are as 

accurate as the mean of the observed data. Finally, when NSE 

is less than 0, the observed mean is a better estimator than the 

model. Unlike the NSE, the more commonly used coefficient 

of determination (R2

) does not expect a 1:1 line fit. However, 

for the model development analyses presented in this report, 

the NSE values are equivalent to calculated R2 values. 

Methods for Calculation of Revised Estimates of 

Historical Streamflows

New regression equations developed in this study were 

used to calculate revised estimates of historical streamflows of 

the St. Croix River at Stillwater and Prescott from measured 

streamflows at St. Croix Falls. Revised historical estimates 

were compared to previous historical estimates derived from 

equations 2 and 5 for Stillwater and Prescott, respectively 

(Kloiber, 2004; Triplett and others, 2009; LaFrancois and others, 2009). Estimates of total monthly streamflow were compared using previous and new regression equations to facilitate 

future comparisons of estimated historical nutrient loads at 

Stillwater and Prescott. 

Methods for Evaluation of Revised Estimates of 

Historical Streamflows

The accuracies of new regression equations for estimating historical streamflows of the St. Croix River at Stillwater 

and Prescott were evaluated by comparing historical estimates 

of streamflow to discrete streamflow measurements made 

before the installation of index-velocity streamgages. Estimated daily streamflows and discrete measured streamflows 

were compared using the percent difference equation:

Percent Difference = * 100 Qe

 – Qm

Qe

(8)

where Qe is the estimated daily mean streamflow and Qm is 

the discrete measured streamflow. Equation 8 was chosen to 

calculate percent differences to obtain negative values when 

estimated streamflows were less than measured streamflows 

and to obtain positive values when estimated streamflows were 

greater than measured streamflows. At Stillwater, nine discrete 

streamflow measurements were made from 2001 through 

2011, and stage data from the USACE stage-only streamgage 

were available for all measurements. Therefore, the ranges 

of stages observed on the days of the discrete measurements 

were used to assess how variability in daily streamflow may 

affect calculated differences between discrete streamflow 

measurements and daily estimates of streamflow. However, 

stage data were not available for discrete streamflow measurements at Prescott, so the potential effects of daily variability 

in streamflow on the calculated differences between discrete 

measurements and daily estimates of streamflow could not be 

examined for that location. At Prescott, 16 discrete streamflow 

measurements were made between 1928 and 2007. However, 

no measurements were made between 1972 and 2000. Little 

information exists about the methods used to make discrete 

measurements in 1972 and earlier, but the methods used to 

measure discharge may not have been as accurate as current 

hydroacoustic methods (Rantz and others, 1982; Levesque and 

Oberg, 2012).

After calculating previous and revised estimates of 

historical daily streamflows for the St. Croix River at Stillwater and Prescott from 1910 to the start of index-velocity 

records (appendixes 2 and 3, respectively), general comparisons were made between previous and revised estimates. 

Comparisons included streamflows during high-flow months 

(March through June) and low-flow months (July through 

September), streamflows during high-flow and low-flow years, 

streamflows throughout 30-year dry and wet periods, and 

mean streamflows throughout the entire period of estimated 

record. Because of serial correlation, additional significance 

tests, confidence intervals, and estimates of variance were not 

determined.
Development of Regression Equations to Estimate Historical Streamflows 7

Development of Regression Equations 

to Estimate Historical Streamflows

New regression equations were developed to improve 

historical streamflow estimates of the St. Croix River at 

Stillwater and Prescott. The new regression equations are 

presented in this section of the report along with an evaluation 

of their accuracy compared to previous equations.

Regression Equations for St. Croix River at 

Stillwater, Minnesota

Published continuous streamflow records (U.S. Geological Survey, 2014) for St. Croix Falls (USGS streamgage 

05340500), Apple River (USGS streamgage 05341500), 

and the recently (2011) installed Stillwater index-velocity 

streamgage (USGS streamgage 05341550) were used to compare previous equations for estimating historical streamflow 

and develop a new linear regression equation that improves 

estimates of historical streamflows of the St. Croix River 

at Stillwater. Several candidate regression equations were 

developed using combinations of the following explanatory 

variables: QSCF , QSCF_lagged_1_day , QApple , and QApple_lagged_1_day. The 

simplest and most accurate regression equation (based on NSE 

values) for estimating streamflow at Stillwater is shown in 

equation 9.

QStillwater = (0.968 * QSCF_lagged_1_day) + 770 (9)

Equation 9 can be used to compute revised estimates of historical streamflows at Stillwater.

The abilities of previous regression equations (equations 

1‒3) and the new regression equation (equation 9) to estimate 

streamflows at Stillwater were compared using the measured 

streamflow record for the index-velocity streamgage at Stillwater (water years 2012 and 2013) and NSE values. The NSE 

values for selected regression equations are presented in 

table 2.

All compared equations did a reasonably good job of 

accurately fitting measured streamflow at Stillwater, with NSE 

values of 0.896 or greater (table 2). Incorporating a 1-day time 

lag (equations 3 and 9) resulted in the largest increase in NSE 

values among equations. However, the developed regression 

equation (equation 9) estimated streamflow at Stillwater better than previous equations (equations 1‒3), which generally 

overestimated peak flows and underestimated base flows 

(fig. 2). In addition, because the period of continuous record 

for the St. Croix Falls streamgage (05340500) extends back 

more to 1910, equation 9 likely produces the most accurate 

estimates for the longest period of time. 

The constant term and 1-day time lag in equation 9 

may produce more accurate streamflow estimates for several 

reasons. The constant term may account for a certain amount 

of base flow or groundwater contributions between St. Croix 

Falls and Stillwater. During low flows, the actual time lag 

may be longer than 1 day, but wind and wave action may 

cause enough variation in flows to mix out the effects of the 

time lag; furthermore, the constant term may help dampen the 

variations caused by wind and wave action at lower flows. 

The improvements in estimation of high flows with the time 

lag indicates that the time lag may be especially important at 

high flows because of the rapid rate of change in flows and 

water levels. Any effects of increased flow at Apple River are 

negated by the more dramatic increases in flow at St. Croix 

Falls. Finally, water velocities and time of travel likely are 

strongly affected by hydrologic damming at Prescott.

Because only 2 full years of measured streamflow data 

from the index-velocity streamgage were available for analysis 

of the goodness-of-fit for the regression equations developed 

for Stillwater, additional analyses were used to evaluate the 

regression method for a variety of flow regimes. Separate 

regression equations were developed for water years 2012 and 

2013 using streamflow at St. Croix Falls with a 1-day time lag 

(QSCF_lagged_1_day) as the explanatory variable and streamflow at 

Stillwater as the response variable. Equation 10 was developed 

using streamflow data from water year 2013 data and was 

used to estimate streamflows at Stillwater for water year 2012 

[QStillwater(WY2012)].

QStillwater(WY2012) = (0.960 * QSCF_lagged_1_day) + 907 (10)

Equation 11 was developed using streamflow data from water 

year 2012 and was used to estimate streamflows at Stillwater 

for water year 2013[QStillwater(WY2013)]. 

QStillwater(WY2013) = (0.971 * QSCF_lagged_1_day) + 680 (11)

The NSE values used to evaluate the abilities of equations 

10 and 11 to estimate streamflow for the other water year are 

presented in table 3, and time-series plots comparing measured 

and estimated streamflows are presented in figure 3. The NSE 

Table 2. Nash-Sutcliffe Efficiencies that evaluate the abilities 

of selected regression equations to fit measured streamflows 

(water years 2012–13) of the St. Croix River at Stillwater, 

Minnesota (U.S. Geological Survey streamgage 05341550).

[Equation number, the number of the corresponding equation in the body of 

the report and listed in table 1‒1; Figure label, the label corresponding to a 

time-series plot in figure 2; A‒D, letters of corresponding time-series plots 

in figure 2; Number of values used, number of data points used to calculate 

NSE values; NSE, Nash-Sutcliffe Efficiency (Nash and Sutcliffe, 1970)]

Equation 

number Figure label Number of values used1 NSE

1 A 444 0.913

2 B 444 0.896

3 C 443 0.970

9 D 444 0.979

1

Estimated values in published streamflow records (U.S. Geological 

Survey, 2014) were omitted from analyses.
8 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

Figure 2. Time-series plots used to visually compare measured streamflows (water years 2012–13) for the St. Croix River 

at Stillwater, Minnesota (U.S. Geological Survey streamgage 05341550) to streamflows estimated using selected regression 

equations from table 1. A, equation 1; B, equation 2; C, equation 3; and D, equation 9.

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

Streamflow, in cubic feet per second

O N D J F M A M J J A S O N D J F M A M J J A S

 A

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

Streamflow, in cubic feet per second

O N D J F M A M J J A S O N D J F M A M J J A S

 B

 Estimated streamflow using equation 1

 Measured streamflow

EXPLANATION

rol15-EMNQ00-0014_fig02

2012 2013

Water year

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]

 Estimated streamflow using equation 2

 Measured streamflow

EXPLANATION

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]
Development of Regression Equations to Estimate Historical Streamflows 9

Figure 2. Time-series plots used to visually compare measured streamflows (water years 2012–13) for the St. Croix River 

at Stillwater, Minnesota (U.S. Geological Survey streamgage 05341550) to streamflows estimated using selected regression 

equations from table 1. A, equation 1; B, equation 2; C, equation 3; and D, equation 9.—Continued

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

O N D J F M A M J J A S O N D J F M A M J J A S

 C

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

O

 D

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

O N D J F M A M J J A S O N D J F M A M J J A S

 C

Streamflow, in cubic feet per second Streamflow, in cubic feet per second

rol15-EMNQ00-0014_fig02 Cont.

2012 2013

Water year

 Estimated streamflow using equation 3

 Measured streamflow

EXPLANATION

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]

 Estimated streamflow using equation 9

 Measured streamflow

EXPLANATION

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]
10 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

values and the time-series plots demonstrate that the regression method is fairly robust throughout varied streamflow 

conditions.

Regression Equations for St. Croix River at 

Prescott, Wisconsin

Published continuous streamflow records (U.S. Geological Survey, 2014) for St. Croix Falls (USGS streamgage 

05340500), Stillwater (USGS streamgage 05341550), 

Willow River (USGS streamgage 05341752), Kinnickinnic River (USGS streamgage 05342000), and the recently 

(2007) installed Prescott index-velocity streamgage (USGS 

streamgage 05344490) were used to compare previous equations for estimating historical streamflows and develop a new 

linear regression equation that can be used to improve historical streamflow estimates for the St. Croix River at Prescott. 

The following explanatory variables were used to construct 

several candidate linear regression equations: QSCF,, QApple, 

QWillow, QKinnickinnic, QMiss_Prescott, QSt.Paul, QMWTF. Candidate regression equations incorporated various combinations of time 

lags for included explanatory variables, and incorporated time 

lags ranged from 1 to 4 days. The simplest and most accurate regression equation (based on NSE values) for estimating streamflow at Prescott used and as the two explanatory 

variables:

QPrescott = (0.585 * QSCF_lagged_1_day) + (0.415 * QSCF_lagged_3_day) + 873 

 (12)

Equation 12 can be used to compute revised estimates of historical streamflows for the St. Croix River at Prescott. The two 

separate time lags may account for responses to precipitation 

events in different parts of the basin and changes in water storage between Stillwater and Prescott. The abilities of previous 

and revised equations to estimate streamflow at Prescott were 

compared using measured streamflow record at Prescott (water 

years 2008 through 2013) and NSE values. The NSE values 

for selected predictor models are presented in table 4.

The NSE values from equations used to estimate streamflows at Prescott ranged from 0.708 to 0.941 (table 4), 

indicating that streamflow at Prescott is more difficult 

to accurately estimate than streamflow at Stillwater (table 

2). In addition to the dynamics of water storage upstream in 

Lake St. Croix, streamflow of the St. Croix River at Prescott 

is affected immediately downstream by the confluence with 

the Mississippi River. The hydrology of the Mississippi River 

is complex because the channel is maintained for commercial 

navigation and controlled by many dams that hold or release 

water for a variety of uses (Upper Mississippi River Basin 

Commission, 1982; Lubinski and others, 1991), including 

Lock and Dam No. 2 (not shown) above Prescott and Lock 

and Dam No. 3 (not shown) below Prescott. In addition, the 

Mississippi River is affected upstream from Prescott by the 

confluence with the Minnesota River (fig. 1). Most of the Minnesota River Basin occupies lower latitudes than the Upper 

Mississippi and St. Croix River Basins (Upper Mississippi 

River Basin Commission, 1982; Lubinski and others, 1991). 

Therefore, precipitation events in the Minnesota River Basin 

can strongly affect the hydrologic damming that occurs at 

Prescott even when precipitation events do not occur in the 

Upper Mississippi and St. Croix River Basins. Furthermore, 

Mississippi River streamflows measured downstream from 

the confluence with the Minnesota River (USGS streamgage 

05331000) can be double the streamflows measured upstream 

from the confluence with the Minnesota River (USGS 

streamgage 05288500; U.S. Geological Survey, 2014). 

The newly developed regression equation (equation 12) 

estimated streamflow at Prescott better than previous equations (equations 4‒6) based on NSE values (table 4) and 

graphical plots (fig. 4). Equation 6 does a reasonable job 

Table 3. Nash-Sutcliffe Efficiencies for regression equations developed using measured 

streamflows of the St. Croix River at St. Croix Falls (U.S. Geological Survey streamgage 05340500) 

from one water year to fit measured streamflows for another water year at the St. Croix River at 

Stillwater, Minnesota (U.S. Geological Survey streamgage 05341550) to validate the regression 

method used to develop equation 9 for estimating historical streamflows at the St. Croix River at 

Stillwater, Minnesota.

[Equation number, the number of the corresponding equation in the body of the report and listed in table 1‒1; 

Figure label, the label corresponding to a time-series plot in figure 3; A‒B, letters of corresponding time-series 

plots in figure 3; Development year, the water year of the measured streamflow data used to develop the corresponding regression equation; Prediction year, the water year of measured streamflow data being estimated 

using the regression equation derived from the development year; Number of values used, the number of daily 

streamflow values in the prediction year that were used in calculating NSE values; NSE, Nash-Sutcliffe Efficiency 

(Nash and Sutcliffe, 1970)]

Equation 

number Figure label Development year

Prediction 

year

Number of 

values used1 NSE

10 A 2013 2012 239 0.981

11 B 2012 2013 223 0.970

1

Estimated values in published streamflow records (U.S. Geological Survey, 2014) were omitted from analyses.
Development of Regression Equations to Estimate Historical Streamflows 11

0

20,000

40,000

Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept.

2012

2013

Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept.

 A

0

20,000

40,000

 B

Streamflow, in cubic feet per second Streamflow, in cubic feet per second

rol15-EMNQ00-0014_fig03

Water year

 Estimated streamflow using equation 10

 Measured streamflow

EXPLANATION

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]

 Estimated streamflow using equation 11

 Measured streamflow

EXPLANATION

[Water year is the 12-month period, October 1 through September 30, 

and is designated by the calendar year in which it ends]

Figure 3. Time-series plots of measured and estimated streamflow for the St. Croix River at Stillwater, Minnesota, using 

regression equations based on streamflow data from the St. Croix River at St. Croix Falls (U.S. Geological Survey streamgage 

05340500) for 2 water years. A, estimated streamflow for water year 2012 based on records from water year 2013 (equation 10, 

table 1–1); and B, estimated streamflow for water year 2013 based on records from water year 2012 (equation 11, table 1–1). 
12 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

estimating peak discharges. However, at low flows, streamflows estimated using equation 6 often were substantially 

overestimated or actually negative (fig. 4C). In contrast, equation 5 consistently underestimates base flow and substantially 

overestimates peak flows (fig. 4B). Because the streamflow 

record for Willow River is brief (only water year 2013) and 

the Stillwater component for equation 4 is estimated before 

September 2011, equation 4 is the least useful predictor equation. Although a measured streamflow record at Stillwater is 

available for water year 2013, equation 2 was used to estimate 

the streamflow at Stillwater used in equation 4 (table 4; fig. 4) 

to match previously used historical estimates of streamflow. 

When the measured streamflow record from Stillwater was 

used in equation 4, the NSE value increased from 0.758 (table 

4) to 0.897. However, even when the measured Stillwater 

streamflows are used, equation 4 generally underestimates 

low flows. Equation 12 most accurately estimates measured 

discharges at Prescott and does not consistently overestimate 

peak flows and underestimate low flows (fig. 4D). In addition, 

because the St. Croix Falls record extends back more than 

100 years, equation 12 likely produces the most accurate estimates for the longest period of time.

Computation and Evaluation of Revised 

Estimates of Historical Streamflows

Revised estimates of historical streamflows for the St. 

Croix River at Stillwater and Prescott were computed and are 

described in this section of the report. The revised estimates of 

historical streamflows were evaluated using discrete streamflow measurements.

Computation of Revised Historical Streamflows 

for the St. Croix River at Stillwater, Minnesota

The newly developed regression equation (equation 9) 

was used to compute revised estimates of historical streamflows for the St. Croix River at Stillwater for water years 

1910‒2011. The revised estimates of historical streamflows for 

Stillwater were compared to previous estimates of streamflows 

that were calculated using equation 2. Although equation 2 

was not the most accurate equation of the previous estimation 

equations (table 2), equation 2 provided a consistent estimation method for the longest period of record and was the equation most used in previous studies (Kloiber, 2004; Triplett and 

others, 2009; LaFrancois and others, 2009). Annual comparisons of previous and revised estimates of historical streamflows at Stillwater are presented in appendix 2. 

Because of the short duration of the continuously measured streamflows at the index-velocity streamgage at Stillwater (water years 2012‒13), some measured streamflows at 

St. Croix Falls (explanatory variable) used in the computation 

of revised historical streamflows at Stillwater (water years 

1910‒2011) were outside the range of streamflows used to 

develop equation 9, which subsequently was used to generate 

revised estimates of historical streamflows at Stillwater. However, the new regression equation still incorporates 95 percent 

of the range of daily streamflows measured at St. Croix Falls 

for the historical streamflow period (water years 1910‒2011) 

(table 5). Although the accuracy of the equation 9 cannot be 

evaluated outside of the range of measured streamflow at St. 

Croix Falls used in the analysis, equation 9 likely provides 

an improvement compared to previous methods of estimating 

historical streamflows (equations 1‒3). Selected statistics used 

to evaluate the regression datasets in relation to the period of 

record at St. Croix Falls for Stillwater are listed in table 5.

Computation of Revised Historical Streamflows 

for the St. Croix River at Prescott, Wisconsin

The newly developed regression equation (equation 12) 

was used to compute revised estimates of historical streamflows for the St. Croix River at Prescott for water years 

1910‒2007. Revised estimates of historical streamflows for 

Prescott were compared to previous estimates of historical 

streamflows that were calculated using equation 5 for several 

reasons. First, equation 5 was the most accurate of the previous regression equations (table 4). Second, equation 5 provided a consistent estimation method for the longest period of 

record. Third, equation 5 was used the most in previous studies (Kloiber, 2004; Triplett and others, 2009; LaFrancois and 

others, 2009). Annual comparisons of previous and revised 

Table 4. Nash-Sutcliffe Efficiencies that evaluate the abilities 

of selected regression equations equations to fit measured 

streamflows (water years 2008–13) of the St. Croix River at 

Prescott, Wisconsin (U.S. Geological Survey streamgage 

05344490 for water years 2008–13 records).

[Equation number, the number of the corresponding equation in the body of 

the report and listed in table 1‒1; Figure label, the label corresponding to a 

time-series plot in figure 4; A‒D, letters of corresponding time-series plots 

in figure 4; Number of values used, the number of daily streamflow values 

used in calculating NSE values; NSE, Nash-Sutcliffe Efficiency (Nash and 

Sutcliffe, 1970)]

Equation 

number Figure label Number of values used1 NSE

4 A 2

212 0.758

5 B 1,733 0.774

6 C 1,255 0.708

12 D 1,752 0.941

1

Estimated values in published streamflow records (U.S. Geological 

Survey, 2014) were omitted from analyses.

2

Only streamflow data for water year 2013 were available for the Willow 

River at Willow River State Park near Burkhardt, Wisconsin (U.S. Geological Survey streamgage 05341752).
Computation and Evaluation of Revised Estimates of Historical Streamflows 13

estimates of historical streamflows at Prescott are presented in 

appendix 3. 

Because of the relatively short duration of continuously 

measured streamflows at the index-velocity streamgage at 

Prescott (water years 2008-2013), some measured streamflows 

at St. Croix Falls (explanatory variable) used in the computation of revised historical streamflows at Prescott (water years 

1910‒2007) were outside the range of streamflows used to 

develop equation 12. However, the new regression equation 

still incorporates 98 percent of daily streamflows measured 

at St. Croix Falls for the historical streamflow period (water 

years 1910‒2007) (table 5). Although the accuracy of equation 12 cannot be evaluated outside the range of measured 

streamflows at St. Croix Falls used in the analysis, equation 

12 likely improves estimates of extreme historical streamflows 

(equations 4‒6). Selected statistics of data used to evaluate 

the regression datasets in relation to the period of record at St. 

Croix Falls for Prescott are listed in table 5.

Evaluation of Revised Estimates of Historical 

Streamflows at Stillwater, Minnesota 

Revised estimates of historical discharges at Stillwater 

computed using equation 9 were compared to nine discrete 

streamflow measurements made from 2001 to 2011. Five 

of the measurements were made by USGS personnel, and 

the data are publicly available in the NWIS database (U.S. 

Geological Survey, 2014). The other four measurements were 

made by MCES staff (Scott Schellhaas, Metropolitan Council 

Environmental Services, written commun., 2014). Discrete 

streamflow measurements generally were made in the span of 

about an hour. The results of these comparisons are presented 

in table 6. 

Percent-difference data in table 6 indicate that equation 

9 does a fairly good job of estimating historical streamflows 

at Stillwater. Revised estimates of historical streamflows 

were within 10 percent of discrete streamflows measured 

by USGS hydrographers (appendix 2). In addition, revised 

estimates were within 18 percent of all discrete streamflow 

measurements. 

Stage data (table 6) provide additional insight into the 

ability of equation 9 to estimate daily streamflows at Stillwater. During periods of falling stage, a discrete streamflow 

measured early in the day, intuitively, would be higher than 

the estimated daily streamflow. Six of the seven measurements 

made during falling stages had discrete streamflow measurements that were greater than daily estimated streamflows 

(table 6). Similarly, during periods of rising stage, a discrete 

streamflow measured early in the day, intuitively, would 

be lower than an estimated daily streamflow. For the single 

measurement made during a rising stage, the discrete streamflow measurement discharge was less than the estimated daily 

streamflow (table 6). For the one measurement made during 

a period of stable stage, the discrete streamflow measurement 

was within 2.1 percent of the estimated daily streamflow 

Table 5. Selected statistics for measured streamflows of the St. Croix River at St. Croix Falls, Wisconsin (U.S. Geological Survey 

streamgage 05340500) used to compute revised estimates of historical streamflows for the St. Croix River at Stillwater, Minnesota and 

Prescott, Wisconsin, and minimum and maximum estimated streamflows at Stillwater and Prescott using new regression equations.

[Site of estimated streamflow, location for which new regression equations were applied to improve estimates of historical streamflows; ft3/s, cubic feet per 

second]

Site of estimated 

streamflow

Minimum 

streamflow 

for explanatory variable1

(ft3

/s)

Percentage 

historical 

streamflow2

 below 

minimum 

streamflow 

for explanatory variable1

Maximum 

streamflow 

for explanatory variable1

(ft3

/s)

Percent of 

historical 

streamflow2

 above 

maximum 

streamflow 

for explanatory variable1

Percent of 

historical 

streamflow2

within range 

of streamflows for 

explanatory 

variable1

Minimum 

estimated 

streamflow 

(ft3

/s)

Maximum 

estimated 

streamflow 

(ft3

/s)

St. Croix River at Stillwater, Minnesota 

3

1,400 35.25 336,800 30.14 394.6 52,120 536,400

St. Croix River at 

Prescott, Wisconsin 

4

1,130 41.44 436,800 40.15 498.4 62,020 636,100

1

Explanatory variable is QSCF_lagged_1_day, which represents streamflow of the St. Croix River at St. Croix Falls, Wisconsin (U.S. Geological Survey 

streamgage 05340500) lagged by 1 day. 

2

Historical streamflow is measured streamflow of the St. Croix River at St. Croix Falls, Wisconsin (U.S. Geological Survey streamgage 05340500) for 

water years 1910‒2011 for St. Croix River at Stillwater, Minnesota, and for water years 1910‒2007 for St. Croix River at Prescott, Wisconsin.

3

Period of record for explanatory variable used in developing regression equation for St. Croix River at Stillwater, Minnesota, was water years 2012‒13.

4

Period of record used for explanatory variable in developing regression equation for St. Croix River at Prescott, Wisconsin, was water years 2008‒13.

5

Period of estimated streamflow is water years 1910‒2011.

6

Period of estimated streamflow is water years 1910‒2007.
14 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

(table 6). The percent-difference comparisons between daily 

estimated streamflows and discrete streamflow measurements provides additional evidence that the newly developed 

equation 9 can be used to accurately back-calculate historical 

streamflows at Stillwater.

The comparisons in table 6 indicate that equation 9 provides reasonable estimates of historical streamflows. However, the ability of equation 9 to accurately estimate historical 

streamflow before 2001 could not be evaluated because all 

previous discrete streamflow measurements were made within 

a relatively short (10-year) period of record. Conversely, the 

channel between St. Croix Falls and Stillwater is completely 

riverine, so factors like water storage within Lake St. Croix 

are less likely to confound the relation between St. Croix Falls 

and Stillwater throughout the period of record. Therefore, 

historical estimates of streamflows computed from equation 9 

likely are reasonably accurate throughout the period of record 

for the St. Croix Falls streamgage (05340500) used to estimate 

streamflows at Stillwater (water years 1910‒2011). 

Comparisons of previous and revised estimates of historical streamflows at Stillwater using equations 2 and 9, respectively (appendix 2), indicate noticeable differences between 

streamflow estimation equations. Monthly and annual comparisons support the findings of the regression analyses that previous regression equations generally underestimated low flows 

and overestimated high flows compared to the new regression 

equation (fig. 2). During the relatively dry, low-flow period 

of water years 1910‒40, the previously estimated mean 

streamflow of 3,600 cubic feet per second (ft3

/s) was 

310 ft3/s less than the revised estimated mean streamflow of 

3,910 ft3/s. Also, during the relatively wet, high-flow period 

of water years 1970‒2000, the previously estimated mean 

streamflow of 5,560 ft3

/s was 60 ft3/s less than the revised 

estimated mean streamflow of 5,620 ft3/s. For the period of 

estimated historical streamflows (water years 1910‒2011), the 

previously estimated mean streamflow of 4,850 ft3/s was 

150 ft3/s less than the revised mean streamflow of 5,000 ft3/s, 

indicating that over the historical period, overestimates of 

peak flows were not high enough to balance the underestimates of low-flow conditions at Stillwater.

The differences between previous and revised estimates 

of historical streamflows may demonstrate the effects of 

groundwater contributions on streamflows at Stillwater. Equation 2 used a simple scalar and generally underestimated low 

flows and overestimated peak flows in comparison to equation 

9. The constant in equation 9 (y-intercept; 770) may represent 

a minimum base-flow value that was not accounted for in 

equation 2. In addition, the constant may better account for 

geologic differences between St. Croix Falls and Stillwater. 

Downstream from St. Croix Falls, increased soil and bedrock 

permeability may allow the surrounding watershed to capture 

a greater portion of the water budget (Juckem, 2007), resulting in less peak runoff and greater base flow. However, the 

hydrograph separation analyses needed to quantify base flow 

and peak flows at St. Croix Falls and Stillwater are beyond the 

scope of this report.

Evaluation of Revised Estimates of Historical 

Streamflows at Prescott, Wisconsin

Revised estimates of historical streamflows for the St. 

Croix River at Prescott computed using equation 12 were 

compared to 16 discrete streamflow measurements made 

from 1928 to 2007. Because daily streamflow estimates were 

Table 6. Comparisons between estimated historical daily mean streamflows and discrete streamflow measurements made before 

the installation of an index-velocity streamgage at the St. Croix River at Stillwater, Minnesota.

[Agency, the agency that made the discrete streamflow measurement with the corresponding date (USGS, U.S. Geological Survey; MCES, Metropolitan 

Council Environmental Services); measurement location (A, streamflow measurement location in a confined portion of the St. Croix River located 3 miles 

upstream from the Stillwater Lift Bridge; B, a streamflow measurement location 50 feet upstream from the Stillwater Lift Bridge); discrete Qm, discrete 

measured streamflow; ft3/s, cubic feet per second; daily Qe, daily mean streamflow estimated using equation 9 in the text and listed in table 1‒1 for the corresponding date; percent difference, the percent difference between the discrete Qm and the daily Qe estimated using equation 8 from the text and listed in 

table 1‒1; range, the range of measured stage (minimum to maximum) for the corresponding date; stage pattern, the general pattern of the stage data for the 

corresponding date]

Date of discrete 

streamflow 

measurement

Time of discrete 

streamflow 

measurement

Agency Measurement 

location

Discrete Qm

(ft3

/s)

Daily Qe

(ft3/s)

Percent 

difference

Stage range 

(feet)

Stage 

pattern

09/18/2001 11:27 USGS A 2,890 3,090 +6.47 0.08 Rising

10/04/2007 11:00 MCES B 4,940 4,220 -17.1 0.18 Falling

10/30/2007 11:00 MCES B 9,000 7,880 -14.2 0.41 Falling

06/17/2008 10:45 MCES B 14,100 14,400 +2.08 0.03 Stable

04/21/2011 10:36 USGS A 10,800 10,600 -1.89 0.33 Falling

05/06/2011 10:17 USGS A 11,700 12,500 +6.40 0.16 Falling

06/06/2011 11:30 MCES B 9,370 8,960 -4.58 0.28 Falling

07/28/2011 8:40 USGS A 4,620 4,320 -6.94 0.22 Falling

07/28/2011 9:37 USGS B 4,740 4,320 -9.72 0.22 Falling
Computation and Evaluation of Revised Estimates of Historical Streamflows 15

−5,000

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

2007 2008 2009 2010 2011 2012 2013

 A

Streamflow, in cubic feet per second

−5,000

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

2007 2008 2009 2010 2011 2012 2013

 B

Streamflow, in cubic feet per second

rol15-EMNQ00-0014_fig04

Year

 Estimated streamflow using equation 4

 Measured streamflow

EXPLANATION

 Estimated streamflow using equation 5

 Measured streamflow

EXPLANATION

Figure 4. Time-series plots used to visually compare measured streamflows (water years 2008–13) for the St. Croix River 

at Prescott, Wisconsin (U.S. Geological Survey streamgage 05344490) to streamflows estimated using selected regression 

equations from table 3. A, equation 4; B, equation 5; C, equation 6; and D, equation 12.
16 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

−5,000

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

2007 2008 2009 2010 2011 2012 2013

 C

−5,000

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

2007 2008 2009 2010 2011 2012 2013

 D

Streamflow, in cubic feet per second Streamflow, in cubic feet per second

rol15-EMNQ00-0014_fig04 Cont.

Year

 Estimated streamflow using equation 6

 Measured streamflow

EXPLANATION

 Estimated streamflow using equation 12

 Measured streamflow

EXPLANATION

Figure 4. Time-series plots used to visually compare measured streamflows (water years 2008–13) for the St. Croix River 

at Prescott, Wisconsin (U.S. Geological Survey streamgage 05344490) to streamflows estimated using selected regression 

equations from table 3. A, equation 4; B, equation 5; C, equation 6; and D, equation 12.—Continued
Computation and Evaluation of Revised Estimates of Historical Streamflows 17

compared to discrete streamflow measurements made in the 

span of about an hour, the discrete measurements may not 

capture all of the variability from precipitation, wind, or dam 

operation changes (on the St. Croix and Mississippi Rivers) 

that occurs in the course of a day at Prescott. The results of 

these comparisons are presented in table 7.

Percent-difference data in table 7 indicate that equation 

12 is less accurate at estimating historical streamflows for 

Prescott than equation 9 is for estimating historical streamflows for Stillwater (table 6). However, estimation methods 

presented in this report indicate that Prescott is a hydrologically more complex location than Stillwater because of water 

storage in Lake St. Croix, contributions from two major 

upstream tributaries, and the confluence with the Mississippi 

River immediately downstream. No historical stage data 

are available for Prescott to help assess daily variations in 

streamflows, but daily estimated streamflows calculated using 

equation 12 overestimated discrete measurements for 10 of the 

16 streamflow measurements. This tendency of overestimating 

measured values may indicate the backwater effects created by 

the confluence with the larger Mississippi River. In addition, 

the complexities of the site at Prescott likely result in larger 

within-day fluctuations in streamflow compared to Stillwater. 

Therefore, discrete streamflow measurements may not capture 

enough of the daily variability to be representative of daily 

mean streamflows. Furthermore, hydroacoustic instruments 

were not available for discrete streamflow measurements made 

during 1928‒72 (table 7). Traditional methods of measuring 

streamflow may not be as accurate for sites like Prescott that 

have extremely variable flows affected by several factors, 

including upstream water storage in Lake St. Croix, backwater from the confluence with the Mississippi River, and wind 

direction (Rantz and others, 1982; Levesque and Oberg, 2012).

Comparisons of previous and revised estimates of 

historical streamflows at Prescott using equations 5 and 

12, respectively (appendix 3), indicate noteable differences 

between the two streamflow estimation equations. Monthly 

and annual comparisons support the findings of regression 

analyses that previous regression equations generally underestimated low flows and overestimated high flows compared 

to the new regression equations (fig. 4). During the relatively 

dry, low-flow period of water years 1910‒40, the previously 

estimated mean streamflow of 3,830 ft3

/s was 290 ft3/s less 

than the revised estimated mean streamflow of 4,120 ft3/s. In 

contrast, during the relatively wet, high-flow period of water 

years 1970‒2000, the previously estimated mean streamflow of 5,910 ft3

/s was 30 ft3/s greater than the revised mean 

streamflow of 5,880 ft3/s. For the period of estimated historical 

streamflows (water years 1910‒2007), the previously estimated mean streamflow of 5,140 ft3

/s was 90 ft3/s less than the 

revised estimated mean streamflow of 5,230 ft3/s, indicating 

that throughout the period of estimated historical streamflows, 

overestimates of peak flows were not high enough to balance 

the underestimates of low-flow conditions. However, the difference between previous and revised mean streamflows was 

less at Prescott than at Stillwater.

The differences between previous and revised estimates 

of historical streamflows may indicate the effects of groundwater contributions on streamflow at Prescott. Equation 5 

used a simple scalar and generally underestimated base flows 

and overestimated peak flows in comparison to equation 12. 

The constant in equation 12 (y-intercept; 873) may represent 

a minimum base-flow value that was not accounted for in 

equation 5. In addition, the constant may better account for 

geologic differences between St. Croix Falls and Prescott. 

Downstream from St. Croix Falls, increased soil and bedrock 

permeability may allow the surrounding watershed to capture 

a greater portion of the water budget (Juckem, 2007), resulting 

in less peak runoff and greater low flows. Furthermore, the difference between previous and revised mean streamflow values 

may be lower at Prescott than Stillwater because of the effects 

of water storage in Lake St. Croix and because of larger 

groundwater contributions to Lake St. Croix from the Willow 

and Kinnickinnic Rivers. However, the hydrograph separation 

analyses needed to quantify low flows and peak flows at St. 

Croix Falls and Prescott are beyond the scope of this report.

Table 7. Comparisons between estimated historical daily mean 

streamflows and discrete streamflow measurements made 

before the installation of an index-velocity streamgage at the 

St. Croix River at Prescott, Wisconsin.

[discrete Qm, discrete streamflow measured by U.S. Geological Survey 

hydrographers; daily Qe, daily mean streamflow estimated using equation 

12 in the text and listed in table 1‒1 for the corresponding date; ft3/s, cubic 

feet per second; percent difference, the percent difference between the 

discrete Qm and the daily Qe using equation 8 from the text and listed in 

table 1‒1]

Date of discrete 

streamflow 

measurement

Discrete Qm

(ft3

/s)

Daily Qe

(ft3/s)

Percent

 difference

05/25/1928 5,620 4,890 -14.9

08/25/1928 2,640 3,430 +23.0

03/05/1930 5,980 3,550 -68.5

06/01/1933 3,970 5,290 +25.0

04/09/1934 5,290 5,640 +6.21

03/21/1935 3,880 5,480 +29.2

09/16/1953 5,200 3,680 -41.3

08/01/1957 4,620 5,400 +14.4

07/24/1962 2,960 3,230 +8.36

07/13/1965 2,590 4,070 +36.4

05/01/1967 5,070 6,150 +17.6

04/01/1969 2,920 5,960 +51.0

06/19/1972 3,960 3,780 -4.76

08/17/2000 3,300 3,820 +13.6

05/02/2006 6,030 5,870 -2.73

07/10/2007 2,470 2,290 -7.86
18 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

Limitations of the Study

Several limitations should be considered when applying results obtained from analyses presented in this report. 

The regression analyses used to develop equations 9 and 12 

for estimating revised historical streamflows at Stillwater and 

Prescott, respectively, violate some assumptions of regression analyses (Helsel and Hirsch, 2002). Because data from 

upstream streamgages are being used to estimate downstream 

streamflows, the explanatory and response variables are serially correlated. The variance of the residuals is not constant 

because the variance is dependent on the explanatory variables 

and time. In addition, the residuals are not normally distributed nor independent. Therefore, the analysis of variance 

(ANOVA) tables generated from the regression analyses are 

not presented in this report, and inferences should not be made 

about the variances of sample populations. The assumptions 

that are not violated in these analyses are that streamflows at 

Stillwater and Prescott are linearly related to the explanatory 

variables, and the data used to fit the models are representative 

of the data of interest. The non-violated assumptions allow 

the estimation of streamflows at Stillwater and Prescott given 

the explanatory variables used. However, the data presented 

in this report cannot be used to obtain a variance for estimated values, obtain an unbiased estimator of streamflow, or 

test hypotheses or estimate confidence intervals (Helsel and 

Hirsch, 2002). Because of the violated assumptions, NSE values (Nash and Sutcliffe, 1970) were used to assess the estimation powers of the hydrological equations used in this report. 

Other limitations of this study are a result of the short 

period of record for the index-velocity streamgages that were 

used to evaluate previous methods of estimating streamflow 

and develop new equations to revise estimates of historical 

streamflows. First, equations 9 and 12 should only be 

considered valid for the range of measured streamflows for 

the explanatory variable used in the regression analyses 

(table 5). Second, the relation between explanatory and 

response variables has not been examined for periods of ice 

cover because no streamflow measurements have been made 

under ice cover at Stillwater (for safety reasons). Third, 

the limited periods of measured streamflow records from 

Stillwater and Prescott used to develop equations 9 and 12, 

respectively, may not capture all the variability in the streamflow records throughout the entire period of record for the 

explanatory variables. Finally, changes to dam operation (on 

the St. Croix and Mississippi Rivers), climate, or land and 

water use within the lower St. Croix River Basin may have 

altered the relation between the explanatory streamflows from 

select streamgages and streamflows at the index-velocity 

streamgages at Stillwater and Prescott.

Summary

A natural dam of glacial-era sediments at the confluence 

of the St. Croix and Mississippi Rivers forms Lake St. Croix, 

a riverine lake that comprises the lowest 25 miles of the St. 

Croix River. In 1993, concerns about future water quality 

in the St. Croix River (including Lake St. Croix) prompted 

several agencies and organizations to form the St. Croix Basin 

Water Resources Planning Team to coordinate research and 

monitoring efforts in the St. Croix Basin. Streamflow measurements for the St. Croix River at the inlet to (near Stillwater, 

Minnesota) and outlet from (near Prescott, Wisconsin) Lake 

St. Croix were needed to estimate nutrient loads. However, 

backwater effects from the Mississippi River prevented the 

use of traditional streamgages for collecting continuous 

streamflow data. Therefore, previous studies used streamgages 

upstream from the inlet to and outlet from Lake St. Croix and 

streamflow-routing equations to estimate streamflows needed 

for nutrient-load calculations. In the late 1990s, advances in 

velocity meters enabled researchers with the U.S. Geological Survey (USGS) to develop index-velocity streamgages to 

measure continuous streamflow in backwater conditions using 

continuously measured velocities at the streamgages. Indexvelocity streamgages were installed on the St. Croix River at 

Prescott and Stillwater in 2007 and 2011, respectively. 

This report was prepared by the U.S. Geological Survey 

in cooperation with the St. Croix Watershed Research Station. 

The purposes of this report are to (1) describe the development 

of regression equations to estimate historical streamflows of 

the St. Croix River at Stillwater and Prescott, (2) describe the 

evaluation of the accuracies of new and previous equations 

used to estimate historical streamflows, and (3) present revised 

estimates and evaluations of estimates of historical streamflows for the St. Croix River at Stillwater, Minn. (water years 

1910‒2011), and at Prescott, Wisc. (water years 1910‒2007). 

Continuous streamflow data from new index-velocity 

streamgages, long-term upstream streamgages, and tributary 

streamgages were used to evaluate previous equations used 

to estimate historical streamflows and develop new regression equations to improve estimates of historical streamflow 

at Stillwater and Prescott. The abilities of previous and new 

equations to accurately estimate (fit) measured streamflows 

were evaluated using Nash-Sutcliffe Efficiency (NSE) values. 

Comparisons of NSE values and time-series plots indicated 

that new regression equations estimated measured streamflows 

more reliably than previous estimation equations. The NSE 

values at Stillwater improved from 0.90 to 0.98, and the NSE 

values at Prescott improved from 0.77 to 0.94. New regression 

equations were used to compute revised estimates of historical 

streamflows for Stillwater and Prescott. Revised estimates of 

historical streamflows were compared to the most commonly 

used previous streamflow estimates for Stillwater and Prescott. 

To assess the accuracies of revised estimates of historical 

streamflow, discrete streamflow measurements made before 

the installation of index-velocity streamgages were compared 

to revised estimates of historical daily streamflows. In general, 
References Cited 19

estimates closely approximated measured discrete streamflows 

at Stillwater. Estimates of streamflows at Prescott were more 

variable when compared to measured discrete streamflows. 

Because daily streamflow estimates were compared to discrete 

streamflow measurements made in the span of about an hour, 

the discrete measurements may not capture all of the variability from precipitation, wind, or dam operation changes (on the 

St. Croix and Mississippi Rivers) that occurs in the course of 

a day at Prescott. In general, Prescott is a hydrologically more 

complex site than Stillwater because of water storage in Lake 

St. Croix directly upstream, the contributions of two major 

tributaries between Stillwater and Prescott, and the confluence 

with the Mississippi River directly downstream.

Additional comparisons were made between previous and 

revised estimates of historical streamflows at Stillwater and 

Prescott to evaluate changes in the estimated historical values. 

Monthly and annual streamflow comparisons indicated that 

previous streamflow estimates generally underestimated low 

flows and overestimated peak flows at Stillwater and Prescott 

when compared to revised estimates. For a 30-year dry period 

(water years 1910‒40), the previously estimated mean streamflows were less than the revised estimated mean streamflows 

at Stillwater and Prescott. In contrast, for a 30-year wet period 

(water years 1970‒2000), the previously estimated mean 

streamflow at Stillwater was slightly less than the revised 

estimated mean streamflow, whereas the previously estimated 

mean streamflow at Prescott was slightly greater than the 

revised estimated mean streamflow. For the entire periods of 

historical streamflow estimates, previously estimated mean 

streamflows were less than revised estimated mean streamflows at Stillwater and Prescott. Constant terms (y-intercepts) 

in the new regression equations may better represent low-flow 

conditions compared to previous equations, and differences in 

the relations between low flows and peak flows among regression equations may indicate changes in groundwater contributions resulting from increased soil and bedrock permeability 

downstream from the streamgage at St. Croix Falls from 

which the measured streamflow record is used as an explanatory variable in the regression equations for Stillwater and 

Prescott.

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Appendixes 1–3
22 Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River

Appendix 1. Summary of all Equations Listed in Text of the Report

Table 1–1. Summary of all the equations listed in the text of the report.

[All streamflow variables are in units of cubic feet per second. QStillwater, estimated streamflow of the St. Croix River at Stillwater, Minnesota; QSCF, St. Croix 

River streamflow at St. Croix Falls, Wisconsin (U.S. Geological Survey streamgage 05340500); QApple, streamflow of the Apple River near Somerset, Wisconsin (U.S. Geological Survey streamgage 05341500); QSCF_lagged_1_day, streamflow of the St. Croix River at St. Croix Falls, Wisconsin (U.S. Geological Survey 

streamgage 05340500) lagged 1 day; QWillow, streamflow of the Willow River at Willow River State Park near Burkhardt, Wisconsin (U.S. Geological Survey 

streamgage 05341752); QKinnickinnic, streamflow of the Kinnickinnic River near River Falls, Wisconsin (U.S. Geological Survey streamgage 05342000); QPrescott, 

estimated St. Croix River streamflow at Prescott, Wisconsin; QMiss_Prescott; streamflow of the Mississippi River at Prescott, Wisconsin (U.S. Geological Survey 

streamgage 05344500); QSt.Paul; streamflow of the Mississippi River at St. Paul, Minnesota (U.S. Geological Survey streamgage 05331000); QMWTF, discharge 

from the Metropolitan Wastewater Treastment Facility; (QSt.Paul + QMWTF)lagged_1_day, sum of variables lagged by 1 day; NSE, Nash-Sutcliffe Efficiency; Qm,i, 

measured daily mean streamflow at the index-velocity streamgage for each day i; n, upper bound of the summation; Qe,i, estimated daily mean streamflow from 

the regression model for each day i; Q̅̅

m, mean of all measured daily streamflows for the examined period of record; QStillwater(WY2012), estimated St. Croix River 

streamflow for water year 2012 at Stillwater, Minnesota; QStillwater(WY2013), estimated St. Croix River streamflow for water year 2013 at Stillwater, Minnesota; 

QSCF_lagged_3_days, St. Croix River streamflow at St. Croix Falls, Wisconsin (U.S. Geological Survey streamgage 05340500) lagged 3 days. 

Equation 

number Equation Reference(s)

1 QStillwater = QSCF + QApples Kloiber, 2004; Triplett and others, 2009; LaFrancois and 

others, 2009.

2 QStillwater = 1.11 * QSCF Kloiber, 2004; Triplett and others, 2009; LaFrancois and 

others, 2009.

3 QStillwater = QSCF_lagged_1_day + QApple_lagged_1_day Greg Mitton, U.S. Geological Survey, written commun, 

2014.

4 QPrescott =QStillwater + QWillow + QKinnickinnic Kloiber, 2004; Triplett and others, 2009; LaFrancois and 

others, 2009.

5 QPrescott =1.18 * QSCF Kloiber, 2004; Triplett and others, 2009; LaFrancois and 

others, 2009.

6 QPrescott = QMiss_Prescott – {1.034[(QSt.Paul + QMWTF )lagged_1_day]} Greg Mitton, U.S. Geological Survey, written commun, 

2014.

7

 

NSE = 1 – (Qm,i – Qe,i)2

(Qm,i – Qm)

2

∑n

i=1

∑n

i=1

—

Nash and Sutcliffe, 1970.

8

 

Percent Difference = * 100 Qe

 – Qm

Qe

This report.

9 QStillwater

 = (0.968 * QSCF_lagged_1_day) + 770 This report.

10 QStillwater(WY2012) = (0.960 * QSCF_lagged_1_day) + 907 This report.

11 QStillwater(WY2013) = (0.971 * QSCF_lagged_1_day) + 680 This report.

12 QPrescott = (0.585 * QSCF_lagged_1_day) + (0.415 * QSCF_lagged_3_day) + 873 This report.
 23

Appendix 2. Previous and Revised Estimates of 

Historical Daily Streamflows for the St. Croix 

River at Stillwater, Minnesota

Previous and revised estimates of historical daily streamflows for the St. Croix River at Stillwater, Minnesota are 

presented as a Microsoft Excel® workbook (http://pubs.usgs.

gov/sir/2014/5239/downloads/Appendix2.xlsx). All streamflow 

values listed in appendix 2 are in cubic feet per second. 

Previous estimates of streamflow were calculated using equation 2 from the text and listed in 1‒1, and revised estimates 

are calculated using equation 9 from the text and listed in 

table 1‒1. The first tab of the Excel worksheet contains estimated streamflow values for the period of record (1910‒2011 

water years), and other tabs contain daily streamflow values 

for an individual water year, organized monthly similar to the 

way daily streamflow values are published in USGS Annual 

Data Reports (http://wdr.water.usgs.gov/).

Appendix 3. Previous and Revised Estimates of 

Historical Daily Streamflows for the St. Croix 

River at Prescott, Wisconsin

Previous and revised estimates of historical daily 

streamflows for the St. Croix River at Prescott, Wisconsin, are 

presented as a Microsoft Excel® workbook (http://pubs.usgs.

gov/sir/2014/5239/downloads/Appendix3.xlsx). All streamflow 

values listed in appendix 3 are in cubic feet per second. Previous estimates of streamflow were calculated using equation 5 

from the text and listed in table 1‒1, and revised estimates 

are calculated using equation 12 from the text and listed in 

table 1‒1. The first tab of the Excel worksheet contains estimated streamflow values for the period of record (1910‒2007 

water years), and other tabs contain daily streamflow values 

for an individual water year, organized monthly similar to the 

way daily streamflow values are published in USGS Annual 

Data Reports (http://wdr.water.usgs.gov/).

Publishing support provided by:

Rolla Publishing Service Center

For more information concerning this publication, contact:

Director, USGS Minnesota Water Science Center

2280 Woodale Drive

Mounds View, Minnesota 55112

(763) 783–3100

Or visit the Minnesota Water Science Center Web site at:

http://mn.water.usgs.gov/


Ziegeweid and Magdalene—Development of Regression Equations to Revise Estimates of Historical Streamflows for the St. Croix River—Scientific Investigations Report 2014–5239

ISSN 2328-0328 (online)

http://dx.doi.org/10.3133/sir20145239