U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2020–5099

Version 1.2, November 2020

Prepared in cooperation with Alabama Power; Alabama Farmers Federation; Alabama Association of 

Conservation Districts; Alabama Association of Resource Conservation and Development Councils; Alabama 

Department of Agriculture and Industries; Alabama Department of Conservation and Natural Resources—

Wildlife and Freshwater Fisheries Division; Alabama Department of Economic and Community Affairs—Office of 

Water Resources; Alabama Department of Environmental Management; Alabama Soil and Water Conservation 

Committee; Choctawhatchee, Pea and Yellow Rivers Watershed Management Authority; Geological Survey of 

Alabama; and The University of Alabama—Alabama Water Institute

Methods for Estimating Selected Low-Flow Frequency 

Statistics and Mean Annual Flow for Ungaged Locations on 

Streams in Alabama
Cover. Photograph showing Chewacla Creek near U.S. Geological Survey streamflow-gaging station 

02418760 Chewacla Creek at Chewacla State Park near Auburn, Alabama, August 22, 2019.
Methods for Estimating Selected Low-Flow 

Frequency Statistics and Mean Annual 

Flow for Ungaged Locations on Streams 

in Alabama

By Toby D. Feaster, Katharine R. Kolb, Jaime A. Painter, and Jimmy M. Clark

Prepared in cooperation with Alabama Power; Alabama Farmers Federation; 

Alabama Association of Conservation Districts; Alabama Association of 

Resource Conservation and Development Councils; Alabama Department 

of Agriculture and Industries; Alabama Department of Conservation and 

Natural Resources—Wildlife and Freshwater Fisheries Division; Alabama 

Department of Economic and Community Affairs—Office of Water Resources; 

Alabama Department of Environmental Management; Alabama Soil and Water 

Conservation Committee; Choctawhatchee, Pea and Yellow Rivers Watershed 

Management Authority; Geological Survey of Alabama; and The University of 

Alabama—Alabama Water Institute

Scientific Investigations Report 2020–5099

Version 1.2, November 2020

U.S. Department of the Interior

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

DAVID BERNHARDT, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia

First release: 2020, online and in print

Revised: September 29, 2020 (ver. 1.1), online

Revised: November 20, 2020 (ver. 1.2), online

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natural hazards, and the environment—visit https://www.usgs.gov or call 1–888–ASK–USGS.

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

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

Government.

Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials 

as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner.

Suggested citation:

Feaster, T.D., Kolb, K.R., Painter, J.A., and Clark, J.M., 2020, Methods for estimating selected low-flow frequency 

statistics and mean annual flow for ungaged locations on streams in Alabama (ver. 1.2, November 20, 2020): 

U.S. Geological Survey Scientific Investigations Report 2020–5099, 21 p., https://doi.org/10.3133/sir20205099.

Associated data for this publication:

Kolb, K.R., Clark, J.M., Feaster, T.D., and Painter, J.A., 2020, Supporting data for estimating selected low-flow 

frequency statistics and mean annual flow for ungaged locations on streams in Alabama: U.S. Geological Survey data 

release, https://doi.org/10.5066/P994UFS7.

ISSN 2328-031X (print)

ISSN 2328-0328 (online)

ISBN 978-1-4113-4378-8
iii

Acknowledgments

The authors would like to thank the numerous cooperators who helped make this investigation 

possible through their cooperative funding. The streamflow data included in this investigation 

were computed from data collected at U.S. Geological Survey (USGS) streamgages operated in 

cooperation with a variety of Federal, State, and local agencies. The authors acknowledge the 

commitment of those cooperators in understanding the importance of having such a long-term 

data-collection program.

We also acknowledge Athena Clark who was the Director of the USGS Alabama Water Science 

Center when this project began and is now the Science Advisor and Tribal Liaison for the USGS 

Southeast Region. Although she was not obligated to do so, Athena continued to meet with 

cooperators and give oversight and guidance throughout the duration of the project. In addition, 

we acknowledge the commitment and vital work of the USGS field-office personnel who collect, 

process, and store the water-resources data needed for this and many other such investigations.

v

Contents

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

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

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

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

Study Area..............................................................................................................................................2

Previous Studies ...................................................................................................................................4

Selection of Streamgages ............................................................................................................................4

Physical and Climatic Basin Characteristics ............................................................................................6

Streamflow-Variability Index ..............................................................................................................6

Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at 

Ungaged Locations in Alabama .....................................................................................................7

Drainage-Area Ratio Method .............................................................................................................8

Regional Regression Analysis ............................................................................................................8

Development of Regional Regression Equations .................................................................10

Final Regression Equations ......................................................................................................11

Accuracy and Limitations .........................................................................................................12

Limitations for Applying the Regional Regression Equations .............................................14

StreamStats ..................................................................................................................................................16

Summary........................................................................................................................................................16

References Cited..........................................................................................................................................16

Appendix 1. U.S. Geological Survey streamgages that were excluded from the regional 

regression analysis for estimating selected low-flow frequency statistics 

and mean annual flow for ungaged locations in Alabama. Available online at 

https://doi.org/10.3133/sir20205099

Appendix 2. U.S. Geological Survey streamgages and independent and dependent 

variables used in the low-flow frequency and mean annual flow regression 

analyses for ungaged locations on streams in Alabama. Available online at 

https://doi.org/10.3133/sir20205099

Figures

1. Map showing U.S. Environmental Protection Agency Level III ecoregions 

in Alabama .....................................................................................................................................3

2. Map showing Level III ecoregions and streamgages considered for inclusion in 

the regional regression analyses for ungaged locations on streams in Alabama ............5

3. Flow-duration curves for U.S. Geological Survey stations 02370700 Pond Creek 

near Milton, Florida, and 02448900 Bodka Creek near Geiger, Alabama .............................8

4. Map showing streamflow-variability index grid for Alabama ...............................................9

5. Diagram showing a generalized workflow for a regression-based 

regionalization study ..................................................................................................................10

6. Example of a correlation matrix plot........................................................................................11

7. Graph showing the relation between the observed and predicted annual 

minimum 1-day flow with a 10-year recurrence interval, annual minimum 7-day 

flow with a 2-year recurrence interval, annual minimum 7-day flow with a 

10-year recurrence interval, and mean annual flow ............................................................13
vi

Tables

1. Basin characteristics tested as explanatory variables in the development of 

low-flow regression equations for ungaged locations in Alabama .....................................7

2. Selected low-flow frequency and mean annual flow regression equations for 

ungaged streams in Alabama ...................................................................................................12

3. Values used to determine prediction intervals for the Alabama low-flow 

regression equations .................................................................................................................15

4. Ranges of explanatory variables used to develop the low-flow frequency and 

mean annual flow regression equations for Alabama .........................................................15

Conversion Factors

U.S. customary units 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) 259.0 hectare (ha)

square mile (mi2) 2.590 square kilometer (km2)

Flow rate

foot per second (ft/s) 0.3048 meter per second (m/s)

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

inch per hour (in/h) 0.0254 meter per hour (m/h)

Datum

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

Abbreviations

1Q10 annual minimum 1-day average flow with a 10-year recurrence interval

7Q2 annual minimum 7-day average flow with a 2-year recurrence interval

7Q10 annual minimum 7-day average flow with a 10-year recurrence interval

APS all-possible-subsets (regression analysis)

EPA U.S. Environmental Protection Agency

GIS geographic information system

MOVE.1 Maintenance of Variance Extension, Type 1

OLS ordinary least squares

SEE standard error of estimate

SVI streamflow-variability index

USGS U.S. Geological Survey

VIF variance inflation factor

Methods for Estimating Selected Low-Flow Frequency 

Statistics and Mean Annual Flow for Ungaged Locations 

on Streams in Alabama

By Toby D. Feaster, Katharine R. Kolb, Jaime A. Painter, and Jimmy M. Clark

Abstract

Streamflow data and statistics are vitally important for 

proper protection and management of the water quality and 

water quantity of Alabama streams. Such data and statistics 

are generally available at U.S. Geological Survey streamflowgaging stations, also referred to as streamgages or stations, but 

are often needed at ungaged stream locations. To address this 

need, the U.S. Geological Survey, in cooperation with numerous Alabama State agencies and organizations, developed 

regional regression equations for estimating selected low-flow 

frequency statistics and mean annual flow for ungaged locations on streams in Alabama that are not substantially affected 

by tides, regulation, diversions, or other anthropogenic influences. A small percentage of the streamgages included in this 

study experience zero flows during certain periods; thus, the 

final low-flow frequency regression equations were developed 

by using weighted left-censored regression analyses to analyze 

the flow data in an unbiased manner, with weights based on 

number of years of record.

The equations developed include the annual minimum 

1- and 7-day average streamflows with a 10-year recurrence 

interval (referred to as the 1Q10 and 7Q10 flows), the annual 

minimum 7-day average streamflow with a 2-year recurrence 

interval (referred to as the 7Q2 flow), and the mean annual 

flow using data from 174 streamgages from Alabama and 

surrounding States. For the 1Q10, 7Q2, and 7Q10 low-flow 

frequency statistics, the regional regression equations are 

functions of drainage area, streamflow-variability index, mean 

annual precipitation, and percentage of the drainage basin 

located in the Piedmont and Southeastern Plains ecoregions. 

The mean annual flow regression equation is a function of 

drainage area, mean annual precipitation, and percentage of 

the drainage basin located in the Southeastern Plains ecoregion. For the mean annual flow regression equation, the 

average standard error of estimate was 11.2 percent. For the 

selected low-flow frequency equations, the average standard 

errors of estimate ranged from 18.1 to 38.8 percent.

The regional regression equations developed from this 

investigation have been incorporated into the U.S. Geological 

Survey StreamStats application for Alabama. StreamStats 

(https://streamstats.usgs.gov/ss/) is a web-based geographic 

information system application that delineates drainage basins 

at selected stream locations and then generates the needed 

basin characteristics for available regional regression equations. Along with the low-flow frequency equations developed 

in this investigation, the StreamStats application also has 

regional regression equations for estimating flood-frequency 

statistics at locations on rural and urban streams in Alabama.

Introduction

As part of their mission to protect public health and 

aquatic ecosystems, Alabama State agencies need accurate 

and representative streamflow statistics to establish realistic 

and applicable criteria for water quality and water quantity. 

Historically, low-flow statistics, such as the annual minimum 

7-day average flow that likely will occur once, on average, 

every 10 years (7Q10), have been used by water-resource 

managers and planners as a threshold criterion for applying 

the chronic aquatic life criteria for such things as determining 

waste-load allocations for point sources, total maximum daily 

loads for streams, and the quantity of water that can be safely 

withdrawn from a particular stream (Alabama Department of 

Environmental Management, 2012). Because of the importance of these applications, it is critical to effectively measure 

and document base-flow data for use in updating low-flow 

frequency relations on a regular basis, preferably every 

10 years (Riggs, 1972), and especially after periods of extreme 

low flow as have occurred over the last decade or so in the 

southeastern United States (Feaster and Lee, 2017).

In 2017, the U.S. Geological Survey (USGS) completed an investigation to update selected low-flow frequency 

and daily duration characteristics at 210 continuous-record 

streamflow-gaging stations (also referred to as streamgages 

or stations) in Alabama and 67 streamgages in the adjoining States of Florida, Georgia, Mississippi, and Tennessee 

by using available daily mean streamflow data through 

March 2014 (Feaster and Lee, 2017). If low-flow statistics are 

needed at a streamgage location or a nearby ungaged location, 

these updated statistics can provide critical information for 
2 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

water-resources managers and planners. However, at locations where streamgages are not available or the record length 

is insufficient to provide reliable estimates, multiple linear 

regression techniques may provide a viable option for estimating streamflow statistics (Hedgecock and Feaster, 2007; 

Farmer and others, 2019). By using multiple linear regression techniques, streamflow statistics computed at USGS 

streamgages can be related to basin and climatic characteristics in a region of interest, resulting in regional regression 

equations that can be used to estimate the streamflow statistic 

at an ungaged location.

Purpose and Scope

The purpose of this report is to present methods for 

estimating low-flow frequency statistics for the annual minimum 1- and 7-day average flows with a 10-year recurrence 

interval (1Q10 and 7Q10, respectively), the annual minimum 

7-day average flow with a 2-year recurrence interval (7Q2), 

and mean annual flow for ungaged locations on Alabama 

streams that are not substantially affected by tides, regulation, 

diversions, urbanization, or other anthropogenic influences. 

The regional regression analysis is based on a subset of the 

low-flow frequency statistics published by Feaster and Lee 

(2017) that are related to basin and climatic characteristics 

that were computed for those streamgages and are presented 

in this report. Mean annual streamflow, which is defined as 

the average of the daily mean flows for a given year, was 

computed from daily mean flows through September 2018 

based on the water year, which is defined as the 12-month 

period from October 1 to September 30 with the year being 

designated by the calendar year in which it ends. The data and 

geospatial datasets used to create the new low-flow and mean 

annual flow regression equations are available from Kolb and 

others (2020).

Study Area

The study area encompasses most of the State of 

Alabama except for coastal areas where low flows are tidally influenced. Shared basins and ecoregions from adjoining 

States are also included in the study area. Alabama encompasses 52,420 square miles (mi2) in the southeastern United 

States (USGS, 2016) and lies within six U.S. Environmental 

Protection Agency (EPA) Level III ecoregions—Piedmont 

(9.3 percent), Southeastern Plains (59.6 percent), Ridge 

and Valley (8.6 percent), Southwestern Appalachians 

(14.6 percent), Interior Plateau (6.4 percent), and Southern 

Coastal Plain (1.5 percent) (fig. 1; EPA, 2016). The ecoregions represent areas of general similarity in ecosystems and 

type, quality, and quantity of environmental resources, and 

they provide a spatial framework for research, assessment, 

management, and monitoring of ecosystems and ecosystem 

components. The ecoregions were determined from an 

analysis of the spatial patterns and the composition of biotic 

and abiotic phenomena that include geology, physiography, 

vegetation, climate, soils, land use, wildlife, and hydrology 

(Omernik, 1987).

The Southwestern Appalachians ecoregion extends from 

Alabama to Kentucky and is composed of open, low mountains containing a mosaic of forest and woodland with some 

cropland and pasture. The eastern boundary of this ecoregion, 

along the more abrupt escarpment where it meets the Ridge 

and Valley ecoregion, is relatively smooth and only slightly 

notched by small, eastward-flowing streams. The Ridge and 

Valley is composed of roughly parallel ridges and valleys of 

various widths, heights, and geologic materials. The western 

boundary of the Southwestern Appalachians ecoregion is 

shared with the Interior Plateau ecoregion, which extends from 

southern Indiana and Ohio to northern Alabama. Elevations in 

the Interior Plateau are lower than those in the Southwestern 

Appalachians. Limestone, chert, sandstone, siltstone, and shale 

of Mississippian to Ordovician age compose the landforms 

of open hills, irregular plains, and tablelands of the Interior 

Plateau (Omernik, 1987).

The Piedmont ecoregion, which extends from Alabama 

to New Jersey, is a transitional area between the mountainous ecoregions of the Appalachians to the northwest and the 

relatively flat coastal plain to the southeast. The Piedmont 

is a complex mosaic of metamorphic and igneous rocks of 

Precambrian and Paleozoic age and contains moderately 

dissected irregular plains and some hills. The soils tend to be 

finer textured than those in the coastal plain. The Piedmont 

was once a largely cultivated region, but much of it has 

reverted to pine and hardwood woodlands, with increasing 

conversion to urban and suburban land cover (Omernik, 1987).

The Fall Line, which extends from Alabama to New 

Jersey, is a geologic feature that separates the higher elevation Interior Plateau, Southwestern Appalachians, Ridge and 

Valley, and Piedmont ecoregions from the lower elevation 

Southeastern Plains ecoregion (fig. 1). The Southeastern 

Plains ecoregion, which extends from Virginia to Louisiana 

and Tennessee, is composed of irregular plains made up of a 

mixture of cropland, pasture, woodland, and forest. The sands, 

silts, and clays of this ecoregion contrast geologically with the 

older rocks of the Piedmont ecoregion. Elevations and relief 

in the Southeastern Plains are greater than in the Southern 

Coastal Plain but generally are less than in much of the 

Piedmont. Streams in the Southeastern Plains have relatively 

low gradients and sandy bottoms. The Southern Coastal Plain 

ecoregion consists of mostly flat plains but is heterogeneous, 

containing barrier islands, coastal lagoons, marshes, and 

swampy lowlands along the Gulf and Atlantic coasts. Relative 

to the Southeastern Plains ecoregion, the Southern Coastal 

Plain ecoregion is lower in elevation, with less relief and wetter soils (Omernik, 1987).
Introduction 3

CLARKE

BALDWIN

MOBILE

PIKE

MONROE

DALE

BARBOUR

COFFEE

ESCAMBIA

WASHINGTON

HENRY

GENEVA

HOUSTON

CHOCTAW

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CONECUH

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EXPLANATION

Level III ecoregion Fall Line

Interior Plateau

Southwestern Appalachians

Ridge and Valley

Piedmont

Southeastern Plains

Southern Coastal Plain

0 20 40 60 80 MILES

0 20 40 60 80 KILOMETERS

88° 87° 86° 85° 84°

35°

34°

33°

32°

31°

Base from Geospatial Data Gateway, U.S. Department of Agriculture, Natural Resources Conservation Service

Level III ecoregion layer from U.S. Environmental Protection Agency (2016)

Universal Transverse Mercator, zone 16N

North American Datum of 1983

FLORIDA

TENNESSEE

GEORGIA MISSISSIPPI

Sipsey

Fork

Alabama River

Mulbe rry Fork

Locust Fork

Alabama Rive r

Tombigbee Ri ver

Coosa River

Mobile

Bay

Figure 1. U.S. Environmental Protection Agency Level III ecoregions (U.S. Environmental Protection Agency, 2016) 

in Alabama.
4 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

Previous Studies

Feaster and Lee (2017) detailed previous studies that 

provided low-flow frequency and flow duration statistics at 

selected USGS streamgages in Alabama going back to the late 

1950s. With respect to previous regionalization studies of lowflow frequency statistics, Bingham (1982) developed low-flow 

frequency regression equations for the 7Q2 and 7Q10 lowflow frequency statistics. The equations were applicable statewide for natural streams that were not substantially altered 

by anthropogenic activities. The equations were functions 

of drainage area, mean annual precipitation, and streamflow 

recession index. The streamflow-recession index is defined in 

days per log cycle, which is the number of days required for 

streamflow to decline one complete log cycle. To compute the 

streamflow recession index at a streamgaging location, streamflow recess curves were plotted on semi-log graph paper for 

several different recessions. Streamflow records for recession 

periods during November through February were generally 

used, and the average slope of the recession curves of base 

flow was used to determine the streamflow-recession index for 

that site. The streamflow-recession index values computed for 

the USGS streamgage locations were used to delineate areas 

determined to have similar values and were provided on a map 

that could be used to estimate the streamflow-recession index 

at ungaged locations.

Selection of Streamgages

As previously noted, the streamgages used in the 

Alabama low-flow regional regression analysis were a subset 

of the 277 streamgages (210 within Alabama and 67 in adjoining States) included in Feaster and Lee (2017), which included 

streamflow data through March 2014 and only included nontidal streamgages (fig. 2). For ease of reference, the map index 

numbers used in this report match the map index numbers 

used by Feaster and Lee (2017). Streamgages with low-flow 

frequency statistics computed for regulated or urbanized 

basins were excluded (appendix 1). In addition, the following 

USGS stations in Alabama were excluded because they are 

springs with indeterminate drainage areas: 02403500 W 12 

Coldwater Spring near Anniston, and 03590500 Tuscumbia 

Spring at Tuscumbia. The following streamgages in Alabama 

were excluded because they have large drainage basins 

(17,095, 25,610, and 30,810 mi2, respectively): 02423000 

Alabama River at Selma, 03575500 Tennessee River at 

Whitesburg, and 03589500 Tennessee River at Florence.

Determining a true natural condition in a stream or 

accurately accounting for all diversions in a basin is generally problematic. Feaster and Lee (2017) noted that diversions from natural flow in a stream can occur for a variety of 

reasons. In some cases, diversions in a stream may only affect 

the flows from a short distance along the stream. For example, 

water may be removed from the river channel, passed through 

a manufacturing plant for use in processing, cooling, or dilution of wastes, and then returned to the river. In such cases, 

consumptive losses from diversions may be negligible (Ries, 

1994). Feaster and Lee (2017) completed a series of quality 

assurance and quality control reviews of the streamflow data 

included in their investigation to assess potential changes that 

might be the result of anthropogenic influences. Based on 

those reviews, if the low flows at a streamgage were considered to be substantially affected by anthropogenic influences, 

the streamgage was excluded from this regionalization study.

Another resource that was used to help assess the 

streamgages for inclusion in the regionalization study was the 

Geospatial Attributes of Gages for Evaluating Streamflow, 

version II (GAGES-II) dataset (Falcone, 2011). The 

GAGES-II dataset provides geospatial data classifications 

for 9,322 streamgages maintained by the USGS and consists 

of streamgages that have 20 or more years of record since 

1950 or are currently active. The GAGES-II dataset classifies streamgages considered “near natural” as “reference” 

gages. GAGES-II also has a disturbance index to help assess 

if the gage is a reference or non-reference streamgage. For 

many of the streamgages in GAGES-II, screening comments 

are included to provide some insight into the classification. 

The GAGES-II documentation notes that the classifications 

presented are not intended to be definitive and, as such, are 

considered another resource to assist in determining whether 

to include a streamgage in a regional regression analysis.

Additional information on potential influences on 

streamflow at the USGS streamgages was obtained from 

annual water data reports (USGS, 2019a). Station description 

information available from the internal USGS site information management system also was reviewed to help determine 

potential influences on streamflow.

For streams with multiple streamgages, an assessment 

was made to determine potential redundancy. Redundancy 

occurs when the drainage basins of two streamgages are 

nested, which is when one basin is contained inside another 

basin and most or all of the streamflow records at the two 

streamgages represent concurrent periods of time (Feaster 

and others, 2014). For this investigation, a streamgage was 

considered redundant if it was on the same stream as another 

streamgage and the drainage area of one streamgage represented more than 50 percent of the drainage area of the 

second streamgage, unless the periods of record for the two 

streamgages do not overlap. In cases where one record represented a relatively long period of time and the other record 

represented a relatively short period of time and was completely or mostly concurrent with the same period in the longer 

record, the streamgage with the short record was excluded. In 

addition, if a streamgage record was extended by Feaster and 

Lee (2017) based on the MOVE.1 record-extension method 

(Hirsch, 1982), the streamgage with the extended record was 

excluded (appendix 1). Most of the streamgages that were 

excluded because the low-flow statistics were based on the 

MOVE.1 record-extension method also were nested with the 
Selection of Streamgages 5

Figure 2. Level III ecoregions and streamgages considered for inclusion in the regional regression analyses for ungaged locations on 

streams in Alabama.

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42 41

40

39

38

37

36

35

34

33

32

31

30 29

28

27 26 25

24

23

22

21 20

19

18

17

16

15

14

13

12 11

10

210

209

208

207

206

205

204

203

202

201

200

199

198

197

196

195

194

193

192

191

190

189

188

187 186

185

184

183

182

181

180

179

178 177

176

175 174

173

172

171

170

169

168

167

166

165

164

163

162

161 160 159

158

157

156

155

154

153

152 151

150

148

149

147

145 144 142 143

146

141

140

139

138

137

136

135 134

133

132

131

130 129

128

127 126

125

124

123

122

121

120

119

118

117

116

115 114

113

112

111

110 109

108 107

106

105

104

103

102

101

100

FLORIDA

TENNESSEE

GEORGIA

MISSISSIPPI

N

AR

MS

TN

SC

NC

LA

FL

AL GA

Gulf of 

Mexico

Atlantic Ocean

Study

area

0 20 40 60 80 MILES

0 20 40 60 80 KILOMETERS

EXPLANATION

Level III ecoregion

Interior Plateau

Southwestern Appalachians

Ridge and Valley

Blue Ridge

Piedmont

Southeastern Plains

Southern Coastal Plain

Fall Line

Streamgage excluded from the

 regional regression analyses 

 and map index number 

 (see appendix 1)

Streamgage included in the 

 regional regression analyses 

 and map index number 

 (see appendix 2)

223

5

Base from Geospatial Data Gateway, U.S. Department of Agriculture, Natural Resources Conservation Service

Level III ecoregion layer from U.S. Environmental Protection Agency (2016)

Universal Transverse Mercator, zone 16N

North American Datum of 1983
6 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

long-term streamgage used in the correlation procedure and, 

therefore, would have been considered redundant even if the 

correlation techniques had not been applied.

Along with using the aforementioned resources, final 

decisions on inclusion or exclusion of streamgages were made 

by using hydrologic judgment. Based on these criteria and 

reviews, 90 streamgages were excluded from the initial dataset 

(appendix 1). An additional 11 streamgages were excluded 

because all or substantial parts of the drainage basin are in the 

Blue Ridge ecoregion, which is not one of the ecoregions in 

Alabama. Also, USGS station 02341800 Upatoi Creek near 

Columbus, Ga., was excluded because it drains 100 percent 

from the Sand Hills ecoregion in Georgia, which is not an 

ecoregion in Alabama. During the final regression analysis, 

one additional station (USGS station 02454055 Lost Creek 

above Parrish, Ala.) was excluded because it was flagged in 

the regression diagnostics as having high influence and (or) 

leverage. Upon review, it was concluded that the low flows at 

USGS station 02454055 were likely being influenced enough 

by anthropogenic sources that it should be removed from the 

regression analysis. After excluding these 103 streamgages, 

174 streamgages were available for inclusion in the regional 

regression analyses.

Physical and Climatic Basin 

Characteristics

Streamflow statistics can be estimated at ungaged sites 

by using multiple linear regression techniques that relate 

streamflow characteristics computed at gaged sites (such as 

the 7Q10 flow) to selected physical and (or) climatic basin 

characteristics computed for the gaged drainage basins 

(table 1). Determining the drainage-basin boundaries for 

each streamgage is the first step for generating other basin or 

climatic characteristics. For this investigation, basin boundaries were generated by using data from The National Map 

3D Elevation Program (USGS, 2019b). Basin characteristics 

chosen for testing as potential explanatory variables were 

selected based on their potential theoretical relation to low 

flows, results of previous studies of low-flow regionalization, 

and the ability to quantify the basin characteristics by using 

geographic information systems (GIS). The use of GIS enables 

automation of the determination of basin characteristics and 

solution of the regional regression equations when using the 

USGS StreamStats application (USGS, 2019c).

Streamflow-Variability Index

The streamflow-variability index (SVI) was originally 

introduced by Lane and Lei (1950) as a useful index in analyzing hydraulic engineering projects and producing synthetic 

flow-duration curves. Areas with similar surface geology could 

be expected to correspond to similar SVI values, suggesting 

SVI would be a beneficial characteristic for regionalizing low 

flows based on geology (Ruhl and Martin, 1991). Given that 

low flows are a groundwater phenomenon, aquifer characteristics would be expected to influence low flows. Aquifer 

characteristics are diverse, and the interaction of aquifers and 

streamflow is complex; therefore, the flow in many streams 

is likely to be affected by several aquifers (Friel and others, 

1989). Consequently, along with bringing a regional component related to geology, the SVI also incorporates the integrated effects of multiple aquifers on low flows within a given 

basin. As such, a generalized SVI has been successfully used 

in many low-flow regionalization studies (Friel and others, 

1989; Martin and Ruhl, 1993; Koltun and Whitehead, 2002; 

Martin and Arihood, 2010; Koltun and Kula, 2013; Southard, 

2013; and Eash and Barnes, 2017).

The SVI is a dimensionless hydrologic characteristic that 

provides a measure of the steepness of the slope of the flowduration curve (Koltun and Whitehead, 2002; Southard, 2013). 

The flow-duration curve is a cumulative frequency curve that 

shows the percentage of time that a specific streamflow is 

equaled or exceeded (fig. 3). The SVI is the standard deviation of the logarithms of the 19 streamflow values at 5-percent 

class intervals from 5 to 95 percent on the flow-duration curve 

of daily mean flows for the analysis period (Searcy, 1959). The 

magnitude of the SVI is inversely related to the capacity of a 

basin to sustain base flow in a stream; for example, smaller 

SVI values are indicative of a higher sustained base flow. The 

SVI is computed as follows:

SVI

D D i i log log ( ) , 10 10

2

5 5

95

18

      

(1)

where

SVI is the streamflow-variability index,

Di is the ith percent duration streamflow (i=5, 

10, 15, ...95), and

log10  D is the mean of the base 10 logarithms of the 

19 streamflow values at 5-percent class 

intervals from 5 to 95 percent on the 

flow-duration curve of the daily mean flow.

For two cases, the 95-percent flow value at a streamgage 

was zero. Because the logarithm of zero is undefined, those 

values were set equal to 0.01.

The SVI values computed at the streamgage locations 

and the ArcMap Natural Neighbor interpolation tool were 

used to create a grid for the State of Alabama, which can be 

used to estimate the SVI at an ungaged location (Esri, 2019; 

fig. 4). The natural neighbors of any point are those associated 

with neighboring Thiessen polygons. The natural neighbor 

method uses the closest subset of input points that surrounds a 

query point and weights them based on proportionate areas of 

the polygons.
Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at Ungaged Locations in Alabama 7

Methods for Estimating Low-Flow 

Frequency Statistics and Mean Annual 

Flow at Ungaged Locations in Alabama

Alabama currently (2020) has 129 streamgages that 

provide real-time daily streamflow data along with a number 

of inactive streamgages for which historical data are available 

through the USGS National Water Information System 

(NWIS) database (USGS, 2020). However, water-resource 

managers and engineers regularly need to determine selected 

streamflow statistics at ungaged locations. Two methods that 

can be used to generate such statistics are (1) a drainage-area 

ratio method on streams with a nearby streamgage, and (2) 

regional regression equations that relate streamflow statistics 

to selected basin characteristics.

Table 1. Basin characteristics tested as explanatory variables in the development of low-flow regression equations for ungaged 

locations in Alabama.

[USGS, U.S. Geological Survey; NLCD, National Land Cover Database; SSURGO, Soil Survey Geographic Database; NOAA, National Oceanic and 

Atmospheric Administration]

Basin characteristic Unit of 

measure

Source

Drainage area Square miles National Map 3D Elevation Program (USGS, 2019b)

Mean elevation Feet National Map 3D Elevation Program (USGS, 2019b)

Minimum elevation Feet National Map 3D Elevation Program (USGS, 2019b)

Maximum elevation Feet National Map 3D Elevation Program (USGS, 2019b)

Relief (maximum elevation − minimum elevation)Feet National Map 3D Elevation Program (USGS, 2019b)

Relief ratio (mean elevation-minimum elevation)/(maximum elevation − minimum elevation)Unitless National Map 3D Elevation Program (USGS, 2019b)

Basin perimeter Feet National Map 3D Elevation Program (USGS, 2019b)

Percentage of basin in each level III ecoregion Percent U.S. Environmental Protection Agency (2016)

Percentage of developed land Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of impervious area Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of forest Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of planted/cultivated (agriculture) Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of woody wetlands (NLCD 2011) Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of herbaceous wetlands (NLCD 

2011)

Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Percentage of storage Percent Multi-Resolution Land Characteristics Consortium (version 2011) 

https://www.mrlc.gov/data

Base-flow index (Wolock, 2003) Unitless https://water.usgs.gov/GIS/metadata/usgswrd/XML/bfi48grd.xml

Groundwater head Feet National Map 3D Elevation Program (USGS, 2019b)

Saturated hydraulic conductivity (K-sat) Inches per 

hour

SSURGO gridded data (U.S. Department of Agriculture, Natural 

Resources Conservation Service, 2017)

Percent clay, silt, sand Percent SSURGO gridded data (https://sdmdataaccess.sc.egov.usda.gov)

Soil drainage index Unitless SSURGO gridded data (https://sdmdataaccess.sc.egov.usda.gov)

Hydrologic soil index Unitless SSURGO gridded data (https://sdmdataaccess.sc.egov.usda.gov)

Mean annual precipitation Inches PRISM 2010 (https://prism.oregonstate.edu/normals/)
8 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

Drainage-Area Ratio Method

The drainage-area ratio method assumes that the streamflow characteristics at the ungaged site are the same on a per 

unit area basis as the nearby streamgage. Consequently, it is 

important to consider the proximity of the streamgage to the 

ungaged site as well as similarities in drainage area size, other 

physical and climatic characteristics, and the magnitude of 

anthropogenic influences (Ries and Friesz, 2000; Watson and 

others, 2005). As such, the drainage-area ratio method is most 

commonly applied, and will tend to be most accurate, when 

the streamgage is on the same stream as the ungaged site.

The drainage-area ratio equation is as follows:

Q DA

DA DARu u Q g

 g

















(2)

where

QDARu is the drainage-area ratio streamflow statistic 

at the ungaged site,

DAu is the drainage area of the ungaged site,

DAg is the drainage area of the gaged site, and

Qg is the streamflow statistic from the gaged site.

With respect to a reasonable drainage-area ratio that is 

appropriate for applying this method, a few researchers have 

tested the procedure and recommended ranges of drainagearea ratios. Koltun and Schwartz (1986) recommended a 

range of 0.85 to 1.15 for the ungaged to gaged drainage areas 

for Ohio streams. For Pennsylvania streams, ratios between 

0.33 and 3.0 were recommended (Ries and Friesz, 2000). 

Ries and Friesz (2000) found that for Massachusetts streams, 

the drainage-area ratio method estimates were generally as 

accurate or more accurate than regression estimates when 

the drainage-area ratios for the ungaged and gaged sites were 

between 0.3 and 1.5. In Idaho, Hortness (2006) found that the 

drainage-area ratios of 0.5 to 1.5 were reasonable, and in Iowa, 

Eash and Barnes (2017) recommended a range from 0.5 to 1.4. 

Southard (2013) found the drainage-area ratio range of 0.4 to 

1.5 to be reasonable for Missouri streams.

Regional Regression Analysis

A regional regression analysis is an iterative process 

(fig. 5; Farmer and others, 2019). For this investigation, the 

data assembly step included updating low-flow frequency 

estimates at USGS streamgages (Feaster and Lee, 2017) and 

computing potential physical and climatic basin characteristics 

by using GIS tools (table 1). Model development involves 

exploratory data analysis to get a general understanding of 

relations among the variables that are to be predicted, which 

can be referred to as the dependent or response variable (for 

example, the 7Q10 streamflow statistic), and certain basin 

characteristics, which can be referred to as the independent or 

0.01

0.1

1

10

100

1,000

10,000

0 10 20 30 40 50 60 70 80 90 100

Daily mean streamflow, in cubic feet per second

Percentage of time indicated streamflow is equaled or exceeded

Streamgage 02370700, SVI = 0.169

Steamgage 02448900, SVI = 1.20

EXPLANATION

Figure 3. Flow-duration curves for U.S. Geological Survey stations 02370700 Pond Creek near Milton, 

Florida, and 02448900 Bodka Creek near Geiger, Alabama. [SVI, streamflow-variability index]
Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at Ungaged Locations in Alabama 9

explanatory variables (for example, drainage area). Often the 

estimation step begins with assessing the complete study area 

by testing all potential independent variables with a dependent 

variable of choice. The results are then evaluated based on 

statistical significance of the independent variables, adherence 

to the assumptions of the regression techniques being applied, 

which are typically assessed based on the residuals computed 

from the difference in the observed and predicted dependent 

variable, and the geographical distribution of those residuals. Mapping the geographical distribution of the residuals 

Base data from Esri, 2010

88° 86°

34°

32°

EXPLANATION

Streamflow-variability index

0.17 to 0.26

Undefined

0.27 to 0.33

0.34 to 0.38

0.39 to 0.42

0.43 to 0.45

0.46 to 0.48

0.49 to 0.53

0.54 to 0.6

0.61 to 0.69

0.70 to 0.82

0.83 to 0.98

0.99 to 1.20

0 25 50 KILOMETERS

0 25 50 MILES

Figure 4. Streamflow-variability index grid for Alabama.
10 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

provides information on potential subregions that might be 

warranted to improve the uncertainty in the predictions. Once 

a set of regression equations has been determined, the final 

step is to document the process with ample detail such that 

users of the equations have a general understanding of the 

proper application of the equations.

Development of Regional Regression Equations

For an exploratory regression analysis, it is best to start 

with the largest reasonable region, which is often the complete 

study area (Farmer and others, 2019). The initial regression 

analysis for this investigation included the complete study 

area, and from there, several potential regions were tested. 

The tested regions included dividing the study area into two 

regions that were differentiated by the Fall Line. The four 

flood regions used in the Alabama flood-frequency study by 

Hedgecock and Feaster (2007) also were tested along with 

various combinations of the EPA Level III ecoregions (fig. 1).

The preliminary regression analysis was done by using 

ordinary least squares (OLS) regression techniques (SAS 

Institute, 2020). The general model for an OLS regression 

analysis is of the form

QT = a Ab Bc Cd ..., (3)

where

QT is a low-flow frequency statistic such as 

the 7Q10;

A, B, C are explanatory (independent) variables; and

a, b, c, and d     are regression coefficients.

If the response and explanatory variables are logarithmically transformed, the regression model has the following 

line form:

 log QT= log a + b(log A) + c(log B) + d(log C) + ..., (4)

where the variables are as previously defined in equation 3. 

The logarithmic and arithmetic relations were used in this 

investigation because the logarithmic transformation of some 

variables did not improve the linear relation with QT.

The OLS regression is an efficient way to explore possible linear relations between the explanatory (basin characteristics) and response (selected low-flow statistics) variables. 

An all-possible-subsets (APS) regression analysis was done 

including all streamgages and the potential basin characteristics (table 1). The initial OLS regression analysis was 

completed by using the SAS statistical software package 9.4 

(SAS Institute, 2020) with the 174 streamgages available for 

inclusion in the regional regression analysis (appendix 2, and 

from Kolb and others, 2020). Various combinations of the 

potential explanatory variables selected from the APS analysis 

were tested. The response variables and some of the explanatory variables were transformed to logarithms (base 10) prior 

to the regression analysis. The purpose of the log transformations was to improve linear relations between the dependent 

and independent variables and to improve the variance of 

the residuals about the regression line. If the log transformation was shown not to substantially improve either of those 

characteristics, the independent variables were used in their 

measured units. Once a smaller subset of potential basin characteristics was determined, stepwise regression also was used 

to assess the strongest potential explanatory variables.

Along with the APS analysis, correlation matrix plots 

were used for a visual assessment of potential correlation 

among the independent variables as well as to assess which 

independent variables could likely be strong explanatory 

variables in the regression analysis (fig. 6). Tabular output 

from the matrix analysis included the Pearson correlation coefficient along with the p-value to assess statistical significance 

of the correlation. It should be noted that the matrix analysis 

and plots are useful for assessing the relations between single 

variables. In the case of a multiple linear regression where 

several independent variables are included, the importance 

of individual independent variables may change; therefore, 

the scatter plots should be considered only as a tool for initial 

assessments (Montgomery and others, 2012).

Data assembly

Exploratory

data analysis

Evaluation Estimation

Model development

Application and documentation

Figure 5. A generalized workflow for a regression-based 

regionalization study (from Farmer and others, 2019).
Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at Ungaged Locations in Alabama 11

Final Regression Equations

At 9 of the 174 streamgages included in the regression analysis, 1 or more of the 3 at-site low-flow frequency 

statistics were equal to zero (appendix 2). As such, weighted 

left-censored regression techniques were used for the final 

regression analyses, with the weights being based on the number of years of record (Lorenz, 2014; Ziegeweid and others, 

2015; Gotvald, 2017). For left-censored regression, a threshold value is imposed on the data that censors low (left) values 

below the threshold, with both the censored and uncensored 

values used in the regression. Censoring and coding data as 

“less than” a threshold value allows the use of a log transformation on the data and, therefore, allows all the data (uncensored and censored) to be used in the regression analysis to 

develop the regional equations (Watson and McHugh, 2014). 

For datasets that do not contain censored values, weighted 

left-censored regression provides the same results as weighted 

least squares regression (Helsel and Hirsch, 2002). Because 

of the uncertainty in measuring low flows and estimating lowflow frequency statistics less than 0.1 cubic foot per second 

(ft3/s), the censoring threshold for the regression analyses in 

this report was set at 0.1 ft3/s. Based on this censoring level, 

five streamgages had one or more of the three at-site low-flow 

frequency statistics that were less than 0.1 ft3/s but greater 

than zero (appendix 2).

Final selection of the independent variables to include 

in the regression models was based on several factors such as 

statistical significance, standard error, coefficient of determination, and ease of measurement of the explanatory variables. 

Correlation among the explanatory variables (multicollinearity) was assessed by using the variance inflation factor (VIF). 

If one or more VIF values exceed 5, the regression coefficients are likely poorly estimated due to multicollinearity 

(Montgomery and others, 2012). In such a case, one of the 

explanatory variables should be removed and the regressions 

reanalyzed to assess the remaining variables.

For both the OLS and left-censored regression analyses, 

multiple regression diagnostics were generated and used to 

identify possible problems with the streamgage data or basin 

characteristics. Along with reviews to ensure that the regression residuals were randomly distributed around zero and that 

they were reasonably distributed geographically, other regression diagnostics were reviewed to determine streamgages that 

Figure 6. Example of a correlation matrix plot.
12 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

had high leverage and (or) high influence. The leverage metric 

is used to measure how unusual the values of independent 

variables at one streamgage are compared to the values of the 

same variables at all other streamgages. The influence metric 

indicates whether the data at a streamgage had a high influence 

on the estimated regression metric values (Eng and others, 

2009). A streamgage may have a high leverage metric indicating that its independent variables are substantially different 

from those at all other streamgages, but the same streamgage 

may not have a high influence on the regression metrics. 

Conversely, a streamgage with a high influence may not have 

a high leverage metric. Sometimes, measurement or transposing errors in reported values of some independent variables 

can produce high leverage or influence metrics. Streamgages 

with high influence or leverage were given additional review 

to determine if such errors had been made or if the streamgage 

should be excluded for other reasons. For the final leftcensored equations for the 1Q10, 7Q2, and 7Q10 low-flow 

frequency statistics, none of the streamgages indicated high 

leverage. Three streamgages indicated high influence: USGS 

station 02410000, 02422000, and 02448900 (appendix 2). 

Additional reviews of the streamgages did not indicate any 

issues that warranted removing them from the analysis. All 

three streamgages had at least one of the low-flow frequency 

statistics being regionalized that was less than 0.1 ft3/s; therefore, the low-flow frequency statistics for those streamgages 

were censored in the analysis (appendix 2). Also, the periods 

of record analyzed for USGS stations 02410000 (1954 to 

1987) and 02422000 (1941 to 1971) included data from 1954, 

which was the driest year of record for the period from 1895 

to 2015 with respect to mean annual precipitation in Alabama 

(Feaster and Lee, 2017). In addition, the period of record 

analyzed for USGS station 02448900 was 1991 to 2014, for 

which 2007 and 2000 were the second and fourth driest years, 

respectively, in Alabama for the period from 1895 to 2015.

For the 1Q10, 7Q2, and 7Q10 low-flow frequency statistics, the combination of drainage area, SVI, mean annual precipitation, percentage of drainage basin in the Piedmont ecoregion, and percentage of drainage basin in the Southeastern 

Plains ecoregion were included in the final left-censored 

regression equations (table 2). All independent variables were 

statistically significant at the probability value (p-value) of 

≤0.05, and the VIF for all independent variables was less 

than 1.60, indicating no issues of multicollinearity among 

the variables. The final regressions were done on a statewide 

basis with the mean annual precipitation, SVI, and percentage of drainage basin in the Piedmont and Southeastern Plains 

ecoregions as the independent variables that would incorporate 

regional differences in the low-flow characteristics. Drainage 

area was the only independent variable that was log transformed for the regression analysis, and for the transformation, 

the natural log was used.

For the mean annual flow, the combination of drainage 

area, mean annual precipitation, and percentage of the drainage basin in the Southeastern Plains ecoregion was included in 

the final regression equations. The mean annual flow data did 

not include any zero flows. All independent variables were statistically significant at a p-value of ≤0.05, and the VIF for all 

independent variables was less than 1.15. Drainage area was 

the only independent variable that was log transformed for the 

regression of the mean annual flow with the base 10 logarithm 

being used.

Figure 7 shows plots of the observed and predicted 1Q10, 

7Q2, 7Q10, and mean annual flow, along with the line of 

equality and for the low-flow frequency results, the censoring 

level of 0.1 ft3/s. Because the plots are shown using a log-log 

scale, zero flows are not shown for the low-flow frequency 

plots. The plots indicate a reasonable scatter distribution 

around the line of equality throughout the range of flows.

Accuracy and Limitations

Users of regional regression equations should be aware 

of the uncertainty in the results. One measure of uncertainty is 

the average standard error of estimate (SEE). In the case of the 

low-flow frequency equations, the average SEE is a measure 

of the average uncertainty in a prediction from the regression equations based on all streamgage data throughout the 

applicable region. However, users typically will be interested 

Table 2. Selected low-flow frequency and mean annual flow regression equations for ungaged streams in Alabama.

[SEE, standard error of estimate; 1Q10, annual minimum 1-day flow with a 10-year recurrence interval, in cubic feet per second; DA, drainage area, in square 

miles; SVI, streamflow-variability index, dimensionless; Precip, mean annual precipitation, in inches; Pied, percentage of drainage basin in the Piedmont ecoregion; SEP, percentage of drainage basin in the Southeastern Plains ecoregion; 7Q2, annual minimum 7-day average flow with a 2-year recurrence interval, in 

cubic feet per second; 7Q10, annual minimum 7-day average flow with a 10-year recurrence interval, in cubic feet per second; Mean annual, mean annual flow, 

in cubic feet per second]

Statistic Regression equation SEE (percent) Number of leftcensored streamgages

1Q10 0.516*DA1.15*e−9.64SVI*e0.0339Precip*e−0.0128Pied*e−0.00560SEP 38.8 14

7Q2 0.784*DA1.06*e−7.28SVI*e0.0249Precip*e−0.00356Pied*e−0.00231SEP 18.1 2

7Q10 0.618*DA1.12*e−9.36SVI*e0.0319Precip*e−0.0115Pied*e−0.00523SEP 35.3 13

Mean annual 

flow

0.174*DA0.996*100.0177*Precip*10−0.000691SEP 11.2 0
Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at Ungaged Locations in Alabama 13

in a measure of uncertainty of the low-flow estimate at a 

specific ungaged location for which the regional regression 

equation is being applied. One such measure of uncertainty 

is the confidence interval of the prediction, also known as 

the prediction interval. The prediction interval is a range in 

values of an estimated response variable, such as the 1Q10, 

7Q2, and 7Q10 statistics, over which the true value of that 

response variable occurs within some stated probability. For 

example, the 90-percent prediction interval for an estimated 

flow value means that there is a 90-percent probability that 

the true value lies with that interval. The USGS StreamStats 

application (https://streamstats.usgs.gov/ss/; USGS, 2019c) 

uses the 90-percent prediction interval estimates as part of 

the computation of low-flow frequency estimates for ungaged 

stream locations. Tasker and Driver (1988) determined that a 

100 (1−α) prediction interval for a flow statistic estimated at 

an ungaged location from a regression equation can be computed as follows:

 Q / C < Q < CQ, (5)

where

Q is the streamflow characteristic for the 

ungaged site; and

C is computed as:

Predicted 1Q10, in cubic feet per second

Observed 1Q10, in cubic feet per second

Predicted 7Q2, in cubic feet per second

Observed 7Q2, in cubic feet per second

Predicted 7Q10, in cubic feet per second

Observed 7Q10, in cubic feet per second

Predicted mean annual flow, in cubic feet per second

Observed mean annual flow, in cubic feet per second

0.01

0.10

1

10

100

1,000

0.01 0.10 1 10 100 1,000

0.01

0.10

1

10

100

1,000

0.01 0.1 1 10 100 1,000

0.01

0.10

1

10

100

1,000

0.01 0.10 1 10 100 1,000

1

10

100

1,000

10,000

1 10 100 1,000 10,000

7Q10

Line of equality

Censoring level

EXPLANATION

1Q10

Line of equality

Censoring level

EXPLANATION

7Q2

Line of equality

EXPLANATION

Censoring level

Mean annual flow

Line of equality

EXPLANATION

A B

C D

Figure 7. The relation between the observed and predicted A, annual minimum 1-day flow with a 10-year recurrence interval (1Q10); 

B, annual minimum 7-day flow with a 2-year recurrence interval (7Q2); C, annual minimum 7-day flow with a 10-year recurrence interval 

(7Q10); and D, mean annual flow.
14 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

C

t SE

n p p,i

    10 / , 2 (6)

where

t(α/2, n−p) is the critical value from the Student’s 

t-distribution at a particular alpha-level 

divided by 2 (α/2) and degrees of freedom 

(n−p) and is equal to 1.68 for an α of 0.10, 

which corresponds to a prediction interval 

of 90 percent; and

SEp,i is the standard error of prediction for 

site i and is computed for a weighted 

left-censored regression as

p i i iT

SE MSE X UX ,

.

  

 

0 5

(7)

where

MSE is the mean square error;

Xi is a row vector of the explanatory variables 

for site i, augmented by a 1 as the 

first element;

U is the covariance matrix for the regression 

coefficients; and

 Xi

T is the transpose of Xi

 (Ludwig and 

Tasker, 1993).

The values for MSE and U are presented in table 3. The 

prediction intervals are provided for users when the statistics 

are computed by using the USGS StreamStats application 

(USGS, 2019c).

Limitations for Applying the Regional Regression 

Equations

The Alabama low-flow regional regression equations 

should be applied considering the following limitations.

1. The equations are not applicable to tidally influenced streams.

2. The range of explanatory variables used to develop 

the Alabama low-flow frequency and mean annual 

flow regional regression equations are listed in table 4. 

Applying the equations at stream locations having 

explanatory variables outside the range of those used in 

this investigation may result in prediction errors that are 

considerably greater than those suggested by the SEEs 

listed in table 2.

3. The low-flow regression equations are applicable to 

streams with minimal anthropogenic influences and, 

therefore, should not be applied for streams that are 

known to be substantially affected by regulation, diversions, or urbanization.

4. Computation of basin characteristics needed to apply 

the regression equations at ungaged locations should be 

computed by using the same GIS datasets and computation methods as those used in this investigation (Kolb 

and others, 2020). The USGS StreamStats application 

includes the same GIS data layers and computation 

methods used in this investigation (Ries and others, 

2008; USGS, 2019c; Kolb and others, 2020).

5. Because of the uncertainty in measuring and estimating streamflows less than 0.1 ft3/s and the relatively 

small percentage of flows below that threshold limit at 

streamgages included in this investigation, a censoring 

threshold of 0.1 ft3/s was used to develop left-censored 

regression equations for Alabama. Consequently, a 

regression analysis at an ungaged location that results in 

a predicted value of less than 0.1 ft3/s should be reported 

as less than 0.1 ft3/s.

6. Caution should be used when applying the regression 

equations in areas known to have karst topography. Low 

flows in karst topography can be substantially affected 

by gains from large springs and losses from sinkholes 

(Eash and Barnes, 2017).
Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow at Ungaged Locations in Alabama 15

Table 3. Values used to determine prediction intervals for the Alabama low-flow regression equations.

[MSE, the mean square error in equation 7; U, the covariance matrix used in equation 7; Intercept, y-axis intercept of regression equation; E, scientific notation 

indicating 10 to the power of; DA, drainage area, in square miles; SVI, streamflow-variability index, dimensionless; Precip, mean annual precipitation, in inches; 

Pied, percentage of drainage basin in the Piedmont ecoregion; SEP, percentage of drainage basin in the Southeastern Plains ecoregion; 1Q10, annual minimum 

1-day flow with a 10-year recurrence interval, in cubic feet per second; 7Q2, annual minimum 7-day average flow with a 2-year recurrence interval, in cubic feet 

per second; 7Q10, annual minimum 7-day average flow with a 10-year recurrence interval, in cubic feet per second; Mean annual, mean annual flow, in cubic 

feet per second]

Statistic MSE U

1Q10 0.140 Intercept DA SVI Precip Pied SEP

Intercept 3.14E−01 −2.27E−03 −6.47E−02 −4.71E−03 −2.68E−04 −1.93E−05

DA −2.27E−03 5.87E−04 −1.11E−03 −6.93E−07 −4.35E−06 −3.44E−06

SVI −6.47E−02 −1.11E−03 4.86E−02 7.84E−04 1.13E−04 5.02E−05

Precip −4.71E−03 −6.93E−07 7.84E−04 7.63E−05 3.53E−06 −3.48E−07

Pied −2.68E−04 −4.35E−06 1.13E−04 3.53E−06 1.29E−06 3.94E−07

SEP −1.93E−05 −3.44E−06 5.02E−05 −3.48E−07 3.94E−07 5.35E−07

7Q2 0.0323 Intercept DA SVI Precip Pied SEP

Intercept 6.80E−02 −6.36E−04 −1.16E−02 −1.03E−03 −5.65E−05 −3.06E−06

DA −6.36E−04 1.24E−04 −1.14E−04 1.64E−06 −7.09E−07 −6.83E−07

SVI −1.16E−02 −1.14E−04 7.84E−03 1.38E−04 2.00E−05 8.36E−06

Precip −1.03E−03 1.64E−06 1.38E−04 1.67E−05 7.51E−07 −8.14E−08

Pied −5.65E−05 −7.09E−07 2.00E−05 7.51E−07 2.83E−07 8.26E−08

SEP −3.06E−06 −6.83E−07 8.36E−06 −8.14E−08 8.26E−08 1.15E−07

7Q10 0.118 Intercept DA SVI Precip Pied SEP

Intercept 2.63E−01 −1.92E−03 −5.35E−02 −3.94E−03 −2.24E−04 −1.64E−05

DA −1.92E−03 4.83E−04 −8.73E−04 −3.44E−08 −3.28E−06 −2.81E−06

SVI −5.35E−02 −8.73E−04 3.98E−02 6.47E−04 9.24E−05 4.13E−05

Precip −3.94E−03 −3.44E−08 6.47E−04 6.38E−05 2.95E−06 −2.88E−07

Pied −2.24E−04 −3.28E−06 9.24E−05 2.95E−06 1.07E−06 3.27E−07

SEP −1.64E−05 −2.81E−06 4.13E−05 −2.88E−07 3.27E−07 4.47E−07

Mean annual 0.0024 Intercept DA Precip SEP

Intercept 3.50E−03 −1.44E−04 −5.67E−05 1.19E−06

DA −1.44E−04 4.72E−05 7.49E−07 −7.38E−08

Precip −5.67E−05 7.49E−07 9.91E−07 −2.42E−08

SEP 1.19E−06 −7.38E−08 −2.42E−08 6.54E−09

Table 4. Ranges of explanatory variables used to develop the low-flow frequency and mean annual flow regression equations for 

Alabama.

Basin characteristic Minimum Maximum

Drainage area (square miles) 2.01 2,469

Streamflow-variability index (dimensionless) 0.169 1.20

Mean annual precipitation (inches) 48.71 67.45

Percentage of drainage basin from the Piedmont ecoregion 0 100

Percentage of drainage basin from the Southeastern Plains ecoregion 0 100
16 Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama

StreamStats

USGS StreamStats is a web-based GIS application that 

provides a range of analytical tools useful for water-resource 

managers, planners, and engineers (Ries and others, 2008). 

The StreamStats application can be used to delineate drainage 

areas for user-selected sites on streams and then generate basin 

characteristics and estimates of streamflow statistics for the 

selected site where the functionality is available. StreamStats 

users also can select USGS data-collection streamgages and, 

where available, get streamflow statistics and other information about the streamgage. StreamStats is currently (2020) 

available for Alabama and can be used to estimate rural and 

urban flood frequency (Hedgecock, 2004; Hedgecock and 

Feaster, 2007; Hedgecock and Lee, 2010; USGS, 2019c).

The regression equations developed in this investigation 

to estimate the 1Q10, 7Q2, and 7Q10 low-flow frequency 

statistics and the mean annual flow have been incorporated 

into the USGS Alabama StreamStats application. Complete 

instructions for using StreamStats are provided on the StreamStats website (USGS, 2019c). The website provides links to 

(1) information about general limitations of the application, 

(2) other State applications, (3) user instructions, (4) definitions of terms, (5) answers to frequently asked questions, (6) 

downloadable presentations and other technical information 

about the application, and (7) contact information.

Summary

Methods for estimating selected low-flow frequency statistics and mean annual flow at ungaged locations on Alabama 

streams were developed as a part of this investigation. Because 

of the uncertainty in measuring and estimating streamflows 

less than 0.1 cubic foot per second (ft3/s) and the relatively 

small percentage of flows below that threshold limit at 

streamgages included in this investigation, a censoring threshold of 0.1 ft3/s was used to develop left-censored regression 

equations for Alabama. The at-site low-flow frequency statistics for the 174 streamgages included in the regression analyses were previously published as part of a U.S. Geological 

Survey investigation updating low-flow frequency statistics 

by using streamflow data through March 2014. The majority of the streamgages were from Alabama, but streamgages 

from Florida, Georgia, Mississippi, and Tennessee that have 

shared drainage basins with Alabama were also included. The 

streamflow data used in the mean annual flow regression were 

computed for the same 174 streamgages by using data through 

September 2018.

Numerous basin characteristics were tested as explanatory variables for the regression equations. The final regression 

equations for the 1Q10, 7Q2, and 7Q10 low-flow frequency 

statistics included drainage area, streamflow-variability index, 

mean annual precipitation, and percentage of the drainage basin located in the Piedmont and Southeastern Plains 

ecoregions. The average standard errors of estimate for the 

1Q10, 7Q2, and 7Q10 low-flow frequency equations were 

38.8, 18.1, and 35.3 percent, respectively. The final regression 

equation for the mean annual flow included drainage area, 

mean annual precipitation, and percentage of the drainage 

basin located in the Southeastern Plains ecoregion as explanatory variables. The average standard error of estimate for the 

mean annual flow was 11.2 percent.

For the low-flow frequency regressions, the analysis 

was done by transforming the low-flow frequency statistics 

and drainage area by using natural logarithms. The remaining 

explanatory variables were used in their measured units. Mean 

annual flows computed at the streamgages and drainage area 

were transformed by using base 10 logarithms. The remaining 

explanatory variables were used in their measured units.

The streamflow data included in the analyses represent streamgages with minimal anthropogenic influences. 

Consequently, the regression equations should be applied 

at locations that are not substantially affected by regulation, 

diversions, or other anthropogenic influences. The equations 

also should not be applied on streams that are known to be 

tidally influenced. The equations should not be applied outside 

of the range of the basin characteristics used in the regression 

analysis because doing so can result in prediction errors that 

are considerably greater than those suggested by the standard 

error of estimates. The ranges of the basin characteristics used 

in the regression analysis are as follows: drainage area, 2.01 

to 2,469 square miles; streamflow-variability index, 0.169 to 

1.20; and mean annual precipitation, 48.71 to 67.45 inches. 

The ranges of percentage of drainage area in the Piedmont and 

Southeastern Plains ecoregions are 0 to 100 percent.

The low-flow frequency and mean annual flow regression 

equations developed in this investigation have been incorporated into the USGS StreamStats application for Alabama. 

The StreamStats application delineates the drainage basin at a 

selected location on a stream and then generates the explanatory variables for the delineated basin needed by the regression equations. Along with the streamflow estimates computed 

by StreamStats, the output from the StreamStats application 

also will provide 90-percent prediction intervals for those 

estimates.

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Appendixes 21

Appendix 1. U.S. Geological Survey streamgages that were excluded from the regional regression analysis 

for estimating selected low-flow frequency statistics and mean annual flow for ungaged locations in 

Alabama. Available online at https://doi.org/10.3133/sir20205099

Appendix 2. U.S. Geological Survey streamgages and independent and dependent variables used in 

the low-flow frequency and mean annual flow regression analyses for ungaged locations on streams in 

Alabama. Available online at https://doi.org/10.3133/sir20205099

For more information about this publication, contact

Director, Lower Mississippi-Gulf Water Science Center

U.S. Geological Survey

640 Grassmere Park, Suite 100

Nashville, TN 37211 

For additional information visit 

https://www.usgs.gov/centers/lmg-water/

Publishing support provided by

Lafayette Publishing Service Center
Feaster and others—Methods for Estimating Low-Flow Frequency Statistics and Mean Annual Flow for Ungaged Locations, Alabama—SIR 2020–5099, ver. 1.2

ISSN 2328-031X (print)

ISSN 2328-0328 (online)

https://doi.org/10.3133/sir20205099

I SBN 978- 1- 4113-4378- 8

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