Evaluation of a Method of Estimating

Low-Flow Frequencies from

Base-Flow Measurements at Indiana Streams

Water-Resources Investigations Report 00–4063

Prepared in cooperation with the

Indiana Department of Natural Resources, Division of Water

U.S. Department of the Interior

U.S. Geological Survey
Evaluation of a Method of Estimating

Low-Flow Frequencies from

Base-Flow Measurements at Indiana Streams

By John T. Wilson

Water-Resources Investigations Report 00-4063

Prepared in cooperation with the

Indiana Department of Natural Resources, Division of Water

Indianapolis, Indiana

2000

U.S. Department of the Interior

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

BRUCE BABBITT, SECRETARY

U.S. Geological Survey

Charles G. Groat, Director

For additional information, write to:

District Chief

U.S. Geological Survey

5957 Lakeside Boulevard

Indianapolis, IN 46278-1996

Copies of this report can be purchased from:

U.S. Geological Survey

Branch of Information Services

Box 25286

Federal Center

Denver, CO 80225-0286
Contents

Contents iii

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

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

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

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Methods of Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Frequency Analysis at Partial-Record Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Evaluation of the Stedinger-Thomas Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Test Cases A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Test Cases C and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Test Case E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Evaluation of All Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Varying the Number of Base-Flow Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Summary and Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

References Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Data Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figures

1. Map showing pairs of streamflow-gaging stations used to evaluate the Stedinger-Thomas method

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when the

index station and partial-record station are located on the same stream or tributaries . . . . . . . . . . . . . . . . . 8

2–5. Graphs showing:

2. Predicted values of 7Q10 and observed values of 7Q10 for Indiana streams in Test Case A and

the relative error of the predicted values of 7Q10 with the observed values of 7Q10 for

Test Case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Predicted values of 7Q2 and observed values of 7Q2 for Indiana streams in Test Case A and

the relative error of the predicted values of 7Q2 with the observed values of 7Q2 for

Test Case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. Predicted values of 7Q10 and observed values of 7Q10 for Indiana streams in Test Case B and

the relative error of the predicted values of 7Q10 with the observed values of 7Q10 for

Test Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5. Predicted values of 7Q2 and observed values of 7Q2 for Indiana streams in Test Case B and

the relative error of the predicted values of 7Q2 with the observed values of 7Q2 for

Test Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6–7. Boxplots showing:

6. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas

method, Test Cases A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas

method, Test Cases A and B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
iv Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

Contents—Continued

Figures—Continued

8. Map showing pairs of streamflow-gaging stations used to evaluate the Stedinger-Thomas method

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when the

index station and partial-record station are located on different streams or basins . . . . . . . . . . . . . . . . . . . 14

9–12. Graphs showing:

9. Predicted values of 7Q10 and observed values of 7Q10 for Indiana streams in Test Case C and

the relative error of the predicted values of 7Q10 with the observed values of 7Q10 for

Test Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

10. Predicted values of 7Q2 and observed values of 7Q2 for Indiana streams in Test Case C and

the relative error of the predicted values of 7Q2 with the observed values of 7Q2 for

Test Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11. Predicted values of 7Q10 and observed values of 7Q10 for Indiana streams in Test Case D and

the relative error of the predicted values of 7Q10 with the observed values of 7Q10 for

Test Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

12. Predicted values of 7Q2 and observed values of 7Q2 for Indiana streams in Test Case D and

the relative error of the predicted values of 7Q2 with the observed values of 7Q2 for

Test Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

13–14. Boxplots showing:

13. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas

method, Test Cases C and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

14. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas

method, Test Cases C and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

15. Map showing sets of streamflow-gaging stations used to evaluate the Stedinger-Thomas method

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when two

index stations and the partial-record station are located on the same stream or tributaries. . . . . . . . . . . . . 19

16–17. Graphs showing:

16. Predicted values of 7Q10 and observed values of 7Q10 for Indiana streams in Test Case E and

the relative error of the predicted values of 7Q10 with the observed values of 7Q10 for

Test Case E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

17. Predicted values of 7Q2 and observed values of 7Q2 for Indiana streams in Test Case E and

the relative error of the predicted values of 7Q2 with the observed values of 7Q2 for

Test Case E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

18–19. Boxplots showing:

18. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas

method, Test Case E compared with Test Cases A and B combined . . . . . . . . . . . . . . . . . . . . . . . . . 21

19. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas

method, Test Case E compared with Test Cases A and B combined . . . . . . . . . . . . . . . . . . . . . . . . . 21

20. Flow-duration curves for a pair of streamflow-gaging stations in Indiana with similar base-flowrecession characteristics and a pair of streamflow-gaging stations in Indiana with different

base-flow-recession characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Contents v

Contents—Continued

Figures—Continued

21–22. Boxplots showing:

21. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas

method, with the number of base-flow measurements varied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

22. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas

method, with the number of base-flow measurements varied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Tables

1. Description of test cases used to evaluate the Stedinger-Thomas method of estimating low-flow

frequencies at partial-record sites in Indiana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Streamflow-gaging stations in Indiana used to evaluate the Stedinger-Thomas method of estimating

low-flow frequencies at partial-record sites, using base-flow measurements . . . . . . . . . . . . . . . . . . . . . . . 32

3. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating low-flow frequencies, using base-flow measurements: Case A . . . . . . . . . . . . . . . . 34

4. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating low-flow frequencies, using base-flow measurements: Case B . . . . . . . . . . . . . . . . 36

5. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating low-flow frequencies, using base-flow measurements: Case C . . . . . . . . . . . . . . . . 38

6. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating low-flow frequencies, using base-flow measurements: Case D . . . . . . . . . . . . . . . . 40

7. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating low-flow frequencies, using base-flow measurements: Case E . . . . . . . . . . . . . . . . 42

8. Low-flow frequencies at selected streamflow-gaging stations in Indiana, estimated with two test cases

of the Stedinger-Thomas method of estimating low-flow frequencies, using base-flow measurements . . 43

9. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 10-year low flows, with the number of base-flow measurements varied:

Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 2-year low flows, with the number of base-flow measurements varied:

Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

11. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 10-year low flows, with the number of base-flow measurements varied:

Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

12. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 2-year low flows, with the number of base-flow measurements varied:

Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

13. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 10-year low flows, with the number of base-flow measurements varied:

Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vi Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

Contents—Continued

Tables—Continued

14. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 2-year low flows, with the number of base-flow measurements varied:

Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 10-year low flows, with the number of base-flow measurements varied:

Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 2-year low flows, with the number of base-flow measurements varied:

Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

17. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 10-year low flows, with the number of base-flow measurements varied:

Case E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

18. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas

method of estimating 7-day, 2-year low flows, with the number of base-flow measurements varied:

Case E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Conversion Factors and Abbreviations

Contents vii

The following abbreviations are used in this report:

Abbreviation Description

7Q10 7-day, 10-year low flow

7Q2 7-day, 2-year low flow

Multiply By To obtain

inch (in.) 25.4 millimeter

foot (ft) 0.3048 meter

mile (mi) 1.609 kilometer

square mile (mi2

) 2.590 square kilometer

cubic foot (ft3

) 0.02832 cubic meter

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

cubic foot per second (ft3

/s) 0.02832 cubic meter per second

Abstract 1

Evaluation of a Method of Estimating

Low-Flow Frequencies from Base-Flow Measurements

at Indiana Streams

By John T. Wilson

Abstract

A mathematical technique of estimating

low-flow frequencies from base-flow measurements was evaluated by using data for streams

in Indiana. Low-flow frequencies at lowflow partial-record stations were estimated

by relating base-flow measurements to concurrent daily flows at nearby streamflow-gaging

stations (index stations) for which low-flowfrequency curves had been developed. A

network of long-term streamflow-gaging

stations in Indiana provided a sample of sites

with observed low-flow frequencies. Observed

values of 7-day, 10-year low flow and 7-day,

2-year low flow were compared to predicted

values to evaluate the accuracy of the method.

Five test cases were used to evaluate

the method under a variety of conditions

in which the location of the index station

and its drainage area varied relative to the

partial-record station. A total of 141 pairs of

streamflow-gaging stations were used in the

five test cases. Four of the test cases used one

index station, the fifth test case used two index

stations. The number of base-flow measurements was varied for each test case to see if

the accuracy of the method was affected by

the number of measurements used.

The most accurate and least variable

results were produced when two index stations

on the same stream or tributaries of the partialrecord station were used. All but one value

of the predicted 7-day, 10-year low flow were

within 15 percent of the values observed for the

long-term continuous record, and all of the predicted values of the 7-day, 2-year low flow were

within 15 percent of the observed values. This

apparent accuracy, to some extent, may be a

result of the small sample set of 15.

Of the four test cases that used one index

station, the most accurate and least variable results were produced in the test case where the

index station and partial-record station were on

the same stream or on streams tributary to each

other and where the index station had a larger

drainage area than the partial-record station. In

that test case, the method tended to over predict, based on the median relative error. In 23 of

28 test pairs, the predicted 7-day, 10-year low

flow was within 15 percent of the observed

value; in 26 of 28 test pairs, the predicted 7-day,

2-year low flow was within 15 percent of the

observed value.

When the index station and partialrecord station were on the same stream or

streams tributary to each other and the index station had a smaller drainage area than

the partial-record station, the method tended
2 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

to under predict the low-flow frequencies.

Nineteen of 28 predicted values of the 7-day,

10-year low flow were within 15 percent of the

observed values. Twenty-five of 28 predicted

values of the 7-day, 2-year low flow were

within 15 percent of the observed values.

When the index station and the partialrecord station were on different streams, the

method tended to under predict regardless

of whether the index station had a larger or

smaller drainage area than that of the partialrecord station. Also, the variability of the

relative error of estimate was greatest for

the test cases that used index stations and

partial-record stations from different streams.

This variability, in part, may be caused by using

more streamflow-gaging stations with small

low-flow frequencies in these test cases. A

small difference in the predicted and observed

values can equate to a large relative error when

dealing with stations that have small low-flow

frequencies.

In the test cases that used one index

station, the method tended to predict smaller

low-flow frequencies as the number of baseflow measurements was reduced from 20

to 5. Overall, the average relative error of

estimate and the variability of the predicted

values increased as the number of base-flow

measurements was reduced.

Introduction

The management and availability of Indiana’s

water resources increase in importance every

year with growing demands for the use and development of Indiana’s waterways. Low-flow

characteristics of streams are needed for management decisions by State and local officials

concerned with water supplies, pollution management of wastewater, and fish and wildlife

preservation. Fowler and Wilson (1996) presented

low-flow characteristics for 229 continuous-record

stations and 285 low-flow partial-record stations

in Indiana. These characteristics included lowflow-frequency analysis and flow-duration analysis

for the continuous-record sites and low-flowfrequency analysis for the partial-record stations.

Continuous-record stations are streamflow-gaging

stations for which daily streamflow is computed

and stored. Low-flow partial-record stations are

sites where discharge measurements are made at

base flow. The term “partial-record station” is used

in this report as a substitute for the longer term

“low-flow partial-record station.”

Frequency analysis for continuous-record

stations is done by a standard procedure of fitting

a frequency curve to observed annual minimum

flows. Frequency curves relate the magnitude of

a variable to the frequency of occurrence (Riggs,

1968). Low-flow-frequency curves developed by

the U.S. Geological Survey (USGS) are defined

by using a mathematical procedure for fitting the

data to a log Pearson type III distribution (Riggs,

1972). In low-flow investigations, the frequency

curve relates the minimum average discharge (Q)

for a given number of consecutive days (N-day)

to the recurrence interval in years (T-year). For

example, the 7-day, 10-year low flow (7Q10) is the

minimum average discharge for 7 consecutive days,

which has a 0.1 probability of not being exceeded

in a given year. The recurrence interval is the reciprocal of the probability of recurrence.

Frequency analysis for low-flow partial-record

stations cannot be done by the frequency-curve

method because a partial-record station does

not have a record of annual minimum flows. Instead, base-flow measurements are related to

the concurrent daily mean flows of a nearby

continuous-record station (index station) for which

a low-flow-frequency curve has been defined. In

Fowler and Wilson (1996), low-flow frequencies

at partial-record stations were estimated by use of

the mathematical technique described in Stedinger

and Thomas (1985) (which will be referred to as

the Stedinger-Thomas method) or by the graphical

correlation method described in Riggs (1972). The

Stedinger-Thomas method was used as the primary

method for estimating low-flow frequencies because it is a mathematical method, which avoids

the bias of drawing best-fit curves through the data.
Methods of Investigation 3

Therefore, the evaluation of the accuracy of lowflow frequencies in this report was based on

the Stedinger-Thomas method. As in Fowler and

Wilson (1996), low-flow-frequency estimates were

limited to the 7Q10 and the 7Q2.

For the purposes of this report, the terms

“flow,” “streamflow,” and “discharge” will be

considered synonymous and will be used interchangeably. All three terms refer to the volume

of water that passes a given point within a given

period of time; all have units of cubic feet per

second (ft3

/s).

Estimates of low-flow frequencies by use of

the Stedinger-Thomas method are predicted from

correlation of base flows and thus are referred to

as “predicted” values in this report. Estimates of

low-flow frequencies by use of log Pearson type III

frequency curves are values observed for longterm continuous record (annual minimum flows)

at streamflow-gaging stations and are referred to

as “observed” values in this report. The observed

values are treated as known or “true” values in

this report for making comparisons to evaluate the

accuracy of the method; however, the “true” lowflow frequencies never are known.

The evaluation of the Stedinger-Thomas

method of estimating low-flow frequencies presented in this report was prepared by the USGS,

in cooperation with the Indiana Department of

Natural Resources, Division of Water.

Purpose and Scope

This report presents an evaluation of the

accuracy of estimating low-flow frequencies of

Indiana streams, using base-flow measurements.

The results of this study provide performance

information for a variety of conditions to users of

low-flow-frequency values in Indiana. The report

provides a general range of the accuracy that can be

expected when estimating low-flow frequencies for

a given set of conditions. The accuracy of estimating low-flow frequencies is highly variable, with

each site being a unique situation. The results of

this study, however, may provide assistance on

how to evaluate the method of relating base-flow

measurements to daily mean flows at an index

station.

This report also provides a detailed evaluation

of the Stedinger-Thomas method of estimating

low-flow frequencies from base-flow measurements (Stedinger and Thomas, 1985), which can

be applied to areas other than Indiana.

Acknowledgments

The author acknowledges the contributions of

three individuals from the USGS. Michael McNally

wrote a computer program that was used to retrieve

daily mean flows from the USGS data base to

match the dates of base-flow measurements at the

partial-record stations. Gary Tasker and Kirk White

provided insightful comments and helpful suggestions in their technical reviews of the manuscript.

Methods of Investigation

A network of long-term streamflow-gaging

stations provides a sample of sites with observed

low-flow frequencies. These observed values

are compared to the values predicted with the

Stedinger-Thomas method to evaluate the accuracy

of the method. Streamflow-gaging stations were

paired for each analysis, in that one station was

treated as a partial-record station with base-flow

measurements and the other station was treated as

the index station. This approach is similar to the

test of the method presented in Stedinger and

Thomas (1985). The Stedinger-Thomas method

was evaluated for five situations or “test cases.”

For descriptive purposes in this report, these five

test cases are referred to as Test Case A through

Test Case E (table 1).

Pairs of streamflow-gaging stations were

selected to provide reasonable index stations for

each other. Pairs of streamflow-gaging stations

were selected primarily on the basis of proximity

to the partial-record station, an important characteristic of a good index station. Ideally, the index

station and the partial-record station should have
4 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

similar base-flow-recession characteristics, such

as similar flow-duration curves or similar hydrographs. The base-flow-recession characteristics at

the partial-record station typically will be unknown,

however, because there is no continuous record of

streamflows. Yet similar base-flow-recession characteristics can be inferred if the watershed for the

index station is of similar terrain, drainage area,

and geologic characteristics as that for the partialrecord station. The pairs of stations that were

selected cover a wide range of drainage areas,

difference in drainage area, and distance between

stations to examine which characteristics, if any,

affect the accuracy of the method. The stations

in each pair are, for the most part, within the same

physiographic region or near the boundaries of

adjacent regions. A total of 141 pairs of streamflowgaging stations were used in the five test cases.

In Case A, the partial-record station and the

index station are on the same stream or the partialrecord station is on a tributary of the stream where

the index station is located. The drainage area

at the index station is greater than at the partialrecord station. In other words, the index station

is downstream from the partial-record station.

Case B is similar to Case A, except that the drainage

area at the index station is less than at the partialrecord station. In other words, the index station is

upstream from the partial-record station. The same

28 pairs of streamflow-gaging stations are used in

Case A and Case B; switching the position of the

index station allows for an evaluation of the effect

of the difference in drainage area between the index

stations and partial-record stations.

In Case C, the partial-record station and the

index station are on different streams, possibly in

different drainage basins, and the drainage area at

the index station is greater than at the partial-record

station. Case D is similar to Case C, except that the

drainage area at the index station is less than at

the partial-record station. The same 35 pairs of

streamflow-gaging stations are used in Case C

and Case D; switching the position of the index

station allows for an evaluation of the effect of

the difference in drainage area between the index

stations and partial-record stations. Cases C and D

allow for an evaluation of how accuracy is affected

by selecting an index station from another basin.

Case E evaluates the Stedinger-Thomas method when two index stations are used. In Case E, the

partial-record station and the two index stations are

on the same stream or tributaries. In 11 instances,

1 index station is upstream from the partial-record

station and 1 index station is downstream from the

partial-record station. In four instances, the partialrecord station is downstream from two tributaries

with index stations. Case E allows for an evaluation

of how accuracy is affected by using two index

stations, as compared to Case A and Case B that

use one index station. The sample size for Case E

Table 1. Description of test cases used to evaluate the Stedinger-Thomas method of estimating low-flow frequencies

at partial-record sites in Indiana

Test

Case

Sample

size

Index-station characteristics

Location relative to partial-record site Drainage area

A

B

C

D

E1

28

28

35

35

15

Same stream or tributary

Same stream or tributary

Different stream or basin

Different stream or basin

Same stream or tributary

Greater than partial-record station

Less than partial-record station

Greater than partial-record station

Less than partial-record station

Greater than or less than partial-record station

1Test Case E uses two index stations.
Frequency Analysis at Partial-Record Stations 5

is smaller than the other test cases because the

distribution of streamflow-gaging stations within

the State provided limited sets of stations suitable

for use as index stations for this case.

Pairs of streamflow-gaging stations selected

for the test cases had to have some period of record

common to both gages because each station served

as an index station and as a partial-record station

(Cases A and B, C and D). Estimates of low-flow

frequencies vary with length of record (Riggs,

1972), so a common period of record was used to

give the method a fair chance at predicting the

observed low-flow frequency. In actual low-flow

investigations, the period of record at the index

station would not be controlled. One would expect

that the longer the period of record at the index

station, the more reliable the estimates at the

partial-record station. Table 2 (at the back of this

report) lists the streamflow-gaging stations used

in this study with their complete periods of record

and most recently published values of 7Q10 and

7Q2. Many of the streamflow-gaging stations listed

in table 2 were used multiple times, often with different periods of record that were dependent on the

streamflow-gaging station with which it was paired.

Low-flow measurements, representing base

flow for each streamflow-gaging station, were

retrieved from the historical files and entered

into the data base if needed. With the base-flow

measurements in the data base, the concurrent daily

flows at the index stations could be retrieved with

a computer program. The base-flow measurements

were plotted against the concurrent daily flows

(using logarithmic scales) to determine a linear

relation between the flows at the two sites. Twenty

measurements, covering a range of flows, were

selected for each analysis to define the linear relation. The measurements were selected to cover

the shortest period of time whenever possible;

however, the measurements were made for routine

maintenance of rating curves, and often only a few

measurements were made at low flow during a year.

In some instances, the measurements cover a span

of 10 years or more. Actual low-flow investigations

would collect base-flow measurements in a much

shorter time span; however, data collected over

such a large time span does provide independent

observations of base flow.

The 20 measurements used to define the

relation between base-flow measurements and

concurrent daily flows are more than most low-flow

investigations will use. Riggs (1972) indicated that

generally 8 to 10 measurements made on different

streamflow recessions, and in more than 1 year,

should adequately define the relation to concurrent

flows at the index station. Because the StedingerThomas method includes a linear regression, it

is recommended that at least 10 measurements

be used (W.O. Thomas, Jr., and others, U.S. Geological Survey, written commun., 1993). Twenty

measurements were used in the analyses for this

study primarily for three reasons: (1) the data

were available because of the long-term record

of the streamflow-gaging stations; (2) choosing

measurements to define the linear relation with

concurrent daily flows has a certain amount of bias

associated with it, which probably increases as

the number of measurements decreases; using

20 measurements generally provided a wider

range of base flows to define the linear relation;

and (3) Stedinger and Thomas (1985) indicated

that the standard error of estimate is not reduced

significantly when the number of observations

exceeds 20.

In a later section of this report, the number of

base-flow measurements used in the analyses is

varied to evaluate how the accuracy of the method

is affected. The number of base-flow measurements

is varied for a subset of the streamflow-gagingstation pairs for each test case, and the results are

compared to those obtained when the original 20

measurements were used.

Frequency Analysis at

Partial-Record Stations

Low-flow frequencies at partial-record stations

were estimated by use of the mathematical technique described in Stedinger and Thomas (1985).

The Stedinger-Thomas method defines the relation
6 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

between base-flow measurements at a partialrecord station and concurrent daily flows at an

index station, using least-squares-regression

analysis of the logarithms of flows. The regression

analysis and low-flow statistics (moments) at the

index station are used to estimate the desired flow

characteristics at the partial-record station. The

least-squares-regression line, based on discharge

measurements at the partial-record station and

concurrent flows at the index station, is defined

by the equation:

y ab += , (1)

where represents the log-transformed flows

at the partial-record station, and

represents the log-transformed flows

at the index station.

From Stedinger and Thomas (1985), the

logarithm of the N-day, T-year low flow at

the partial-record station, , is estimated by

the equation:

Y , T μy Kyσy += (2)

where denotes the mean,

denotes the standard deviation, and

is the frequency factor for the

value of skew coefficient at

the T-year recurrence interval

for the partial-record station.

Because the partial-record station has no

record of N-day low flows, the logarithms of baseflow measurements at the partial-record station

and concurrent daily flows at the index station are

used to estimate the parameters in equation 2. One

of the key assumptions of the Stedinger-Thomas

method is that the frequency factor for the index

station, , and the frequency factor for the partialrecord station, , are the same. It is assumed that

the frequency factors will be approximately equal

if the sites are in similar hydrologic settings and

have similar drainage areas. This is an important

assumption of the method that must be considered

when choosing index stations.

Given that the frequency factors for the index

station and partial-record station are assumed to

be the same, in equation 2 can be estimated by

, which can be substituted with . The

relation in equation 2 then can be approximated by:

where is the sample mean of the logarithms

of the annual N-day low flows at the

index station,

is the sample standard deviation of

the logarithms of the annual N-day

low flows at the index station,

is the standard error of estimate of

the least-squares-regression equation (eq. 1),

is the number of base-flow measurements and concurrent daily flows

used to estimate the regression

equation,

is the standard deviation of the logarithm of the concurrent daily flows

at the index station used in the regression equation, and

 is the logarithm of the N-day, Tyear low flow for the index station.

x

y

x

YT

μ

σ

Ky

Kx

Ky

Ky

Kx

XT mx –

sx

------------------- 

 

YT a bmx

XT mx –

sx

-------------------     b

2

sx

2 se2 1 sx

2

( ) L 1– s ̃

x

2 – ---------------------- 



 +





 1 2⁄

= + + , (3)













mx

sx

se

L

s ̃ x

XT
Evaluation of the Stedinger-Thomas Method 7

The factor

was used to obtain an unbiased estimator of

(Stedinger and Thomas, 1985, p. 4).

Stedinger and Thomas (1985) recommend

that the correlation coefficient exceed 0.70. In this

study, there were 7 instances, out of 141, in which

the correlation coefficient was less than 0.70. Because the method uses the logarithms of flows,

zero flows cannot be used. Many small streams in

southern Indiana often have zero flow (Stewart

and others, 1999; Arihood and Glatfelter, 1991);

therefore, many streamflow-gaging stations in

southern Indiana could not be used in this study.

The Stedinger-Thomas method has been automated by the USGS and has the capability of using

multiple index stations (W.O. Thomas and others,

U.S. Geological Survey, written commun., 1993).

The user is responsible for selecting which measurements are used in the analysis. The method

will fit a least squares regression to the data, so the

user must determine if the base-flow measurements

have an adequate linear relation with the concurrent

daily flows at the index station.

Evaluation of the

Stedinger-Thomas Method

The Stedinger-Thomas method of estimating

low-flow frequencies at partial-record stations was

evaluated through five test cases designed to cover

a variety of conditions. In these test cases the location of the index station was varied to evaluate the

effects of drainage-area differences, using an index

station from a different stream or drainage basin

and using two index stations. A discussion of the

results of each test case follows.

Test Cases A and B

Test Cases A and B are designed to evaluate

the Stedinger-Thomas method when the index

station and the partial-record station are on the

same stream or when the upstream station is on

a tributary of the downstream station. In Case A,

the drainage area of the index station is greater

than at the partial-record station. In Case B, the

drainage area of the index station is less than at

the partial-record station. The pairs of streamflowgaging stations used in Cases A and B are shown

in figure 1. The pair numbers in figure 1 correspond

to those in tables 3 and 4 (at the back of this report).

Cases A and B use the same 28 pairs of streamflowgaging stations, with the positions of the index

and partial-record stations reversed for Case B.

Low-flow frequencies listed in the tables include

one more significant figure than normally would

be reported. The extra significant figure is used

for comparison purposes to help in evaluating the

method.

Figure 2a shows the predicted 7Q10 and the

observed 7Q10 for Case A. The relative error of predicted 7Q10 over the range of observed values is

shown in figure 2b. The observed values are based

on log Pearson type III frequency curves of the

annual 7-day minimum flows at the partial-record

station for the same period of record as the index

station. The relative error of estimate was determined by the equation:

(Observed 7QT - Predicted 7QT) / Predicted 7QT

where T is the recurrence interval (in years) of

the annual 7-day minimum flows.

There were 17 instances in Case A in which

the Stedinger-Thomas method predicted higher

values of 7Q10 than the values observed for the

continuous record. There were seven instances in

which the Stedinger-Thomas method predicted

lower values than the observed values. The average

relative error of estimate was -1.0 percent, with a

standard deviation of 16.2 (table 3). In Case A, the

relative error of estimate ranged from -28.0 percent

to 47.6 percent. A negative value of relative error

(fig. 2b) indicates that the predicted value is greater

than the observed value.

A predicted value within 15 percent of the observed value was selected arbitrarily as the criterion

that represented an excellent estimate. The 15-

percent error band is included in the plots of relative

error. In Case A, 23 of 28 predicted values of 7Q10

1 sx

2

(L 1– s ̃

x

2 – ---------------------- 





) 





σy

2

 , (4)
Ea st

Fork

19

19

20

20

24 24 23

21

22

22 21

23

25

25

26

27

26

28 27

28

7

7

5 4

4

5

6 6

1

1

2

3

3

13

2

12 12

13

8

8 9

10

10

11

9

11

14

14,15

15

16

16

17

17

28

18

18

Figure 1. Map showing pairs of streamflow-gaging stations used to evaluate the Stedinger-Thomas method 

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when the 

index station and partial-record station are located on the same stream or tributaries. 

Wabash

River

Base from U.S. Geological Survey digital data, 1:100,000 1986

Albers Equal-Area projection

Standard parallels 29 30' and central meridian -86

38

39

40

41

87 86 85

EXPLANATION

DRAINAGE BASIN BOUNDARY

STREAMFLOW-GAGING

STATION NUMBER

STREAMFLOW-GAGING STATION

WITH PAIR NUMBER

Ka nkakee

River

Lake Michigan

0

0

10

10

20

20

30

30

40

40

50

50

KILOMETERS

MILES

03274650

03274750

03275000

03275500

03275600

03322500

03322900

03324200

03324300

03325500

03326000

03328000

03328500

03334500

03335000

03335500

03336000

03339500

03340000

03340500

03341500

03349000 03348000

03361000

03361500

03362000

03362500

03363000

03363500

03363900

03364000

03365500

03365500

03371500

03373500

04099510

04099750

04179500

04180000

05515000

05515400

05515500

05516000

05516500

05517000

05517530 05518000

05522000

05522500

05524500

White

River

White

River

White

River
Evaluation of the Stedinger-Thomas Method 9

0.1 10,000 1 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

0.1

10,000

0.2

0.5

1

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q10, IN CUBIC FEET PER SECOND

0.1 10,000 1 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

0.1 10,000 1 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

0.1

10,000

0.2

0.5

1

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q2, IN CUBIC FEET PER SECOND

0.1 10,000 1 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=28

N=28

7 under predicted

17 over predicted

12 under predicted

16 over predicted

Figure 2. Predicted values of 7Q10 and observed values of 7Q10 (a) for Indiana streams in Test Case A and (b) the

relative error of the predicted values of 7Q10 with the observed values of 7Q10 for Test Case A.

Figure 3. Predicted values of 7Q2 and observed values of 7Q2 (a) for Indiana streams in Test Case A and (b) the

relative error of the predicted values of 7Q2 with the observed values of 7Q2 for Test Case A.

a b

a b
10 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

were within 15 percent of the observed 7Q10. The

relative error of estimate shows no direct relation

with the magnitude of the 7Q10; however, the largest errors are for those stations with a 7Q10 less than

10 ft3/s (fig. 2b). At small flows, a relatively small

difference equates to a large error in percent. For

example, in the first pair listed in table 3, a difference of 0.1 ft3

/s equates to an error of 47.6 percent.

Overall, the predicted values of 7Q2 for Case A

were slightly more accurate, with less variability

than the predicted values of 7Q10 (fig. 3). The relative error of estimate ranged from -16.4 percent to

15.8 percent. All but two predicted values of 7Q2

were within 15 percent of the observed value.

In Case A, there were 16 instances in which the

Stedinger-Thomas method predicted higher values

of 7Q2 than the values observed for the continuous record. There were 12 instances in which the

Stedinger-Thomas method predicted lower values

than the observed values (fig. 3b). The average

relative error of estimate was -1.0 percent, with

a standard deviation of 7.5 percent—less than half

the standard deviation for the predicted values of

7Q10 (table 3).

The positions of the index and partial-record

stations in Case B are reversed from the positions

in Case A. In other words, the index station of

Case B is the partial-record station of Case A.

Figure 4 shows the predicted 7Q10 and the observed

7Q10 for Case B, and the relative error of predicted

7Q10 over the range of observed values. In Case B,

the Stedinger-Thomas method tended to predict

lower values of 7Q10 than the values observed for

the continuous record. The 7Q10 was under predicted 19 times and over predicted 9 times. The

average relative error of estimate was 9.6 percent,

with a standard deviation of 16.8 percent (table 4,

at the back of this report). In Case B, 19 of 28 predicted values of 7Q10 were within 15 percent of

the observed 7Q10. The relative error of estimate

ranged from -12.8 percent to 55.1 percent. The

relative error of estimate shows no direct relation

with the size of the observed 7Q10; however, all but

one of the observations with greater than 15-percent

error have index stations with a 7Q10 of less than

10 ft3/s. The index stations for Case B, and their

low-flow frequencies, are listed in table 3 as the

partial-record stations of Case A. The pair numbers

in tables 3 and 4 correspond to the same pairs of

streamflow-gaging stations.

The predicted values of 7Q2 for Case B were

slightly more accurate, with less variability than

the predicted values of 7Q10 (fig. 5). The relative

error of estimate ranged from -8.6 percent to

31.1 percent. All but three predicted values of 7Q2

were within 15 percent of the observed value, but

there was a tendency to under predict (fig. 5b).

The method under predicted the 7Q2 17 times

and over predicted 10 times. The average relative error of estimate was 4.5 percent, with a

standard deviation of 8.7 percent—almost half

the standard deviation of the estimates of 7Q10

(table 4).

Boxplots of the relative error were used to

illustrate the performance of the Stedinger-Thomas

method in Cases A and B (figs. 6 and 7). In Case A,

the method tended to over predict (based on the

median relative error) the 7Q10 and the 7Q2, but

more so for the 7Q10. In Case B, the method tended

to under predict the 7Q10 and the 7Q2, once again,

more so for the 7Q10. The Case B estimates of 7Q10

also show more variability than for Case A, based

on the spread of the interquartile range (fig. 6). The

difference between the two cases is the location of

the index station; the index station in Case B has

a smaller drainage area and smaller flows than for

Case A. The smaller flows probably are associated

with more variability and reduced accuracy. When

Cases A and B are combined, the median relative

error is close to zero for both 7Q10 (fig. 6) and 7Q2

(fig. 7), which indicates that there is no bias to over

predict or under predict. The combined average

relative error of estimate is 4.3 percent for 7Q10

and 1.7 percent for 7Q2.

Test Cases C and D

Test Cases C and D are designed to evaluate

the Stedinger-Thomas method when the index

station and partial-record station are on different

streams or in different basins. In Case C, the drainage area of the index station is greater than at the

partial-record station. In Case D, the drainage area
Evaluation of the Stedinger-Thomas Method 11

1 10,000 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

1

10,000

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q10, IN CUBIC FEET PER SECOND

1 10,000 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

1 10,000 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

1

10,000

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q2, IN CUBIC FEET PER SECOND

1 10,000 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=28

N=28

19 under predicted

9 over predicted

17 under predicted

10 over predicted

Figure 4. Predicted values of 7Q10 and observed values of 7Q10 (a) for Indiana streams in Test Case B and (b) the

relative error of the predicted values of 7Q10 with the observed values of 7Q10 for Test Case B.

Figure 5. Predicted values of 7Q2 and observed values of 7Q2 (a) for Indiana streams in Test Case B and (b) the

relative error of the predicted values of 7Q2 with the observed values of 7Q2 for Test Case B.

a

a

b

b
12 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

CASE A CASE B CASES A & B

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=28 N=28 N=56

CASE A CASE B CASES A & B

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=28 N=28 N=56

over predicted

over predicted

under predicted

under predicted

Figure 6. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas method,

Test Cases A and B.

Figure 7. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas method,

Test Cases A and B.

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to

interquartile range outside the quartile

Outlier data value more than 3 times the

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to
Evaluation of the Stedinger-Thomas Method 13

of the index station is less than at the partial-record

station. The pairs of streamflow-gaging stations

used in Cases C and D are shown in figure 8. The

pair numbers in figure 8 correspond to those in

tables 5 and 6 (at the back of this report). Cases C

and D use the same 35 pairs of streamflow-gaging

stations, with the positions of the index and partialrecord stations reversed for Case D.

Figure 9a shows the predicted 7Q10 and the

observed 7Q10 for Case C. The relative error of

predicted 7Q10 over the range of observed values

is shown in figure 9b. In Case C, there were 14 instances in which the Stedinger-Thomas method

predicted higher values of 7Q10 than the values

observed for the continuous record. There were

20 instances in which the predicted values of 7Q10

were lower than the values observed for the continuous record. The average relative error of estimate

was 20.6 percent, with a standard deviation of

45.5 percent (table 5). The noticeably large average relative error, compared to Cases A and B, is a

result of a few stations with small flows that have

large relative errors (fig. 9b). For example, in pair

number 33 listed in table 5, the difference between

the observed and predicted 7Q10 is only 0.06 ft3/s;

however, this equates to a relative error of 150 percent.

In Case C, 15 of 35 predicted values of 7Q10

were within 15 percent of the observed 7Q10. The

relative error of estimate ranged from -40.0 percent

to 150 percent. The relative error of estimate tends

to increase as the observed 7Q10 decreases; the

largest errors are for those stations with a 7Q10 less

than 2 ft3/s (fig. 9b). The large variability in relative

error of Case C compared to Case A may in part

be related to the increased number of stations with

small values of 7Q10. At extremely low flows in

natural channels, it becomes more difficult to maintain the accuracy of discharge measurements and to

account for shifting controls at streamflow-gaging

stations, which can result in reduced accuracy of

the streamflow record.

The predicted values of 7Q2 for Case C were

slightly more accurate with less variability than

the predicted values of 7Q10. The relative error

of estimate ranged from -34.1 percent to 96.1 percent (table 5). There were 23 instances in which

the method predicted lower values of 7Q2 than the

values observed for the continuous record. There

were 10 instances in which the predicted values

of 7Q2 were higher than the values observed for

the continuous record (fig. 10). In Case C, 19 of

35 predicted values of 7Q2 were within 15 percent

of the observed value. The average relative error

of estimate was 14.2 percent, with a standard deviation of 27.6 percent (table 5).

The positions of the index station and partialrecord station in Case D are reversed from the

positions in Case C so that the drainage area of

the index station is less than at the partial-record

station. Figure 11 shows the predicted 7Q10 and

the observed 7Q10 for Case D and the relative error

of the predicted 7Q10 over the range of observed

values. There were 17 instances in Case D in

which the predicted value of 7Q10 was lower than

the value observed for the continuous record. There

were also 17 instances in which the predicted value

was higher than the observed value. The average

relative error of estimate was 14.6 percent, with a

standard deviation of 36.6 percent (table 6). The

large average relative error, compared to Cases A

and B, is a result of a few stations with small flows

that have large relative errors (fig. 11b).

In Case D, 18 of 35 predicted values of 7Q10

were within 15 percent of the observed 7Q10. The

relative error of estimate ranged from -28.6 percent

to 123 percent. Although there is no direct relation

between the two, the relative error of estimate tends

to increase as the observed 7Q10 decreases. As in

Case B, all but one of the observations with greater

than 15-percent error have index stations with a

7Q10 of less than 10 ft3/s. The index stations for

Case D, and their low-flow frequencies, are listed

in table 5 as the partial-record stations of Case C.

The predicted values of 7Q2 for Case D were

slightly more accurate, with less variability than

the estimates of 7Q10. The relative error of estimate

ranged from -16.6 percent to 81.8 percent (table 6).

There were 20 instances in Case D in which the

predicted value of 7Q2 was lower than the value

observed for the continuous record. There were 13

instances in which the predicted value was higher

than the observed value (fig. 12). In Case D, 25 of

35 predicted values of 7Q2 were within 15 percent
EXPLANATION

DRAINAGE BASIN BOUNDARY

STREAMFLOW-GAGING

STATION NUMBER

STREAMFLOW-GAGING STATION 

WITH PAIR NUMBER

03365500

Base from U.S. Geological Survey digital data, 1:100,000 1986

Albers Equal-Area projection

Standard parallels 29 30' and central meridian -86

0

0

10

10

20

20

30

30

40

40

50

50

KILOMETERS

MILES

Fork

White

River

Ea st

Figure 8. Map showing pairs of streamflow-gaging stations used to evaluate the Stedinger-Thomas method

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when the 

index station and partial-record station are located on different streams or basins.

White

River

White

River

03274750

03275000

03275600

03322500

03324000

03324200

03324300

03325500

03326070

03326500

03327520

03328000

03328430

03329700

03334000

03334500

03339500

03340800

03341200

03348000

03348350 03348020

03350700

03351500

03352200 03353120

03353180

03353620

03357500

03358000

03359500

03361500

03362500

03363000

03363900

03364200

04094000

04094500

04095300

04096100

04099750

04099808

04100252

04100295

04100500

04178000

0417950

04180000

04182590

05515400

05519500

Lake Michigan

05515000

05519000

26

35

35

8,10

26

27

34

34

28 28,29

29

30

33

9

9

8

4

1

1,2

2

18

23

12

11

13

12,13

20,21

20,22

23

24

25

35

24,25

17

19

18,19

15

6

14

16

16,17

14,15

7

3,4,7

5 3

5,6

33

30 31

32

31,32

27

21,22

0

10,11

03322900 

Ri ver

87 86 85

38

39

40

41

Kankakee

Wabash

River
Evaluation of the Stedinger-Thomas Method 15

0.01 100 0.1 1 10

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

0.01

100

0.02

0.05

0.1

0.2

0.5

1

2

5

10

20

50

PREDICTED 7Q10, IN CUBIC FEET PER SECOND

0.01 100 0.1 1 10

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

-200

200

-100

0

100

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

0.01 100 0.1 1 10

0BSERVED 7Q2, IN CUBIC FEET PER SECOND

0.01

100

0.02

0.05

0.1

0.2

0.5

1

2

5

10

20

50

PREDICTED 7Q2, IN CUBIC FEET PER SECOND

0.01 100 0.1 1 10

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

-100

100

-50

0

50

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=35

N=35

20 under predicted

14 over predicted

23 under predicted

10 over predicted

Figure 9. Predicted values of 7Q10 and observed values of 7Q10 (a) for Indiana streams in Test Case C and (b) the

relative error of the predicted values of 7Q10 with the observed values of 7Q10 for Test Case C.

Figure 10. Predicted values of 7Q2 and observed values of 7Q2 (a) for Indiana streams in Test Case C and (b) the

relative error of the predicted values of 7Q2 with the observed values of 7Q2 for Test Case C.

a b

a b
16 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

0.1 1,000 1 10 100

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

0.1

1,000

0.2

0.5

1

2

5

10

20

50

100

200

500

PREDICTED 7Q10, IN CUBIC FEET PER SECOND

0.1 1,000 1 10 100

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

-150

150

-100

-50

0

50

100

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

0.1 1,000 1 10 100

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

0.1

1,000

0.2

0.5

1

2

5

10

20

50

100

200

500

PREDICTED 7Q2, IN CUBIC FEET PER SECOND

0.1 1,000 1 10 100

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

-100

100

-50

0

50

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=35

N=35

17 under predicted

17 over predicted

20 under predicted

13 over predicted

Figure 11. Predicted values of 7Q10 and observed values of 7Q10 (a) for Indiana streams in Test Case D and (b) the

relative error of the predicted values of 7Q10 with the observed values of 7Q10 for Test Case D.

Figure 12. Predicted values of 7Q2 and observed values of 7Q2 (a) for Indiana streams in Test Case D and (b) the

relative error of the predicted values of 7Q2 with the observed values of 7Q2 for Test Case D.

a b

a b
Evaluation of the Stedinger-Thomas Method 17

of the observed value. The average relative error of

estimate was 7.6 percent, with a standard deviation

of 19.9 percent (table 6).

Boxplots of the relative error were used

to illustrate the performance of the StedingerThomas method in Cases C and D (figs. 13 and 14).

In Case C, the method tended to under predict

(based on the median relative error) the 7Q10 and

the 7Q2. In Case D, the method tended to under

predict the 7Q2 but not the 7Q10. Predicted values

of 7Q10 and 7Q2 in Case C show more variability

than for Case D, based on the spread of the interquartile range and whiskers. When Cases C and D

are combined, the median relative error of the

estimates of 7Q10 is 2.5 percent (fig. 13) and the

median relative error of the estimates of 7Q2 is

5.5 percent (fig. 14), which indicates that there is a

slight bias to under predict. The combined average

relative error of estimate is 17.6 percent for 7Q10

and 10.9 percent for 7Q2.

Test Case E

Test Case E is designed to evaluate the

Stedinger-Thomas method when two index stations

are used. In Case E, the partial-record station and

the two index stations are on the same stream or

tributaries. For 11 of the 15 sets of streamflowgaging stations, one index station is upstream from

the partial-record station and one index station

is downstream from the partial-record station.

For the remaining four sets, the partial-record

station is downstream from the confluence of

two tributaries with index stations. The 15 sets

of streamflow-gaging stations used in Case E are

shown in figure 15. The set numbers in figure 15

correspond to those in table 7 (at the back of this

report). The results of Case E can be compared

to the combined results of Cases A and B because

each test uses stations that are on the same stream

or tributaries.

Figure 16 shows the predicted values of 7Q10

and the observed values of 7Q10 for Case E and

the relative error of the predicted 7Q10 over the

range of observed values. In Case E, there were

six instances in which the predicted value of 7Q10

was higher than the value observed for the continuous record. There were nine instances in which

the predicted value was lower than the observed

value. The average relative error of estimate was

2.0 percent, with a standard deviation of 7.9 percent (table 7). All but one predicted value of 7Q10

was within 15 percent of the observed value.

The relative error of estimate ranged from -9.5 percent to 20.8 percent. The average relative error

of estimate is similar to that for Cases A and B

combined—4.3 percent, with a standard deviation

of 17.2 percent.

Figure 17 shows the predicted 7Q2 and the

observed 7Q2 for Case E and the relative error

of the predicted 7Q2 over the range of observed

values. The accuracies of the predicted values of

7Q2 for Case E were similar to those for the 7Q10.

There were six instances in which the method

predicted higher values of 7Q2 than the values observed for the continuous record. There were eight

instances in which the predicted value was lower

than the observed value. All of the predicted values

of 7Q2 were within 15 percent of the observed

value. The relative error of estimate ranged from

-6.2 percent to 11.5 percent. The average relative

error of estimate was 1.7 percent, with a standard

deviation of 5.2 percent (table 7). The average

relative error of estimate is similar to that for

Cases A and B combined—1.7 percent, with a

standard deviation of 8.5 percent.

Boxplots of the relative error of estimate were

used to illustrate the performance of the StedingerThomas method in Case E and to compare it to

the combined results of Cases A and B (figs. 18

and 19). The median and interquartile range for

Case E are comparable to those for Cases A and B

combined for the 7Q10 and the 7Q2. The small

range in relative error of estimate for Case E probably is a result of the small sample set and the high

degree of correlation in the flows between the

stations that were used. The apparent accuracy of

the method in Case E also can be explained, in part,

by the lack of small low-flow frequencies; only

three of the stations have an observed 7Q10 less

than10 ft3/s (table 7). The largest relative error is

associated with the smallest 7Q10 (fig. 16b). The

sample size for Case E is smaller than the other
18 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

CASE C CASE D CASES C & D

-200

200

-150

-100

-50

0

50

100

150

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=35 N=35 N=70

CASE C CASE D CASES C & D

-150

150

-100

-50

0

50

100

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=35 N=35 N=70

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to

Figure13. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas method,

Test Cases C and D.

Figure 14. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas method,

Test Cases C and D.

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to

interquartile range outside the quartile

Outlier data value more than 3 times the

over predicted

under predicted

under predicted

over predicted
Figure 15. Map showing sets of streamflow-gaging stations used to evaluate the Stedinger-Thomas method

of estimating low-flow frequencies from base-flow measurements at Indiana streams, when two index

stations and the partial-record station are located on the same stream or tributaries.

Base from U.S. Geological Survey digital data, 1:100,000 1986

Albers Equal-Area projection

Standard parallels 29 30' and central meridian -86

0

0

10

10

20

20

30

30

40

40

50

50

KILOMETERS

MILES

EXPLANATION

DRAINAGE BASIN BOUNDARY

STREAMFLOW-GAGING

STATION NUMBER

STREAMFLOW-GAGING STATION 

WITH SET NUMBER

03365500

03274650

03274750

03275000

03325500 03326000

03326500

03334000

03334500

03335000

03335500

03336000

03340500

03341500

03361000

03361500

03362000

03362500

03363000

03363900

03364000

03365500

03371500 03373500

05515000

05515500

05516000

05516500

05517000

05517500

05522000

05522500

05524500

Lake Michigan

12

12

14

15

15

2

2 2

1

1

6

1

5

9

15

4,5

4,5

4

3

3 3

15

12,14 13,14

13

13

6,8

7

7,8

6,7,8,9

9,10

10,11

11

10,11

River

Wabash

Ka nkakee

White

River

White

River

Ea st

Fork

White

River

38

39

40

41

87 86 85

River
20 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1 10,000 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

1

10,000

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q10, IN CUBIC FEET PER SECOND

1 10,000 10 100 1,000

OBSERVED 7Q10, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

1 10,000 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

1

10,000

2

5

10

20

50

100

200

500

1,000

2,000

5,000

PREDICTED 7Q2, IN CUBIC FEET PER SECOND

1 10,000 10 100 1,000

OBSERVED 7Q2, IN CUBIC FEET PER SECOND

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=15

N=15

9 under predicted

6 over predicted

8 under predicted

6 over predicted

Figure 16. Predicted values of 7Q10 and observed values of 7Q10 (a) for Indiana streams in Test Case E and (b) the

relative error of the predicted values of 7Q10 with the observed values of 7Q10 for Test Case E.

Figure 17. Predicted values of 7Q2 and observed values of 7Q2 (a) for Indiana streams in Test Case E and (b) the

relative error of the predicted values of 7Q2 with the observed values of 7Q2 for Test Case E.

a b

a b
Evaluation of the Stedinger-Thomas Method 21

CASE E CASES A & B

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=15

N=56

CASE E CASES A & B

-100

100

-80

-60

-40

-20

0

20

40

60

80

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

N=15

N=15 N=56

Figure 18. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas method, Test Case E

compared with Test Cases A and B combined.

Figure 19. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas method, Test Case E

compared with Test Cases A and B combined.

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to

interquartile range outside the quartile

Outlier data value more than 3 times the

EXPLANATION

25th percentile

Median

75th percentile

the interquartile range outside the quartile

Data value less than or equal to 1.5 times

interquartile range outside the quartile

3 and more than 1.5 times the

Outlier data value less than or equal to

under predicted

under predicted

over predicted

over predicted
22 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

cases because the distribution of streamflow-gaging

stations within the State provided limited sets of

stations suitable for use as index stations.

Case E uses many of the same streamflowgaging stations as Cases A and B. The results

of Case E can be compared directly to the results of

Case A or B when streamflow-gaging stations with

a common period of record and a common index

station are used. Table 8 shows the low-flow frequencies that were predicted for Test Case E and

either Test Case A or B. The average percent difference between the two predicted values of 7Q10 is

7.5 percent, with a standard deviation of 6.2 percent. The average percent difference between the

two predicted values of 7Q2 is 4.9 percent, with a

standard deviation of 4.8 percent. These relatively

small differences indicate that using two index

stations does not produce significantly different

results than using one index station under these

limited and specific conditions.

Evaluation of All Test Cases

The previous sections discussing each test

case indicate that the Stedinger-Thomas method

provides the best estimate when the index station is on the same stream as the partial-record

station or on one of its tributaries. In Case A,

in which the drainage area of the index station

is greater than the partial-record station, the

Stedinger-Thomas method provided the best results among the four test cases that used one index

station, but there was a tendency to predict higher

values than observed for the continuous record

(figs. 6 and 7). In Case B, in which the position

of the index station and partial-record station are

reversed, the Stedinger-Thomas method also performed well but not as well as in Case A—and there

was a tendency to under predict. The tendency

of the method to predict high or low values, relative to the observed values, would not be a concern

in actual low-flow investigations if the average

relative error of estimate were as small as in these

test cases. The combined results of Cases A and B

produced unbiased results, which indicate an equal

chance of over estimating or under estimating

the low-flow frequencies (figs. 6 and 7). Unbiased

estimates are important for actual low-flow investigations because index stations will be upstream

from some partial-record stations and downstream

from others.

The results of Test Cases C and D indicate

that estimates of low-flow frequencies are less

accurate when the index station and the partialrecord station are on different streams or in

different basins. Relating flows between streams

that are not tributary to each other inherently poses

more uncertainty. Because droughts and low-flow

conditions usually are of regional extent, however,

good correlations are often possible between stations on different streams. Many of the test pairs

in Cases C and D had relative errors of less than

15 percent, a criterion that was selected to indicate

excellent estimates of low-flow frequency.

The large variability in relative error of Case C,

as compared to Case A, may be related to the increased number of stations with small values of

7Q10. In these instances, a small difference in

flow equates to a large relative error. Overall, the

Stedinger-Thomas method tended to under predict

in Cases C and D (figs. 13 and 14), but the median

relative error was well below 10 percent (tables 5

and 6). In Cases C and D, there were 12 instances

(17 percent) in which the relative error of estimate

of 7Q10 was greater than 50 percent.

Of the five test cases, the Stedinger-Thomas

method performed best when two index stations

from the same stream or tributaries were used

(Case E). The results of Test Case E were similar

to the combined results of Test Cases A and B, but

with much less variability (figs. 18 and 19). The

small range in relative error for Case E probably is

because of the small sample set and the high degree

of correlation in the flows between the stations that

were used.

Several factors or characteristics of the partialrecord stations were evaluated in an attempt to

explain and predict the variability of the errors

of estimate. These characteristics include drainage

area, difference in drainage area between the

partial-record station and the index station, correlation coefficient of the least-squares-regression

part of the method, and distance between the two
Evaluation of the Stedinger-Thomas Method 23

sites. None of these characteristics showed a direct

relation with the relative error of estimate and,

therefore, could not explain the variability of the

error nor be used to predict the relative error.

The only characteristic that showed some relation to the relative error of estimate was the size

of the observed low-flow-frequency value at the

partial-record station or the index station (“observed” refers to the value for the continuous record

from the log Pearson type III frequency curves).

A small low-flow-frequency value often was associated with the largest relative errors of estimate.

This is partly a result of simple mathematics—

a small difference between the predicted value and

the observed value equates to a large relative error

when the observed value is small. The quality of

streamflow record also could explain why large

relative errors are associated more typically

with small low-flow frequencies. At extremely

low flows in natural channels, it becomes more

difficult to maintain the accuracy of discharge

measurements and to keep track of shifting controls

at streamflow-gaging stations, which can result in

reduced accuracy of the streamflow record.

Estimates of low-flow frequencies at partialrecord stations are most accurate when the index

station and partial-record station have similar

base-flow-recession characteristics. Base-flowrecession characteristics can be determined for

streamflow-gaging stations from hydrographs

and flow-duration curves of the daily mean flows.

Because partial-record stations do not have a

continuous record of the daily flows, however,

base-flow-recession characteristics cannot be determined. When choosing an index station, one is

left to infer the similarity in base-flow-recession

characteristics through geographic proximity

and the similarity in terrain, drainage area, and

geologic characteristics. Because the number of

continuous-record streamflow-gaging stations is

limited, geographic proximity is usually the first

consideration when choosing an index station.

Figure 20 shows flow-duration curves for two

pairs of streamflow-gaging stations that were used

in Test Cases A and B. The flow-duration curves

for each pair are based on the same period of record,

and the daily mean flows have been normalized

to account for drainage area. These flow-duration

curves may help explain the performance of the

Stedinger-Thomas method. Figure 20a shows similar base-flow-recession characteristics for South

Fork Wildcat Creek near Lafayette and Wildcat

Creek near Lafayette. The flow-duration curves

have similar values near the low-flow end—where

flow is equaled or exceeded 98 to 99 percent of

the time. This pair of streamflow-gaging stations

produced good estimates of the 7Q10 and 7Q2 in

Test Cases A and B (pair 8, tables 3 and 4). In Test

Case A, the relative error of estimate was 1.5 percent for the 7Q10 and -2.2 percent for the 7Q2, with

a correlation coefficient of 0.86. In Test Case B,

the relative error of estimate was -8.5 percent

for the 7Q10 and 0.8 percent for the 7Q2, with a

correlation coefficient of 0.77.

Figure 20b shows flow-duration curves for

Cedar Creek at Auburn and Cedar Creek near

Cedarville, which indicate different base-flowrecession characteristics at the two sites. The

flow-duration curves diverge from each other at

the low-flow end, indicating that Cedar Creek

has a more sustained supply of ground water near

Cedarville during base-flow conditions than it

does near Auburn. This pair of streamflow-gaging

stations produced poor estimates of the 7Q10

and relatively poor estimates of the 7Q2 in Test

Cases A and B (pair 20, tables 3 and 4). In

Test Case A, the relative error of estimate was

-28.0 percent for the 7Q10 and -9.5 percent for the

7Q2, with a correlation coefficient of 0.85. In Test

Case B, the relative error of estimate was 55.1 percent for the 7Q10 and 14.7 percent for the 7Q2,

with a correlation coefficient of 0.87.

Arihood and Glatfelter (1991) developed an

equation for estimating low-flow characteristics of

ungaged streams in Indiana. One of the significant

basin characteristics that was used in their equation

was flow-duration ratio, which is the 20-percent

flow duration divided by the 90-percent flow duration (using daily mean flow, in ft3/s). In that report,

Arihood and Glatfelter (1991) subdivided the State

into areas of common flow-duration ratios. The

flow-duration ratios for the two stations in figure

20a are similar, 8.1 for South Fork Wildcat Creek

and 8.7 for Wildcat Creek. The flow-duration ratios
24 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

Figure 20. Flow-duration curves for (a) a pair of streamflow-gaging

stations in Indiana with similar base-flow-recession characteristics and

(b) a pair of streamflow-gaging stations in Indiana with different baseflow-recession characteristics.

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

0.01

10

0.02

0.03

0.04

0.05

0.07

0.1

0.2

0.3

0.4

0.5

0.7

1

2

3

4

5

7

NORMALIZED DAILY MEAN FLOW, IN CUBIC FEET PER SECOND PER SQUARE MILE

South Fork Wildcat Creek near Lafayette

(03334500)

243 square miles

Wildcat Creek near Lafayette

(03335000)

794 square miles

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

PERCENT OF TIME THAT FLOW WAS EQUALED OR EXCEEDED

0.01

10

0.02

0.03

0.04

0.05

0.07

0.1

0.2

0.3

0.4

0.5

0.7

1

2

3

4

5

7

Cedar Creek at Auburn

(04179500)

87.3 square miles

Cedar Creek near Cedarville

(04180000)

270 square miles

a

b
Evaluation of the Stedinger-Thomas Method 25

for the two stations in figure 20b are different, 20.3

for Auburn and 10.7 for Cedarville. A map of flowduration ratios, like that presented in Arihood and

Glatfelter (1991), is helpful for selecting index

stations with base-flow-recession characteristics

similar to the partial-record station. By plotting

the location of the partial-record station, one could

look for potential index stations within the same

subarea or in an adjacent subarea with a similar

flow-duration ratio.

Some of the error in estimating low-flow

frequencies for the sites on Cedar Creek may be

attributed to the relatively small values of 7Q10

and 7Q2 for Cedar Creek at Auburn (pair 20,

table 3). In three of the four instances (7Q10

and 7Q2 for Test Cases A and B), however, the

difference between the predicted and observed

low-flow frequency was significantly larger for

the Cedar Creek stations than for the Wildcat

Creek stations. The Cedar Creek stations have a

much smaller difference in drainage area (table 3),

but the Wildcat Creek stations are closer to each

other (fig. 1).

Correlation coefficients (coefficient of determination) for the linear regression part of the

method are included in tables 3 to 7. There was no

apparent relation between the relative error of estimate and the correlation coefficient; however, not

many of the test pairs had correlation coefficients

below 0.70. Stedinger and Thomas (1985) recommended that the correlation coefficient exceed 0.70.

The average correlation coefficient for Test Cases A

and B was about 0.88, with a minimum of 0.64. The

average correlation coefficient for Test Cases C and

D was about 0.84, with a minimum of 0.50. Test

pairs with low correlation coefficients were just as

likely to produce accurate estimates of the low-flow

frequencies (for example, test pair 7 in table 4 and

test pair 9 in table 6).

Varying the Number of Base-Flow Measurements

In previous sections of this report, the

Stedinger-Thomas method is evaluated for a

variety of test conditions, with the number of

base-flow measurements held constant at 20.

As previously mentioned, most low-flow investigations will not have 20 base-flow measurements

with which to work. In this section, the number

of base-flow measurements is varied to see how

the accuracy of low-flow-frequency estimates is

affected. Because the Stedinger-Thomas method

includes a linear regression, it is recommended that

at least 10 measurements be used (W.O. Thomas,

Jr., and others, U.S. Geological Survey, written

commun., 1993). The automated method, however,

will work with less than 10 measurements. Also,

for practical application of low-flow investigations, many sites may not have 10 base-flow

measurements available. Riggs (1972) indicated

that, generally, 8 to 10 measurements made on

different streamflow recessions and in more than

1 year should define the relation to concurrent

flows at the index station. A study of estimating

low-flow characteristics (7Q10) in Massachusetts

and Rhode Island indicated that using more than

six or eight base-flow measurements added little

confidence to the estimates (Tasker, 1975).

A subset of 10 station pairs was selected from

each of the five test cases. These subsets were made

up of station pairs that had a low relative error

for the original test cases that used 20 base-flow

measurements. The low-flow frequencies then

were estimated, with the number of base-flow

measurements reduced to 15, then 10, and finally 5.

Each test started with a scatter plot of the original

20 base-flow measurements plotted against

the concurrent daily flows at the index station.

Measurements then were dropped from the data set

while trying to maintain a linear relation between

the base-flow measurements and the daily flows.

The predicted values of 7Q10 and 7Q2 and the relative errors of estimate for each pair of stations are

listed in tables 9 to 18.

Boxplots of the relative error of estimate

were used to illustrate the performance of the

Stedinger-Thomas method when the number of

base-flow measurements was varied (figs. 21 and

22). Figure 21 shows the relative error in predicted

values of 7Q10 for all five test cases, and figure 22

shows the relative error of predicted values of 7Q2

(n=10 for all test cases). In Case A, the method

tended to over predict the 7Q10 and the 7Q2 when

the original 20 measurements were used (the
26 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N

-140

140

-120-100

-80-60-40-20

0

20406080

100120

7Q10 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

CASE A CASE B CASE C CASE D CASE E

under predicted

Figure 21. Relative error in predicted values of 7Q10 for Indiana streams, using the Stedinger-Thomas method, with the number o

base-flow measurements varied.

over predicted
Evaluation of the Stedinger-Thomas Method 27 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 N=20 N=15 N=10 N=5 -80

80

-60-40-20

0

204060

7Q2 RELATIVE ERROR, IN PERCENT

[(OBSERVED - PREDICTED) / PREDICTED] x 100

CASE A CASE B CASE C CASE D CASE E

under predicted

Figure 22. Relative error in predicted values of 7Q2 for Indiana streams, using the Stedinger-Thomas method, with the number of

base-flow measurements varied.

over predicted
28 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

median value in the boxplots can be used as a

reference). As the number of measurements was

reduced, the method predicted smaller values

of 7Q10 (table 9) and 7Q2 (table 10). In Case B,

there was the same trend to predict increasingly

smaller values as the number of measurements

was reduced from 20 to 5 (tables 11 and 12). As the

median and average relative error increased with

decreasing number of measurements in Cases A

and B, so did the variability.

Case C (tables 13 and 14) and Case D

(tables 15 and 16) show the same trends as

Cases A and B—on average, the method tends to

predict smaller values of 7Q10 and 7Q2 as the number of measurements decreases. This tendency is

probably the result of the changing slope in the

linear-regression part of the method. As more

measurements are dropped from the analysis, the

regression line is influenced more by each measurement. With 15 or 20 measurements, the regression

line is influenced more by an averaging effect.

In Case E, the method did not consistently predict smaller low-flow frequencies as the number

of measurements decreased. The method tended to

under predict the 7Q10 when 20, 15, or 10 measurements were used; when 5 measurements were

used, the method tended to over predict the 7Q10

(table 17). In Case E, the method tended to

under predict the 7Q2, regardless of the number

of measurements used; there was not much change

in the median, average, or variability in the relative

error as the number of measurements was reduced

(table 18).

Summary and Conclusions

A mathematical technique of estimating

low-flow frequencies at partial-record stations was

evaluated, using streams in Indiana. The StedingerThomas method estimates low-flow frequencies

at low-flow partial-record stations by relating

base-flow measurements to daily flows at a nearby

streamflow-gaging station (index station) for

which a low-flow-frequency curve has been developed. A least-squares-regression analysis of the

logarithms of flows is used to define the relation

between base-flow measurements at the partialrecord station and concurrent daily flows at the

index station. The regression analysis and lowflow statistics (moments) at the index station are

used to predict the desired low-flow frequency at

the partial-record station.

A network of long-term streamflow-gaging

stations in Indiana provided a sample of sites with

observed low-flow frequencies. Observed values

of the 7-day, 10-year low flow (7Q10) and 7-day,

2-year low flow (7Q2) were compared to values

predicted with the Stedinger-Thomas method to

evaluate the accuracy of the method. Streamflowgaging stations were paired for each analysis—one

station was treated as a partial-record station with

base-flow measurements, and the other station was

treated as the index station. Low-flow frequencies

for each pair of streamflow-gaging stations were

based on a common period of record.

Five test cases were used to evaluate the

Stedinger-Thomas method under a variety of conditions. In four test cases, the location of the index

station was varied to evaluate the effect of differences in drainage area and the effect of using an

index station on a different stream or drainage

basin. The fifth test case evaluated the effectiveness

of using two index stations. A total of 141 pairs of

streamflow-gaging stations were used in the five

test cases.

The most accurate and least variable results

were produced when two index stations on the same

stream or tributaries of the partial-record station

were used (Test Case E). All but one value of the

predicted 7Q10 were within 15 percent of the values

observed for the long-term continuous record.

The relative error of the predicted 7Q10 ranged

from -9.5 percent to 20.8 percent, with a median

of 2.9 percent (a positive value of relative error

indicates that the predicted flow is lower than

the observed flow). All of the predicted values of

7Q2 were within 15 percent of the observed values.

The relative error of the predicted 7Q2 ranged

from -6.2 percent to 11.5 percent, with a median
Summary and Conclusions 29

of 1.3 percent. This apparent accuracy and small

variability may be a result of the small sample

set of 15 that included only a few stations with

low-flow frequencies less than 10 ft3/s.

Of the four test cases that used one index

station, the most accurate and least variable results

were produced when the index station and partialrecord station were on the same stream or streams

tributary to each other and when the index station

had a larger drainage area than did the partialrecord station (Test Case A). In this case, the

method tended to over predict the low-flow frequencies, based on the median relative error. In

23 of 28 test pairs, the predicted 7Q10 was within

15 percent of the observed value. The relative error

of the predicted 7Q10 ranged from -28.0 percent

to 47.6 percent, with a median of -4.0 percent. In

26 of 28 test pairs, the predicted 7Q2 was within

15 percent of the observed value. The relative error

of the predicted 7Q2 ranged from -16.4 percent to

15.8 percent, with a median of -1.4 percent.

When the index station was on the same stream

or a tributary of the partial-record station and had a

smaller drainage area than the partial-record station

(Test Case B), the method tended to under predict

the low-flow frequencies. Nineteen of 28 predicted

values of 7Q10 were within 15 percent of the observed values. The relative error of estimate ranged

from -12.8 percent to 55.1 percent, with a median of

5.6 percent. Twenty-five of 28 predicted values of

7Q2 were within 15 percent of the observed values.

The relative error of estimate ranged from -8.6 percent to 31.1 percent, with a median of 1.2 percent.

When the index station and the partial-record

station were on different streams (Test Cases C and

D), the method tended to under predict regardless

of whether the index station had a larger or smaller

drainage area than that of the partial-record station.

Also, the variability of the relative error of estimate

was much higher for the test cases that used index

stations and partial-record stations on different

streams. When the index station had a larger drainage area than that of the partial-record station (Test

Case C), 15 of 35 predicted values of 7Q10 were

within 15 percent of the observed values. The relative error of estimate ranged from -40.0 percent to

150 percent, with a median of 5.7 percent. Nineteen

of 35 predicted values of 7Q2 were within 15 percent of the observed values. The relative error of

estimate ranged from -34.1 percent to 96.1 percent,

with a median of 6.2 percent.

When the index station and partial-record

station were on different streams and the index

station had a smaller drainage area than that of the

partial-record station (Test Case D), 18 of 35 predicted values of 7Q10 were within 15 percent of

the observed values. The relative error of estimate

ranged from -28.6 percent to 123 percent, with a

median of 0.0 percent. In 25 of 35 test pairs, the predicted 7Q2 was within 15 percent of the observed

value. The relative error of estimate ranged from

-16.6 percent to 81.8 percent, with a median of

2.1 percent.

The only characteristic that showed some

relation with the relative error of estimate was the

size of the observed low-flow-frequency value at

the partial-record station or the index station. There

was not a direct, or linear, relation but a small lowflow-frequency value often was associated with the

largest relative errors of estimate. This is partly

a result of the simple mathematics of computing

the relative error or of a reduced accuracy in the

measurement of streamflow and computation of

streamflow records at very low flows.

In the test cases that used one index station,

the Stedinger-Thomas method tended to predict

smaller low-flow frequencies as the number of

base-flow measurements was reduced from 20 to 5.

In most of these cases, the variability of the predicted values also increased as the number of

base-flow measurements was reduced.
30 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

References Cited

Arihood, L.D., and Glatfelter, D.R., 1991, Method for

estimating low-flow characteristics of ungaged

streams in Indiana: U.S. Geological Survey WaterSupply Paper 2372, 18 p.

Fowler, K.K., and Wilson, J.T., 1996, Low-flow characteristics of Indiana streams: U.S. Geological Survey

Water-Resources Investigations Report 96-4128,

313 p.

Riggs, H.C., 1968, Frequency curves: Techniques

of water-resources investigations of the

United States Geological Survey, book 4, chap. A2,

15 p.

_____1972, Low-flow investigations: Techniques of

water-resources investigations of the United States

Geological Survey, book 4, chap. B1, 18 p.

Stedinger, J.R., and Thomas, W.O., Jr., 1985, Lowflow frequency estimation using base-flow

measurements: U.S. Geological Survey OpenFile Report 85-95, 22 p.

Stewart, J.A., Keeton, C.R., Hammill, L.E., Nguyen,

H.T., and Majors, D.K., 1999, Water resources

data—Indiana—water year 1998: U.S. Geological

Survey Water-Data Report IN-98-1, 452 p.

Tasker, G.D., 1975, Combining estimates of low-flow

characteristics of streams in Massachusetts and

Rhode Island: Journal of Research of the U.S. Geological Survey, v. 3, no. 1, p. 107–112.
Data Tables
32 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

Table 2. Streamflow-gaging stations in Indiana used to evaluate the Stedinger-Thomas method of estimating low-flow

frequencies at partial-record sites, using base-flow measurements

[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; 7Q10 and 7Q2 are for the

period of record through 1993, from Fowler and Wilson, 1996; ft3/s, cubic feet per second; c, current year as of September 1998—these stations

were still active as of September 1998 (Stewart and others, 1999)]

Station

number Station name

Drainage area

(square miles)

Period of

record1

7Q10

(ft3/s)

7Q2

(ft3/s)

03274650

03274750

03275000

03275500

03275600

03322500

03322900

03324000

03324200

03324300

03325500

03326000

03326070

03326500

03327520

03328000

03328430

03328500

03329700

03334000

03334500

03335000

03335500

03336000

03339500

03340000

03340500

03340800

03341200

03341500

03348000

03348020

03348350

03349000

03350700

03351500

03352200

03353120

03353180

03353620

03357500

03358000

Whitewater River near Economy

Whitewater River near Hagerstown

Whitewater River near Alpine

East Fork Whitewater River at Richmond

East Fork Whitewater River at Abington

Wabash River near New Corydon

Wabash River at Linn Grove

Little River near Huntington

Salamonie River at Portland

Salamonie River near Warren

Mississinewa River near Ridgeville

Mississinewa River near Eaton

Big Lick Creek near Hartford City

Mississinewa River at Marion

Pipe Creek near Bunker Hill

Eel River at North Manchester

Weesau Creek near Deedsville

Eel River near Logansport

Deer Creek near Delphi

Wildcat Creek at Owasco

South Fork Wildcat Creek near Lafayette

Wildcat Creek near Lafayette

Wabash River at Lafayette

Wabash River at Covington

Sugar Creek at Crawfordsville

Sugar Creek near Byron

Wabash River at Montezuma

Big Raccoon Creek near Fincastle

Little Raccoon Creek near Catlin

Wabash River at Terre Haute

White River at Anderson

Killbuck Creek near Gaston

Pipe Creek at Frankton

White River at Noblesville

Stony Creek near Noblesville

Fall Creek near Fortville

Mud Creek at Indianapolis

Pleasant Run at Arlington Avenue at Indianapolis

Bean Creek at Indianapolis

Lick Creek at Indianapolis

Big Walnut Creek near Reelsville

Mill Creek near Cataract

10.4

58.7

529

121

200

262

453

263

85.6

425

133

310

29.2

682

159

417

8.87

789

274

396

243

794

7,267

8,218

509

670

11,118

139

133

12,263

406

25.5

113

858

50.8

169

42.4

7.58

4.40

15.6

326

245

1970-c

1970-c

1928-c

1949-78

1965-c

1951-88

1964-c

1943-c

1959-93

1957-c

1946-c

1952-71

1971-c

1923-c

1968-c

1929-c

1970-c

1943-c

1943-c

1943-73, 88-c

1943-c

1954-c

1969-c

1969-c

1938-c

1940-71

1969-c

1957-c

1957-71

1969-c

1931-93

1968-91

1968-c

1946-c

1967-c

1941-c

1958-76

1959-c

1970-93

1970-c

1949-c

1949-c

0.3

7.6

51

4.2

18

2.3

5.6

4.4

1.0

7.6

1.2

2.9

.5

19

4.9

37

.4

100

12

19

20

60

868

1,040

8.0

22

1,210

2.4

4.6

1,390

28

1.1

4.3

81

2.9

15

.5

.1

.6

.2

4.5

1.7

0.6

13

86

10

30

5.9

11

12

1.9

15

3.5

6.6

1.0

38

9.1

59

.8

145

23

34

30

98

1,320

1,580

24

45

1,970

6.8

7.1

2,300

58

2.5

7.9

122

6.1

28

2.0

.4

1.0

1.0

19

7.5
 Estimating Low-Flow Frequencies at Partial-Record Sites, Using Base-Flow Measurements 33

1

Wabash River stations downstream from Huntington Reservoir are partially regulated by upstream reservoirs. Low-flow statistics are

calculated for the regulated period, 1969–93. The period of record shown does not include the years prior to 1969.

Table 2. Streamflow-gaging stations in Indiana used to evaluate the Stedinger-Thomas method of estimating low-flow

frequencies at partial-record sites, using base-flow measurements—Continued

Station

number Station name

Drainage area

(square miles)

Period of

record1

7Q10

(ft3/s)

7Q2

(ft3/s)

03359500

03361000

03361500

03362000

03362500

03363000

03363500

03363900

03364000

03364200

03365500

03371500

03373500

04094000

04094500

04095300

04096100

04099510

04099750

04099808

04100252

04100295

04100500

04178000

04179500

04180000

04182590

05515000

05515400

05515500

05516000

05516500

05517000

05517500

05517530

05518000

05519000

05519500

05522000

05522500

05524500

Deer Creek near Putnamville

Big Blue River at Carthage

Big Blue River at Shelbyville

Youngs Creek near Edinburgh

Sugar Creek near Edinburgh

Driftwood River near Edinburgh

Flatrock River at St. Paul

Flatrock River at Columbus

East Fork White River at Columbus

Haw Creek near Clifford

East Fork White River at Seymour

East Fork White River near Bedford

East Fork White River at Shoals

Little Calumet River at Porter

Salt Creek near McCool

Trail Creek at Michigan City

Galena River near LaPorte

Pigeon Creek near Angola

Pigeon River near Scott

Little Elkhart River at Middlebury

Forker Creek near Burr Oak

Rimmell Branch near Albion

Elkhart River at Goshen

St. Joseph River near Newville

Cedar Creek at Auburn

Cedar Creek near Cedarville

Harber Ditch at Fort Wayne

Kankakee River near North Liberty

Kingsbury Creek near LaPorte

Kankakee River at Davis

Yellow River near Bremen

Yellow River at Plymouth

Yellow River at Knox

Kankakee River at Dunns Bridge

Kankakee River near Kouts

Kankakee River at Shelby

Singleton Ditch at Schneider

West Creek near Schneider

Iroquois River near North Marion

Iroquois River at Rensselaer

Iroquois River near Foresman

59.0

184

421

107

474

1,060

303

534

1,707

47.5

2,341

3,861

4,927

66.2

74.6

54.1

17.2

106

361

97.6

19.2

10.7

594

610

87.3

270

21.9

174

7.08

537

135

294

435

1,352

1,376

1,779

123

54.7

144

203

449

1954-65, 67-72

1950-c

1943-c

1942-c

1942-c

1940-92

1930-c

1967-c

1947-c

1967-91

1927-c

1939-c

1923-c

1945-c

1945-91

1969-94

1969-c

1945-c

1968-c

1979-c

1969-c

1979-c

1931-c

1946-c

1943-73

1946-c

1964-91

1951-c

1970-86

1924-c

1955-73

1948-c

1943-c

1948-c

1974-c

1922-c

1948-c

1948-51, 54-72

1948-93

1948-c

1948-c

0.1

27

40

1.5

20

94

2.3

31

133

.6

172

258

265

21

19

24

8.1

7.1

89

30

.2

.2

85

20

1.8

21

.1

57

1.2

189

6.3

21

76

348

381

417

7.4

4.6

4.4

5.9

11

0.6

47

66

3.4

40

151

11

58

230

1.7

285

443

464

25

26

30

9.8

15

127

40

.6

.4

139

39

3.8

29

.3

71

1.9

257

9.0

33

106

476

507

574

15

6.9

11

14

24
34 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams Table 3. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case A—the paired stations are on the same stream or tributaries, and the drainage area at the index station is gr[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; ft3/s, cubic feet per second; R2, coefficien

explained by a regression of the base-flow measurements with the concurrent daily flows at the index station] Pair Partial- record station Index station Drainage- area difference (square miles) Years of record Period of record Predicted1 7Q10 (ft3/s) Observed2 7Q10 (ft3/s) Relative error3 of predicted 7Q10 (percent) Predicted 7Q2 (ft3/s) 

123456789

1011121314151617181920

0327465003274750032755000332250003324200033255000332800003334500033355000333950003340500033480000336100003362000033625000336350003364000033715000409951004179500

0327475003275000032756000332290003324300033260000332850003335000033360000334000003341500033490000336150003362500033630000336390003365500033735000409975004180000

48.3

470

79.0

191339177372551951161

1,145

452237367586231634

1,066

255183

2222122333194938233023464542504925382426

1972-931972-931967-781966-881961-931953-711945-931956-931971-931942-711971-931948-931949-931952-931944-931944-921969-931956-931970-931948-73

0.218.07.64.01.0

.96

42.119.9

827

7.7

1,217

42.828.4

1.4

21.6

3.6

152273

10.0

2.5

0.317.67.34.01.0

.86

42.220.2

868

6.7

1,208

40.426.6

1.5

20.3

4.9

133258

9.61.8

47.6

-5.0-3.9

00

-10.4

.2

1.55.0

-13.0

-.7

-5.6-6.3

7.1

-6.0

36.1

-12.5

-5.5-4.0

-28.0

0.57

13.213.2

6.22.02.4

62.231.9

1,270

21.2

2,012

64.443.9

3.9

38.415.2

232432

17.7

4.2
Estimating Low-Flow Frequencies, Using Base-Flow Measurements, Case A 35 1Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985). 2Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown. 3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 3. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatinmeasurements: Case A—the paired stations are on the same stream or tributaries, and the drainage area at the index station is grContinued

Pair

Partialrecord

station

Index

station

Drainagearea

difference

(square

miles)

Years of

record

Period of

record

Predicted1

7Q10

(ft3/s)

Observed2

7Q10

(ft3/s)

Relative

error3 of

predicted

7Q10

(percent)

Predicted

7Q2

(ft3/s)

2122232425262728

0551500005515400055160000551650005517530055220000552200005522500

0551550005515500055165000551700005518000055225000552450005524500

363530159141403

59.0

305246

4215174418444444

1952-931972-861957-731950-931976-931950-931950-931950-93

65.0

1.24.7

22.1

391

4.45.77.0

56.6

1.26.3

20.8

381

4.44.46.0

-12.9

0

34.0

-5.9-2.6

0

-22.8-14.3

85.2

1.78.8

36.5

558

10.511.814.5

AverageStandard deviationMedianMinimumMaximum

382286322

48.3

1,145

3212321250

112279

7.45

.31

1,208

-1.0

16.2

-4.0

-28.0

47.6
36 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams Table 4. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case B—the paired stations are on the same stream or tributaries, and the drainage area at the index station is le[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; ft3/s, cubic feet per second; R2, coefficien

explained by a regression of the base-flow measurements with the concurrent daily flows at the index station] Pair Partial- record station Index station Drainage- area difference (square miles) Years of record Period of record Predicted1 7Q10 (ft3/s) Observed2 7Q10 (ft3/s) Relative error3 of predicted 7Q10 (percent) Predicted 7Q2 (ft3/s) 

123456789

1011121314151617181920

0327475003275000032756000332290003324300033260000332850003335000033360000334000003341500033490000336150003362500033630000336390003365500033735000409975004180000

0327465003274750032755000332250003324200033255000332800003334500033355000333950003340500033480000336100003362000033625000336350003364000033715000409951004179500

48.3

470

79.0

191339177372551951161

1,145

452237367586231634

1,066

255183

2222122333194938233023464250492545382426

1972-931972-931967-781966-881961-931953-711945-931956-931971-931942-711971-931948-931952-931944-931944-921969-931949-931956-931970-931948-73

8.5

47.715.5

6.18.62.4

95.565.0

1,092

16.6

1,429

85.433.217.091.729.5

175347

90.512.7

7.6

62.519.3

6.27.52.9

100

59.5

1,040

21.9

1,393

80.540.420.593.230.6

191333

89.319.7

-10.6

31.024.5

1.6

-12.8

20.8

4.7

-8.5-4.8

31.9

-2.5-5.7

21.720.6

1.63.79.1

-4.0-1.3

55.1

13.088.226.6

9.4

16.3

6.9

145

97.2

1,598

40.0

2,299

126

67.630.9

159

58.8

298562131

23.8
Estimating Low-Flow Frequencies, Using Base-Flow Measurements, Case B 37 1Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985). 2Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown. 3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 4. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case B—the paired stations are on the same stream or tributaries, and the drainage area at the index station is leContinued

Pair

Partialrecord

station

Index

station

Drainagearea

difference

(square

miles)

Years of

record

Period of

record

Predicted1

7Q10

(ft3/s)

Observed2

7Q10

(ft3/s)

Relative

error3 of

predicted

7Q10

(percent)

Predicted

7Q2

(ft3/s)

2122232425262728

0551550005515500055165000551700005518000055225000552450005524500

0551500005515400055160000551650005517530055220000552200005522500

363530159141403

59.0

305246

4215174418444444

1952-931972-861957-731950-931976-931950-931950-931950-93

184218

20.472

405

5.57.88.4

195239

17.975.8

441

6.0

10.810.8

6.09.6

-12.3

5.38.99.1

38.528.6

230279

28.9

101568

13.822.421.7

AverageStandard deviationMedianMinimumMaximum

165319

50.0

2.9

1,393

9.6

16.8

5.6

-12.8

55.1
38 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams Table 5. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case C—the paired stations are on different streams or basins, and the drainage area at the index station is great[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; ft3/s, cubic feet per second; R2, coefficient 

explained by a regression of the base-flow measurements with the concurrent daily flows at the index station] Pair Partial- record station Index station Drainage- area difference (square miles) Years of record Period of record Predicted1 7Q10 (ft3/s) Observed2 7Q10 (ft3/s) Relative error3 of predicted 7Q10 (percent) Predicted 7Q2 (ft3/s) 

123456789

1011121314151617181920

0327475003275600033242000332420003324300033243000332607003327520033284300332970003334500033408000334120003348020033483500335070003352200033531800335318003358000

0327560003275000033225000332550003322900033265000332420003329700033280000333400003334000033395000334080003348350033480000335150003351500033531200335362003357500

141329176

47.428.0

257

56.4

115408122153370

6.0

87.5

293118127

3.18

11.281.0

2227283328362124223333351323242517222243

1972-931967-931961-881961-931966-931958-931973-931970-931972-931945-771945-771959-931959-711970-921970-931969-931960-761972-931972-931951-93

9.5

15.8

.51.48

8.76.8

.22

3.3

.47

10.412.7

1.55.11.13.12.2

.66.6.44

2.0

7.6

18.4

.98

1.07.97.6

.48

4.9

.39

10.318.9

2.44.61.14.32.9

.46.65.65

1.7

-20.0

16.592.2

108

-9.2

11.8

118

48.5

-17.0

-1.0

48.860.0

-9.8

0

38.731.8

-30.3

8.3

47.7

-15.0

14.825.5

1.31.2

15.412.9

.51

7.0

.87

18.724.0

4.77.32.46.45.21.91.0

.9

7.1
Estimating Low-Flow Frequencies, Using Base-Flow Measurements, Case C 39

1Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).2Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown.3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 5. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case C—the paired stations are on different streams or basins, and the drainage area at the index station is greatContinued

Pair

Partialrecord

station

Index

station

Drainagearea

difference

(square

miles)

Years of

record

Period of

record

Predicted1

7Q10

(ft3/s)

Observed2

7Q10

(ft3/s)

Relative

error3 of

predicted

7Q10

(percent)

Predicted

7Q2

(ft3/s)

212223242526272829303132333435

033595000335950003361500033639000336420004094000040961000409980804099808041002950417950004180000041825900551540005519500

033575000335800003362500033630000336390004094500040953000409975004100500041002520417800004178000033240000551500005519000

267186

53.0

526486

8.4

36.9

263496

8.5

523340241167

68.3

141449242345231313122646261520

1956-721956-721945-931969-921969-911947-911971-931981-931981-931982-931948-731948-931966-911972-861950-69

0.2

.19

40.232.1

.79

19.4

8.7

29.030.7

.11

1.6

16.6

.04.99

4.8

0.12

.12

40.430.0

.63

20.5

8.1

30.130.1

.21

1.8

20.9

.1

1.24.6

-40.0-36.8

.5

-6.5

-20.3

5.7

-6.9

3.8

-2.0

90.912.525.9

150

21.2

-4.2

0.68

.91

69.649.7

2.3

25.010.439.741.2

.26

3.1

27.5

.15

1.47.2

AverageStandard deviationMedianMinimumMaximum

188162141

3.18

526

2510241249

8.2

10.9

2.9

.1

40.4

20.645.5

5.7

-40.0

150
40 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams Table 6. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case D—the paired stations are on different streams or basins, and the drainage area at the index station is less t[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; ft3/s, cubic feet per second; R2, coefficient 

explained by a regression of the base-flow measurements with the concurrent daily flows at the index station] Pair Partial- record station Index station Drainage- area difference (square miles) Years of record Period of record Predicted1 7Q10 (ft3/s) Observed2 7Q10 (ft3/s) Relative error3 of predicted 7Q10 (percent) Predicted 7Q2 (ft3/s) 

123456789

1011121314151617181920

0327560003275000033225000332550003322900033265000332420003329700033280000333400003334000033395000334080003348350033480000335150003351500033531200335362003357500

0327475003275600033242000332420003324300033243000332607003327520033284300332970003334500033408000334120003348020033483500335070003352200033531800335318003358000

141329176

47.428.0

257

56.4

115408122153370

6.0

87.5

293118127

3.18

11.281.0

2227283328362124223333351323242517222243

1972-931967-931961-881961-931966-931958-931973-931970-931972-931945-771945-771959-931959-711970-921970-931969-931960-761972-931972-931951-93

12.669.8

3.5

.98

3.5

28.6

.91

14.552.421.123.612.7

2.53.6

54.519.114.2

.13.14

5.3

17.562.3

2.51.75.6

25.8

1.2

17.050.619.119.111.8

2.44.2

43.617.115.9

.29.26

5.7

38.9

-10.7-28.6

73.560.0

-9.8

31.917.2

-3.4-9.5

-19.1

-7.1-4.0

16.7

-20.0-10.5

12.0

123

85.7

7.5

25.0

109

6.43.29.2

50.3

2.3

25.768.437.536.028.9

4.37.5

78.140.325.3

.37.55

20.6
Estimating Low-Flow Frequencies, Using Base-Flow Measurements, Case D 41

1Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).2Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown.3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 6. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatinmeasurements: Case D—the paired stations are on different streams or basins, and the drainage area at the index station is less Continued

Pair

Partialrecord

station

Index

station

Drainagearea

difference

(square

miles)

Years of

record

Period of

record

Predicted1

7Q10

(ft3/s)

Observed2

7Q10

(ft3/s)

Relative

error3 of

predicted

7Q10

(percent)

Predicted

7Q2

(ft3/s)

212223242526272829303132333435

033575000335800003362500033630000336390004094500040953000409975004100500041002520417800004178000033240000551500005519000

033595000335950003361500033639000336420004094000040961000409980804099808041002950417950004180000041825900551540005519500

267186

53.0

526486

8.4

36.9

263496

8.5

523340241167

68.3

141449242345231313122646261520

1956-721956-721945-931969-921969-911947-911971-931981-931981-931982-931948-731948-931966-911972-861950-69

4.41.0

20.4

107

27.421.624.289.8

115

.37

10.321.112.163.2

5.0

5.61.9

20.4

105

31.119.424.595.0

111

.31

18.020.1

9.1

58.9

6.1

27.390.0

0

-1.9

13.5

-10.2

1.25.8

-3.5

-16.2

74.8

-4.7

-24.8

-6.8

22.0

13.5

3.8

35.2

184

48.926.529.0

129170

.76

24.036.318.382.811.7

AverageStandard deviationMedianMinimumMaximum

24.329.517.1

.26

111

14.636.6

0

-28.6

123
42 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).2Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown.3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 7. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatin

measurements: Case E—the grouped stations are on the same streams or tributaries, and there are two index stations

[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; ft3/s, cubic feet per second; R2, coefficient 

explained by a regression of the base-flow measurements with the concurrent daily flows at the index stations]

Set

Partialrecord

station

Index

station 1

Index

station 2

Years of

record

Period of

record

Predicted1

7Q10

(ft3/s)

Observed2

7Q10

(ft3/s)

Relative

error3 of

predicted

7Q10

(percent)

Predicted

7Q2

(ft3/s)

123456789

101112131415

032747500332600003335000033360000334050003361500033625000336300003364000033655000337150005515500055165000551750005522500

032746500332550003334500033355000333600003361000033620000336150003363000033640000336550005515000055160000551550005522000

032750000332650003334000033405000334150003363000033630000336250003363900033715000337350005517500055170000551700005524500

221922232341494824353542174244

1972-931953-711956-931971-931971-931952-921944-921945-921969-921959-931959-931952-931957-731952-931950-93

8.42.4

57.8

1,0551,234

38.519.887.3

149182276185

19.2

316

6.5

7.62.9

54.8

1,0401,208

40.220.392.9

156193284195

17.9

348

6.0

-9.5

20.8

-5.2-1.4-2.1

4.42.56.44.76.02.95.4

-6.8

10.1

-7.7

12.9

6.0

92.1

1,6132,001

67.735.7

150251309451239

32.2

482

14.4

AverageStandard deviationMedianMinimumMaximum

3211351749

244374

92.9

2.9

1,208

2.07.92.9

-9.5

20.8
Estimating Low-Flow Frequencies, Using Base-Flow Measurements, Two Test Cases 43

1Observed 7Q10 and 7Q2 were determined by frequency analysis of the annual 7-day low flows for the period of record shown.2Predicted 7Q10 and 7Q2 were estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).

Table 8. Low-flow frequencies at selected streamflow-gaging stations in Indiana, estimated with two test cases of the Stedinger-Th

frequencies, using base-flow measurements

[7Q10 and 7Q2 are the minimum average discharge for 7 consecutive days with a recurrence interval of 10 and 2 years; Case E, uses two index stations that are on t

station; Case A and B use one index station that is on the same stream or tributary as the partial-record station; ft3/s, cubic feet per second]

Pair

Partialrecord

station

Years of

record

Period of

record

Observed1

7Q10

(ft3/s)

Case E

predicted2

7Q10

(ft3/s)

Case A or B

predicted

7Q10

(ft3/s)

Percent

difference

in 7Q10

estimates

Observed

7Q2

(ft3/s)

Ca

pred

7

(ft

123456789

032747500332600003336000033405000336250005515500055165000552250005522500

221923234942174444

1972-931953-711971-931971-931944-921952-931957-731950-931950-93

7.62.9

1,0401,208

20.3

195

17.9

6.06.0

8.42.4

1,0551,234

19.8

185

19.2

6.56.5

8.02.0

1,0921,217

21.6

184

20.4

7.05.5

4.9

18.2

3.41.48.7

.5

6.17.4

16.7

12.9

6.6

1,5841,972

39.8

263

30.213.913.9

12

6

1,6132,00

35

239

321414

MeanStandard deviationMedianMinimumMaximum

3113231749

7.56.26.1

.5

18.2
44 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1Observed 7Q10 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q10 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 9. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimati

with the number of base-flow measurements varied: Case A—the paired stations are on the same stream or tributaries, and the drai

is greater than at the partial-record station

[7Q10 is the minimum average discharge for 7 consecutive days with a recurrence interval of 10 years; ft3/s, cubic feet per second; n, number of base-flow measur

Pair

Partialrecord

station

Index

station

Observed1

7Q10

(ft3/s)

Predicted2

7Q10

n=20

(ft3/s)

Predicted

7Q10

n=15

(ft3/s)

Predicted

7Q10

n=10

(ft3/s)

Predicted

7Q10

n=5

(ft3/s)

Relative

error3 of

predicted

7Q10

n=20

(percent)

Relativ

error o

predicte

7Q10

n=15

(percen

23479

1314192426

03274750032755000332250003328000033355000336100003362000040995100551650005522000

03275000032756000332290003328500033360000336150003362500040997500551700005522500

7.67.34.0

42.2

868

26.6

1.59.6

20.8

4.4

8.07.64.0

42.1

827

28.4

1.4

10.022.1

4.4

7.57.33.8

38.2

793

27.2

1.18.8

19.2

4.5

7.36.73.6

37.6

768

26.5

.73

8.4

20.5

4.7

6.66.13.3

36.5

678

26.2

.64

8.4

19.9

4.6

-5.0-3.9

0

.2

5.0

-6.3

7.1

-4.0-5.9

0

1.305.3

10.5

9.5

-2.2

36.4

9.18.3

-2.2

MeanStandard deviationMedianMinimumMaximum

99.2

270

8.61.5

868

-1.3

4.6

-2.0-6.3

7.1

7.6

11.2

6.8

-2.2

36.4
Estimating 7-Day, 2-Year Low Flows, with the Number of Base-Flow Measurements Varied, Case A 45

1Observed 7Q2 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q2 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 10. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estima

with the number of base-flow measurements varied: Case A—the paired stations are on the same stream or tributaries, and the drstation is greater than at the partial-record station

[7Q2 is the minimum average discharge for 7 consecutive days with a recurrence interval of 2 years; ft3/s, cubic feet per second; n, number of base-flow measure

Pair

Partialrecord

station

Index

station

Observed1

7Q2

(ft3/s)

Predicted2

7Q2

n=20

(ft3/s)

Predicted

7Q2

n=15

(ft3/s)

Predicted

7Q2

n=10

(ft3/s)

Predicted

7Q2

n=5

(ft3/s)

Relative

error3 of

predicted

7Q2

n=20

(percent)

Relativ

error o

predict

7Q2

n=15

(percen

23479

1314192426

03274750032755000332250003328000033355000336100003362000040995100551650005522000

03275000032756000332290003328500033360000336150003362500040997500551700005522500

12.912.6

5.9

63.1

1,318

46.5

3.4

18.233.210.7

13.213.2

6.2

62.2

1,270

43.9

3.9

17.736.510.5

12.813.2

5.9

59.7

1,226

43.8

3.4

15.932.910.5

12.612.4

5.6

60.1

1,170

44.0

2.7

15.633.410.4

12.411.3

5.5

59.0

1,063

43.6

2.6

15.633.210.3

-2.3-4.5-4.8

1.43.85.9

-12.8

2.8

-9.0

1.9

0.8

-4.5

05.77.56.20

14.5

.9

1.9

MeanStandard deviationMedianMinimumMaximum

152410

15.6

3.4

1,318

-1.8

6.0

-.5

-12.8

5.9

3.35.31.4

-4.5

14.5
46 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1Observed 7Q10 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q10 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 11. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimat

with the number of base-flow measurements varied: Case B—the paired stations are on the same stream or tributaries, and the drstation is less than at the partial-record station

[7Q10 is the minimum average discharge for 7 consecutive days with a recurrence interval of 10 years; ft3/s, cubic feet per second; n, number of base-flow measu

Pair

Partialrecord

station

Index

station

Observed1

7Q10

(ft3/s)

Predicted2

7Q10

n=20

(ft3/s)

Predicted

7Q10

n=15

(ft3/s)

Predicted

7Q10

n=10

(ft3/s)

Predicted

7Q10

n=5

(ft3/s)

Relative

error3 of

predicted

7Q10

n=20

(percent)

Relativ

error o

predict

7Q10

n=15

(percen

4789

111215192426

03322900033285000333500003336000033415000334900003363000040997500551700005522500

03322500033280000333450003335500033405000334800003362500040995100551650005522000

6.2

100

59.5

1,0401,393

80.593.289.375.8

6.0

6.1

95.565.0

1,0921,429

85.491.790.572

5.5

5.8

85.861.8

1,0681,374

86.487.790.367.3

5.4

4.3

75.360.8

1,0331,458

88.982.392.653.7

4.9

4.3

76.049.2

910

1,446

86.474.689.750.6

5.8

1.64.7

-8.5-4.8-2.5-5.7

1.6

-1.3

5.39.1

6.9

16.6

-3.7-2.6

1.4

-6.8

6.3

-1.1

12.611.1

MeanStandard deviationMedianMinimumMaximum

294494

84.9

6

1,393

-.1

5.5

.2

-8.5

9.1

4.17.83.9

-6.8

16.6
Estimating 7-Day, 2-Year Low Flows, with the Number of Base-Flow Measurements Varied, Case B 47

1Observed 7Q2 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q2 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 12. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estima

with the number of base-flow measurements varied: Case B—the paired stations are on the same stream or tributaries, and the drastation is less than at the partial-record station

[7Q2 is the minimum average discharge for 7 consecutive days with a recurrence interval of 2 years; ft3/s, cubic feet per second; n, number of base-flow measurem

Pair

Partialrecord

station

Index

station

Observed1

7Q2

(ft3/s)

Predicted2

7Q2

n=20

(ft3/s)

Predicted

7Q2

n=15

(ft3/s)

Predicted

7Q2

n=10

(ft3/s)

Predicted

7Q2

n=5

(ft3/s)

Relative

error3 of

predicted

7Q2

n=20

(percent)

Relativ

error o

predicte

7Q2

n=15

(percen

4789

111215192426

03322900033285000333500003336000033415000334900003363000040997500551700005522500

03322500033280000333450003335500033405000334800003362500040995100551650005522000

9.9

145

98.0

1,5842,305

122152127108

13.9

9.4

145

97.2

1,5982,299

126159131101

13.8

9.5

135

94.4

1,5712,226

123157131

95.713.7

8.3

125

92.7

1,5412,230

129154130

80.613.2

8.2

122

84.0

1,4542,232

128149126

76.513.4

5.30

.8

-.9

.3

-3.2-4.4-3.1

6.9

.7

4.27.43.8

.8

3.5

-.8

-3.2-3.1

12.9

1.5

MeanStandard deviationMedianMinimumMaximum

466799124

9.9

2,305

.2

3.6

.2

-4.4

6.9

2.74.92.5

-3.2

12.9
48 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1Observed 7Q10 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q10 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 13. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatwith the number of base-flow measurements varied: Case C—the paired stations are on different streams or basins, and the drainagreater than at the partial-record station[7Q10 is the minimum average discharge for 7 consecutive days with a recurrence interval of 10 years; ft3/s, cubic feet per second; n, number of base-flow measu

Pair

Partialrecord

station

Index

station

Observed1

7Q10

(ft3/s)

Predicted2

7Q10

n=20

(ft3/s)

Predicted

7Q10

n=15

(ft3/s)

Predicted

7Q10

n=10

(ft3/s)

Predicted

7Q10

n=5

(ft3/s)

Relative

error3 of

predicted

7Q10

n=20

(percent)

Relativ

error o

predict

7Q10

n=15

(percen

26

1013232426272829

03275600033243000332970003341200033615000336390004094000040961000409980804099808

03275000033265000333400003340800033625000336300004094500040953000409975004100500

18.4

7.6

10.3

4.6

40.430.020.5

8.1

30.130.1

15.8

6.8

10.4

5.1

40.232.119.4

8.7

29.030.7

16.1

7.29.94.8

35.829.517.7

8.6

28.930.8

16.4

7.09.04.1

31.527.516.8

8.0

27.431.2

16.1

6.38.83.9

28.927.115.7

7.4

26.231.3

16.511.8

-1.0-9.8

.5

-6.5

5.7

-6.9

3.4

-2.3

14.3

5.64.0

-4.2

12.8

1.7

15.8

-5.8

3.8

-2.6

MeanStandard deviationMedianMinimumMaximum

20.012.219.5

4.6

40.4

1.18.4

-.3

-9.8

16.5

4.57.73.9

-5.8

15.8
Estimating 7-Day, 2-Year Low Flows, with the Number of Base-Flow Measurements Varied, Case C 49

1Observed 7Q2 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q2 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 14. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estima

with the number of base-flow measurements varied: Case C—the paired stations are on different streams or basins, and the draina

greater than at the partial-record station[7Q2 is the minimum average discharge for 7 consecutive days with a recurrence interval of 2 years; ft3/s, cubic feet per second; n, number of base-flow measurem

Pair

Partialrecord

station

Index

station

Observed1

7Q2

(ft3/s)

Predicted2

7Q2

n=20

(ft3/s)

Predicted

7Q2

n=15

(ft3/s)

Predicted

7Q2

n=10

(ft3/s)

Predicted

7Q2

n=5

(ft3/s)

Relative

error3 of

predicted

7Q2

n=20

(percent)

Relativ

error o

predicte

7Q2

n=15

(percen

26

1013232426272829

03275600033243000332970003341200033615000336390004094000040961000409980804099808

03275000033265000333400003340800033625000336300004094500040953000409975004100500

30.315.021.3

7.1

65.757.025.0

9.8

40.040.0

25.512.918.7

7.3

69.649.725.010.439.741.2

25.612.918.8

7.2

66.748.323.810.239.340.9

26.012.218.4

7.0

62.248.423.2

9.7

38.441.5

25.811.718.6

7.2

60.747.822.8

9.0

37.041.5

18.816.313.9

-2.7-5.6

14.7

0

-5.8

.8

-2.9

18.416.313.3

-1.4-1.5

18.0

5.0

-3.9

1.8

-2.2

MeanStandard deviationMedianMinimumMaximum

31.119.627.7

7.1

65.7

4.89.9

.4

-5.8

18.8

6.49.13.4

-3.9

18.4
50 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams

1Observed 7Q10 was determined by frequency analysis of the annual 7-day low flows.2Predicted 7Q10 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Table 15. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatwith the number of base-flow measurements varied: Case D—the paired stations are on different streams or basins, and the drainagless than at the partial-record station[7Q10 is the minimum average discharge for 7 consecutive days with a recurrence interval of 10 years; ft3/s, cubic feet per second; n, number of base-flow measur

Pair

Partialrecord

station

Index

station

Observed1

7Q10

(ft3/s)

Predicted2

7Q10

n=20

(ft3/s)

Predicted

7Q10

n=15

(ft3/s)

Predicted

7Q10

n=10

(ft3/s)

Predicted

7Q10

n=5

(ft3/s)

Relative

error3 of

predicted

7Q10

n=20

(percent)

Relativ

error o

predicte

7Q10

n=15

(percen

26

1013232426272829

03275000033265000333400003340800033625000336300004094500040953000409975004100500

03275600033243000332970003341200033615000336390004094000040961000409980804099808

62.325.819.1

2.4

20.4

105

19.424.595.0

111

69.828.621.1

2.5

20.4

107

21.624.289.8

115

66.425.420.7

2.3

18.7

106

21.523.688.9

111

62.325.621.2

1.9

18.0

104

20.923.686.6

107

63.521.019.9

2.0

18.6

103

20.823.388.9

107

-10.7

-9.8-9.5-4.0

0

-1.9

-10.2

1.25.8

-3.5

-6.2

1.6

-7.7

4.39.1

-.9

-9.8

3.86.90

MeanStandard deviationMedianMinimumMaximum

48.541.125.2

2.4

111

-4.3

5.7

-3.8

-10.7

5.8

.1

6.3

.8

-9.8

9.1
Estimating 7-Day, 2-Year Low Flows, with the Number of Base-Flow Measurements Varied, Case D 51

Table 16. Pairs of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimawith the number of base-flow measurements varied: Case D—the paired stations are on different streams or basins, and the drainaless than at the partial-record station[7Q2 is the minimum average discharge for 7 consecutive days with a recurrence interval of 2 years; ft3/s, cubic feet per second; n, number of base-flow measurem

Pair

Partialrecord

station

Index

station

Observed1

7Q2

(ft3/s)

1Observed 7Q2 was determined by frequency analysis of the annual 7-day low flows.

Predicted2

7Q2

n=20

(ft3/s)

2Predicted 7Q2 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).

Predicted

7Q2

n=15

(ft3/s)

Predicted

7Q2

n=10

(ft3/s)

Predicted

7Q2

n=5

(ft3/s)

Relative

error3 of

predicted

7Q2

n=20

(percent)

3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

Relativ

error o

predict

7Q2

n=15

(percen

26

1013232426272829

03275000033265000333400003340800033625000336300004094500040953000409975004100500

03275600033243000332970003341200033615000336390004094000040961000409980804099808

103

46.633.8

5.1

40.6

173

25.729.6

137170

109

50.337.5

4.3

35.2

184

26.529.0

129170

103

45.737.6

4.1

33.8

180

26.328.5

129164

96.346.037.9

3.6

33.4

177

25.428.2

127160

97.141.937.1

3.6

32.5

172

25.028.0

123160

-5.5-7.4-9.9

18.615.3

-6.0-3.0

2.16.20

02.0

-10.1

24.420.1

-3.9-2.3

3.96.23.7

MeanStandard deviationMedianMinimumMaximum

76.463.543.6

5.1

173

1.09.7

-1.5-9.9

18.6

4.4

10.5

2.9

-10.1

24.4
52 Evaluation of a Method of Estimating Low-Flow Frequencies from Base-Flow Measurements at Indiana Streams Table 17. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimat

number of base-flow measurements varied: Case E—the grouped stations are on the same streams or tributaries, and there are tw[7Q10 is the minimum average discharge for 7 consecutive days with a recurrence interval of 10 years; ft3/s, cubic feet per second; n, number of base-flow measu

Pair Partial- record station Index station 1 Index Station 2 Observed1 7Q10 (ft3/s)

1Observed 7Q10 was determined by frequency analysis of the annual 7-day low flows.

Predicted2

7Q10

n=20

(ft3/s)

2Predicted 7Q10 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).

Predicted

7Q10

n=15

(ft3/s)

Predicted

7Q10

n=10

(ft3/s)

Predicted

7Q10

n=5

(ft3/s)

Relative

error3 o

predicte

7Q10

n=20

(percent

3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

1346789

121315

03274750033350000333600003361500033625000336300003364000055155000551650005522500

03274650033345000333550003361000033620000336150003363000055150000551600005522000

03275000033340000334050003363000033630000336250003363900055175000551700005524500

7.6

54.8

1,040

40.220.392.9

156195

17.9

6.0

8.4

57.8

1,055

38.519.887.3

149185

19.2

6.5

8.6

57.4

984

38.017.792.1

151185

20.3

6.5

8.3

55.7

1,020

37.719.885.4

152188

21.6

5.7

7.9

57.5

1,098

42.619.768.8

146188

23.2

6.7

-9.5-5.2-1.4

4.42.56.44.75.4

-6.8-7.7

MeanStandard deviationMedianMinimumMaximum

163315

47.5

6.0

1,040

-.7

6.1

.6

-9.5

6.4
Estimating 7-Day, 2-Year Low Flows, with the Number of Base-Flow Measurements Varied, Case E 53

Table 18. Sets of streamflow-gaging stations in Indiana used to evaluate the accuracy of the Stedinger-Thomas method of estimatnumber of base-flow measurements varied: Case E—the grouped stations are on the same streams or tributaries, and there are tw[7Q2 is the minimum average discharge for 7 consecutive days with a recurrence interval of 2 years; ft3/s, cubic feet per second; n, number of base-flow measurem

Pair

Partialrecord

station

Index

station 1

Index

Station 2

Observed1

7Q2

(ft3/s)

1Observed 7Q2 was determined by frequency analysis of the annual 7-day low flows.

Predicted2

7Q2

n=20

(ft3/s)

2Predicted 7Q2 was estimated, using the Stedinger-Thomas method (Stedinger and Thomas, 1985).

Predicted

7Q2

n=15

(ft3/s)

Predicted

7Q2

n=10

(ft3/s)

Predicted

7Q2

n=5

(ft3/s)

Relative

error3 o

predicte

7Q2

n=20

(percent

3Relative error, in percent, is computed as: [(observed - predicted) / predicted] x 100.

1346789

121315

03274750033350000333600003361500033625000336300003364000055155000551650005522500

03274650033345000333550003361000033620000336150003363000055150000551600005522000

03275000033340000334050003363000033630000336250003363900055175000551700005524500

12.994.5

1,584

65.939.8

152256263

30.213.9

12.992.1

1,613

67.735.7

150251239

32.214.4

13.092.0

1533

67.632.0

153249242

32.614.3

12.990.0

1,584

67.334.8

143248241

34.213.3

12.892.9

1,698

70.736.5

131251239

33.213.6

02.6

-1.8-2.7

11.5

1.32.0

10.0

-6.2-3.5

MeanStandard deviationMedianMinimumMaximum

251477

80.212.9

1,584

1.35.7

.7

-6.2

11.5