In cooperation with the

VERMONT AGENCY OF TRANSPORTATION

A Geographic Information System Tool to 

Solve Regression Equations and Estimate

Flow-Frequency Characteristics of 

Vermont Streams

Open-File Report 02-494

U.S. Department of the Interior

U.S. Geological Survey
A Geographic Information System Tool to Solve 

Regression Equations and Estimate FlowFrequency Characteristics of Vermont Streams

By Scott A. Olson, Gary D. Tasker, and Craig M. Johnston

Prepared in cooperation with the

VERMONT AGENCY OF TRANSPORTATION

Pembroke, New Hampshire

2003

Open-File Report 02-494

U.S. GEOLOGICAL SURVEY
For additional information write to:

District Chief

U.S. Geological Survey

New Hampshire/Vermont District

361 Commerce Way

Pembroke, NH 03275-3718

http://nh.water.usgs.gov

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does 

not imply endorsement by the U.S. Government.

U.S. DEPARTMENT OF THE INTERIOR

GALE A. NORTON, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director
Contents iii

CONTENTS

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

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

Estimating Flow-Frequency Characteristics of Unregulated, Ungaged, Rural Vermont Streams....................... 2

Regression Equations.................................................................................................................................. 2

Allowable Range of Basin Characteristics for Use in the Regression Equations....................................... 3

Standard Error of Residuals and Prediction of the Regression Equations.................................................. 3

Equivalent Years of Record of Results of the Regression Equations ......................................................... 5

Prediction Intervals of Results of the Regression Equations ...................................................................... 8

Vermont Flow-Frequency Tool............................................................................................................................ 8

System Requirements.................................................................................................................................. 9

Data Requirements...................................................................................................................................... 9

Installation of the Vermont Flow-Frequency Tool ..................................................................................... 12

Using the Vermont Flow-Frequency Tool .................................................................................................. 12

Additional Utilities of Vermont Flow-Frequency Tool ....................................................................................... 15

Vermont Flow-Frequency Tool Options Menu........................................................................................... 15

Use of Regression Equations at or near Stream-Gaging Stations............................................................... 15

Expected Probability Adjustment to Flood Discharges .............................................................................. 16

National Flood Frequency Program............................................................................................................ 17

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

Selected References.............................................................................................................................................. 18

Appendix 1. Use of Regression Equations with Metric Units ............................................................................. 22

Selected References.............................................................................................................................................. 23
iv Figures

FIGURES

Figure 1. Directory structure of the data files required by the Vermont Flow-Frequency Tool. 

Hydrologic unit codes 01080105 and 01080106 are shown as examples. ........................................ 10

Figure 2. ArcView window with the Vermont Flow-Frequency Tool installed. .............................................. 13

Figure 3. ArcView window displayed when a point on a stream is selected.................................................... 14
Tables v

TABLES

Table 1. Range of basin characteristics used in development of the regression equations for estimating 

flow-frequency characteristics of ungaged, unregulated streams in Vermont .................................. 4

Table 2. Standard error of residuals of the regression equations for estimating flow frequencies on 

ungaged, unregulated streams in rural drainage basins in Vermont .................................................. 4

Table 3. Model error variance and the (XT Λ-1 X)-1 matrices for the regression equations ........................... 6

Table 4. Average standard error of prediction and average equivalent years of record of the regression 

equations for estimating peak flow-frequency characteristics on ungaged, unregulated streams 

in rural drainage basins in Vermont .................................................................................................. 7

Table 5. Buttons that run the functions of the Vermont Flow-Frequency Tool ............................................. 13
vi Conversion Factors and Web Sites

CONVERSION FACTORS AND WEB SITES

CONVERSION FACTORS

WEB SITES USEFUL FOR OBTAINING DATA AND INFORMATION PERTINENT TO THE 

VERMONT FLOW-FREQUENCY TOOL

Multiply By To obtain

inch (in.) 25.4 millimeter (mm)

foot (ft) 0.3048 meter (m)

square mile (mi2) 2.589 square kilometer (km2)

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

Web Site Owner Source of

http://www.cits.rncan.gc.ca Natural Resources Canada Canadian elevation datasets

http://www.vcgi.org Vermont Center for Geographic Information Topographic images of Vermont

http://nhd.usgs.gov U.S. Geological Survey National Hydrography Dataset

http://edcnts12.cr.usgs.gov/ned U.S. Geological Survey National Elevation Dataset

http://vt.water.usgs.gov/CurrentProjects/

vtfloodfreq/index.htm

U.S. Geological Survey, Vermont District Vermont Flow-Frequency Tool

http://water.usgs.gov U.S. Geological Survey, Water Resources Division National Flood Frequency Program

http://www.climatesource.com Climatesource PRISM Precipitation Datasets
Introduction 1

A Geographic Information System Tool to Solve 

Regression Equations and Estimate FlowFrequency Characteristics of Vermont Streams

By Scott A. Olson, Gary D. Tasker, and Craig M. Johnston

ABSTRACT

Estimates of the magnitude and frequency 

of streamflow are needed to safely and 

economically design bridges, culverts, and other 

structures in or near streams. These estimates also 

are used for managing floodplains, identifying 

flood-hazard areas, and establishing floodinsurance rates, but may be required at ungaged 

sites where no observed flood data are available 

for streamflow-frequency analysis. This report 

describes equations for estimating flow-frequency 

characteristics at ungaged, unregulated streams in 

Vermont. 

In the past, regression equations developed 

to estimate streamflow statistics required users to 

spend hours manually measuring basin 

characteristics for the stream site of interest. This 

report also describes the accompanying 

customized geographic information system (GIS) 

tool that automates the measurement of basin 

characteristics and calculation of corresponding 

flow statistics. The tool includes software that 

computes the accuracy of the results and 

adjustments for expected probability and for 

streamflow data of a nearby stream-gaging station 

that is either upstream or downstream and within 

50 percent of the drainage area of the site where 

the flow-frequency characteristics are being 

estimated. The custom GIS can be linked to the 

National Flood Frequency program, adding the 

ability to plot peak-flow-frequency curves and 

synthetic hydrographs and to compute adjustments 

for urbanization.

INTRODUCTION

Estimates of the magnitude and frequency of 

streamflow are needed to safely and economically 

design bridges, culverts, and other structures in or near 

streams. These estimates also are used for managing 

floodplains, identifying flood-hazard areas, and 

establishing flood-insurance rates, but may be required 

at ungaged sites where no observed flood data are 

available for streamflow-frequency analysis. The 

U.S. Geological Survey (USGS), in cooperation with 

the Vermont Agency of Transportation (VTrans), 

developed regression equations for estimating peakflow frequency for recurrence intervals of 2, 5, 10, 25, 

50, 100, and 500 years and estimates of the 25, 50, and 

75 percent flow duration (daily discharges exceeded 

25, 50, and 75 percent of the time) for ungaged, 

unregulated streams in Vermont. Generalized least 

squares regression (Tasker, 1987) was used to relate 

basin characteristics to flow-frequency characteristics 

for the peak-flow equations. Ordinary least squares 

regression was used to develop the flow-duration 

equations. A report by Olson (2002) documents the 

development of those equations. 

To solve the flow-frequency equations accurately 

and quickly, a computer application was developed 

to automate the delineation of drainage-basin 

characteristics required by the equations. A 

geographical information system (GIS) application, 

named the Vermont Flow-Frequency Tool, was 

created to automate basin-characteristic delineation. 

The National Hydrography Dataset (NHD) 

(U.S. Geological Survey and U.S. Environmental 

Protection Agency, 2002) tools are the foundation for 

the GIS tool customized for Vermont. The tool includes 

software that was written in the computer language, 
2 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

FORTRAN, to solve the regression equations, compute 

the accuracy of the results, and determine adjustments 

for expected probability and for data from a nearby 

stream-gaging station. A nearby station is defined as a 

station that is either upstream or downstream and 

within 50 percent of the drainage area of the site where 

the flow-frequency characteristics are being estimated. 

The Vermont Flow-Frequency Tool can be linked to the 

National Flood Frequency (Ries and Crouse, 2002) 

program, providing the abilities to plot peak-flowfrequency curves and synthetic hydrographs and 

compute adjustments for urbanization.

The purpose of this report is to describe 

(1) the equations for estimating flow-frequency 

characteristics, (2) an error analysis of the equations 

and (3) the Vermont Flow-Frequency Tool.

The authors wish to thank the following for 

their help or guidance in developing the GIS datasets 

and(or) programming of the computer application: 

Jennifer R. Hill of Horizon Systems Corporation, 

Lynn C. Bjorklund, Laura Hayes, David S. Nail, 

Kernell G. Ries, Keven S. Roth, and Peter A. Steeves of 

the U.S. Geological Survey.

ESTIMATING FLOW-FREQUENCY 

CHARACTERISTICS OF UNREGULATED, 

UNGAGED, RURAL VERMONT STREAMS

Generally, the most reliable estimates of flowfrequency characteristics are obtained by frequency 

analysis of flow data from gaged streams. Estimates, 

however, often are required at stream locations where 

no data are available. To solve this problem, regression 

equations were developed for estimating flood 

discharges at ungaged, unregulated streams in rural 

drainage basins in Vermont by relating flow-frequency 

characteristics to basin characteristics at gaged 

locations. Details of the development of the equations 

can be found in the report, “Flow-Frequency 

Characteristics of Vermont Streams” by Olson (2002).

Regression Equations

The equations for estimating flood-flow 

frequencies were developed using generalized least 

squares (GLS) regression methods (Tasker, 1987) and 

the GLSNET software (G.D. Tasker, K.M. Flynn, 

A.M. Lumb, W.O. Thomas, Jr., U.S. Geological 

Survey, written commun., 1995). Tasker and Stedinger 

(1989) demonstrated that GLS regression provides the 

most appropriate results for hydrologic purposes 

because the principles of GLS allow weight to be given 

to each station in the analysis to compensate for 

differences in length of record and for streamflow data 

that are correlated spatially and in time. The equations 

for estimating flow duration (daily-mean discharges 

exceeded a given percentage of time) were developed 

using ordinary least squares (OLS) regression 

procedures (SAS Institute Inc., 1990) because periodof-record flow-duration statistics lack the annual time 

series required by GLSNET software. 

The equations used to estimate peak flow at 

recurrence intervals of 2, 5, 10, 25, 50, 100, and 

500 years and the magnitude of daily-mean discharges 

exceeded 25, 50, and 75 percent of the time are

, (1)

, (2)

, (3)

, (4)

, (5)

, (6)

, (7)

, (8)

, and (9)

. (10)

Q2 38.1A0.914L–0.294E0.0776Y–0.180 =

Q5 61.8A0.902L–0.295E0.0835Y–0.253 =

Q10 79.7A0.897L–0.302E0.0890Y–0.298 =

Q25 106A0.883L–0.316E0.104Y–0.349 =

Q50 129A0.874L–0.327E0.115Y–0.385 =

Q100 153A0.865L–0.336E0.125Y–0.420 =

Q500 217A0.846L–0.355E0.148Y–0.497 =

D75 0.000627A1.08P1.55E0.101 =

D50 0.00152A1.04P1.58E0.0603 =

D25 0.00431A1.01P1.55E0.0438 =
Estimating Flow-Frequency Characteristics of Unregulated, Ungaged, Rural Vermont Streams 3

where

The equations with metric units are described in 

Appendix 1.

Allowable Range of Basin Characteristics for Use 

in the Regression Equations

Regression equations 1–10, developed to 

estimate flow-frequency characteristics, are limited to 

sites on ungaged, unregulated streams in rural Vermont 

drainage basins and to sites with basin characteristics 

that are within the range of basin characteristics used in 

the development of the equations. The ranges of basin 

characteristics used in the regression analysis are 

shown in table 1. If basin characteristics from an 

ungaged site are outside of the ranges, the accuracies of 

the predictions of flow frequency are unknown. The 

Vermont Flow-Frequency Tool indicates if basin 

characteristics are outside the allowable ranges.

Basin characteristics used for developing 

equations 1–10 were determined with a GIS using 

datasets and methods previously explained in the 

description of each of the equation input variables. 

Determining the basin characteristics for use in the 

regression equations without a GIS, or with data 

sources or methods that deviate from those described in 

this report, may introduce bias and produce 

streamflow-frequency estimates with unknown error.

Standard Error of Residuals and Prediction of the 

Regression Equations

Standard error of residuals and prediction are 

measures of the accuracy of a regression equation. 

Residuals are the difference between observations and 

predictions; therefore, the standard error of residuals is 

a measure of how well the regression estimates agree 

with the observed data. The standard error of residuals 

for each equation is shown in table 2.

The standard error of residuals is an indication of 

how well the observed data are being modeled, thus, it 

is only an approximation of model error or how well 

the equations will predict streamflow at ungaged 

locations. The average standard error of prediction has 

a model error component and a sampling error 

component. The sampling error is a measure of the 

error in the regression prediction due to the regression 

equation being developed with observed (sampled) 

streamflow characteristics instead of true streamflow 

characteristics. Thus, the average standard error of 

prediction is a measure of the expected accuracy 

to which a regression equation can estimate a

Qn is the calculated peak flow for recurrence 

interval n, in cubic feet per second;

Dn is the estimated daily discharge exceeded n

percent of the time, in cubic feet per second;

A is the drainage area of the watershed 

delineated using a GIS (boundaries for the 

drainage areas were from a 12-digit 

hydrologic unit coverage (Natural Resources 

Conservation Service, 1996) or digitized 

when a watershed’s boundary was not 

defined by this GIS coverage), in square 

miles;

L is the area of lakes and ponds in a watershed 

as a percent of drainage area, plus 1 percent, 

determined using a GIS from the 1:24,000 

scale National Hydrography Dataset (U.S. 

Geological Survey and U.S. Environmental 

Protection Agency, 2002) using all features 

in the dataset that were lake or pond 

boundaries including ponds as small as 

several hundreths of an acre and pooled areas 

of streams; 

E is the percent of the watershed that is at or 

greater than 1,200 ft altitude, plus 1 percent, 

computed using a GIS from the National 

Elevation Dataset (U.S. Geological Survey, 

2001b);

Y is the northing of the centroid of the drainage 

basin determined with a GIS, in the Vermont 

State Plane Coordinate System, divided by 

100,000, then increased by 1.0; and

P is the basinwide mean of the mean annual 

precipitation, in inches, determined using a 

GIS and the Parameter-elevation Regressions 

on Independent Slopes Model (PRISM) 

dataset (Daly, 2000) resampled with bilinear 

interpolation to a 30-m cell resolution using 

the GIS, ARC/INFO RESAMPLE command 

(Environmental Systems Research Institute, 

Inc., 1994). Canadian precipitation data 

(Ghislain Jacques, Environment Quebec, 

written commun., January 17, 2002) were 

merged with the PRISM precipitation data 

where needed along the northern Vermont 

border.
4 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

Table 1. Range of basin characteristics used in development of the regression equations for estimating flow-frequency 

characteristics of ungaged, unregulated streams in Vermont

Regression 

equation Basin characteristic Minimum Maximum Mean

Peak flow Drainage area (square miles) 0.211 850 96.8

Peak flow Percent lake and pond area 0 6.86 0.641

Peak flow Percent of basin greater than 1,200 feet altitude 0 100 58.9

Peak flow Centroid northing (Vermont State Plane) -87 296,194 151,420

Flow duration Drainage area (square miles) 2.09 850 148

Flow duration Mean annual precipitation (inches) 32.8 63.8 44.5

Flow duration Percent of basin above 1,200 feet altitude 0 100 63.8

Table 2. Standard error of residuals of the regression equations for estimating flow frequencies 

on ungaged, unregulated streams in rural drainage basins in Vermont

Flow-frequency characteristic

Standard error of residuals

(in log10 units) (in percent)

Peak flow with 2-year recurrence interval 0.171 48.4 to –32.6

Peak flow with 5-year recurrence interval .170 48.0 to –32.4

Peak flow with 10-year recurrence interval .175 49.7 to –33.2

Peak flow with 25-year recurrence interval .182 52.0 to –34.2

Peak flow with 50-year recurrence interval .190 54.9 to –35.4

Peak flow with 100-year recurrence interval .200 58.4 to –36.9

Peak flow with 500-year recurrence interval .227 69.0 to –40.8

Daily discharge exceeded 75 percent of the time .0540 13.2 to –11.7

Daily discharge exceeded 50 percent of the time .0832 21.1 to –17.4

Daily discharge exceeded 25 percent of the time .124 33.2 to –24.9
Estimating Flow-Frequency Characteristics of Unregulated, Ungaged, Rural Vermont Streams 5

flow-frequency characteristic at an ungaged location 

with basin characteristics similar to the ones used to 

develop the regression equation. 

Hodge and Tasker (1995) describe the 

mathematical equation for computing the standard 

error of prediction, SEpred, of a flow-frequency 

estimate as 

, (11)

where

The standard error of prediction of a streamflow 

estimate at an ungaged location can be computed, 

because xi is not required to be a row of X, The 

Vermont Flow-Frequency Tool accompanying this 

report computes and displays the standard error of 

prediction for each flow-frequency estimate. To find 

the average standard error of prediction of the 

equations that are shown in table 4, the standard error 

of prediction was computed for each of the stations 

used in the development of the flood-flow-frequency 

regression equation then averaged.

The standard error of prediction of estimates of 

flow duration at an ungaged site also is computed with 

equation 11, but with 2 substitutions for variables. This 

substitution is necessary because OLS regression was 

used instead of GLS regression to develop the 

equations for estimating flow duration. The first 

substitution is for model error, γ, which is 

approximated with the standard error of residuals 

(table 2). The second substitution is for the covariance 

matrix, Λ, used for weighting the station data. This 

matrix is simplified to a matrix with diagonal elements 

equal to γ

2 and off-diagonal elements equal to zero.

The standard error of prediction is converted to 

positive and negative percent errors with the following 

formulas:

, and (12)

. (13)

where

Equations 12 and 13 apply not only to the standard 

error of prediction, but also to the average standard 

error of prediction and standard error of residuals by 

substituting the appropriate error term in logarithmic 

units. The probability that the true value of streamflow 

at a given frequency is between the positive-percent 

and negative-percent standard error of prediction is 

approximately 68 percent. The standard error of 

prediction is computed and included in the results each 

time a flow-frequency estimate is made with the 

Vermont Flow-Frequency Tool.

Equivalent Years of Record of Results of the 

Regression Equations

The equivalent years of record (Hardison, 1971) 

shown in table 4 and computed by the provided 

Vermont Flow-Frequency Tool is another measure of 

accuracy of the regression equations. The equivalent 

years of record are the number of years of data 

collection at an ungaged site that would be required to 

achieve frequency results with accuracy equal to the 

regression equations. It is computed by use of the 

equation

, (14)

γ

2 is the model-error variance (table 3);

xi is a row vector of the logarithms of 

the basin characteristics for the study 

site i, augmented by a 1 as the first 

element;

T is the matrix algebra symbol for 

transposing (Grossman, 1986) a 

matrix; and

(XTΛ−1X)-1 is the (p x p) matrix for the 

regression equations (table 3) with X

being a (n x p) matrix that has a row 

of logarithmically transformed basin 

characteristics augmented by a 

leading 1 and Λ being the (n x n) 

covariance matrix used for weighting 

sample data in the GLS regression; n

is the number of stream-gaging 

stations used in the regression 

analysis and p is the number of basin 

characteristics, plus 1.

SEpred γ

2

xi XTΛ–1 ( ) X –1xi

T [ ] +

1 2⁄

=

Spos is the positive percent error of prediction,

Sneg is the negative percent error of prediction, 

and

SEpred is the standard error of prediction in 

logarithmic units.

Spos 100 10

SEpred = ( ) – 1

Sneg 100 10

SEpred – = ( ) – 1

E s2 1 kng 0.5kn

2 1 0.75g

2 [ ] + + ( ) + SEpred

2 = ⁄
6 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

Table 3. Model error variance and the (XT Λ -1 X)

-1 matrices for the regression equations

Flow-frequency characteristic

Modelerror 

variance, 

γ2

 (XT Λ-1 X)

-1 matrix

Peak flow with a 2-year recurrence interval 0.0284 0.39106E-02 -0.22912E-03 -0.97874E-03 -0.10037E-02 -0.36445E-02

-0.22912E-03 0.38268E-03 -0.63663E-03 -0.11393E-03 -0.63470E-04

-0.97874E-03 -0.63663E-03 0.83023E-02 -0.33102E-04 0.10605E-02

-0.10037E-02 -0.11393E-03 -0.33102E-04 0.74670E-03 -0.15507E-03

-0.36445E-02 -0.63470E-04 0.10605E-02 -0.15507E-03 0.10302E-01

Peak flow with a 5-year recurrence interval .0271 0.41068E-02 -0.26008E-03 -0.10128E-02 -0.97846E-03 -0.39752E-02

-0.26008E-03 0.40128E-03 -0.61274E-03 -0.12005E-03 -0.86724E-04

-0.10128E-02 -0.61274E-03 0.82284E-02 -0.45938E-04 0.10812E-02

-0.97846E-03 -0.12005E-03 -0.45938E-04 0.74762E-03 -0.15726E-03

-0.39752E-02 -0.86724E-04 0.10812E-02 -0.15726E-03 0.11254E-01

Peak flow with a 10-year recurrence interval .0277 0.45209E-02 -0.29160E-03 -0.10952E-02 -0.10313E-02 -0.45038E-02

-0.29160E-03 0.43815E-03 -0.63798E-03 -0.13321E-03 -0.10916E-03

-0.10952E-02 -0.63798E-03 0.87145E-02 -0.60592E-04 0.11726E-02

-0.10313E-02 -0.13321E-03 -0.60592E-04 0.80298E-03 -0.17208E-03

-0.45038E-02 -0.10916E-03 0.11726E-02 -0.17208E-03 0.12762E-01

Peak flow with a 25-year recurrence interval .0282 0.50699E-02 -0.32950E-03 -0.11943E-02 -0.10999E-02 -0.52262E-02

-0.32950E-03 0.48450E-03 -0.67035E-03 -0.15032E-03 -0.14061E-03

-0.11943E-02 -0.67035E-03 0.93168E-02 -0.80941E-04 0.12900E-02

-0.10999E-02 -0.15032E-03 -0.80941E-04 0.87551E-03 -0.19373E-03

-0.52262E-02 -0.14061E-03 0.12900E-02 -0.19373E-03 0.14825E-01

Peak flow with a 50-year recurrence interval .0294 0.56002E-02 -0.36166E-03 -0.12972E-02 -0.11833E-02 -0.58946E-02

-0.36166E-03 0.53107E-03 -0.71790E-03 -0.16740E-03 -0.16736E-03

-0.12972E-02 -0.71790E-03 0.10048E-01 -0.97417E-04 0.14115E-02

-0.11833E-02 -0.16740E-03 -0.97417E-04 0.95469E-03 -0.21490E-03

-0.58946E-02 -0.16736E-03 0.14115E-02 -0.21490E-03 0.16725E-01

Peak flow with a 100-year recurrence interval .0313 0.62124E-02 -0.39699E-03 -0.14203E-02 -0.12880E-02 -0.66509E-02

-0.39699E-03 0.58615E-03 -0.78119E-03 -0.18749E-03 -0.19653E-03

-0.14203E-02 -0.78119E-03 0.10972E-01 -0.11479E-03 0.15561E-02

-0.12880E-02 -0.18749E-03 -0.11479E-03 0.10506E-02 -0.23911E-03

-0.66509E-02 -0.19653E-03 0.15561E-02 -0.23911E-03 0.18869E-01

Peak flow with a 500-year recurrence interval .0378 0.79249E-02 -0.49238E-03 -0.17810E-02 -0.16035E-02 -0.87217E-02

-0.49238E-03 0.74532E-03 -0.98171E-03 -0.24512E-03 -0.27330E-03

-0.17810E-02 -0.98171E-03 0.13796E-01 -0.15917E-03 0.19741E-02

-0.16035E-02 -0.24512E-03 -0.15917E-03 0.13314E-02 -0.30477E-03

-0.87217E-02 -0.27330E-03 0.19741E-02 -0.30477E-03 0.24721E-01

Table 3. Model error variance and the (X T Λ -1 X)

-1 matrices for the regression equations

[A (XTΛ-1X)

-1 matrix is the (p x p) matrix for the regression equations with X being a (n x p) matrix that has a row of logarithmically transformed basin 

characteristics augmented by a leading 1 and Λ being the (n x n) covariance matrix used for weighting sample data in the GLS regression; n is the number of 

stream-gaging stations used in the regression analysis and p is the number of basin characteristics, plus 1; Numbers in the (XT

Λ-1X)

-1 matrices are in scientific 

notation; --, no data]
Estimating Flow-Frequency Characteristics of Unregulated, Ungaged, Rural Vermont Streams 7

Daily flow exceeded 25 percent of the time .00292 0.54316E-01 -0.24661E-03 -0.34488E-01 0.17354E-02 --

-0.24661E-03 0.83665E-04 0.76129E-04 -0.15047E-04 --

-0.34488E-01 0.76129E-04 0.22208E-01 -0.13052E-02 --

0.17354E-02 -0.15047E-04 -0.13052E-02 0.25861E-03 --

Daily flow exceeded 50 percent of the time .00692 0.12877E-00 -0.58470E-03 -0.81768E-01 0.41146E-02 --

-0.58470E-03 0.19836E-03 0.18049E-03 -0.35675E-04 --

-0.81768E-01 0.18049E-03 0.52653E-01 -0.30946E-02 --

0.41146E-02 -0.35675E-04 -0.30946E-02 0.61314E-03 --

Daily flow exceeded 75 percent of the time .0153 0.28853E-00 -0.13100E-02 -0.18320E-00 0.92191E-02 --

-0.13100E-02 0.44444E-03 0.40441E-03 -0.79931E-04 --

-0.18325E-00 0.40441E-03 0.11797E-00 -0.69336E-02 --

0.92191E-02 -0.79931E-04 -0.69336E-02 0.13737E-02 --

Table 3. Model error variance and the (XT Λ -1 X)

-1 matrices for the regression equations--Continued

Flow-frequency characteristic

Modelerror 

variance, 

γ2

 (XT Λ-1 X)

-1 matrix

Table 4. Average standard error of prediction and average equivalent years of record of the regression equations for estimating peak flow-frequency 

characteristics on ungaged, unregulated streams in rural drainage basins in Vermont

Flow-frequency characteristic

Average standard error of prediction

Average equivalent years of record

(in log10 units) (in percent)

Peak flow with 2-year recurrence interval 0.173 48.8 to –32.8 1.4

Peak flow with 5-year recurrence interval .169 47.6 to –32.2 2.3

Peak flow with 10-year recurrence interval .172 48.4 to –32.6 3.2

Peak flow with 25-year recurrence interval .174 49.1 to –32.9 4.6

Peak flow with 50-year recurrence interval .178 50.5 to –33.5 5.5

Peak flow with 100-year recurrence interval .184 52.6 to –34.5 6.3

Peak flow with 500-year recurrence interval .202 59.2 to –37.2 7.6
8 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

where

Because equation 14 requires variables that do not 

apply to flow duration estimates such as the frequency 

factor, kn, equivalent years of record cannot be 

computed for the flow-duration estimates. Equivalent 

years of record are computed and output each time a 

flood-frequency estimate is made with the Vermont 

Flow-Frequency Tool.

Prediction Intervals of Results of the Regression 

Equations

Prediction intervals indicate the uncertainty in 

the use of the equations. For example, one can be 

90-percent confident that the true value of a flowfrequency characteristic lies within the 90-percent 

prediction interval. The prediction interval can be 

computed as follows:

Let

, (15)

then

, (16)

where

Prediction intervals at the 50-, 67-, 90-, and 95-percent 

confidence levels are computed and output each time a 

flow-frequency estimate is made with the Vermont 

Flow-Frequency Tool.

VERMONT FLOW-FREQUENCY TOOL

To determine the basin characteristics required 

by the flow-frequency estimating equations for streams 

in Vermont, a GIS application was developed. The 

application, called the Vermont Flow-Frequency Tool, 

allows the user to interactively select a point on a 

stream of interest. The tool automatically delineates the 

required basin characteristics and solves the regression 

equations. In addition, a link between the Vermont 

Flow-Frequency Tool and the National Flood 

Frequency (NFF) program (Ries and Crouse, 2002) 

provides the ability to plot frequency curves and 

synthetic hydrographs, and adjust flow-frequency 

curves for urbanization. The application customized for 

Vermont also calculates standard error of prediction, 

prediction intervals, and expected probability 

adjustments.

The Vermont Flow-Frequency Tool was 

developed in the Environmental Systems Research 

Institute, Inc. (ESRI) ArcView version 3.2 software 

(1996a) using ESRI Avenue programming language 

(Environmental Systems Research Institute, Inc., 

1996b) and converted to an ESRI ArcView extension. 

Extensions are plug-in programs that can be easily 

E is the equivalent years of record;

s is the standard deviation of annual events 

estimated from a regression between the 

standard deviation and drainage area, in 

square miles, A, at stations used in the 

regression analysis (s = 0.0339A-0.102);

kn is the log-Pearson type III frequency 

factor for the n-year event;

g is the skew coefficient used in the 

computation of the frequency curve 

(assumed to be zero when computing 

equivalent years of record for an estimate 

at an ungaged site); and

SEpred is the standard error of prediction.

V 10

tα ⁄ 2, n p – SEpred ( ) =

( ) 1 ⁄ V Qpred < < Qtrue ( ) V Qpred

tα/2,n-p is the critical value from a Students-t

distribution (found in many introductory 

statistics textbooks) at alpha level α

(α = 0.10 for a 90-percent prediction 

interval) with n – p degrees of freedom, 

n is the number of stations used in the 

regression analysis, and p is the number 

of basin characteristics in the regression 

equation, plus 1;

SEpred is the standard error of prediction of a 

flow-frequency estimate;

Qpred is the computed flow at a selected 

frequency from the regression equation; 

and

Qtrue is the true value of flow at a selected 

frequency.
Vermont Flow-Frequency Tool 9

loaded or unloaded into ArcView as needed. The 

drainage-basin delineation done by the Vermont FlowFrequency Tool is based on the NHD and the 

NHD Toolkit (U.S. Geological Survey and U.S. 

Environmental Protection Agency, 2002) and can be 

downloaded from the USGS New Hampshire/Vermont 

District website at http://vt.water.usgs.gov

/CurrentProjects/vtfloodfreq/index.htm.

System Requirements

The Vermont Flow-Frequency Tool requires the 

following software and toolkits:

1. ESRI ArcView version 3.1 or higher except 

versions 8.x, which do not support the Avenue 

programming language;

2. The ESRI ArcView Spatial Analyst extension; and

3. The NHD Toolkit (U.S. Geological Survey and 

U.S. Environmental Protection Agency, 2002).

The NHD Toolkit is available for download from 

the NHD website, http://nhd.usgs.gov/. The NHD 

Toolkit contains a suite of extensions some of which 

are used to process NHD data and some of which the 

Vermont Flow-Frequency Tool is dependent upon for 

execution.

Although the computer and operating system 

requirements are defined by the ESRI ArcView 

software, the Vermont Flow-Frequency Tool does limit 

the operating system to the Microsoft Windows 

operating system. The Vermont Flow-Frequency Tool 

successfully runs in Microsoft Windows98, NT, and 

2000. Other operating systems have not been tested. 

Notepad, a program for viewing text files commonly 

packaged with the Microsoft Windows operating 

system is required. Required disk space for the entire 

suite of topographic map images, NHD data, and other 

supporting datasets for computing basin characteristics 

throughout Vermont is estimated to be 5.3 gigabytes. 

Data such as the NHD and the topographic map 

images, however, can be installed in different locations 

on a local network. One gigabyte of virtual memory is 

required for delineating the full range of drainage basin 

sizes encountered in Vermont.

The Vermont Flow-Frequency Tool can be linked 

to the NFF Program (Ries and Crouse, 2002), which is 

discussed in greater detail in the section “Utilities of 

the National Flood Frequency Program.” Although the 

Vermont Flow-Frequency Tool does not require NFF, 

users may find the utilities provided by NFF to be 

valuable. NFF is available on the Internet from the 

USGS (http://water.usgs.gov/software/nff.html). After 

NFF is installed, a copy of the NFF executable file 

needs to be placed in the Vermont Flow-Frequency 

Tool utilities folder.

Data Requirements

The Vermont Flow-Frequency Tool requires 

several gigabytes of data to work statewide. For the 

Tool to function properly, the data directory must be 

structured and use the naming conventions as shown in 

figure 1. For each hydrologic unit, there are three 

directories whose name begins with the hydrologic unit 

code and ends in either “_24shp,” “ _ff,” or “_ws.” The 

folder with the “_24shp” suffix contains the 1:24,000 

scale NHD for Vermont projected into Vermont State 

Plane coordinates and converted to the ESRI ArcView 

shapefile format using the NHD Arc2shape extension. 

The naming convention of the subdirectories in the 

“_24shp” directory are generic to the NHD and are 

explained in the helpfiles of NHD extensions. The 

instructions for the NHD Watershed extension explain 

the processing steps in detail (Steeves, 2002). The 

NHD for Vermont and the processing instructions can 

be downloaded from the NHD Web site, 

http://nhd.usgs.gov/.

The folders with the “_ff” suffix (fig. 1) contain 

the data for computing basin characteristics in a 

hydrologic unit. The hydrologic unit is identified by the 

prefix of the folder name. Each of these “_ff” folders 

contain three grids projected into Vermont State Plane 

coordinates. The first grid dataset is the elevation grid 

clipped to the respective hydrologic unit and named 

“dem_grid.” The source of the grid is the National 

Elevation Dataset (NED) (U.S. Geological Survey, 

2001b). The second grid, “grid_1200ft,” is for 

computing drainage basin areas greater than 1,200 ft 

altitude. This grid was computed from the NED with 

cells greater than 1,200 ft having a value of 2, and cells 

at or less than 1,200 ft having a value of 1. The final 

grid in the “_ff” folder, “precip_grd,” contains the 

mean-annual-precipitation data. The Vermont FlowFrequency Tool only requires the precipitation grid for 

computing daily discharges exceeded 25, 50, and 

75 percent of the time. The precipitation data are not 

needed for estimating flood-flow frequency. The source 

of the precipitation grid is the PRISM dataset 

(Daly, 2000), which is available through Climate 
10 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

DataDirectoryName

01080105_24shp

openme.txt

metadata

nhd

nhdduu

nhdpt

tables

01080105_ff

dem_grid

grid_1200ft

info

precip_grd

01080105_ws

accum

dendrite

dir_grd

info

shed_cov

01080106_24shp

01080106_ff

01080106_ws

DataLocation.txt

Data in ESRI ARC/INFO grid format, an ESRI 

 ARC/INFO coverage and a folder containing a 

 shapefile use for delineating drainage-basin

 boundaries

EXPLANATION

 Folder

 ARC/INFO coverage or grid

 Text file

Data in ESRI ARC/INFO grid format needed 

 for computing basin characteristics

Folders containing National Hydrography Dataset 

 data converted to ArcView shapefiles

Text file used by the National Hydrography Dataset

 (NHD) Toolkit to facilitate locating the NHD data

Text file used by the Vermont Flow-Frequency 

 Tool to locate the data directories

Figure 1. Directory structure of the data files required by the Vermont Flow-Frequency Tool. Hydrologic unit codes 01080105 

and 01080106 are shown as examples.
Vermont Flow-Frequency Tool 11

Source at http://www.climatesource.com/. The PRISM 

data are projected into Vermont State Plane coordinates 

and resampled with bilinear interpolation to a 30-m cell 

resolution using the GIS ARC/INFO command 

RESAMPLE (Environmental Systems Research 

Institute, Inc., 1994). Obtaining, projecting, and 

processing the PRISM precipitation data covering 

Vermont is the responsibility of the user.

The PRISM precipitation data do not cover 

Canadian parts of basins shared between Vermont and 

Canada. Canadian mean-annual-precipitation data 

(Ghislain Jacques, Environment Quebec, written 

commun., January 17, 2002), therefore, were obtained 

for precipitation stations in areas adjacent to the 

northern Vermont border. A mean-annual-precipitation 

grid with cells at 30-m spacing covering the areas north 

of the Vermont border was interpolated from this pointprecipitation data and merged with the PRISM 

precipitation data along the northern Vermont border.

The folders with the “_ws” suffix (fig. 1) contain 

data required for delineating a drainage basin. Detailed 

instruction for developing the data in the “_ws” folders 

can be downloaded from the Applications section of 

the NHD Web site, http://nhd.usgs.gov/. Within the 

“_ws” folder is a folder named “dendrite,” which 

contains the dendritic stream network used in the 

selection of a point on a stream. This dendritic dataset 

is a copy of the NHD-reach route system downloaded 

from the NHD Web site, projected to Vermont State 

Plane coordinates and converted to ESRI ArcView 

shapefiles; however, this dendritic dataset has some 

stream reaches removed. The purpose of removing 

some of the reaches is to prevent users from selecting a 

stream reach in which the flow-frequency estimating 

equations do not apply. For example, artificial 

channels, stream braids, and the Connecticut River are 

all reaches where the equations do not apply and are 

removed from the dendrite dataset. The creation of the 

dendrite dataset is further explained in instructions for 

the NHD Watershed extension downloadable from the 

NHD Web site.

The other datasets in the “_ws” folder include 

the flow accumulation grid, “accum;” the flow 

direction grid, “dir_grd;” and a coverage of drainage 

basin catchments for each stream reach in the NHD, 

“shed_cov.” These datasets are processed from the 

NED (U.S. Geological Survey, 2001b). First, the 

ARC/INFO command TOPOGRID (Environmental 

Systems Research Institute, Inc., 1994) and the 

ARC/INFO script, “agree.aml,” which is available at 

the Web site http://www.ce.utexas.edu/prof/maidment/

gishydro/ferdi/, are applied to the NED to produce an 

elevation grid with a 10-m cell size that is 

hydrologically correct and horizontally matches the 

NHD network. Additional raster processing of the 

elevation grid is done by use of ARC/INFO to 

exaggerate basin boundaries taken from the Natural 

Resources Conservation Service (NRCS) 12-digit 

hydrologic unit coverage (Natural Resources 

Conservation Service, 1996) to enforce agreement of 

the elevation grid to these boundaries. Next, the 

ARC/INFO commands FILL, FLOWDIRECTION, 

and FLOWACCUMULATION are used to create the 

flow direction and accumulation grids. Last, the 

watershed catchment coverage, “shed_cov,” is created 

using the ARC/INFO command WATERSHED and 

each catchment is attributed with the corresponding 

NHD reach code, matching each catchment to a stream 

segment in the NHD. The creation of these three 

datasets is explained further in the instructions for the 

NHD Watershed extension downloadable from the 

NHD Web site.

The utilities folder, named Utilities, also is 

important to the functionality of the Vermont FlowFrequency Tool. The utilities folder contains some of 

the data that is displayed by the tool to make locating 

streams easier, such as roads and basin, town, and 

county boundaries. The utilities folder contains two 

shapefiles, “dams.shp” and “huc8.shp,” which are used 

by the tool to check for upstream flood control 

structures and to load hydrologic unit data, 

respectively. The program (vtreg.exe) and datasets for 

solving the regression equations and analyzing the 

prediction intervals also are located in this directory. 

Each user of the Vermont Flow-Frequency Tool must 

have a copy of the utilities directory because this 

directory is used as a temporary workspace for saving 

files created during intermediate steps in computing 

flow frequency. A copy of the required utilities folder 

can be downloaded from the Web site 

http://vt.water.usgs.gov/CurrentProjects/vtfloodfreq/in

dex.htm.

The last dataset used by the Vermont FlowFrequency Tool is the topographic map images in 

electronic format at 1:24,000 scale. These maps are 

known also as Digital Raster Graphs. The images will 

make the process of accurately locating streams simple. 

The topographical images can be obtained from the 

Vermont Center for Geographic Information 

(http://www.vcgi.org/).
12 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

Installation of the Vermont Flow-Frequency Tool

Once the required ESRI software is installed and 

the datasets and utilities are loaded in the appropriate 

directory structure, and the Vermont Flow-Frequency 

Tool extension has been copied to the ESRI ArcView 

extensions directory, the Vermont Flow-Frequency 

Tool is ready for use. First, the ESRI ArcView software 

needs to be executed and the ESRI ArcView Spatial 

Analyst extension loaded and set as default. To load an 

extension, select the File menu in the ArcView window 

and select Extensions. A window of available 

extensions will open for the user to choose from. With 

the ESRI ArcView Spatial Analyst extension loaded, 

open the extension window again and load the 

extensions titled “NHD Load/Unload Workspace,” 

“NHD Navigate,” and “Vermont FF Tool.”

After the Vermont Flow-Frequency Tool 

extension is loaded, a menu named “VTSETUP” will 

appear in the project window. This set-up routine 

should be executed when the extension is first loaded 

into each new ESRI ArcView project file to identify the 

locations of required data and utilities folder and create 

the map of Vermont in the display window. Once the 

set-up routine is run, it will not be needed again in the 

ESRI ArcView project file and the “VTSETUP” menu 

will disappear. Saving and using only this project file 

will eliminate the need to run the set-up routine again.

The set-up routine first prompts the user for the 

location of the utilities folder. The user must browse for 

the “utilitiesLocation.txt” file located in the user’s 

utilities directory. The location of the data folders is 

requested in a similar manner, but the user must browse 

for the “dataLocation.txt” file (fig. 1) located in the 

directory that contains the data folders. The location of 

the topographic map database file (topo.dbf) also is 

requested; however, including the topographical maps 

is not required and search for this file can be canceled. 

The set up for the Vermont Flow-Frequency Tool is 

complete and the routine then opens a window 

displaying a map of Vermont. The Vermont FlowFrequency Tool menu and buttons are now displayed in 

the ESRI ArcView menubar and toolbar, respectively. 

The ESRI ArcView window may need to be enlarged to 

display the entire menubar and toolbar. An example of 

an ESRI ArcView project window with the Vermont 

Flow-Frequency Tool loaded and set up is shown in 

figure 2.

Using the Vermont Flow-Frequency Tool

This section of the report includes instructions 

for using the Vermont Flow-Frequency Tool. To be 

successful, the user must be familiar with the basic 

functions of ESRI ArcView such as pan and zoom, 

making themes active, interactively identifying 

features, and turning on and off the display themes, 

such as the topographical maps.

With the set up complete, the Vermont FlowFrequency Tool is ready for use. There are five buttons 

on the toolbar that run the major functions of the tool. 

These buttons are displayed in table 5. The first button, 

the load hydrologic unit button , must be used to 

load the appropriate hydrography data and locate the 

appropriate support files for computing basin 

characteristics before a basin can be delineated. With 

the load hydrologic unit button depressed, the cursor is 

used to select the hydrologic unit where the stream of 

interest is located. The hydrologic unit boundaries are 

outlined in green by default (fig. 2).

With a hydrologic unit selected, the Vermont 

Flow-Frequency Tool button can be depressed 

and a point on a stream selected. For a point to be 

selected on a stream, the “HUC_24shp – Reach” 

theme, where HUC is the hydrologic unit code, must be 

active. The basin above this point is delineated by 

implementing the functionality of the NHD. The basin 

characteristics upstream of this point and the flowfrequency estimates for the selected point are 

automatically computed and displayed as shown in the 

example in figure 3. The tabular results are displayed 

and can be saved to disk using Notepad.

The Vermont Flow-Frequency Tool offers an 

option for computing channel characteristics, such as 

slope, for a drainage basin already delineated with the 

Flow-Frequency Tool. With the Channel 

Characteristics button depressed, the user must 

reselect the point from which the currently displayed 

drainage basin was delineated. The tool uses the NHD 

to determine the main channel. It then locates the 

upstream end of the main channel, delineates the 

catchment above this point, and extends the channel to 

the point that provides the longest flow length between 

the draingage divide and the upstream end of the 

channel identified on a USGS topographical map. 

Results are output in tabular form to a Notepad 

window. The tables include the elevation at the 

downstream end of the main channel, the elevation at 

the upstream end of the main channel, the elevation at 
Vermont Flow-Frequency Tool 13

Table 5. Buttons that run the functions of the Vermont Flow-Frequency Tool

[Buttons are shown in figures 2 and 3; NHD, National Hydrography Data Set]

Button Name Function

Load hydrologic unit Loads NHD data and locates support data sets for the hydrologic unit interactively selected.

Flow-frequency tool Delineates a drainage basin by use of NHD for a point on a stream interactively selected, 

computes the basin characteristics, and solves the regression equations.

Channel characteristics Estimates selected channel characteristics.

Elevation query Displays the elevation for a point interactively selected.

Optional flow-frequency tool Delineates a drainage basin by use of the flow direction and accumulation grids for a point 

interactively selected, computes the basin characteristics, and solves the regression 

equations.

Vermont Flow-Frequency 

 Tool menu

Stream-gaging

 station

White River Basin 

 data loaded

Hydrologic unit 

 boundaries

Vermont Flow-Frequency 

 Tool buttons

Figure 2. ArcView window with the Vermont Flow-Frequency Tool installed.
14 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

the upstream end of the main channel extended to the 

drainage divide, the elevation of points 10 and 

85 percent up the main channel from the selected 

location to the drainage divide, the length of the main 

channel, the length of the main channel extended to the 

drainage divide, and the slope of the main channel 

between the 10 and 85 percent points described above.

If the user is not satisfied with the Vermont 

Flow-Frequency Tool attempt to extended the channel 

to the drainage divide, a manual option is available. By 

depressing and holding down the Channel 

Characteristics button, an optional button appears 

for selection. With the optional Channel Characteristics 

button selected, a point along the inside of the drainage 

divide where the main channel should be extended to 

can be selected. The Vermont Flow-Frequency Tool 

determines the new extended channel using the flow 

direction grid, computes the channel characteristics, 

and outputs the results to a new Notepad window.

The Elevation Query button is for 

identifying elevations on a map. With the Elevation 

Query button depressed, the cursor is used to select a 

point on a map for which the elevation will be 

displayed in feet.

The Optional Flow-Frequency Tool 

functions in the same manner as the FlowFrequency Tool, except the basin delineation is done 

with a flow-direction and flow-accumulation grid. A 

flow-direction grid is a grid in which each cell contains 

information on the direction of flow at that cell. A 

flow-accumulation grid is a grid in which each cell 

contains the number of cells that flow into it. Basin 

delineation with these grids takes more time and 

computer resources than basin delineation using the 

NHD approach. Attempting to delineate large basins, 

such as those greater than a few hundred square miles 

with the optional Flow-Frequency Tool may exceed the 

capabilities of a personal computer.

Selected point on stream

Delineated drainage-basin 

 boundary

Tabular output

Figure 3. ArcView window displayed when a point on a stream is selected.
Additional Utilities of Vermont Flow-Frequency Tool 15

Two technical difficulties are known to exist in 

the tool. The first is related to braided streams. User’s 

should review the boundary across the braided reach 

and confirm that the Vermont Flow-Frequency Tool 

delineated this part of the boundary correctly. It is best 

to avoid a braided reach by selecting a point of interest 

either upstream or downstream of the braid. The 

second problem can occur if the point chosen for 

delineation is at a confluence. For example, if a 

drainage basin above a confluence is desired and the 

point selected is too close to the confluence, both 

catchments related to the reaches entering the 

confluence may be delineated; however, only the 

headwater catchments above the reach that the selected 

point is on will be included in the total drainage area. 

To avoid the problem, the Vermont Flow-Frequency 

Tool requires selected points to be farther than 100 ft or 

about 30 m in the display’s scale upstream or 

downstream from the confluence.

ADDITIONAL UTILITIES OF VERMONT 

FLOW-FREQUENCY TOOL

Additional utilities are provided in the Vermont 

Flow-Frequency Tool. These include items such as 

adjusting the results with data from a nearby streamgaging station, adjusting the results for expected 

probability, and executing the National Flood 

Frequency program.

Vermont Flow-Frequency Tool Options Menu

A menu on the ESRI ArcView menubar titled 

“VTFF_Options” is visible when the Vermont FlowFrequency Tool extension is installed. This menu has 

13 selections. Of the 13 selections, three—”Adjust 

Results for Nearby Gage,” “Run NFF Program using 

English Units,” and “Run NFF Program using Metric 

Units”—are discussed in more detail in the following 

sections. Ten of the selections need only the following 

brief descriptions. The first two selections in this menu 

are the “Select Utilities Location” and “Select Data 

Location” menu options. These can be used if datasets 

or the utilities folder are moved and the user does not 

want to run the set-up routines again. When one of 

these options is selected, the user must browse for the 

respective “utilitiesLocation.txt” or “dataLocation.txt 

file,” as described in the previous section “Installation 

of the Vermont Flow-Frequency Tool.”

Two options under the “VTFF_Options” menu 

are used to locate features and move around on the 

displayed map of Vermont. The “Zoom to Vermont 

Map” option zooms in or out so that the window 

displays the entire State of Vermont. The “Locate 

Stream by Name” option opens a window with all the 

names of the streams in the currently loaded hydrologic 

unit. Once the user selects a stream, the map display 

zooms to that stream. Another option, “Watershed 

display properties,” allows the user to toggle the 

display of the watershed polygon from shaded to 

unshaded.

Options “Recompute Watershed Tool Results” 

and “Recompute Flow Accumulation Tool Results,” 

compute the basin characteristics and estimated flowfrequency characteristics of a basin polygon that is 

already displayed. These options are helpful if the 

delineated basin boundary identified by the Watershed 

Tool or the Flow Accumulation Tool must be manually 

edited. The user must be proficient in hydrology and 

with the ESRI ArcView editing tools before using these 

options. The option “Display Gage Information” 

provides streamflow statistics for selected streamgaging stations used for developing the flow-estimating 

equations. The station must first be selected with the 

ESRI ArcView Selection tool.

Finally, a menu item called “Clean Workspace” 

deletes many of the temporary work files that are left in 

the utilities directory. This option, however, does not 

delete files from the default ArcView Temp directory. 

A menu item called “About” provides a short 

explanation of the Vermont Flow-Frequency Tool.

Use of Regression Equations at or near StreamGaging Stations

A flow-frequency estimate from the regression 

equations can be improved with actual streamflow data 

if the estimate is made at or near a stream-gaging 

station. This is done by adjusting the regression 

equation results with the frequency curve computed 

from the streamflow data. A database of the streamflow 

data needed for this adjustment (through the 2000 

water year, which ends on September 30) is included in 

the Vermont Flow-Frequency Tool. This method may 

be particularly useful when a limited number of years 
16 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

of record at a gaged site are available for frequency 

analysis. The procedure recommended in Bulletin 17B 

of the U.S. Interagency Advisory Committee on Water 

Data (1982) is to compute a weighted average of the 

regression estimate and station estimate using the 

following equation:

, (17)

where

Flow frequency at sites that are not at but relatively 

near a stream-gaging station and on the same stream, 

can be calculated by combining the results of the 

regression equations and the nearby station data. This 

method is adapted from Sauer (1974) and is applicable 

to statistics of peak flow and flow duration. For sites 

with drainage area within 50 percent of the drainage 

area of the gaged site on the same stream, the following 

equations are used. For the gaged site, the ratio, R, is 

computed as 

. (18)

This ratio is linearly decayed to 1.0 at plus or 

minus 50 percent of the gaged drainage area. Thus, for 

the ungaged site a modified ratio, R', is computed as

, (19)

where

To calculate the adjusted discharge estimate at the 

ungaged site, R' is applied to the logarithmically 

transformed regression estimate using the following 

equation:

, (20)

where

After a flow-frequency estimate had been made for a 

point on a stream with the Vermont Flow-Frequency 

Tool, the above computations (equations 17–20) can be 

made to adjust for a nearby station by selecting “Adjust 

Results for Nearby Gage” under the “VTFF_Options” 

menu. When this menu option is selected, a streamgaging station number can be entered or the application 

can search for potential stations. The adjusted flowfrequency estimates are output in table format to a new 

Notepad window.

Expected Probability Adjustment to Flood 

Discharges

There is error in all flood-frequency curves 

because of the limited years of record available for 

analysis. These errors are magnified when estimating 

low-frequency floods, such as a 100-year event, from a 

finite series of annual peaks as few as 10 years. 

Furthermore, the distribution of errors around the true 

flood-frequency characteristics is not symmetrical. For 

example, a 100-year flood is a flood that will be 

exceeded on average one time in a hundred years. This 

flood estimate could actually be exceeded 2, 3, 4, or 

more times in a hundred years, but it can never be 

exceeded fewer than zero times in a hundred years. 

Because the distribution is not symmetrical, the median 

and mean of the distribution are not the same.

Exceedence probabilities computed on the basis 

of guidelines in Bulletin 17B of the U.S. Interagency 

Advisory Committee on Water Data (1982) are 

median-probability estimates. Expected probability 

(Beard, 1960) conforms to the mean of the distribution 

and is defined as the average of all estimated true 

QT,w is the weighted discharge for flowfrequency characteristic T,

QT,s is flow-frequency characteristic T

computed from the streamflow data,

QT,r is the regression estimate for flowfrequency characteristic T,

N is the number of years of streamflow data 

used to compute QT,s, and

E is the equivalent years of record for QT,r.

 ∆A is the absolute value of the difference 

between the drainage area at the ungaged 

site and drainage area at the gaged site, and

Ag is the drainage area of the gaged site.

LogQT w,

QT s, ( ) N QT r, log + log ( ) E

N E + = --------------------------------------------------------------

R QT w, log QT r, = ⁄ log

R' R ∆A

0.5Ag

= – -------------( ) R – 1

QT,a is the adjusted discharge estimate of flowfrequency characteristic T at the ungaged 

site, and

QT,u is the regression estimate of flow-frequency 

characteristic T at the ungaged site.

QT a, 10

R' logQT u, ( )

=
Additional Utilities of Vermont Flow-Frequency Tool 17

probabilities for a specified flow frequency that might 

be made from successive samples of a specified size 

(U.S. Interagency Advisory Committee on Water Data, 

1982). Thomas (1976) documented that the meanprobability estimate exceeds the median-probability 

estimate and the median-probability estimate 

approaches the true probability of a time series as the 

length of the streamflow record increases.

There are limitations and advantages in the use 

of median- and mean-probability estimates, but the use 

of the flood-frequency estimate will define which is 

appropriate. For example, when a user wants the result 

to have an equal likelihood of being too high or too 

low, the median probability estimate is appropriate. If 

the average number of years likely to experience 

various flood discharges over a number of sites are 

desired, the mean probability (expected probability) 

estimate is appropriate (Thomas, 1976). According to 

the U.S. Interagency Advisory Committee on Water 

Data (1982), expected probability is most often used in 

estimates of annual flood damages and in establishing 

design flood criteria.

Hardison and Jennings (1972) provide a method 

for estimating expected probabilities at ungaged sites. 

This method involves the use of a table in their report 

that converts exceedence probability to expected 

probability for selected record lengths (Hardison and 

Jennings, 1972, table 3, p. 422). For a normal 

distribution, the expected probabilities in their table 

can be approximated by the use of the following 

equation:

, (21)

where

Thus, expected probability discharges can be computed 

as follows:

Let

Then, the expected probability frequency factor, K'p, is 

estimated as

, (22)

and the logarithm of the expected probability 

discharge, qe,T, is estimated by

. (23)

The expected probability discharge, Qe,T, is 

computed by raising 10 to the power qe,T. The software 

attached to this report automatically computes the 

expected-probability discharge for each estimated 

peak-flow frequency.

National Flood Frequency Program

The National Flood-Frequency (NFF) program 

(Ries and Crouse, 2002) was developed as a 

compilation of regression equations used across the 

United States for estimating flood frequency. In 

addition to solving regression equations, the program 

also contains options for plotting flood-flow-frequency 

curves and synthetic hydrographs, computing estimates 

of flood-flow frequency for urban basins, and 

displaying results in English or metric units. The NFF 

program and these options can be accessed through the 

menu options titled “Run NFF Program using English 

Units” and “Run NFF Program using Metric Units” 

under the “VTFF_Options” menu of the Vermont 

Flow-Frequency Tool. Some of the highlights of the 

 n is the number of year of annual peaks,

Kp is the standard normal frequency factor for 

exceedence probability p, and 

tn-1 is the Student’s t-statistic with n – 1 degrees 

of freedom. 

Prob tn – 1 Kp[ ] n n ⁄ ( ) + 0.25 0.5 { } >

qT represent the base 10 logarithm of the 

estimated T-year flood from the 

regression,

Ne represent the computed equivalent years 

for the estimate rounded up to nearest 

integer, and

tNe-1,p the Student’s t-statistics for Ne – 1 

degrees of freedom and probability, 

p = 1/T. 

K'

p tNe – 1, p Ne ( ) + 0.25 Ne = [ ] ⁄

qe T, qT q2 ( ) – K'

p Kp ( ) ⁄ q = + 2
18 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

NFF program utilities are listed in the following 

paragraphs. Instructions and a complete description of 

the NFF program’s functions can be found in the help 

files.

1. The NFF program contains equations for 

estimating the magnitude and recurrence intervals 

for floods in urbanized areas based on Sauer and 

others (1983). The equations, which are 

considered valid for use throughout the United 

States, have seven input parameters—drainage 

area, channel slope, the 2-hour 2-year rainfall, 

basin storage, percent impervious surface, a basindevelopment factor, and the discharge at the 

respective recurrence interval of an equivalent 

rural basin. The Vermont Flow-Frequency Tool 

does not determine the additional basin 

characteristics required for computing estimates of 

urban flood frequency.

2. The NFF program also contains routines for 

plotting the frequency curve and a synthetic 

hydrograph. For the synthetic hydrograph, the user 

is required to input the basin lag time—the elapsed 

time, in hours, from the center of the mass of 

rainfall excess to the peak of the hydrograph. The 

NFF program then computes the synthetic 

hydrograph using a dimensionless hydrograph 

method developed by Stricker and Sauer (1982).

The NFF program is not required by Vermont 

Flow-Frequency Tool. The Vermont Flow-Frequency 

Tool writes a file called “current.nff” to the Utilities 

folder (see section titled “Data Requirements” for 

description of the Utilities folder), which the NFF 

program will read when executed through the 

“VTFF_Options” menu. The NFF program can be 

downloaded from 

http://water.usgs.gov/software/nff.html

SUMMARY

This report describes a tool that has been 

developed by the U.S. Geological Survey, in 

cooperation with the Vermont Agency of 

Transportation, for applying regression equations that 

estimate the peak-flow frequency for recurrence 

intervals of 2, 5, 10, 25, 50, 100, and 500 years and 

flow duration for daily discharges exceeded 25, 50, and 

75 percent of the time for ungaged, unregulated 

streams in Vermont. The tool automates the delineation 

of basin characteristics required by the equations and 

solves the equations, significantly reducing the 

processing time normally required when using 

regression equations to determine flow-frequency 

manually.

The customized geographical information 

system (GIS) application, Vermont Flow-Frequency 

Tool, is adapted from the National Hydrography 

Dataset toolkit. The functionality of the National 

Hydrography Dataset provides the basin boundary 

delineation. Additional programming capabilities 

included in the Vermont Flow-Frequency Tool evaluate 

basin characteristics, solve the regression equations, 

and compute the accuracy of the results and an 

expected probability adjustment to the streamflow 

estimates.

Several options are made available in the 

Vermont Flow-Frequency Tool. One option in the tool 

allows the recomputation of flow-frequency 

characteristics adjusted for data from a nearby streamgaging station that may be on the same stream. Another 

option provides a link to the utilities of the National 

Flood-Frequency Program. The National FloodFrequency program utilities include plotting of 

frequency curves, plotting of synthetic hydrographs, 

and computing peak-flow-frequency estimates for 

urban drainage basins.

SELECTED REFERENCES

Beard, L.R., 1960, Probability estimates based on small 

normal-distribution samples: Journal of Geophysical 

Research, v. 65, no. 7, p. 2, 143-2, 148.

Benson, M.A., 1962, Factors influencing the occurrence of 

floods in a humid region of diverse terrain: U.S. 

Geological Survey Water-Supply Paper 1580-B, 64 p.

Daly, Christopher, 2000, PRISM (Parameter-elevation 

Regressions on Independent Slopes Model), U.S. 

Department of Agriculture and Natural Resources 

Conservation Service Climate Mapping Project: 

National Water and Climate Center, Oregon State 

University Partnership.

Environmental Systems Research Institute, Inc. (ESRI), 

1994, ARC/INFO users guides, version 7: Redlands, 

Calif.

———1996a, ArcView GIS: The geographic information 

system for everyone: Redlands, Calif., 350 p.

———1996b, Avenue: Customization and application 

development for ArcView: Redlands, Calif., 259 p.
Selected References 19

Grossman, S.I., 1986, Multivariable calculus, linear algebra, 

and differential equations, Second Edition: New York, 

Academic Press, Inc., 876 p.

Harbeck, G.E., 1966, Reservoirs in the United States: 

U.S. Geological Survey Water-Supply Paper 1838, 

115 p.

Hardison, C.H., 1971, Prediction error of regression 

estimates of streamflow characteristics at ungaged sites: 

U.S. Geological Survey Professional Paper 750-C, 

p. 228-236.

Hardison, C.H., and Jennings, M.E., 1972, Bias in computed 

flood risk, Proceedings: Journal of the Hydraulics 

Division, American Society of Civil Engineers, v. 98, 

no. HY3, p. 415-427.

Harvey, C.A., and Eash, D.A., 1995, Description, 

instructions, and verification for Basinsoft, a computer 

program to quantify drainage-basin characteristics: 

U.S. Geological Survey Water-Resources Investigations 

Report 95-4287, 25 p.

Helsel, D.R., and Hirsch, R.M., 1992, Statistical methods in 

water resources: New York, Elsevier Science 

Publishing Co., 522 p.

Hodge, S.A., and Tasker, G.D., 1995, Magnitude and 

frequency of floods in Arkansas: U.S. Geological 

Survey Water-Resources Investigations Report 95-

4224, p. 37-39.

Jennings, M.E., Thomas, W.O., Jr., and Riggs, H.C., 1994, 

Regional regression equations for estimating magnitude 

and frequency of floods for ungaged sites, 1993: 

U.S. Geological Survey Water-Resources Investigations 

Report 94-4002, 189 p.

Johnson, C.G., and Tasker, G.D., 1974, Progress report on 

flood magnitude and frequency of Vermont streams: 

U.S. Geological Survey Open-File Report 74-130, 37 p.

Maidment, D.R., 1992, Handbook of hydrology: New York, 

McGraw-Hill, Inc., 29 chap.

Natural Resources Canada, 2002, National Topographic 

Data Base, 1:50:000-scale, accessed May 13, 2002, at 

URL<http://www.cits.rncan.gc.ca/.

Natural Resources Conservation Service, 1996, Watershed 

boundaries of Vermont based on U.S. hydrologic units, 

accessed December 21, 2001, at URL < 

http://www.vcgi.org/pub/dataware/gisdata/layers_vcgi/

WaterHydro_WATSHED.exe

Olson, S.A., 2002, Flow-frequency characteristics of 

Vermont streams: U.S. Geological Survey WaterResources Investigations Report 02-4238, 47 p.

Randall, A.D., 1996, Mean annual runoff, precipitation, and 

evapotranspiration in the glaciated northeastern United 

States, 1951-80: U.S. Geological Survey Open-File 

Report 96-395, 2 plates, scale 1:1,000,000.

Ries, K.G., III, and Crouse, M.Y., 2002, The National Flood 

Frequency Program, version 3: A computer program for 

estimating magnitude and frequency of floods for 

ungaged sites: U.S. Geological Survey WaterResources Investigations Report 02-4168, 41 p.

Riggs, H.C., 1968a, Some statistical tools in hydrology: 

U.S. Geological Survey Techniques of Water-Resources 

Investigations, book 4, chap. A1, 39 p.

———1968b, Frequency curves: U.S. Geological Survey 

Techniques of Water-Resources Investigations, book 4, 

chap. A2, 15 p.

———1968c, Regional analyses of streamflow 

characteristics: U.S. Geological Survey Techniques of 

Water-Resources Investigations, book 4, chap. B3, 15 p.

SAS Institute, Inc., 1990, SAS/STAT User’s guide (vers. 6, 

4th ed.): Cary, N.C., v. 1 and v. 2, 1,686 p.

Sauer, V.B., 1974, Flood characteristics of Oklahoma 

streams: U.S. Geological Survey Water-Resources 

Investigations Report 52-73, 301 p.

Sauer, V.B., Thomas, W.O., Jr., Stricker, V.A., and 

Wilson, K.V., 1983, Flood characteristics of urban 

watersheds in the United States: U.S. Geological 

Survey Water-Supply Paper 2207, 63 p.

Stedinger, J.R., and Tasker, G.D., 1985, Regional hydrologic 

analysis 1. Ordinary, weighted, and generalized least 

squares compared: Water Resources Research, v. 21, 

no. 9, p. 1,421-1,432.

Steeves, P.A., 2002, NHD Applications: NHD Watershed, 

accessed September 2002, at URL 

<http://nhd.usgs.gov/app_page.htm

Stricker, V.A., and Sauer, V.B., 1982, Techniques for 

estimating flood hydrographs for ungaged urban 

watersheds: U.S. Geological Survey Open-File Report 

82-365, 24 p.

Tasker, G.D., 1986, Generating efficient gaging plans for 

regional information, in Moss, M.E., ed., Integrated 

design of hydrologic networks, Budapest Symposium, 

July 1986, Proceedings: International Association of 

Hydrologic Sciences, Publication 158, p. 269-281.

———1987, Theory and application of generalized least 

squares, in Medina, K.D., Analysis of surface-water 

data network in Kansas for effectiveness in providing 

regional streamflow information: U.S. Geological 

Survey Water-Supply Paper 2303, p. 24-26.

Tasker, G.D., and Stedinger, J.R., 1989, An operational GLS 

model for hydrologic regression: Journal of Hydrology, 

v. 3, p. 361-375.

Thomas, D.M., 1976, Flood frequency: Expected and 

unexpected probabilities: U.S. Geological Survey 

Open-File Report 76-775, 7 p.
20 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

U.S. Geological Survey, 2001a, STATSGO soil 

characteristics for the conterminous United States: 

U.S. Geological Survey Open-File Report 656, 

accessed December 21, 2001, at URL <http:// 

water.usgs.gov/GIS/metadata/usgswrd/muid.html

———2001b, National elevation dataset, accessed 

December 21, 2001, at URL 

<http://edcnts12.cr.usgs.gov/ned/.

U.S. Geological Survey and U.S. Environment Protection 

Agency, 2002, National hydrography dataset, accessed 

June 18, 2002, at URL <http://nhd.usgs.gov/.

U.S. Interagency Advisory Committee on Water Data, 1982, 

Guidelines for determining flood-flow frequency, 

Bulletin 17-B of the Hydrology Subcommittee: 

Reston, Va., U.S. Geological Survey, Office of Water 

Data Coordination, 183 p.
APPENDIX 1. USE OF REGRESSION EQUATIONS 

WITH METRIC UNITS
22 A Geographic Information System Tool to Solve Regression Equations and Estimate Flow-Frequency Characteristics of Vermont Streams

APPENDIX 1. USE OF REGRESSION 

EQUATIONS WITH METRIC UNITS

The equations used to estimate peak flow at 

recurrence intervals of 2, 5, 10, 25, 50, 100, and 

500 years and the magnitude of daily-mean discharges 

exceeded 25, 50, and 75 percent of the time when using 

metric units for drainage area and precipitation are

, (1)

, (2)

, (3)

, (4)

, (5)

, (6)

, (7)

, (8)

, and (9)

. (10)

where

Qn is the calculated peak flow for recurrence 

interval n, in cubic meters per second;

Dn is the estimated daily discharge exceeded n

percent of the time, in cubic meters per 

second;

Q2 0.452A0.914L–0.294E0.0776Y–0.180 =

Q5 0.742A0.902L–0.295E0.0835Y–0.253 =

Q10 0.962A0.897L–0.302E0.0890Y–0.298 =

Q25 1.30A0.883L–0.316E0.104Y–0.349 =

Q50 1.59A0.874L–0.327E0.115Y–0.385 =

Q100 1.90A0.865L–0.336E0.125Y–0.420 =

Q500 2.75A0.846L–0.355E0.148Y–0.497 =

D75 4.28 –8 ( ) ×10 A

1.08

P1.55E0.101 =

D50 9.73 –8×10 A

1.04

P1.58E0.0603 =

D25 3.08 –7 ( ) ×10 A

1.01

P1.55E0.0438 =

A is the drainage area of the watershed 

delineated using a GIS (boundaries for the 

drainage areas from a 12-digit hydrologic 

unit coverage (Natural Resources 

Conservation Service, 1996) or digitized 

when a watershed’s boundary was not 

defined by this GIS coverage), in square 

kilometers;

L is the area of lakes and ponds in a watershed 

as a percent of drainage area, plus 1 percent, 

determined using a GIS from the 1:24,000 

scale National Hydrography Dataset (U.S. 

Geological Survey and U.S. Environmental 

Protection Agency, 2002) using all features 

in the dataset that were lake or pond 

boundaries including ponds as small as 

several hundreths of an acre and pooled 

areas of streams; 

E is the percent of the drainage basin that is at 

or greater than 365.7 meters (1,200 ft) 

altitude, plus 1 percent, computed using a 

GIS from the National Elevation Dataset 

(U.S. Geological Survey, 2001);

Y is the northing of the centroid of the 

drainage basin determined with a GIS, in 

the Vermont State Plane Coordinate System, 

divided by 100,000, then increased by 1.0; 

and

P is the basinwide mean of the mean-annual 

precipitation, in millimeters, determined 

using a GIS and the PRISM dataset (Daly, 

2000) resampled with bilinear interpolation 

to a 30-meter-cell resolution using the ESRI 

ARC/INFO RESAMPLE command 

(Environmental Systems Research Institute, 

Inc., 1994). Canadian precipitation data 

(Ghislain Jacques, Environment Quebec, 

written commun., January 17, 2002) was 

merged with the PRISM precipitation data 

where needed along the northern Vermont 

border.
Selected References 23

SELECTED REFERENCES

Daly, Christopher, 2000, PRISM (Parameter-elevation 

Regressions on Independent Slopes Model): 

U.S. Department of Agriculture and Natural Resources 

Conservation Service Climate Mapping Project, 

National Water and Climate Center, Oregon State 

University Partnership.

Environmental Systems Research Institute, Inc. (ESRI), 

1994, ARC/INFO users guides, version 7: Redlands, 

Calif.

Natural Resources Conservation Service, 1996, Watershed 

boundaries of Vermont based on U.S. hydrologic units, 

accessed December 21, 2001, at URL 

http://www,vcgi.org/pub/dataware/gisdata/layers_vcgi/

WaterHydro_WATSHED.exe.

U.S. Geological Survey, 2001, National elevation dataset, 

accessed December 21, 2001, at URL 

http://edcnts12.cr.usgs.gov/ned/.

U.S. Geological Survey and U.S. Environmental Protection 

Agency, 2002, National hydrography dataset, accessed 

June 18, 2002, at URL http://nhd.usgs.gov/.