U.S. Department of the Interior

U.S. Geological Survey

The National Flood-Frequency Program—Methods for Estimating 

Flood Magnitude and Frequency in Rural Areas in Utah

Introduction

Estimates of the magnitude and frequency of flood-peak discharges and flood 

hydrographs are used for a variety of purposes, such as for the design of bridges, 

culverts, and flood-control structures; and 

for the management and regulation of 

flood plains. To provide simple methods of 

estimating flood-peak discharges, the U.S. 

Geological Survey (USGS) has developed 

and published equations for every State, 

the Commonwealth of Puerto Rico, American Samoa, and a number of metropolitan 

areas in the United States. In 1993, the 

USGS, in cooperation with the Federal 

Emergency Management Agency and the 

Federal Highway Administration, compiled all current USGS statewide and metropolitan area equations into a computer 

program, titled “The National Flood-Frequency (NFF) Program” (Jennings and 

others, 1994). 

Since 1993, new or updated equations 

have been developed by the USGS for various areas of the Nation. These new equations have been incorporated into an 

updated version of the NFF Program. 

Fact sheets that describe application 

of the updated NFF Program to various 

areas of the Nation are available. This fact 

sheet describes the application of the 

updated NFF Program to streams that drain 

rural areas in Utah.

Overview

The State of Utah is wholly located 

within a regional flood study area that 

encompasses the arid lands of the southwestern United States (Thomas and others, 

1997). The study area is divided into 16 

hydrologic flood regions, of which 7 

include portions of Utah (fig. 1). These 

regions were delineated on the basis of 

regional flood sources (snowmelt, summer 

thunderstorms, or cyclonic rainfall), elevation, and analysis of flood yields and residuals of preliminary regional flood114

37

38

39

40

41

42

113 112

110

111

EXPLANATION

INTERSTATE HIGHWAY

U.S. HIGHWAY

STATE HIGHWAY

BOUNDARY OF FLOOD

REGIONS

7 FLOOD-REGION NUMBER

7

6

4

3

9

8

Great

Salt

Lake

0

0 50 KILOMETERS

50 MILES

80

40

89

50

191

40

191

6

89

50

6

6

50

89

91

191

40

70

15

15

80

84 80

15 84

6

Figure 1. Hydrologic regions for Utah.

°

°

°

°

°

°

°

°°°

°

666

163 191

Green

River

Colorado

River

San Juan

River

USGS Fact Sheet 124–98

September, 1999
frequency relations. Within Utah, sites 

greater in elevation than a threshold that is 

based upon latitude (fig. 2) are considered 

to be in region 1. Sites located at elevations 

of 7,500 feet or less may belong to regions 

3, 4, 6, 7, 8 or 9 on the basis of geographic 

location (fig. 1). 

Thomas and others (1997) developed 

regression equations for estimating peak 

discharges (QT), in cubic feet per second, 

that have recurrence intervals that range 

from 2 to 100 years for ungaged, unregulated rural streams. The NFF Program provides estimates of the 500-year discharge 

on the basis of extrapolation. Although 

some sites with drainages greater than 200 

square miles were used to develop the 

equations, applications are best limited to 

200 square miles or less.

Procedure

The equations are based on the inchpound system of units, but the NFF Program will accept and report either the inchpound or metric system of units. The 

explanatory watershed variables used in the 

regression equations are as follows:

Drainage area (AREA), in square miles, is 

the total area that contributes runoff 

upstream of the location of the stream site 

of interest.

Mean annual precipitation (PREC), in 

inches, is the average mean annual precipitation for the basin as determined from isohyetal maps developed by the U.S. Weather 

Bureau (1963). The average is best determined by use of grid sampling techniques. 

Lines of equal precipitation from the 

Weather Bureau map are intersected with 

(drawn on to) a map of the drainage basin, 

a grid with equal-size cells is overlaid on 

the map, the mean annual precipitation is 

determined at each grid intersection, and 

the values are averaged.

Mean basin elevation (ELEV), in feet 

above sea level (National Vertical Datum of 

1929), is also determined by grid sampling 

techniques. The elevations of a minimum 

of 20 equally spaced points are determined, 

and the average of the points is taken. As 

many as 100 points may be needed for 

large basins.

The regression equations, the average 

standard errors of prediction, and the 

equivalent years of record for regions 1, 3, 

4, 7, 8, and 9 are given in table 1. The average standard errors of prediction are an 

average measure of the accuracy of the 

regression equations when estimating peakTable 1. Flood-peak discharge regression equations and associated statistics for regions 

1, 3, 4, 7, 8, and 9 in Utah (modified from Thomas and others, 1997)

[ , peak discharge for recurrence interval T, 2 to 100 years, in cubic feet per second; AREA, drainage area, in 

square miles; PREC, mean annual precipitation, in inches; ELEV, mean basin elevation, in feet above sea level 

(NGVD of 1929)] 

Regression equation

Average

standard error

of prediction,

in percent

Equivalent years 

of record

Region 1 (For sites located at elevations greater than elevation threshold from figure 2) 

-165 stations

Q2 = 0.124AREA0.845PREC1.44 59 0.16

Q5 = 0.629AREA0.807PREC1.12 52 .62

Q10 = 1.43AREA0.786PREC0.958 48 1.34

Q25 = 3.08AREA0.768PREC0.811 46 2.50

Q50 = 4.75AREA0.758PREC0.732 46 3.37

Q100 = 6.78AREA0.750PREC0.668 46 4.19

Region 3 -35 stations

Q2 = 0.444AREA0.649PREC1.15 86 0.29

Q5 = 1.21AREA0.639PREC0.995 83 .49

Q10 = 1.99AREA0.633PREC0.924 80 .77

Q25 = 3.37AREA0.627PREC0.849 78 1.23

Q50 = 4.70AREA0.625PREC0.802 77 1.57

Q100 = 6.42AREA0.621PREC0.757 78 1.92

Region 4 -108 stations

Q2 = 0.0405AREA0.701(ELEV/1,000)2.91 64 0.39

Q5 = 0.408AREA0.683(ELEV/1,000)2.05 57 .95

Q10 = 1.26AREA0.674(ELEV/1,000)1.64 53 1.76

Q25 = 3.74AREA0.667(ELEV/1,000)1.24 51 3.02

Q50 = 7.04AREA0.664(ELEV/1,000)1.02 52 3.89

Q100 = 11.8AREA0.662(ELEV/1,000)0.835 53 4.65

Region 7 -28 stations

Q2 = 0.0150AREA0.697(ELEV/1,000)3.16 56 0.25

Q5 = 0.306AREA0.590(ELEV/1,000)2.22 45 1.56

Q10 = 1.25AREA0.526(ELEV/1,000)1.83 45 3.07

Q25 = 122AREA0.440 49 4.60

Q50 = 183AREA0.390 53 5.27

Q100 = 264AREA0.344 59 5.68

Region 8 -108 stations

Q2 = 598AREA0.501(ELEV/1,000)-1.02 72 0.37

Q5 = 2,620AREA0.449(ELEV/1,000)-1.28 62 1.35

Q10 = 5,310AREA0.425(ELEV/1,000)-1.40 57 2.88

Q25 = 10,500AREA0.403(ELEV/1,000)-1.49 54 5.45

Q50 = 16,000AREA0.390(ELEV/1,000)-1.54 53 7.45

Q100 = 23,300AREA0.377(ELEV/1,000)-1.59 53 9.28

Region 9 -43 stations

Q2 = 0.0204AREA0.606(ELEV/1,000)3.5 68 0.14

Q5 = 0.181AREA0.515(ELEV/1,000)2.9 55 .77

Q10 = 1.18AREA0.488(ELEV/1,000)2.2 52 1.70

Q25 = 18.2AREA0.465(ELEV/1,000)1.1 53 2.81

Q50 = 248AREA0.449 57 3.36

Q100 = 292AREA0.444 59 3.94

QT
discharge values for ungaged watersheds 

similar to those that were used to derive the 

regression equations. The equivalent years 

of record is the number of years of streamflow record needed to achieve the same 

accuracy as the regression equation. 

The regression equation for region 6 

was developed using an iterative regression 

method (Hjalmarson and Thomas, 1992) 

and a modified form of the station year statistical analysis method (Fuller, 1914). The 

regression equation, the estimated average 

standard error of regression, and the equivalent years of record for region 6 are given 

in table 2. The average standard error of 

regression is an estimate of the predictive 

accuracy of these regression equations and 

is determined by a direct sampling method. 

The approximate ranges of the 

explanatory watershed variables over 

which the equations are applicable are 

shown in table 3. Thomas and others 

(1997) presented the actual ranges of applicability as two-dimensional clusters or 

clouds of explanatory variables plotted 

against one another. The ranges shown in 

table 3 define a rectangular space that 

brackets the clouds and, therefore, include 

pairs of values of the explanatory variables 

near the corners of the rectangle that are 

outside of the clouds. Application of the 

equations for values of the variables near 

the extremes of a range should be done cautiously. The standard errors increase appreciably when any explanatory watershed 

variable is near or outside the quoted range.

Improving Estimates with 

Gaged Data

The U.S. Water Resources Council 

(1981, appendix 8) described weighting 

techniques to improve estimates of peak 

discharge at gaged locations by combining 

the estimates derived from analysis of gage 

records with estimates derived by other 

means, including regression equations.

The weights for these two estimates 

are based on the length of the stream gage 

record (in years) and the equivalent years 

of record of the applicable regression equation. The weighted estimate of peak discharge is computed as:

where

QT(W) is the weighted estimate for recurrence interval T at the gaged site,

is the estimate of derived from 

analysis of the gage records,

QT(R) is the estimate of QT derived from 

application of the regression equation,

N is the number of years of stream 

gage record, and

EQ is the equivalent years of record 

(table 1 and 2).

The accuracy of the weighted discharge estimate, in equivalent years of 

record, is equal to N + EQ. The NFF Program contains the appropriate algorithms 

for this computation, which differs slightly 

from that described by Thomas and others 

(1997).

Sites Near Gaged Sites on the 

Same Stream

Thomas and others (1997) showed 

how the weighted estimate of peak discharge can be used to improve estimates of 

peak discharge of an ungaged site on the 

same stream that has a drainage area that is 

between 50 and 150 percent of the drainage 

area of the gaged site. The weighted estimate is computed as:

,

where

QT(u) is the weighted peak-discharge 

estimate for the recurrence interval T at the ungaged site,

QT(W) is the weighted estimate of peak 

discharge at the gaged site,

AREAungaged and AREAgaged are the drainage areas of the ungaged and 

gaged sites, respectively, and

b is an exponent for each region as 

follows:

Region Exponent

1 0.8

3 .7

4 .7

6 .6

7 .5

8 .4

9 .5

QT log ( ) W

N QT ⋅ log ( ) G EQ QT + ⋅ log ( ) R

N EQ + = ----------------------------------------------------------------------------

QT( ) G QT

QT( ) u QT( ) W

Areaungaged

Areagaged

-----------------------------     ⋅

b

=

Table 2. Flood-peak discharge equations and associated statistics for region 6 in Utah 

(modified from Thomas and others, 1997)

[QT, peak discharge for recurrence interval T, 2 to 100 years, in cubic feet per second; AREA, drainage area, in 

square miles; ELEV, mean basin elevation, in feet above sea level (NGVD of 1929)]

Estimated average standard error of regression for these equations includes much of the within-station residual 

variance and therefore is not comparable to standard error of estimate from an ordinary-least-squares regression. 

Table 3. Range of explanatory variables for which regression equations are applicable

[--, not applicable.]

Regression equation

Estimated average

standard error of

regression,

in log units

Equivalent

years of record

Region 6 -80 stations

Q2 = 0 -- --

Q5 = 32AREA0.80(ELEV/1,000)-0.66 1.47 0.233

Q10 = 590AREA0.62(ELEV/1,000)-1.6 1.12 .748

Q25 = 3,200AREA0.62(ELEV/1,000)-2.1 .796 2.52

Q50 = 5,300AREA0.64(ELEV/1,000)-2.1 1.10 1.75

Q100 = 20,000AREA0.51(ELEV/1,000)-2.3 1.84 .794

Hydrologic

study region

Drainage area,

in square miles1

1For best results, applications should be limited to basins of less than 200

square miles.

Mean basin

elevation,

in feet above

sea level2

2NGVD of 1929.

Mean annual

precipitation,

in inches

Region 1 0.30–1,0601 -- 10–45

Region 3 2.35-1,4501 -- 10-46

Region 4 2.08-1,2301 5,740-10,700 --

Region 6 .20-1,6701 -- --

Region 7 5.58-3401 6,070-9,560 --

Region 8 .20-1,9901 4,300-10,200 --

Region 9 4.00-1,1801 6,550-10,060 --
The adjustment to the weighted estimate of peak discharge at the gaged site can 

be used when the drainage area at the 

ungaged site is within 50 to 150 percent of 

the drainage area of the gaged site. Otherwise, the estimate at the ungaged site 

should be based on the appropriate regression equation only.

Sites in Transition Zones

When the drainage area of the site of 

interest is in more than one of the regions 3, 

4, 6, 7, 8, or 9, a weighted estimate of the 

peak discharge should be computed. The 

equations for the appropriate regions 

should be applied independently by using 

basinwide estimates of the required explanatory variables. The weighted estimate is 

then computed by multiplying each 

regional estimate against the fraction of the 

drainage area in that region and summing 

the products. The NFF Program provides 

an algorithm for this computation.

When the elevation of the stream site 

of interest is between 6,800 and 7,500 feet, 

a weighted estimate of the peak discharge 

should be computed by using the equations 

for region 1 and the other regions in which 

the basin is located. The applicable equations are each applied by using basinwide 

estimates of the required explanatory variables, and the region estimates are 

weighted as a function of elevation as follows:

QT( ) W QT( ) u 7 500 , – E

700 ------------------------

QT( ) Region1 1 7 500 , – E

700 – ------------------------     + ⋅

= ⋅

QT(W) is the weighted of peak-discharge 

estimate for the recurrence interval T at the site of interest,

QT(u) is the estimate of peak discharge 

using the equations for regions 3, 

4, 6, 7, 8 or 9 as appropriate,

QT(Region 1) is the estimate of the peak 

discharge using the equations for 

region 1, and

E is the elevation of the stream site 

of interest.

The NFF Program does not provide an 

algorithm for this weighting computation.

Thomas and others (1997) summarized the basin characteristics, the estimates 

of peak discharge, and the weighted estimates of peak discharge for most of the 

1,323 sites used in the study, including 212 

sites in Utah. 

—Prepared by Robert R. Mason, Jr., and 

Kernell G. Ries III, of the U.S. Geological 

Survey; and Jeffrey N. King and Wilbert O. 

Thomas, Jr., of Michael Baker, Jr., Inc.

References

Fuller, W.E., 1914, Flood flows: American 

Society of Civil Engineers Transactions, v. 77, no. 1293, p. 564-617.

Hjalmarson, H.W., and Thomas, B.E., 

1992, A new look at regional floodfrequency relations for arid lands: 

American Society of Civil Engineers, 

Journal of Hydraulic Engineering, v. 

118, no. 6, p. 868-886.

Jennings, M.E., Thomas, W.O., Jr., and 

Riggs, H.C., comps., 1994, Nationwide summary of U.S. Geological 

Survey regional regression equations 

for estimating magnitude and frequency of floods for ungaged sites, 

1993: U.S. Geological Survey WaterResources Investigations Report 

94–4002, 196 p.

Thomas, B.E., Hjalmarson, H.W., and 

Waltemeyer, S.D., 1997, Methods for 

estimating magnitude and frequency 

of floods in the southwestern United 

States: U.S. Geological Survey WaterSupply Paper 2433, 195p.

U.S. Water Resources Council, 1981, 

Guidelines for determining flood flow 

frequency: U.S. Water Resources 

Council Bulletin 17B, 28p. 14 appendixes.

U.S. Weather Bureau, 1963, Normal annual 

precipitation (1931-60) for the States 

of Arizona, Colorado, New Mexico, 

Utah: U.S. Department of Commerce, 

Weather Bureau Map, scale 

1:500,000.

For more information contact:

U.S. Geological Survey

Office of Surface Water

415 National Center

Reston, Virginia 20192

(703) 648-5301

USGS hydrologic analysis software is 

available for electronic retrieval through 

the World Wide Web (WWW) at 

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

and through anonymous File Transfer Protocol (FTP) from water.usgs.gov 

(directory: /pub/software). The WWW 

page and anonymous FTP directory from 

which the National Flood-Frequency software and user documentation can be 

retrieved are 

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

/pub/software/surface_water/nff, 

respectively.

Additional earth science information is 

available from the USGS through the 

WWW at http://www.usgs.gov/ 

or by calling 1-888-ASK-USGS.

ELEVATION ABOVE SEA LEVEL, IN FEET

10,000

9,000

8,000

7,000

6,000

5,000

28 30 32 34 36 38 40 42 44 46 

LATITUDE, IN DEGREES

BOUNDARY OF REGION 1

Figure 2. Estimated elevation threshold for region 1 (from Thomas and others, 1997)