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 Arizona

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, 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 Arizona.

Overview

The State of Arizona 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 Arizona (fig. 1). 

These regions were delineated on the basis 

of regional flood sources (snowmelt, sumFigure 1. Hydrologic flood regions for Arizona.

114° 37° 113° 112° 111° 110°

36°

35°

34°

33°

32°

EXPLANATION

INTERSTATE HIGHWAY

U.S. HIGHWAY

STATE HIGHWAY

BOUNDARY OF FLOOD

REGIONS

Gila Riv er

Sa

Gila

Riv er

Colo rado

Lit tle

Riv er

Riv er

Lake

Mead

Colo

ra do

Riv er

11 FLOOD-REGION NUMBER

11

8

12

10

10

13 14

Virgin River

0

0 50 KILOMETERS

50 MILES

10

8

10

19

17

40

40

93

15

89

163

80

191

160

180

191

95

60

93

89

89

70

60

60

10

89

LINE OF EQUAL FREE WATERSURFACE EVAPORATIONÑ

Interval 5 inches (Farnsworth

and others, 1982)

ALT

89

ALT

89

60

89

666

60

60

60

60 60

60

65

70

70

65

70

60

60

70

75

75

70

65

65

65

70

80

75

70

65

75

80

55

65

55

60

70

75

60

75

85

55

55

55

50

50

50

40 50

45

55

60

60

lt

rado Colo

Riv er

65

65

60

180

mer thunderstorms, or cyclonic rainfall), 

elevation, and analysis of flood yields and 

residuals of preliminary regional flood-frequency relations. Within Arizona, sites 

greater in elevation than 7,500 feet above 

sea level [National Geodetic Vertical 

Datum of 1929 (NGVD of 1929)] are considered to be in region 1. Sites located at 

elevations of 7,500 feet or less may belong 

to regions 8, 10, 11, 12, 13, or 14 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.

USGS Fact Sheet 111-98

January, 1999
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, 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.

Mean annual evaporation (EVAP), in 

inches, is the mean annual free water-surface evaporation at the study site. This variable should be estimated for the stream site 

of interest by linear interpolation between 

the lines of free surface-water evaporation 

shown in figure 1.

The regression equations, the average 

standard errors of prediction, and the 

equivalent years of record for regions 1, 8, 

12, 13, and 14 are given in table 1. The 

average standard errors of prediction are an 

average measure of the accuracy of the 

regression equations when estimating peakdischarge 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 equations for regions 

10 and 11 were 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 equations, 

the estimated average standard errors of 

regression, and the equivalent years of 

record for regions 10 and 11 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 

Table 1. Flood-peak discharge regression equations and associated statistics for regions 

1, 8, 12, 13, and 14 in Arizona (modified from Thomas and others, 1997)

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

in square miles; PREC, 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 7,500 feet above sea level1) -165 stations

1NGVD of 1929

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 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 12 -68 stations

Q2 = 41.1AREA0.629 105 0.23

Q5 = 238AREA0.687(ELEV/1,000)-0.358 68 1.90

Q10 = 479AREA0.661(ELEV/1,000)-0.398 52 6.24

Q25 = 942AREA0.630(ELEV/1,000)-0.383 40 17.8

Q50 = 10 (7.36 - 4.17AREA-0.08)

(ELEV/1,000)-0.440 37 27.5

Q100 = 10 (6.55 - 3.17AREA-0.11)

(ELEV/1,000)-0.454 39 32.1

Region 13 - 73 stations

Q2 = 10 (6.38 - 4.29AREA-0.06) 57 2.0

Q5 = 10 (5.78 - 3.31AREA-0.08) 40 6.25

Q10 = 10 (5.68 - 3.02AREA-0.09) 37 11.1

Q25 = 10 (5.64 - 2.78AREA-0.10) 39 15.0

Q50 = 10 (5.57 - 2.59AREA-0.11) 43 15.9

Q100 = 10 (5.52 - 2.42AREA-0.12) 48 16.1

Region 14- 22 stations

Q2 = 583AREA0.588(ELEV/1,000)-1.3 74 1.69

Q5 = 618AREA0.524(ELEV/1,000)-0.70 63 3.54

Q10 = 361AREA0.464 65 4.95

Q25 = 581AREA0.462 63 7.75

Q50 = 779AREA0.462 64 9.65

Q100 = 1,010AREA0.463 66 11.2

QT
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 from other 

means including regression equations.

The weights for these two estimates 

are based on the length of the 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 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).

QT log ( ) W

N QT ⋅ log ( ) G EQ QT + ⋅ log ( ) R

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

QT( ) G QT

Sites Near Gaged Sites on the 

Same Stream

Thomas and others (1997) showed 

how the weighted estimate of peak discharge at a gaged site can be used to estimate the 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:

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 

8, or 10-14, 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 estiQT( ) u QT( ) W

Areaungaged

Areagaged

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

b

=

Region Exponent

1 0.8

8 .4

10 .6

11 .6

12 .6

13 .5

14 .5

Table 2. Flood-peak discharge equations and associated statistics for regions 10 and 11 

in Arizona (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; EVAP, mean annual evaporation, in inches]

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. 

Regression equation

Estimated average

standard error of

regression,

in log units

Equivalent

years of record

Region 10 -104 stations

Q2 = 12 AREA0.58 1.14 0.618

Q5 = 85 AREA0.59 .602 3.13

Q10 = 200 AREA0.62 .675 3.45

Q25 = 400 AREA0.65 .949 2.49

Q50 = 590 AREA0.67 .928 3.22

Q100 = 850 AREA0.69 1.23 2.22

Region 11 -46 stations

Q2 = 26 AREA0.62 0.609 0.428

Q5 = 130 AREA0.56 .309 2.79

Q10 = 0.10 AREA0.52EVAP2.0 .296 4.63

Q25 = 0.17 AREA0.52EVAP2.0 .191 17.1

Q50 = 0.24 AREA0.54EVAP2.0 .294 9.20

Q100 = 0.27 AREA0.58EVAP2.0 .863 1.32
mate 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) is the weighted 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 8, or 10-14 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 259 sites in Arizona. 

Prepared by Robert R. Mason, Jr., of the 

U.S. Geological Survey; and Jeffrey N. 

King and Wilbert O. Thomas, Jr., of 

Michael Baker, Jr., Inc.

References

Farnsworth, R.K., Thompson, E.S., and 

Peak, E.L., 1982, Evaporation atlas 

QT( ) W QT( ) u 7 500 , – E

700 = ⋅ ------------------------

QT + )( ) Region1 1 7 500 , – E

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

for the contiguous 48 United States: 

National Oceanic and Atmospheric 

Administration Technical Report 

NWS 33, 26 p.

 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 

Water-Resources 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 Water-Supply 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 1-800-426-9000.

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

applicable

[--, not applicable.]

Hydrologic

study region

Drainage area,

in square miles1

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

Mean annual

evaporation,

in inches

Mean basin

elevation,

in feet above

sea level2

2

NGVD of 1929.

Mean annual

precipitation,

in inches

Region 1 0.6–1,061 -- -- 11–43

Region 8 1.0–1,990 -- 4,300–10,200 --

Region 10 0.1–1,000 -- -- --

Region 11 0.2–890 44.1–55.7 -- --

Region 12 0.6–1,520 -- 1,730–8,700 --

Region 13 0.1–1,780 -- -- --

Region 14 0.8–1,860 -- 3,350–8,950 --