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 Arkansas

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 Arkansas.

Overview

Arkansas is divided into four hydrologic regions (fig. 1) on the basis of drainage boundaries and physiography. Hodge 

and Tasker (1995) developed regression 

equations for estimating peak discharges 

(QT), in cubic meters per second, that have 

recurrence intervals (T) that range from 2 

to 500 years for ungaged, unregulated, Figure 1. Hydrologic regions for Arkansas.

Digital base from U.S. Geological Survey 1:2,000,000 Albers projection.

Standard parallels: 29°30' and 45°30', Zone 15. Region boundary

EXPLANATION

A Region identification

rural streams with drainage areas of less 

than 7,770 square kilometers.

Procedure

The updated equations presented in 

this report are in metric units; however, the 

NFF Program will accept input and report 

results in either the inch-pound or the metric system of units. The explanatory watershed variables used in the regression 

equations are as follow:

Drainage area (A), in square kilometers, is 

the total area that contributes runoff 

upstream of the stream site of interest.

Main channel slope (S), in meters per kilometer, is the difference in elevations at 

points 10 and 85 percent of the distance 

along the main channel (measured 

upstream from the location of the desired 

peak discharge to the drainage divide) 

divided by the distance between the points.

Mean basin elevation (E), in meters above 

USGS Fact Sheet 128-97

December, 1998
sea level [National Geodetic Vertical 

Datum of 1929 (NGVD of 1929)], is the 

mean elevation of the drainage basin determined from topographic maps by the gridsampling method. The grid-sampling 

method averages the point value of the 

variable in question at approximately 50 

regularly spaced nodes across the drainage 

area.

Basin shape factor (SH), a dimensionless 

number, is the drainage area divided by the 

square of the main channel length.

The main channel length, in kilometers, is 

the distance between the gaged site and the 

basin divide (used in computation of the 

basin shape factor only).

The regression equations, the average 

standard errors of prediction, and the 

equivalent years of record are shown in 

table 1. The average standard errors of prediction are an average measure of the accuracy of the regression equations when 

estimating peak-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 ranges of the 

explanatory watershed variables over 

which the equations are applicable are 

shown in table 2. The standard error of 

prediction increases appreciably when any 

explanatory watershed variable is outside 

the quoted range of data used to develop 

the equations. 

The regression equations were developed by using peak-discharge data for 

streamflow-gaging stations with at least 10 

years of record collected at 189 locations 

in Arkansas and 15 locations in Missouri, 

Oklahoma, and Louisiana. 

Improving Estimates with 

Gaged Data

The U.S. Water Resources Council 

(1981, appendix 8) described, and Hodge 

and Tasker (1995) applied, weighting techniques to improve estimates of peak discharge at gaged locations. The improved 

estimates are obtained by combining the 

estimates derived from analysis of gage 

records with estimates derived by other 

means, including regression equations. 

The weights of the two independent estiTable 1. Flood-peak discharge regression equations and associated statistics for 

streams that drain rural areas in Arkansas (modified from Hodge and Tasker, 1995)

[QT, peak discharge for recurrence interval T, 2 to 500 years, in cubic meters per second; A, drainage area, in 

square kilometers; S, main channel slope, in meters per kilometer; E, mean basin elevation, in meters above 

sea level (NGVD of 1929); SH, basin shape factor] 

Table 2. Range of explanatory variables for which regression equations are applicable

[--, not applicable] 

Regression equation

Average standard 

error of prediction,

in percent

Equivalent

years of

record

Region A

Q2 = 1.88A0.795S0.113 42 4

Q5 = 3.26A0.802S0.157 32 7

Q10 = 4.31A0.804S0.178 31 10

Q25 = 5.74A0.804S0.197 32 13

Q50 = 6.89A0.808S0.209 34 14

Q100 = 8.05A0.803S0.210 37 14

Q500 = 10.9A0.803S0.241 45 13

Region B

Q2 = 0.041A0.745E0.927SH0.342 42 4

Q5 = 0.113A0.735E0.875SH0.396 39 5

Q10 = 0.182A0.729E0.854SH0.422 38 7

Q25 = 0.288A0.723E0.839SH0.446 38 9

Q50 = 0.377A0.718E0.832SH0.460 39 11

Q100 = 0.471A0.715E0.827SH0.472 40 12

Q500 = 0.714A0.708E0.823SH0.494 43 14

Region C

Q2 = 4.18A0.669 42 3

Q5 = 7.23A0.676 38 5

Q10 = 9.44A0.680 37 6

Q25 = 12.3A0.684 36 9

Q50 = 14.6A0.687 35 11

Q100 = 16.7A0.690 35 12

Q500 = 22.1A0.694 37 15

Region D

Q2 = 2.56A0.667S0.211SH0.208 33 3

Q5 = 4.16A0.688S0.289SH0.252 31 4

Q10 = 5.25A0.699S0.334SH0.270 33 4

Q25 = 6.67A0.710S0.385SH0.286 37 4

Q50 = 7.72A0.717S0.418SH0.295 41 4

Q100 = 8.76A0.723S0.448SH0.302 45 4

Q500 = 11.2A0.735S0.510SH0.316 56 4

Hydrologic

study

region

Drainage area,

in square 

kilometers

Main channel

slope,

in meters per 

kilometer

Mean basin

elevation,

in meters above 

sea level1

1NGVD of 1929

Basin shape 

factor

Region A 0.23–1,980 0.56–70.3 -- --

Region B 0.13–6,890 -- 43.3–381 0.058–0.642

Region C 0.21–5,280 -- -- --

Region D 0.52–4,530 0.086–7.73 -- 0.015–0.533
mates 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,

QT(S) is the estimate of QT 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).

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

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

computations, which differ slightly from 

the equation described by Hodge and 

Tasker (1995).

Ungaged Sites Near Gaged 

Sites on the Same Stream

Hodge and Tasker (1995) showed 

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

improve estimates of peak discharge for an 

ungaged site on the same stream that has a 

drainage that is between 50 and 150 percent of the drainage area of the gaged site. 

The regression estimate for the ungaged 

site is multiplied by an adjustment factor, 

which is computed as:

,

where

AF is the adjustment factor,

∆A is the difference in drainage area 

(Agaged -Aungaged), and

R is the ratio of the weighted peakdischarge estimate at the gaged 

site to the regression estimate at 

the ungaged site, QT(W)/QT(R).

The equations are used without adjustment 

where the drainage area of the ungaged site 

is not within 0.5 to 1.5 times the drainage 

area of the gaged site.

Sites in Transition Zones

When the drainage area of the site of 

interest is in more than one region, a 

weighted estimate of the peak discharge 

should be computed. The equations for the 

appropriate regions should be applied 

independently using basinwide estimates 

of the required explanatory variables as if 

the entire basin was in each region. 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.

Estimates of peak discharge (QT) 

derived from gaged records and the lengths 

of the gaged records (N) are summarized 

by Hodge and Tasker (1995). They also 

presented annual peak-flow data, and 

explanatory watershed variables for the 

204 streamflow-gaging stations used to 

develop the regression equations. Hodge 

and Tasker (1995) did not include data 

from large main stem stations on the Red, 

the Arkansas, the White, the Black, the St. 

Francis, the Mississippi, and the Ouachita 

Rivers in the regression analysis because 

of their unique, large river floodflow characteristics.

Region of Influences Technique

Hodge and Tasker (1995) included an 

additional regression analysis that is based 

on the region of influence regression technique. The region of influence regression 

technique generates an independent regression equation for a site by analyzing data 

from a set of sites with similar explanatory 

watershed characteristics. The sites 

included in the set are not necessarily in 

the same hydrologic region as the site in 

question. On the basis of a root-meansquare error comparison, Hodge and 

Tasker (1995) concluded that the independent regression equations based on the 

region of influence regression technique 

outperforms the regional regression equations in regions A, B, and C; the regional 

regression equations are more accurate in 

region D. Hodge and Tasker (1995) stated, 

“The region of influence method is still 

being improved and is to be considered 

only as a second alternative to the regional 

regression equations.”

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

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, 52 p.

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.

U.S. Water Resources Council, 1981, 

Guidelines for determining flood flow 

frequency: U.S. Water Resources 

Council Bulletin 17B, 28p. 14 appendixes.

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.

QT log ( ) W

N QT ⋅ log ( ) S EQ QT + ⋅ log ( ) R

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

AF R ∆A R( ) – 1

0.5Agaged

= – -------------------------