XV

Flood-Frequency 

Analyses

Manual of Hydrology: Part 3. Flood-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1543-A
Flood-Frequency 

Analyses

By TATE DALRYMPLE

Manual of Hydrology: Part 3. Flood-Flow Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1543-A

Methods and practices 

of the Geological Survey

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S Government Printing Office 

Washington 25, D.C.
CONTENTS

Page

Abstract__ -___--_-_-__---___----_______________________________ 1

Introduction.___--------___-_-_----_-___-___-_____________________ 1

Flood frequency at a gaging station..________________________________ 5

Two kinds of flood series.______________________________________ 5

Annual floods.____________________________________________ 5

Partial-duration series____________________________________ 6

Relation between the two series.____________________________ 6

Flood-frequency curves_-__-----_______--______________-_____- 7

Discharge curves__-_-__------____-__---_____-______-__-___ 7

Stage curves._____________________________________________ 7

Flood-volume curves_-----_---_______-_----_____-_-________ 8

Listing of data._______________________________________________ 8

Annual-flood peaks._______________________________________ 8

Partial-duration series peaks-----_-_-_--------_____----_____ 10

Flood volumes-__-_---------________-_-_-_________________ 12

Historical data._______________________________________________ 13

Plotting positions._____________________________________________ 14

Methods.____----_____----_____-_-_-_____________________ 15

Plotting historical data___________________________________ 16

Plotting paper._______________________________________________ 19

Fitting frequency graphs.____---_-____-__-____-____-___-_-_____ 20

Single-station analysis________________________________________ 21

Regional flood frequency.__________________________________________ 25

Basic frequency curve__________________________________________ 27

Homogeneity test_______________________________________ 27

Mean annual flood___________________________________________ 29

Physiographic factors._____________________________________ 29

Meteorologic factors-_-----------_--_-------_-___-_-------- 30

Composite factors_________________________________________ 30

Computation procedure._______________________________________ 31

Selection of stations______________________________________ 31

Selection of base period.___________________________________ 32

Adjustment of records to base period.______________________ 34

Preliminary frequency curve. _______________________________ 37

The mean annual flood.____________________________________ 37

Homogeneity test_______________________________________ 38

Computation of median flood ratios..________________-_-_---_ 39

Definition of regional frequency curve__-------------_-------- 40

Estimation of mean annual flood. ___________________________ 41

Definition of a frequency curve-_______-------_-_---_-------- 45

Summary of procedure.______.________-____-_____-_-----_-_ 45

Special treatment for large streams._____________________----_--- 46

m
IV CONTENTS

Page

Plotting positions in frequency analysis, by W. B. Langbein. ___________ 48

Median ______________________________________________________ 49

Mean ________________________________________________________ 49

Mode __________-____--___________--_---__------------_-.______ 50

Characteristics of frequency curves based on a theoretical 1,000-year

record, by M. A. Benson________--_----__--------_-_--------_-_-_ 51

Base data____________________-____-__-____-------_----_-_____ 51

Mean annual flood____________________________________________ 54

Short periods of record.___________-_-____-______--------__-____ 54

Periods used____________________________---_-------_______ 54

Drawing of frequency curves..------------------------------ 54

Arithmetic versus graphical mean annual flood___----_--______ 56

Reliability of mean annual-flood values_____________________ 61

"Working range" of mean annual floods._____---------_______ 61

Reliability of flood magnitudes-__________--_---------_______ 63

Composite flood-frequency curves______________________________ 65

Periods combined____-_-_---------------------------------- 65

Results from composite graphs-__-____--______------_-______ 67

Effect of number of stations combined.---------------------- 67

Median versus average ratios-___________--_----------______ 71

Theoretical distribution of composite curves__________________ 71

Conclusions ___________________-_--______----___------------- 73

Selected references._____________________________--_-----------___-_ 74

Index ________________________-___-_---------_-_--------------_-_- 79

ILLUSTRATIONS

FIGURE 1. Flood data listing- _____________________________________ 9

2. Tabulation of flood-volume data.____---_---_------------ 13

3. Flood-frequency curve, rectangular scale__________________ 22

4. Flood-frequency curve, logarithmic scale __________________ 23

5. Partial-duration series flood-frequency curve.______________ 24

6. Annual flood volume-frequency curves__________-_--_-_-__ 25

7. Youghiogheny and Kiskiminetas River basins, showing

location of gaging stations.____________________________ 33

8. Bar graph showing period of record of maximum annual peaks

at gaging stations_____----_.________--__-------____-_ 34

9. Tabulation of flood data, Big Piney Run near Salisbury, Pa__ 35

10. Correlation of annual floods, Big Piney Run versus Stony

Creek.______________________________________________ 36

11. Preliminary frequency curve, Big Piney Run near Salisbury,

Pa_______.__________________________________________ 37

12. Homogeneity test chart-__-----_-__--------------------- 39

13. Regional frequency curve____-______-----_----_----_---_- 41

14. Variation of mean annual flood with drainage area________ 42

15. Determination of mean annual flood from a short record- _ _ _ 44

16. Flood-frequency curve for a 100-square-mile drainage basin__ 45

17. Variation of mean annual flood with distance along stream __ 46

18. Relation of flood frequencies to drainage area ______________ 47
CONTENTS V

Page

FIGURE 19. Theoretical frequency curve._____-_____-____-___---____- 53

20. Frequency curves for 10-year periods_____________________ 57

21. Frequency curves for 25-year periods.___-_--------------- 58

22. Frequency curves for 50-year periods _____________________ 59

23. Frequency curves for 100-year periods ____________________ 60

24. Recurrence intervals of 2.33-year flood-___________________ 62

25. Computing range for mean annual flood.__________________ 63

26. Recurrence intervals of 10-year flood._____________________ 66

27. Composite frequency curves based on ten 10-year periods__ 68

28. Composite frequency curves based on ten 25-year periods._ 69

29. Composite frequency curves based on ten 50-year periods..-. 70

30. Theoretical distribution of composite curves______--__-_._- 72
MANUAL OF HYDROLOGY: PART 3. FLOOD-FLOW TECHNIQUES

FLOOD-FREQUENCY ANALYSES

By TATE DALRYMPLE

ABSTRACT

This report describes the method used by the U.S. Geological Survey to determine the magnitude and frequency of momentary peak discharges at any place 

on a stream, whether a gaging-station record is available or not. The method 

is applicable to a region of any size, as a river basin or a State, so long as the 

region is hydrologically homogeneous.

The analysis provides two curves. The first expresses the flood dischargetime relation, showing variation of peak discharge, expressed as a ratio to 

the mean annual flood, with recurrence interval. The second relates the mean 

annual flood to the size of drainage area alone, or to the size area and other 

significant basin characteristics.

A frequency curve may be defined for any place in the region by use of 

these two curves. The procedure is: (a) measure the drainage area and other 

appropriate basin characteristics from maps; (b) from the second curve, select 

the mean annual flood corresponding to the proper drainage area factors; (c) 

from the first curve, select ratios of peak discharge to mean annual flood for 

selected recurrence intervals, as 2, 10, 25, and 50 years; and (d) multiply 

these ratios by the mean annual flood and plot the resulting discharges of 

known frequency to define the frequency curve.

Two reports not previously given general circulation are included as sections 

of this report. These are "Plotting Positions in Frequency Analysis" by W. B. 

Langbein, and "Characteristics of Frequency Curves Based on a Theoretical 

1,000-Year Record" by M. A. Benson.

INTRODUCTION

This report on flood-frequency analyses reflects work done by many 

investigators. Of the Geological Survey investigators, special acknowledgment is due to W. B. Langbein, who has made many contributions to the techniques described. Acknowledgment is due also 

to M. A. Benson of the Washington office staff, and to many others 

in the field offices who helped prepare about 25 regional studies for 

all parts of the United States.

Techniques used by the Surface Water Branch, Water Resources 

Division, of the Geological Survey in making flood-frequency studies
2 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

are described. The report incorporates material previously released 

for official use only, as part of the Handbook for Hydrologists, chapter 1, "Instructions for Flood Frequency Compilations," revised October 1949, and chapter 8, "Flood Frequency Analyses," revised May 

1950.

Knowledge of the magnitude and probable frequency of recurrence 

of floods is necessary to the proper design and location of structures 

such as dams, bridges, culverts, levees, highways, waterworks, sewagedisposal plants, and industrial buildings. An engineer often must 

design a structure which may be damaged or destroyed by occasional 

floods of varying magnitude. The frequency with which such damage may occur must be considered in determining the size or strength 

of the structure, its location, or the feasibility of building it at all. 

The problem is an economic one, involving computation of the total 

annual cost of maintaining a structure of a given design compared 

to the cost for other designs.

Either overdesign or underdesign of structures involves excessive 

costs on a long-time basis. The initial cost of a bridge that is designed to pass only a 5-year flood may be small, but the cost of rebuilding it at an average 5-year interval would be large. A bridge 

at the same site which is built to pass the 100-year flood might be 

extremely costly. Intermediate design would provide a bridge at 

that site for the lowest average annual cost.

Knowledge of flood frequency is necessary also to flood insurance 

and flood zoning, activities which are now considered on a broad 

scale. Without such knowledge, these activities would be seriously 

handicapped if not prohibited. Problems of flood insurance and 

flood zoning are economically important and they make the development of sound flood-frequency methods imperative.

The use of the flood-frequency method has met some criticism, 

largely because it has been abused. The method has little place in 

determining maximum limits of flood design, that is, "the maximum 

possible flood." With the ordinary streamflow record of 25-year 

length, errors of sampling introduce large errors in judging the magnitude of the greater floods., However, if properly computed and 

conservatively interpreted, flood-frequency analysis is a valuable 

hydrologic tool. The subject attracts many students of hydrology 

and it has benefited by their extensive writings.

Foster (1924) described the application of frequency curves to 

engineering problems of the flexible skew-distribution curves devised 

by Karl Pearson for frequency distributions of annual floods. The 

fitting of these curves requires selection of type and calculation of 

mean and coefficients of variation and skew. Fisher and Tippett 

(1928) developed frequency distributions of maximum values, sub-
FLOOD-FREQUENCY ANALYSES 3

sequently applied by Gumbel (1945a) to floods. Hazen (1930) published a general treatise on the determination of frequency and magnitude by the use of logarithmic skew-frequency curves.

Jarvis (1936) edited a comprehensive source book on flood frequency containing chapters by Saville (p. 398-420) on the methods 

of Fuller, Foster, Hazen, Goodrich, and Slade; by Slade (p. 421- 

432) giving an analysis of the errors inherent in calculations of the 

mean, coefficient of variation, and the coefficient of skew with small 

samples; by Horton (p. 433-450) presenting his integral-frequency 

formula, in which flood magnitudes always continue to increase as 

the recurrence interval increases, but they increase towards a finite 

limit, not towards infinity; and by Bernard (p. 451-461), who discusses the determination of floodflow by the unit-hydrograph method.

Gumbel (1941) presented a paper on the return period of floodflows. Powell (1943) introduced the Gumbel method to engineers. 

Kinnison and Colby (1945) correlated peak discharges with measurable drainage-basin characteristics.

The viewpoints and theories expressed by these writers, although 

instructive, have not always been consistent. Knowledge gained 

from a study of these works of statisticians, hydrologists, and engineers, plus what has been learned in the preparation of 24 statewide 

or regional flood-frequency reports, has led to a method of analysis 

that is presented in this report. The method reflects the latest developments based on a continuing study of the subject by engineers of 

the Water Resources Division of the Geological Survey. The method 

has been revised several times in recent years and probably will be 

revised in the future.

The discharge records collected by the Geological Survey and other 

Federal, State, and private agencies at about 7,000 places in the 

United States are the records upon which flood-frequency studies are 

based. Most of the records are short and the sampling errors are 

correspondingly large, or the records for different streams are for 

different periods of time. Few records are for as long as 60 years, 

and most are for less than 30 years. It is doubtful that a rigid 

mathematical treatment is justified for such short records.

An important element in statistical analysis is the skewness of the 

data; Slade (1936, p. 426) has aptly remarked "* * * that skewness 

is never a truly significant characteristic when the sample from which 

it is computed has less than about 140 items * * * and it is quite meaningless to use this measure when there are 50 or fewer items." The 

hopelessness of obtaining excellent results by rigid application of 

complex statistical analysis is obvious. For Geological Survey reports only graphical definition of the frequency curve is contemplated.

540570 O - 60 - 2
4 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

A curve based on a gaging-station record applies only to the site 

of the station; generally the information is wanted for an ungaged 

point. Investigations have been made of the possibilities of combining the flood data for all gaging stations in a drainage basin or a 

larger region, and of relating the resulting flood-frequency function 

to measurable characteristics of the drainage basin. This procedure 

would, in the first instance, reduce the larger sampling errors and 

in the second instance, give the data regional significance and so make 

the flood-frequency studies applicable to ungaged areas. In general, 

concern is with floods not exceeding 50- to 100-year recurrence intervals, as these satisfy the needs of most engineering studies.

The ultimate objective of the analysis generally is to prepare a regional flood-frequency report. First, an analysis must be made of 

each gaging-station record, resulting in a frequency curve for the 

station, and second, these station- frequency curves must be combined 

to give results applicable to any stream in a region. The need is for 

as many gaging-station records as possible, and each for as long a 

period of time as possible. Records used in the analysis should not 

be appreciably affected by works of man. The effect on floodflows 

by works of man should be studied independently, and these effects 

added to natural-flow frequency relations. ,

The regional flood-frequency method developed by the Geological 

Survey provides two basic curves. The first expresses the flood discharge-time relation, showing variation of peak discharge, expressed 

as a ratio to the mean annual flood, with recurrence interval. The 

second relates the mean annual flood of the size of the drainage area 

alone, or to the size and other significant basin characteristics. A 

frequency curve may be defined for any place in the region by use 

of these two curves.

Significant features of the flood-frequency method described in this 

report are as follows:

1. It is concerned with momentary peak discharges.

2. Recurrence intervals are computed by the formula

m

3. Curves are fitted graphically.

4. The mean annual flood is defined as the flood having a recurrence 

interval of 2.33 years.

5. A means is provided for computing flood frequencies of natural

flow on any stream, gaged or ungaged, in a region. 

In addition to describing the Geological Survey's flood-frequency 

method, this report includes two sections on reports by Survey au-
FLOOD-FREQUENCY ANALYSES 5

thors on related subjects; these reports have not been published 

previously in a generally distributed journal. One of these reports, 

"Plotting Positions in Frequency Analysis," prepared in November 

1954, by W. B. Langbein, gives a derivation of the formula adopted 

for computing plotting positions. The other report, "Characteristics 

of Frequency Curves Based on a Theoretical 1,000-Year Record" by 

M. A. Benson, furnishes a measure of the reliability of the adopted 

flood-frequency method.

FLOOD FREQUENCY AT A GAGING STATION

An analysis must be made of each gaging-station record before a 

regional study can be started. Three kinds of flood-frequency curves 

may be prepared: (a) discharge; (b) stage; and (c) volume. Discharge-frequency curves are the most common; these are the basis 

for the most often desired regional reports. Stage-frequency curves 

apply only to one gage and therefore are not suitable for regional 

treatment; these are basic to flood-insurance and flood-zoning problems. Volume-frequency curves, with proper caution, may be regionalized, and are useful in the study of water supply and storage 

requirements. This report is concerned mostly with discharge-frequency analysis, although the same techniques may be applicable to 

stage and volume analyses.

The time scale for frequency curves is the recurrence interval. 

This term is defined as the average interval of time within which a 

flood of a given magnitude will be equaled or exceeded once. A flood 

having a recurrence interval of 10 years is one that has a 10 percent 

chance of recurring in any year; likewise a 50-year flood has a 2 

percent chance, and a 100-year flood has a 1 percent chance of recurring in any year.

TWO KINDS OF FLOOD SERIES

Two methods of treating flood data for studying the frequency of 

floods are in common use. The first is the annual-flood array and the 

second is the partial-duration series. Although most analysts take a 

tolerant view, a few are active protagonists for one method as against 

the other. The differences are largely a matter of definition.

ANNUAL FLOODS

An annual flood is defined as the highest mometary peak discharge 

in a water year. The use of only one flood in each year is the most 

frequent objection to the use of annual floods. Infrequently, the 

second highest flood in a given year, which is omitted in the above 

definition, may outrank many annual floods.
6 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES 

PARTIAlrDTJRATION SERIES

The objection noted under annual floods is resolved by listing all 

floods that are greater than a selected base without regard to number 

within any given time period. The base is generally selected as equal 

to the lowest annual flood so that at least one flood in each year is 

included, however, in a long record, the base is generally raised so 

that on the average only 3 or 4 floods a year are included. The only 

other criterion followed in selecting the floods is that each peak be 

individual; that is, be separated by substantial recession in stage and 

discharge.

An objection to the use of the partial-flood series is that the floods 

listed may not be fully independent events; closely consecutive flood 

peaks may actually be one flood.

The greater number of floods listed in the partial-duration series 

might be an advantage, particularly if the record is short. However, 

most of the additional floods are of low discharge and plot where 

the curve is well denned; the high-discharge floods are generally 

identical with those in the annual-flood series.

RELATION BETWEEN THE TWO SERIES

A definite relation between values in the two series exists (Langbein, 1949; Chow, 1950). The following table shows comparative 

values of recurrence intervals by the two methods:

Recurrence intervals in year?

Partial-duration series Annual floods

0.5

1.0

1.45

2.0

5.0

10

20

50

100

1. 16

1.58

2.00

2. 54

5. 52

10. 5

20. 5

50.5

100. 5

There is distinction in meaning between the recurrence interval of 

these floods. In the annual-flood series the recurrence interval is the 

average interval in which a flood of given size will recur as an annual 

maximum. In the partial-duration series, the recurrence interval is 

the average interval between floods of a given size regardless of their 

relation to the year or any other period of time. This distinction 

remains, although for large floods the two approach numerical 

equality.

The simplicity of the annual flood method is an attractive statistical feature, and this method is used by the Geological Survey. Where 

a frequency curve derived from the partial-duration series is desired,
FLOOD-FREQUENCY ANALYSES 7

an annual flood curve is prepared and converted to the partial-duration series by the relation expressed in the preceding table.

FLOOD-FREQUENCY CURVES

Three kinds of flood-frequency curves may be prepared from gaging-station records. These are: (a) discharge curves, derived from 

momentary peak rates of flow; (b) stage curves, relating momentary 

peak stages to time; and (c) volume curves, relating time to the 

maximum 1-day, 2-day, 5-day, 10-day, or other-day, discharge.

DISCHARGE CURVES

Discharge-frequency curves are the most common and their preparation will be described in detail. Discharge curves derived from 

gaging-station records are the basis for establishing regional relation ; this subject will be treated in another section of this report.

STAGE CURVES

Where it is necessary to compute frequencies of stage occurrence 

for purposes of prediction, thought must be given to the nature of the 

stage-discharge relation. If the stage-discharge relation has remained virtually stable throughout the period of record, frequencies 

can either be computed directly from stages, or can be computed 

from discharges first, then transformed to stages by means of the 

stage-discharge relation. If there has been shifting of the stage-discharge relation, past stages may have little connection with expected 

stages. In such case it may be best to assume the most recent stagedischarge relation will hold in the future. After frequencies for 

discharge are computed, they can be transformed to stage by means 

of the most recent rating. Sandy channels may shift frequently. If 

the shifting is entirely random, it is necessary to work on the basis 

of past stages alone. However, if such shifting is in one direction, 

either a successive aggradation or degradation of the channel, then 

the best approach is to compute first the discharge frequencies, and 

to transform to stage by means of an assumed stage-discharge relation based on the previous trend. Frequencies based on such assumptions would be valid only for a definite period of time, and would 

have to be revised periodically.

Ice-affected streams involve other problems yet unsolved. If peaks 

due to ice jams are among the events included in an array of high 

stages, it is not possible to transform from discharge to stage frequencies. If a stream has either a constant stage-discharge relation 

or frequent random shifts, a stage-frequency study can be made 

based on stages alone. Where the stage-discharge relation is not constant, but shifting either progressively or at infrequent intervals,
8 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

some method must be devised which treats the effects of ice jams 

separately, then combines the resultant frequencies.

FLOOD-VOLUME CURVES

There are certain problems in which flood volumes are of as great 

concern as flood peaks. These problems involve reservoir design, 

where it is desired to provide a measure of flood-control storage. The 

frequency of flood volume can be determined by the same method as 

the frequency of flood peaks. An example is given using the anualflood method.

LISTING OF DATA

A form (Survey form 9-179) has been prepared for listing data, 

and is reproduced as figure 1. Columns 1-4 of this form should be 

filled in when data are listed. Column 1 is the water year; list the 

year as "1956" not "1955-56" for the year ending September 30, 1956. 

Show dates in column 2 as "Nov. 14, 1956" or "June 19, 1957"; and 

show the calendar year to avoid any chance for mistake when recording peaks occurring during months of October, November, and 

December. List the gage height in column 3 and the discharge in 

column 4; a dash in the discharge column, as shown for the February 

26, 1941 peak, indicates the discharge was less than the base.

A horizontal line should be shown across: (a) column 1 if there 

is a break in the record; (b) column 3 where there has been a change 

in datum and gage heights have not been adjusted to one datum; 

(c) column 4 if a change in location of gage has affected the stagedischarge relation; and (d) columns 1-4 where both gage heights and 

discharges are not exactly comparable, as when records for two adjacent stations are combined.

ANNUAL-FLOOD PEAKS

List all complete years of record. No selection should be made of 

part of a record to be used except to the extent of using the largest 

continuous period or using great historical floods.

If an annual peak may have occurred while the recorder was out 

of operation, make an estimate of the peak discharge, if at all possible. Do not omit any years during the period of record.

A record may begin in April a few days before a large flood which 

is not exceeded for the remainder of the year and examination of 

adjacent station records indicates that there was little flood activity 

prior to the April flood. The recorded flood may then be accepted 

as an annual flood.

Fragmentary historical flood data should be listed. Data of this 

kind may prove to be valuable, and should be obtained wherever 

possible.
FLOOD-FREQUENCY ANALYSES

UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL SURVEY 

WATCR RnOUItCC* DIVISION

File.

Flood data for ............................

Drainage area __SS>A.......... square miles. Period of record .......J.?3jS.r..f-7.

Flood data for momentary peak discharges greater than -.3rs£jQQ ~ cfs.

OAOE

HEIGHT

(F«et)

ANNUAL FLOODS

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.SI. ...,31. 

J..4A

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Graphical mean annual flood (Qi.«) ______ ofs tor period .___..___.._._...................

She* ..£... of...... Ltoted by ....:£&................ Date&/*7/$» Cheeked by .._

FIGURE 1. Flood data listing.

Floods are to be listed in chronologic order. Peak stages and discharges are both to be listed. The peak stages should be those of 

the stream, that is, on outside gage and not necessarily on inside gage 

unless data show the stages to be reasonably the same. Peaks only 

are included in this table. Daily discharges are excluded although 

previous reports (Jarvis, 1936) show daily discharges almost exclusively; that situation probably reflected the practice adopted by private practitioners to whom only daily discharges were available.
10 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Special effort should be made to determine flood peaks from the 

original observations or graphs except possibly on large rivers where 

the daily discharges and the peaks are about the same.

Stages are included for their own intrinsic value and to enable 

comparison with discharges a test of consistency of the record. Flood 

stages in many places are more useful than discharges, especially 

where it is desirable to place valuable property above flood levels of 

specified recurrence interval.

For streams with loop ratings, or those subject to rate of change, 

or backwater or ice effects, peak stages are not concurrent with peak 

discharges. For such streams peak stages should be listed independently of peak discharges. For areas where ice is a factor add a 

column showing ice effect in feet, for each peak listed (Prior, 1949, 

P. 12).

List the highest observed peak in each water year in chronologic 

order. The list of annual flood peaks will not necessarily be the same 

as the yearly maximums published in the annual water-supply papers. Where maximums are shown which occur near the end or the 

beginning of a water year, examine them, first to see whether they 

represent peaks, and second to see that they are included only once 

in the list of peaks.

For example, on the Suwanee Kiver at White Springs, Fla., the 

maximum discharge for the 1907 water year was not a peak it 

occurred on a rising stage at 12 p.m., September 30. The peak of 

this rise came 4 days later on October 4. Use the highest peak during 

the year, that of August 24-25, 1907, in the list of annual flood peaks, 

although the discharge is less than the maximum yearly. The 

peak of October 4 is surpassed by that of January 1-2 in the 1908 

water year, and so is not included. For this same station, the maximum discharge for both the 1928 and the 1929 water years occurred 

on the same rise, the peak of which lasted through September 30 

and October 1, 1928, at the turn of the water year. In a listing of 

annual peak discharges, do not include this peak twice. The peak 

for April 30, 1928, which is the largest peak, should be included 

except the one for September 30-October 1, in the 2 water years, 

and exclude a listing of that for September 30, 1928. The September 

30-October 1 peak is assigned to the 1929 water year to include the 

April 30, 1928 peak which was the second highest in the 2 water 

years.

PARTIAL-DURATION SERIES PEAKS

For partial-duration series, list all peaks in which the discharge 

exceeds a chosen base discharge, regardless of the number of peaks 

occurring in a year. Peak discharges, other than the annual maximum, for many gaging stations are published in the annual series
FLOOD-FREQUENCY ANALYSES 11

of water-supply papers that present gaging-station data. The base 

for listing flood peaks for the frequency-report tabulations should 

be the same as that for listing flood peaks in the annual reports. 

Rules taken from instructions for preparation of annual reports are 

given below:

1. Generally, list peaks for all stations for which a recording-gage 

record is available.

2. List peaks for nonrecording-gage stations if the frequency of 

gage readings is such that reasonably accurate graphs can be 

drawn.

3. Do not list peaks for canals, ditches, drains, or any stream where 

the peaks are subject to substantial control by man. For stations that are affected by a few or minor manmade regulation, 

either list all the peaks, regardless of whether some may be 

affected by regulation, or list none at all.

4. Omit peaks for stations where the record is incomplete and where 

peaks above the base may have occurred during the period of 

nonoperation. If, however, the period of nonoperation was 

one of low flow with no possibility of any floods, the list of 

peaks can be shown exactly like any complete-year station.

5. Omit peaks for streams where the crests are so flat that the 

peaks generally are either the same as the daily mean discharge 

or exceed it by such a small amount that the daily mean discharge for all practical purposes is the same as the peak. If 

the peaks, in general, do not exceed the daily mean by more 

than 5 percent, list none although an occasional peak may substantially exceed the daily mean.

For those stations for which peak discharge will be published, the 

following rules should be used for determining which peaks to publish and in what form to publish them:

1. Publish all peaks for which discharge equals or exceeds a chosen 

base discharge, regardless of the number of peaks occurring in 

a year. The gage height corresponding to the peak discharge 

will also be given. The number of peaks should average about 

three per year if the base is chosen properly. (Suggestions for 

selecting the base discharge are given on page 12.)

2. Indicate the selected base in the manuscript and in the published 

reports each year so that the user of the record will know what 

the base is at each station for which peaks are published.

3. Publish only the highest peak of 2 or more occurring within 48 

hours of each other, unless it is probable that the peaks in that 

period are independent, a condition which will occur at times 

on some streams. The peaks are independent if the hydrograph recedes to well-defined troughs as defined in paragraph

540570 O - 60 - 3
12 MANUAL OF HYDROLOGY! FLOOD-FLOW TECHNIQUES

4. List only the first, if 2 dependent peaks occurring within 

48 hours of each other happen to be equal.

4. Do not publish a peak unless the discharge of the trough between 

it and the adjacent higher peak is 25 percent or more below 

the discharge of the lower peak.

5. Publish for periods of diurnal peaks caused by snowmelt, only 

the highest occurring during each distinct period of melting 

regardless of the fact that other peaks may fulfill the preceding 

requirements.

6. List all maximum annual peaks, although they may be less than 

the base discharge.

7. At recording-gage stations peaks may occur while the recorder 

is out of operation, also peaks sometimes occur during iceaffected periods. It would be misleading to exclude mention 

of peaks that otherwise fit the criteria because complete data 

are lacking. In such cases, estimates must be made or the 

whole year eliminated. 

Criteria for selecting base discharge. The following instructions

for selecting the base discharge are followed in the preparation of

annual reports.

1. For stations for which flood-frequency data have been compiled, 

use the same base if it has been found satisfactory. A discharge that is exceeded on an average of three times per year 

is low enough.

2. For stations with records of more than 5 years but for which no 

flood-frequency data have been compiled, list the annual flood 

peaks, compute their recurrence interval, and choose as the base 

a discharge whose recurrence interval is 1.15 years.

3. For stations with records of 5 years or less, select a base guided 

by judgment and by comparison with other stations. The base 

selected can be modified as more data become available. A base 

should be selected that is a little low so that if it does become 

necessary to change the base, a higher one can be selected.

4. Changes in the base are not desirable; after once selecting a base 

retain it unless it proves entirely unsatisfactory. A revision 

upward in base does not affect the continuity of the array of 

peaks above the new base. However, a revision downward in 

the base means that all the lists of peaks above the higher base 

as previously published are incomplete so far as the new base 

is concerned.

FLOOD VOLUMES

Flood-volume data may be listed on a standard computation form 

(Survey form 9-230) or other suitable form. Separate lists should 

be prepared of the maximum 1-day discharge, maximum 2-day (con-
FLOOD-FREQUENCY ANALYSES 13

ei

$-[

II

yar

secutive) discharge, 3-day, 4-day, 5-day, 7-day, 10-day, 15-day, or 

other period, discharge in each water year. A tabulation of flood 

volumes for Licking Eiver at Toboso, Ohio, is presented as figure 2.

HISTORICAL DATA

Historical floods provide probably the most effective data available 

on which to base flood-frequency determinations, and where the data 

are reliable this information should be given the greatest weight in
14 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

constructing the flood-frequency graph. Effort should be expended 

to search out historical data from newspapers, local historical society 

records, local history reports and other sources.

Historical data are particularly valuable where there is an account 

of all floods, above a certain stage, over a long period antedating 

the beginning of stream gaging. The minimum or base stage is the 

stage where damage begins or threatens. A list of historical floods 

is of the nature of a partial-duration series above a high base, but 

because there is generally only 1 flood of such magnitude in any 1 

year, it may also be viewed as a partial list of annual floods. Where 

there is only one historical flood, the "maximum known," the base is 

the same as the flood.

An example of historical floods above a base is given by the record 

for the Susquehanna River at Harrisburg, Pa., where floods above 

18.0 feet date from 1786 (Benson, 1950). If combined with the continuous record obtained in recent years, a period of record for 172 

years is available.

A good example of the use made of newspaper sources for historical information is given by Woolley (1946). Most flood reports 

published by the Geological Survey contain data on historical floods; 

the report by Grover (1937) has information dating back to 1639.

Much historical information has been obtained in some areas, and 

incorporated in the annual reports on streamflow. (See reports on 

"Surface Water Supply of the United States, part 8, Western Gulf 

of Mexico Basins.") Examples are: (a) San Antonio River at San 

Antonio, Tex. (gaging-station record began 1915), "Maximum stage 

known since at least 1819, that of Sept. 10, 1921; flood of July 5, 

1819, equalled or exceeded that of Sept. 10, 1921."; and (b) Nueces 

River below Uvalde, Tex. (gaging-station record began 1939), "Maximum stage known since at least 1836, 40.4 ft June 14, 1935, from 

floodmarks (discharge, 616,000 cfs, by slope-area measurement at former site.)" The value of such information in a flood-frequency study 

is obvious.

PLOTTING POSITIONS

The analysis of a series of flood data starts with a listing of the 

peaks under consideration; for example, all the annual peaks at a 

gaging station. These are ranked according to magnitude, customarily starting with the highest as 1 (col. 5, fig. 1). Some measure of 

frequency must then be computed so that a "plotting position" is
FLOOD-FREQUENCY ANALYSES 15

obtained for the frequency scale. This plotting position is in terms 

of years. The objective of the frequency analysis is to determine 

the magnitude of the flood which will be equaled or exceeded once 

in a specified period of years; this specified period of years is known 

as the recurrence interval.

Based on considerations of probability, several methods have been 

used to compute the recurrence interval.

METHODS

"California" method. This is the simplest form of computation,

in which the recurrence interval, Tr= , where n is the number of

m

years of record and m is the rank starting with the highest as 1. 

(See Calif. Dept. Public Works, 1923; Jarvis and others, 1936). A 

minor objection to this method, where considering the probability of 

the event, is the reciprocal of the recurrence interval. The probability of the lowest flood occurring is computed as 1, which precludes the occurrence of any flood lower than this (such a point 

could not be plotted on probability paper). Also, this method gives 

no weight to the probability that the highest flood of record has a 

recurrence interval of something over n years. 

Hazen method. Hazen (1930) computed the "return period,"

2?i Tp, as , T-. . This equation results in a recurrence interval of 2n

for the highest flood of record, which is an artificial lengthening of 

the period of record.

Gumbel method. The theoretical plotting by Gumbel (1945a) is 

based on the assumption that the observed mth value is the most 

probable, or modal, value of this rank of flood. Its return period is 

therefore skewed towards the mode of the theoretical distribution.

The Gumbel theory does not apply strictly to floods for the following reasons:

1. It is assumed that the same treatment derived for daily discharges 

can be applied to peaks.

2. The daily discharges are not independent events.

3. The 365 daily discharges in a year do not constitute a "large" 

number as predicated by the theory.

4. The annual peaks under consideration do not come from the same 

statistical population. Some peaks are caused by ordinary seasonal rains, some by snowmelt, others by hurricane conditions. 

Entirely different physical factors influence each type. There-
16 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

fore, although other considerations fitted the assumptions, the 

different peaks at one station would not lie on a straight line.

In spite of this, the Gumbel plotting has some value because in 

certain ranges, particularly the low range, the frequency curve tends 

towards a straight line, so that an index flood chosen in that range 

is better defined.

Beard method. This method (Beard, 1952) is based on the assumption that the mth value is the median value of this rank of flood, or 

that the probability of this value occurring is 50 percent.

Geological Survey method. The formula used by the Geological 

Survey is:

m 

where

T= recurrence interval, in years

n= number of years of record, and

m= magnitude of flood, the highest being 1.

This formula gives essentially the same results as Gumbel's computed values, and is much simpler to use. It has been adopted by 

Gumbel and by many adherents of his theories. (Gumbel, 1945b; 

Chow, 1953; Velz, 1952.) An explanation of the derivation of the 

formula is given by Langbein. (See p. 48.;

Recurrence intervals computed by this formula give results similar 

to those computed by the California method, but lack the theoretical 

deficiencies of the latter. For example, consider a record to contain 

20 floods, numbered from 1 for the highest to 20 for the lowest. The 

California method would consider that the highest flood would be 

equaled or exceeded % 0 > °r 5 percent, of the time, a logical procedure. 

But suppose the numbering is reversed, then all floods will be equal 

to or less than the highest 2 %0 or 100 percent of the time. Obviously 

the sum of the percentages that a flood can occur, be exceeded, or 

fallen short of, must be equal to 100. The Survey method shows the 

highest flood equaled or exceeded, %± or 4.76 percent of the time, and 

that all floods are equal to or less than the highest 2 %i or 95.24 percent of the time. Thus it shows that a flood can occur, be exceeded, 

or fallen short of 100 percent of the time (Cross, 1946, p. 17).

The formula adopted by the Survey is simple to compute, is applicable both to annual flood data and to the partial-duration series, 

and gives results that acceptably conform with some of the latest 

theories.

PLOTTING HISTORICAL. DATA

In computing plotting positions by any formula there are times 

when the computations must be modified. For example, the highest 

flood of a 40-year record should not always have a recurrence inter-
FLOOD-FREQUENCY ANALYSES 17

val of 41 years. There is frequently additional historical knowledge 

which might indicate, for example, that the highest flood in a 40-year 

record is the highest in 300 years. Its plotting position should then 

be computed as though it were the highest in a series of 300 items 

or as 301 instead of 41 years. The second flood (in 40 years) would 

then be computed as usual.

Plotting positions for historical floods may be computed for three 

cases:

1. Where there is one historical flood, higher than any for the period 

of record, with no intimation of what may have occurred between the date of the historical flood and the beginning of the 

record except that the historical flood is higher than any other 

flood known. In this case, the recurrence interval of the historical flood should be recorded as equal to one plus the period 

for which it is the greatest. An example of this is the 1876 

flood, listed in figure 1.

2. Where there is one historical flood known to be the highest until 

a greater flood occurs during the period of record. As an 

example, assume a discharge of 1,000 cfs occurred in 1863 and 

that the record began in 1923; the 1863 flood was the "maximum 

known" until 1938 when a discharge of 2,000 cfs was recorded.

Plotting positions, up to and including the 1957 Hood

Floods

Maximum flood in 1863-1957_. ____________

2d highest in 1863-1957_-_____-________-__

2d highest in 1923-57 __________________

3d highest in 1923-57________--_________._

35th highest in 1923-57_-__-________-_.___

Formula

95+1

1 

95+1

2 

35 + 1

2 

35+1

3

35+1

35

Years

48

18

12

1.03

Cubic feet 

per second

2,000

1,000

800

600

100

3. Where there is a historical record of all floods above a high base, 

as above "bankful" stage, and it may be assumed that lesser 

floods follow the same distribution as for period of record. 

Order numbers, and recurrence intervals, for annual floods for 

the period of record can be adjusted to the longer period for 

which historical data are available by a method described by 

Benson (1950). The general formula for the transformed order numbers is:
18 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

(m A)

where

m= order number, where the highest flood is 1, the second highest 2,

and, for all floods both those for period of record and those from the

historical record ; 

mi = order number of floods below base of historical record, adjusted

to the time base of the historical record ; 

,4= the number of annual floods equalling or exceeding the lowest

historical flood;

H=the length of the historical record, in years; and 

T=the total number of items, historical and recent, in the array.

To illustrate, assume the same record as shown for example 2. 

There would be a total of 36 items (1863, 1923-57). Of these, the 

first 2 are above the base of 1,000 cfs and are the 2 highest in 95 years 

of record (1863-1957). The other 93 years of record since 1863, 

excluding these 2, are below 1,000 cfs. The other 34 items of known 

record below 1,000 cfs are assumed to represent the distribution of 

frequencies during the 93 years below 1,000 cfs. Each of the 34 years 

are considered equal to 93/34 or 2.735 years of recofd. The transformed orders of all items after the first two are therefore computed 

by the formula mi=2+93/34(m 2). The order of the last item is 

95, and the total array now represents the frequency distribution in 95 

years of record. The recurrence intervals for all 36 items are computed by using the formula '

llv\

Plotting positions

Year

1938_---_. --.---__--___.-__

1863__ _--___--__-___--_-_____

Discharge

(cfs)

2,000

1,000

800 

600 

100

ra

1

2

3

4 

36

Order

7R-2

1 

2 

34

mi

1

2

4. 74 

7.47 

95.0

T

(years)

96.0

48.0

20. 3 

12.9 

1.01

An example for a record beginning in 1786 and continuous since 

1874 (Susquehanna River at Harrisburg, Pa.) is given by Benson 

(1950).
FLOOD-FREQUENCY ANALYSES 19

PLOTTING PAPER

For the general purposes of flood-frequency graphs, the kind of 

graduations on the paper is not important. However uniformity is 

desirable, and if a choice is to be made, the chart based on the theory 

of largest values has much to offer. A time scale has been devised by 

Powell (1943) that tends to make the frequency curve plot as a straight 

line. The equation for the time graduation is:

y= loj

where

y=a linear distance

T= recurrence interval.

The zero value of y occurs at T=1.582, and other values must be 

measured from this zero point.

Values of T and y 

(Powell, 1943)

T

1.01

1.05

1. 25

1.67

2. 00

2.33

2. 50

3. 33

4.00

v

-1. 53

-1. 10

-.48

.09

.37

. 58

. 67

1.03

1.25

T

5.0

10

20

25

50

100

200

500

1,000

v

1. 50

2. 25

2.97

3. 20

3. 90

4. 60

5. 30

6.21

6.91

These values of T and y may be used to prepare plotting paper; 

generally recurrence intervals are measured along the horizontal axis. 

The y values may be measured in inches, or in inches times a constant 

to adjust to different sizes of paper.

Two discharge scales are used for plotting annual floods, one rectangular and the other logarithmic. Standard Survey forms have 

been printed for each of these two scales; these are not available generally. Survey form 9-179a is to rectangular ordinate scale and 

form 9-l79b is to logarithmic ordinate scale; these are designed so 

that the time scale may be extended by attaching standard 3-inch 

logarithmic graph paper.

Annual flood peaks are plotted to either rectangular or logarithmic 

discharge scales, whichever provides the nearest to a straight-line 

plot. (A sample of these forms, giving examples of data plotted to 

both scales, is shown on figures 3 and 4.) See pages 22, 23.

For partial-duration series plots, 3-cycle semilog graph paper generally is used. The linear scale (ordinate) is used for the discharge

540570 O - 60 - 4
20 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

data and the logarithmic scale (abscissa) for recurrence interval. 

(A duration-series form giving an example of a data plot, is shown 

on figure 5.) See page 24.

FITTING FREQUENCY GRAPHS

When a frequency diagram is plotted there is a need for fitting a 

curve to the data. The fact that most streamflow records are less 

than 25 years in length does not, however, satisfy the demand for 

estimates of long-term destructive floods. The tendency is to use the 

frequency graph for purposes of extrapolation; this tendency cannot 

be encouraged. The linear distance from 25 to 200 years seems very 

short on most graphs, but extrapolation can only be justified when 

the phenomena have been proven to conform to underlying law. The 

error of a curve fitted by whatever method may be extremely great 

at its outer end. Since no known fitted curve can serve any use in 

extrapolation, its main purpose would be merely to provide a smoothing or interpolation formula. The value of an analytically fitted 

function therefore seems doubtful.

Many distrust any method that allows for personal judgment and 

prefer a mathematical fitting of curves. In order to use mathematics, 

it is first necessary to know the mathematical law to which flood data 

must conform such a law has never been demonstrated. The many 

studies of flood-frequency relations made so far have failed to define 

a specific distribution that is typical of all locations.

Again, the use of a least-squares method of fitting a straight line 

to flood data may lead to absurd results, because of the neglect of 

hydrologic factors. A good example of this is the short period of 

record containing a high flood the single high flood may outweigh 

all other floods and lead to a straight line which is away from any 

of the plotted points. In such a case, although the lines drawn graphically by a number of men might vary, each would be better than a 

mathematically fitted line. A method is not better because it leads 

to uniform answers, if those answers are uniformly wrong.

Only graphical treatment is contemplated for Survey reports. The 

several plotted points should be indicated by circles.

Unless a very long record is being analysed, as at major stations 

on large rivers, it is not advisable, except in an emergency, to prepare 

a frequency curve derived from a single station. The arra$ of peaks 

at any one station is a random sample, and, as such, may be far different in character, particularly in a short record, from those of nearby 

stations of otherwise like characteristics. Flood frequencies should 

be generalized and related to a common period of record where possible. Frequency characteristics of individual stations should then 

be based on or related to these generalized frequency curves.
FLOOD-FREQUENCY ANALYSES 21

For certain engineering applications, where single station analysis 

only is done, a simple curve of best fit may be drawn. The recurrence interval for the highest flood is doubtful, and those next highest probably decreasingly so, hence the topmost points cannot always 

be used directly in drawing the curve. The design of the plotting 

paper tends to straighten out the shape of the frequency curves, but, 

with the above exception, the base data remains the best indication 

of the shape.

If it becomes necessary to extrapolate any appreciable distance 

beyond the extent of the usable data, then methods other than mere 

extension of a curve become necessary. These methods might include 

study of the ratios of the 25- or 50-year flood to the mean annual flood 

at several long-term stations. They might include a compilation and 

graphing of extreme discharges in an area, plotting discharge against 

drainage area. From an envelope curve for such data, to which some 

frequency might be assigned, based on composite length of experience or frequencies attributed to some of the data, a figure might be 

obtained which could be used either as a point towards which to extend the curve or as a value which it might approach asymptotically.

Examples of frequency curves based on the annual flood series, 

plotted to both rectangular and logarithmic scale's, and the partialduration series, plotted to semilogarithmic scales, are presented as 

figures 3,4, and 5, respectively.

An example of volume-frequency curves, plotted from data of figure 2 is presented as figure 6.

SINGLE-STATION ANALYSIS

The foregoing discussion on methods of frequency analysis applies 

to the analysis of the records at an individual site. The results of 

such a study represent an exact description of what has happened 

at the site in the past, for some definite period of time.

However, the purpose of frequency analysis is generally the prediction of what will happen in the future. In using past records to 

predict the future, it must be assumed that there is no change in the 

nature of the factors causing floods. The past record is then considered a sample of the total statistical population consisting of past 

and future floods. However, when used for predicting the future, 

the sample must be considered as only an approximation, because 

all samples may vary from the group as a whole. The extent of the 

variation, that is, the probability of any extent of variation from 

the norm, is known from statistical theory.

In general, flood records represent relatively short samples, statistically speaking. For this reason, the record at any individual station may depart considerably from a true representation of the over-
22 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

all long-time flood-frequency relation.

A study was made to demonstrate the variability inherent in short 

records by Benson (1952). Basically the same report is reproduced 

on pages 51-74. This study was made starting with a theoretical

< c

UJ _g

QNO03S d3d 133J OISHO JO SONVSOOHi Nl '30MVHOSIQ
FLOOD-FREQUENCY ANALYSES 23

QN003S tOd 133J OISHO JO SONVSHOHl Nl '3OyVHOSIQ
24 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

0.1 1 10

RECURRENCE INTERVAL, IN YEARS, FOR FLOODS ABOVE 3500 CUBIC FEET PER SECOND 

FIGCEE 5. Partial-duration series flood-frequency curve.

1.000 years of record with annual peaks distributed so as to exactly 

define a straight-line graph on the standard frequency graph paper. 

Individual peaks were then drawn at random to form groups of 

samples equivalent to 10-year, 25-year, 50-year, and 100-year periods 

of record. Frequency curves for these short-term periods give a 

graphic picture of the variation in frequency curves for short-term 

samples (figs. 20-23). Even the 100-year records show more variation than might be expected. The significance is that an actual 

gaging-station record of 25 years, for example, could take the position of any one of the curves in figure 21. The range in variation 

from the long-term value at any frequency can be judged from these 

figures. Even the mean annual flood (recurrence interval, 2.33 years) 

can vary significantly from the overall value. The study shows that 

12 years of record are required to define the mean annual flood within 

25 percent (with expectation of correct results 95 percent of the 

time).

Because of the unreliability of the single-station curve, more extensive methods have been developed by the Survey. These include 

a study of frequency relations on a regional basis. The results of 

such studies reduce the large sampling errors in individual records 

and are applicable to all points within the region studied.
FLOOD-FREQUENCY ANALYSES 25

135 570

15 days 10 days

1.01 1.1 1.2 1.5 2 3 4 5 6 7 8 9 10 20 30 40 50

RECURRENCE INTERVAL, IN YEARS 

FIGURE 6. Annual flood volume-frequency curves.

REGIONAL FLOOD FREQUENCY

Flood records at gaging stations are generally short, the sampling 

errors correspondingly large, and the records represent different 

periods of time. In addition it is only rarely that flood-frequency 

information is needed at a gaging-station site. More often it is required at an ungaged site.

A method of combining records within a region has been developed 

which reduces the sampling error, bases the results on a uniform 

period of experience, and produces flood-frequency relations generally applicable within the region. Experience has proved that such 

a procedure is not only possible but leads to results of acceptable 

accuracy for design and planning.

It has been suggested that combining records by the station-year 

method of analysis could be applied profitably to flood-frequency 

studies as it has been to rainfall-intensity frequencies. By this
26 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

method, for example, 5 records of 20 years each could be combined 

to obtain a 100-year record. Such a method requires that floodfrequency characteristics be comparable and that the data be independent. Studies indicate that the first of these requirements is met 

by some drainage basins or regions but the second requirement is not 

met, at least by gaging stations with drainage areas of over 100 

square miles. Five stations in a hydrologically homogeneous area 

measure the same flood 5 times rather than measuring 5 floods each 

year; in other words, a 20-year period cannot yield more than a 20- 

year record, and cannot be expanded so as to become a 100-year 

record.

Rather than add several short records to produce a long-term record, they may be averaged thereby providing several measurements 

of each year's event. By this method, 5 records of 20 years each 

when combined give only a 20-year record, but it is considered that 

each year's flood has been measured 5 times. The median of these 

five values is assumed to give a better measure of the frequency characteristics of that event.

In order to test this assumption, a study was made of a theoretical 

1,000-year record (Benson, 1952). A perfect 1,000-year flood history 

was assumed, and divided into 100 10-year periods, each randomly 

distributed. One hundred frequency curves were drawn, each based 

on a 10-year record. The variation of these 100 curves from the basic 

1,000-year curve gives a measure of possible variation of curves based 

on short-term records of natural events. These 100 10-year frequency 

curves scattered greatly from the true curve.

By combining only 5 or 10 of the 10-year records, and selecting 

the median,, it was found that the resultant composite curve reduced 

the chance for error greatly. The study indicated that by combining 

5 records the chances are better than 50-50 that the composite curve 

will lie within the central one-fifth of the spread of all the individual 

curves and by combining 10 records, chances are 3 out of 4 that it 

will do so.

Regional flood-frequency study consists of two major parts. The 

first is the development of basic, dimensionless frequency curves representing the ratio of the flood of any frequency to an index flood 

(the mean annual flood). The second part is the development of 

relations between topographic characteristics of drainage areas and 

the mean annual flood, to enable the mean annual flood to be predicted at any point within the region. The combination of the mean 

annual flood with the basic-frequency curve, which is in terms of the 

mean annual flood, provides a frequency curve for any location.
FLOOD-FREQUENCY ANALYSES 27 

BASIC-FREQUENCY CURVE

Throughout large regions, which are homogeneous with respect to 

flood-producing characteristics, individual streams having a wide 

range of drainage area will have frequency curves of about equal 

slope or steepness. If the peak floods at each gaging station are 

reduced to dimensionless ratios by dividing by an index flood, the 

curves plotted using the flood ratios can be superimposed and will 

nearly coincide.

These curves will all pass through the 2.33-year frequency at the 

ratio of 1.0, but will have somewhat different slopes. The variation 

of these slopes can be tested to see whether the spread could occur by 

chance among samples from the same population (this is the "homogeneity test," which uses the ratio of the 10-year flood to the 2.33-year 

flood as the slope). If so, the assumption of a homogeneous region 

containing all the stations is satisfactory. Within such a homogeneous region, the best representation of the flood-frequency relation can 

be obtained by combining all dimensionless curves. The resulting 

average-frequency curve is then applicable throughout the region, 

and is called the regional frequency curve.

If the spread in slope is greater than can be attributed to chance 

alone, then further separation of the region is made to produce two 

or more homogeneous regions. Statewide studies have commonly 

found 2 to 3 such regions within a State; however, these will correlate with like regions in adjacent States, so that the area of a particular homogeneous region can be large.

Combining individual curves to obtain the regional frequency curve 

is accomplished by taking the median of station-flood ratios at the 

same recurrence intervals. These median values are then plotted to 

define the regional frequency curve.

HOMOGENEITY TEST

The homogeneity test was developed by Langbein of the Geological 

Survey and is explained below.

The question whether the records in a group are homogeneous may 

be answered in a statistical sense by determining whether they differ 

from one another by amounts that cannot reasonably be expected by 

chance. Naturally no two records precisely represent the same experience nor have exactly comparable characteristics. On the other 

hand, where these differences are no more than those due to the operations of chance, we can readily conclude that they merely represent 

different aspects of the same thing and thereby group them.

A chart such as is shown for the homogeneity test could be constructed as follows: If a 1,000-year record were divided into 100 10-

540570 O - 60 - 5
28 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

year records in which the highest flood in each of the 10-year periods 

represented an estimate of a 10-year flood, the distribution of those 

10-year floods could then be studied. They would not all be the 

same but the differences that exist would be merely due to chance. 

From this record, it could be decided what ranges of variations could 

be accepted and what variations would represent results other than 

chance. The same test could be done by studying a 2,000-year record 

in which 100 20-year records could be charted. In each chart the 

second highest flood would represent the 10-year flood. The spread 

of the 10-year flood as estimated for 20-year records would be less 

than in the second chart. From this the kind of spread due to chance 

could be determined.

There are no records of such length, but such a record is not necessary because if it is agreed that floods can occur fortuitously, then 

it is possible to calculate from theory the distribution of floods in a 

1,000-year or 10,000-year record. This cannot be done in terms of 

discharge but it can be done in terms of probability. Such calculations are the basis of the test for homogeneity.

The "true" position of a frequency graph for a station can differ 

greatly from the position indicated by the plotted points. Although 

the true position is unknown, the chances that it will lie within certain distances from the plotted position can be calculated. Therefore, if several frequency graphs are plotted, most of them will 

group together, but there will be certain ones that plot apart. The 

chance that these variants represent can be calculated. If the calculations show that the spread is too great, those graphs are set aside 

as not being a part of the group being tested. The problem is to 

calculate the spread to be expected, and to set a limit to the spread 

that will be acceptable.

The standard deviation of the reduced variate, y, of the Gumbel 

distribution is equal to:

where

T= recurrence interval

n= number of years of record

y= function of the recurrence interval T

This means that in a large number of different but homogeneous 

records, each n years long, probably two-thirds of the estimates of 

the T-year flood will be within ay of their most probable value of T.

A wider range of variation has been decided on and therefore two 

standard deviations have been selected as the permissible range. 

This means that 95 percent of the estimates will lie within 2 a of the
FLOOD-FREQUENCY ANALYSES 29

most probable value of T. It was also decided to make the test on 

the 10-year flood because this is the longest recurrence interval for 

which most records will give dependable estimates. The following 

calculations are derived (T=10 years; y=2.25; and e"=9.49.):

2ev I i f IT 

2 <ry= -/= %/ yd;' or 2 (ry==2-V^' verv

=T = 2 X9.49 J^=6.33. [or 2 <ry=2v/To=6.32]

A table giving values of y corresponding to T can be found in 

Gumbel (1942; also Powell, 1943).

Values of y corresponding to T

[Columns headed TL and Tv respectively give the lower and upper limits of the chart for the

homogeneity test]

n 

(in years)

5

10

20

50

100

200

500

1000

V

2.25

2.25

2.25

2.25

2.25

2.25

2.25

2.25

6.33

-fn

2.84

2.00

1.42

.90

.63

.45

.28

.20

Lower limit

y 2o-

-0.59

+.25

.83

1.35

1.62

1.80

1.97

2.05

TL

1.2

1.85

2.8

4.4

5.6

6.5

7.7

8.3

Upper limit

jr+2<r

5.09

4.25

3.67

3.15

2.88

2.70

2.53

2.45

TV

160

70

40

24

18

15.5

13

12

MEAN ANNUAL FLOOD

The mean annual flood for a gaging station is by definition, the 

2.33-year flood from the graphic-frequency curve defined by points 

which are referred to the same base-time period.

The magnitude of the mean annual flood may be affected by many 

factors, which can be classed as either physiographic or meteorologic. 

The problem is: Given a drainage basin of certain physical characteristics, located in a region subject to certain meteorologic conditions, what mean annual flood can be expected? The answer is obtained by correlating the known mean annual floods of drainage areas 

within a region with the known characteristics of the basin and the 

region.

PHYSIOGRAPHIC FACTORS

The physiographic factors which may influence the mean annual 

flood at a given point are: (a) The size of drainage area, (b) channel 

storage (c) artificial or natural storage in lakes or ponds, (d) slope 

of streams, (e) land slope, (f) stream density, (g) stream pattern, 

(h) elevation, (i) aspect, (j) orographic position, (k) underlying 

geology, (1) soil cover, (m) cultivation, and others.
30 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Practical methods of computing some of the important physical 

factors are described by Langbein and others, (1947). Some of the 

factors listed are fairly simple and can be expressed by a definite 

figure. Others, such as geology, channel storage, or orographic pattern are difficult to evaluate and have not yet been successfully used 

in correlations with peak floods. Many of these factors are interdependent.

Development of the relations, if they exist, requires much work. 

It also requires much work to compute them for ungaged areas. In 

many cases, topographic maps are not available for computing topographic characteristics.

METEOROLOGIC FACTORS

Meteorologic factors are concerned with the magnitude and distribution pattern of the precipitation falling on a drainage area. 

Some of the elements involved are: (a) Type of region, whether humid 

or arid, (b) storm directions, (c) storm patterns, (d) storm volumes, 

(e) precipitation intensities, (f) effect of snowmelt, (g) extent of 

ice jams, and probably others.

The evaluation, treatment, and use of the meteorologic elements 

are generally less certain than for the physiographic factors. The 

difficulty lies in determining what precipitation figures to use. Total 

annual precipitation has been used, but this is related only generally 

to storm rainfall. Kainfall intensities would be more directly related to peak discharges, but intensities must be expressed by both 

a definite period of time and a frequency of occurrence, as for example, a 50-year, 5-minute intensity. The possible combinations are 

many, and since this is only one of many other factors, both topographic and meteorologic, the selection of the best parameter becomes difficult. There is a great deal of opportunity for original 

investigation in this field.

COMPOSITE FACTORS

Many physiographic and meteorologic factors make demonstrating 

significant correlation difficult except for those factors that are outstandingly influential.

Various combinations of previously listed factors have been used 

in correlation with mean annual floods. One such combination is the 

mean annual runoff. This is a general index of the amount of precipitation available, and also an indication of the runoff-inducing 

characteristics of the basin. The mean annual rainfall is another 

factor which has been used, although not as successfully as the runoff.

In the Illinois flood-frequency study by Mitchell (1954) the basin 

lag was used. This is the time lag between the center of rainfall and 

the center of runoff. This time lag represents the composite effect
FLOOD-FREQUENCY ANALYSES 31

of most or all of the topographic factors. It is therefore a very useful figure, but, it cannot always be defined from known basin characteristics. It must be measured during actual storms.

Another method equal to using a composite factor is that of dividing the study region into several parts, called hydrologic areas. 

Within each area a separate curve of mean annual flood is correlated 

with the drainage area and perhaps some other significant factor. 

In each of these areas such factors as rainfall, geology, and other 

features probably have the same overall effect.

If satisfactory relations for several areas can be found by using 

drainage area alone, it may not be worthwhile to include other factors 

which would improve the correlation only slightly. The data are 

generally limited, and the additional factors cause a loss of degrees 

of freedom, so that no improvement results.

If, after all practicable factors have been considered in the correlation, then the residuals from the average relation may be analyzed 

for geographic location. A pattern may result which can be associated with orographic effect, soils, or some other factor. If isograms 

of the residuals are plotted, the use of a mean coefficient may improve 

the accuracy of the mean annual flood.

COMPUTATION PROCEDURE 

SEIjECTION OF STATIONS

The first step in beginning a flood compilation is to list the gaging 

stations. A record should be included if it is 5 or more years in 

length, although generally recurrence intervals should not be computed for records shorter than 10 years. Old, discontinued records 

that otherwise qualify should be included. Storage or other artificial 

factors which would tend to modify flood discharges significantly 

should be a minimum. Exclude canals, ditches, and grains, in which 

discharges are subject to substantial control by man. Always include 

the total usable storage capacity in the basin above the gage.

If the records for 2 drainage areas on the same stream show differences in area of less than 25 percent, the 2 may be treated as 1 record. 

If both records are for the same period of time, use the better one or 

use both and give each a weight of one-half; if the records are for 

different periods of time, combine them into one longer record. This 

is not always done for the larger rivers, such as in the lower reaches 

of the Missouri; see the treatment given to the Mississippi River 

(Searcy, 1955, p. 12) and a later discussion on page 46.

Include stations maintained by other agencies, such as the U.S. 

Weather Bureau; Corps of Engineers, Department of the Army; 

U.S. Soil Conservation Service; State agencies; private agencies, such
32 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

as power companies, water companies, and milling companies; and 

any others. Include stations at which only stage records have been 

kept.

Use all records representing natural flow. Do not discard a record 

because it does not fit into a pattern defined by other records.

List the stations not included in the compilation because they have 

too short a record, are not rated, or for other reasons; show why 

these stations were excluded.

The step-by-step procedure for making a regional flood-frequency 

analysis will be illustrated by the study made in the Youghiogheny 

and Kiskiminetas River basins, in Pennsylvania and Maryland 

(Noecker, 1952).

In the 3,800 square miles drained by the Youghiogheny and Kiskiminetas Rivers, 15 gaging stations had annual flood records of 

11 to 69 years when the analysis was made in 1951; these were all the 

records of adequate length that were not seriously affected by artificial regulation. These records were used in the analysis; the stations 

are:

1. Stony Creek at Ferndale, Pa.

2. Conemaugh River at Seward, Pa.

3. Conemaugh River at Tunnelton, Pa.

4. Kiskiminetas River at Avonmore, Pa.

5. Little Conemaugh River at East Conemaugh, Pa.

6. Blacklick Creek at Blacklick, Pa.

7. Loyalhanna Creek at Kingston, Pa.

8. Loyalhanna Creek at New Alexandria, Pa.

9. Youghiogheny River at Ohiopyle, Pa.

10. Youghiogheny River at Connelsville, Pa.

11. Youghiogheny River at Sutersville, Pa.

12. Casselman River at Markleton, Pa.

13. Big Piney Run near Salisbury, Pa.

14. Laurel Hill Creek at Ursina, Pa.

15. Green Lick Run at Green Lick Reservoir, Pa.

The locations of these stations are shown on figure 7.

Records for six other stations in the area were not used because of 

excessive backwater (making the discharge record doubtful or fragmentary) or because they were of too short duration (5 years or less).

SELECTION OF BASE PERIOD

Any statistical analysis of flood data must proceed on the assumption that the data analyzed are of a random nature. Yet there are 

persisting patterns or cyclic changes in weather, so the storms or 

floods of one period of time will not equal in general magnitude or 

distribution those of another period. Although the weather pattern 

may be random when considering a long period of time already past 

or in forecasting the future weather pattern, any definite time period 

in the past may be either high or low with respect to the average.
FLOOD-FREQUENCY ANALYSES 33

20 MILES

FIGURE 7. Youghiogheny and Kiskiminetas River basins, showing location of

gaging stations.

The relation of any particular period of time to the average is difficult or impossible to define. In order to combine records, comparable 

data must be analyzed. This eliminates the variability with time, so 

that the effect of other factors on flood peaks may be analyzed more 

easily. For the purpose of defining the long-term relation the period 

of time used should be as long as possible; for the purpose of separating the various factors affecting flood peaks, a common period of 

time for all records is desirable.

The base- or common-time period which is chosen is derived from 

the longest available records. Although all records will not be
34 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

equally long, missing years may be filled in by correlation procedures. 

Such estimated peaks are not used directly, but serve the purpose of 

allowing the correct order numbers to be assigned to the peaks of record, with respect to the base time period. The records then can properly be compared or combined. A bar graph furnishes a compact 

record of what data are available, and is an aid in selecting a base 

period.

Bar graph. A bar graph showing graphically the length of usable 

records in the example area is presented as figure 8.

The bar graph shows that, to avoid excessive adjustment of records, selection of two base periods would be desirable. One period 

was selected as 1914-50 and the other 1884 1950. The procedure was 

to extend all records to the short period, five records to the long 

period, and then to adjust from the short to the long period in one 

step. The procedure for making this adjustment is described on 

pages 40 and 42.

ADJUSTMENT OF RECORDS TO BASE PERIOD

The problem is to compute recurrence intervals for the floods in a 

partial record (not complete for the base period) that will be the 

same for these floods if the record was complete. The steps in the 

adjusting process follow.

List data on work sheets. Annual flood peaks for each gaging 

station should be copied on work sheets, such as Survey form 9-179, 

that has a line for each year of the base period. Tabulate only the 

year and the discharge, as shown in columns 1 and 4 (for 1933-50) 

of figure 9. A similar form should be made for each gaging station 

in the region under study.

1900 1910 1920 1930 1940 1950

FIGURE 8. Bar graph (showing period of record of maximum annual peaks at 

gaging stations.
FLOOD-FREQUENCY ANALYSES 35

UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL SURVEY 

WATER RESOURCES DIVISION

File

Flood data for ...i

Drainage area -. &4*-£........ square miles. Period <

Flood data for momentary peak discharges greater than _.._...~__..__ cfs.

WATER 

YEAH

/

J5/4..

JJUS..

/?/£

...W.7.- .j.m.

...IIM.

...L9Z0.

...1921.

...1.92.2.. 

..122.3 

JU24.

..L9Z5.

..1326.

.J3Z.7.

.J1&.

..L5Z1. 

.-L1.9.0

.-193.1 ..mi

ItS 3

-.133.4

.Jtt.$ 

_ZMt

..L3.3.7.

-1Z3&

..1939. 

..L9.40.

..J.94L

..134.2.

..19-43

...1.7.44

..&4£. 

...1.946.

1947

...1940

...1949. (990

OATK

2

OAOB 

HEIGHT 

Cert)

3

DISCHARGE

CFB

4-

.......pfio

ffifC 18.73^ ..ZLftfia

...Ut.70.ol

....&/.£<?}

LS4O

-JJL&ASQ.

fa3o}

.JQSJBft

L31O\ Lz 3.09}

..-.....(13Q\

(.190

.JL*a&

iLsze) .....J.6M

(i&ti

(6Jo) I Qoo

IZ3&L 474-

4 loo

......4..3t.e.o

......L.M.O..

......U-&Q~

......1..S.LO.

L&faP

......L.2.lo_

.. .2,460

628

......J+.e-f& 

........36.8-

634

.........&7t

47&

7.96

RATIO TO 

Qui

&

...........

ANNUAL FLOODS

°asr

6

3$

22

/£

7

..41-

3.3

17

2,1

A

27

4

£5

26-

Ift

1

31

23

27

32

ff

It

34

2

/

£

14-

10

13

/2

3

$0

..1.5. 

20

2&

91

3(.

24

RlCUBBlNCK *&

7

/

T

2"5

1.f*Q

....2..0.0.

.....L.I.&.

..13j?.....

..3&tO.....

6-33

....2.I.I..

...3.3.Q..

....2..ZZ

3.17

..LiJ....

LZ7

.J2..33. 

Irfo

L3£

....L81.

...Mo.6

/. 98

PARTIAL DORATION SERIES

OKDU 

<M>

6

CnnTiL (Yara)

f

REMARKS

10

....................................

Graphical mean annual flood (Q,.a) ./.&&£.. ofs for period .....

Sheet........of........ Itotedby.......^!!^.-............ Date £/&$&* Checked by .. .&$ !..-............. Date &.

n«« omci 1» KSM-»

FIGURE 9. Tabulation of flood data, Big Piney Run near Salisbury, Pa.

Estimate years of no record. There is no record for the period 

1914-32 at Big Piney Run. To complete this period, plot annual 

floods of record against annual floods at another station where the 

record is complete. Plot annual flood versus annual flood, not a 

flood at one station versus a flood of the same date at the other station, unless the annual floods occur at the same time. For example, 

figure 10 shows a correlation of annual peaks for Big Piney Run

540570 O - 60 - 6
36 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

50

O

8

V)

K.

UJ 40

s>u. °-

0 «

i!

O <- -an

THOUSANDS IN

Creek Stony a 

5 {

UJ 

C9 

(E

X

O

o 

10

0

 

} '

V

^f

^p

^f

/ .

^ 

'X

'

%

0 1000 2000 3000 4000 5CX

DISCHARGE, IN CUBIC FEET PER SECOND 

Big Piney Run near Salisbury, Pa.

FIGURE 10. Correlation of annual floods, Big Piney Run versus Stony Creek.

and Stony Creek. The correlation is not perfect; if it were, there 

would be no independence and there would be only one record that 

had been measured twice.

From the curve shown on figure 10, record the estimates for the 

years 1914-32 for Big Piney Run from the record at Stony Creek. 

These estimates are shown in parenthesis in column 4 of figure 9. 

Although discharge records have been used to obtain the estimates, 

do not plot or use them as discharges, but consider them as computation figures.

Compute order numbers. There now is either an annual peak 

discharge or an estimate for each year of the base period. Order 

numbers should be assigned to all of these figures; the largest should
FLOOD-FREQUENCY ANALYSES 37

be numbered 1, the second largest 2, and so forth. These order 

numbers are shown in column 6 of figure 9.

Compute recurrence intervals. Next, compute recurrence intervals, 

(n+1) from the formula T=-

m

for each flood of record. Do not compute

recurrence intervals for the estimated figures. Record them as shown 

in column 7 of figure 9.

The floods for 1933-50 have now been adjusted to the base period 

1914-50.

PRELIMINARY FREQUENCY CURVE

Plot discharge against recurrence intervals for each gaging station. 

Plot historical data in accordance with the discussion on page 17. 

Plot on forms similar to those illustrated in figures 3 and 4 (Survey 

form 9-179a and 9-179b).

Draw a frequency curve as a curve of visual best fit; do not compute mathematically. Extend the curve as far as the data warrants. 

A preliminary-frequency curve for Big Piney Run near Salisbury, 

Pa., is shown as figure 11; this curve bends up at the outer end, but 

if the data are plotted to a logarithmic-discharge scale a straight-line 

curve will be defined.

THE MEAN ANNUAL FLOOD

The mean used for each frequency distribution is the graphical 

mean determined by the intersection of the visually best fitting frequency line with the line corresponding to the 2.33-year recurrence 

interval. The graphical mean is more stable and dependable than 

an arithmetic mean. This method of determining the mean gives 

greater weight to the medium floods than to the extreme floods with 

large sampling errors, and for this reason is not influenced adversely

6000

5000

4000

o 3000

- 2000

1000

1.1 2 5 10 20 

RECURRENCE INTERVAL, IN YEARS

50 100

FIGURE 11. Preliminary frequency curve, Big Plney Run near Salisbury, Pa.
38 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

by the chance inclusion or exclusion of a major flood, as is the arithmetic mean.

In determining the mean annual flood, the curve should be carefully fitted to the points plotting near the 2.33-year line. In the 

example, the mean annual flood for Big Piney Run is 1,060 cfs according to the curve of figure 11. (The arithmetic mean, computed 

from the 18 years of record, is 1,540 cfs.)

HOMOGENEITY TEST

The test setup requires a study of the 10-year floods as estimated 

at each station. Each 10-year flood should be divided by the 

mean flood to get the 10-year ratio, and an average of these ratios 

should be obtained. Tabulate for each station the length of record 

in years and the recurrence interval corresponding to a discharge 

equal to the average flood ratio times the mean flood. Data should 

be tabulated as shown in the following table:

Data for homogeneity test, base period 1914-50

No.

1

1

2 

3 

4

5

6

7- 

8 

9 

10 

11 

12

13

14 

15

Stream

2

Stony Creek ______

Conemaugh River ...... 

Conemaugh River __ . 

Kiskiminetas River. ... 

Little Conemaugh 

River _________

Blacklick Creek.. ......

Loyalhanna Creek ......

Loyalharina Creek __ . 

Youghiogheny River. .. 

Youghiogheny River. .. 

Youghiogheny River. . .

Laurel Hill Creek...... 

Green Lick Run .......

Drainage 

area 

(square 

miles)

3

451

715 

1,358 

1,723

183

390

168 

265 

1,062 

1,362 

1,715 

382

24.5

121 

3.07

Mean 

annual 

flood,

Q2.33

(cfs)

4

m sno

19,000 

34,000 

42,000

6,250

11,500

6,000 

7,400 

26,300 

31,500 

38,200 

11,200

1,060

4,850 

230

10-year 

flood,

Qio 

(cfs)

5

22,400

31,500 

54,000 

65,000

9,800

18,800

9,600 

11,800 

43,100 

58,100 

62,500 

19,200

2,260

7,400 

500

Ratio

Qio

Q2.33

6

2.07

1.66 

1.59 

1.55

1.57

1.64

1.60 

1.60 

1.64 

1.84 

1.64 

1.71

2.13

1.53 

1.79

Q2.33

X1.70 

(cfs)

7

18,700

32,300 

57,800 

71,400

10,600

19,600

10,200 

12,600 

44,700 

53,500 

65,000 

19,000

1,800

8,250 

476

Tfor 

Qof 

column?

(years)

8

7

11 

14 

16

13

12

13 

13 

11 

8 

12 

10

6

17 

8

Period 

of record, 

adjusted 

(years)

9

37

25 

24 

32

24

37

24 

28 

33 

33 

33 

36

27

37 

30

Average ratio..__________________________.___ 1.70

The adjusted period of record, column 9, is the number of years of 

actual record plus one-half the number of years the record was 

extended.

A set of test curves has been devised that shows within what range 

of recurrence intervals an estimate of a 10-year flood should be for a 

specified length of record; a range of 2 standard deviations is allowed. 

It is appropriate to base the test on the 10-year flood, because this is 

the longest recurrence interval for which many records will give 

dependable estimates. The test curves may be drawn on a standard 

chart, such as Survey form 9-179a, or a chart prepared from data
FLOOD-FREQUENCY ANALYSES 39

given in the table under the section on "Homogeneity test" page 29. 

Plot the values of TL and Tv.

As an example, data from columns 8 and 9 of the preceding table 

have been plotted on the chart as shown in figure 12. All points plot 

within the limits, therefore the records probably are acceptably 

homogeneous, and records from the 15 stations may be grouped together to define a regional flood-frequency curve. Had some points 

plotted outside the limits, it would have been necessary to separate 

the records into two or more groups and define a separate frequency 

curve for each.

COMPUTATION OF MEDIAN FLOOD RATIOS

The preliminary frequency curves (fig. 11) for all stations in a 

homogeneous region should be assembled. The ratios of several 

floods of different recurrence intervals to the mean annual flood 

should be tabulated for each station. Select only enough recurrence 

intervals to define the curve. Tabulate the flood ratios as shown in 

the following table. The ratios are easily computed as recorded, by 

slide-rule division.

150

100

80

60

50

40

$ 30 

z 20

r

£ 10 

u 8

5 6

g 5

I 4 

3

\

1.5 

1.3 

1.2

f10 20 30 40 

EFFECTIVE LENGTH OF RECORD, IN YEARS

50 60

FIGURE 12. Homogeneity test chart.
40 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES 

Median flood ratios, 1914-50 period

Station

!___-_ --------------

2. ---.-_.___-.__..__

3

4---------- -----

5_--_ -- - ---__

6

7-_--.--_ ---------

8----_-------------

9--.-. --------------

10-_---.-_ --------

11---------.-. --__--

12------------------

13----__._-__-_---_

14

15------------------

Recurrence intervals, in years

1.1

0 49 

57 

54 

57 

56 

49 

50 

45 

69 

62 

58 

63 

47 

46 

54

0. 54

1.5

0 75 

78 

79 

80 

79 

78 

75 

76 

83 

81 

78 

80 

74 

76 

75

0. 78

5

1. 46 

1.36 

1. 32 

1.33 

1.34 

1.37 

1. 34 

1.39 

1.37 

1.39 

1.32 

1.36 

1.47 

1.35 

1. 35

1.36

10

1. 93 

1.74 

1.55 

1.62 

1. 63 

1.65 

1.70 

1.76 

1.80 

1.77 

1.67 

1. 80 

2. 13 

1.64 

1. 73

1.73

20

2.55 

2. 18 

1. 79 

1. 94 

1.97 

1. 93 

2.08 

2. 21 

2.38 

2.28 

2. 10 

2. 39 

3.02 

1. 91 

2. 22

2. 18

50

3.03 

2.73 

2.09 

2.48 

2. 52 

2. 28 

2. 57 

3.02 

3.00 

2. 92 

2. 90 

2. 91 

4.57 

2. 25 

2. 80

2. 80

After tabulating the flood ratios, compute median ratios, as tabulated in the bottom line of the table. This median is the midvalue 

of an odd number of events or the mean of the two central values of 

an even number of events.

DEFINITION OF REGIONAL FREQUENCY CURVE

Each median flood ratio should be plotted to its corresponding recurrence interval on a frequency chart and an average frequency 

curve should be drawn. The regional frequency curve for the 

Youghiogheny and Kiskiminetas basins, based on the period 1914- 

50, is shown as figure 13.

The regional frequency curve, showing flood discharge in ratio to 

the mean annual flood, is based on all significant discharge records 

available and represents the most likely flood-frequency values for 

all areas in the region.

Adjustment from short to long period. If more than one base 

period has been selected, the above process must be repeated for each 

period. If 2 base periods have been selected, 3 computations must 

be made:

1. Compute a regional frequency curve, as above, for the short period, using data for all stations.

2. Compute a regional curve for the long period, using records from 

all stations that are complete or have been filled in to the long 

period.

3. Compute a regional curve for the short period for only those stations that have records for the long period.
FLOOD-FREQUENCY ANALYSES 41

1.01 1.5 2 5 10 20 100

RECURRENCE INTERVAL, IN YEARS 

FIGURE 13. Regional frequency curve.

Adjust the short-term curve computed in 1. to the long-term period 

by the ratio of the other two curves; that is, the frequency curve adjusted to the long-term period =1. X2./3. This can easily be done by 

adjusting 6 or 8 points such as ratios for the 1.1-, 1.5-, 5-, 10-, 20-, and 

50-year recurrence intervals.

If the short period curves extend to the 50-year interval, the longperiod curves extend to the 100-year interval, and the adjustment 

below 50 years is appreciable, the 100-year point on the adjusted 

curve should extend the same amount above the 50-year point because the two are on the long-period curve.

ESTIMATION OF MEAN ANNUAL, FLOOD

In order to apply the regional flood-frequency curve to an ungaged 

drainage basin estimate the mean flood for that basin. This involves 

a correlation analysis of the observed mean floods with drainagebasin characteristics.

If it is assumed that the region is hydrologically homogeneous, the 

factors which affect the mean flood are size, topography, shape of the 

drainage basin, and channel storage. Of these size is the most important, and the factor most readily available. Measuring the other 

factors is more difficult, and it may be impossible unless good topographic maps are available. Channel storage undoubtedly has an 

important effect, but cannot be directly measured.
42 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

For practical engineering use, a correlation of mean flood with 

drainage area may suffice. This simple correlation may require more 

than one curve for the region under study but eliminates tedious and 

lengthy computations that often may not be practicable. Mean 

annual-flood discharge as ordinate and drainage area as abscissa generally are plotted to log-log scales. A mean curve should be fitted 

graphically to the points. The mean annual-flood discharge for any 

stream in the region can be selected from this curve if the size of its 

drainage area is known.

A curve showing the variation of mean annual flood with drainage 

area is shown as figure 14. Data plotted on this figure were taken 

from columns 3 and 4 of the table shown in the section on "Homogeneity test".

Instead of plotting a point representing the discharge, plot a range 

line using the chance range in the mean annual flood based on the 

length of record. An average curve passing through all the range 

lines may be considered as the best determination of the relation 

curve. A discussion of this range line is given in the section " 'Working range' of mean annual floods," and figure 25.

Adjustment from short to long period. If more than one base 

period has been selected, the mean annual floods derived from the 

short period must be adjusted to the long period before plotting. This

40,000

Q

o

a

£ 20,000

£ 

fc

£ 10,000

i

0

z 

g 4000

3Li.

S 2000

3-

00

«

Si

_, 1000 

| 800

< 600

z 

| 400

200

l 

>

t/

/

/

/

/

/

^

X

/

A

/

/

^

?t fs

J

/

-£r^

/

4 6 8 10 20 40 100 200 400 1000 2000 401

DRAINAGE AREA, IN SQUARE MILES 

FIGURE 14. Variation of mean annual flood with drainage area.
FLOOD-FREQUENCY ANALYSES 43

adjustment may be made by computing the average ratio determined 

from the 2.33-year floods computed for the two periods for the same 

gaging stations.

Correlation with physical factors. Correlation of the mean annual 

flood with basin or meteorologic factors, other than the size of the 

drainage basin, can be made either by using statistical correlation 

techniques or by using graphical methods. Techniques of multiple 

correlation are covered in most textbooks on statistics. Eesults of 

correlation studies can be shown in entirely graphical form, such as 

a set of curves covering all the necessary relations. This has been 

the method used in most of the regional studies made so far. It is 

the simplest method of applying the results to ungaged areas as far 

as users of the report are concerned.

Results may be shown in the form of nomographs, as was done 

in a study for western Washington. (Bodhaine and Eobinson, 1952, 

p. 28, 29). Or they may be shown in the form of equations, as was 

done for Connecticut (Bigwood and Thomas, 1955, p. 15). Equations are less convenient for the users; they present an opportunity 

for serious error in that they are apt to be used far beyond the limits 

of the data from which they were originally developed.

Regional frequency studies have been made in about half of the 

States; the factors that significantly correlate with the mean annual 

flood are: size of the drainage area (by far the most significant); 

slope of the principal streams; channel storage; area of lakes, ponds, 

and swamps; mean altitude; mean distance to outlet; annual runoff; 

basin lag time; soil types; and degree of urbanization.

Use of short-term records. A long-term gaging station record is 

necessary for completely satisfactory flood frequencies. However, 

there are situations in which information is wanted in a brief time. 

In such situations, obtaining a record of all flood peaks above a base 

can be useful, provided there is a long-term record of a station 

nearby. The problem involves estimating of the mean annual flood; 

discharges of the rarer floods can be estimated from an appropriate 

regional frequency curve, such as the one shown as figure 13.

From a reasonably short record (2-6 years), barring an uncommon 

sequence of drought years, a short partial-duration series of floods 

can be developed, at least in the lower range. Graphical comparison 

of such discharges at a short-term station with the corresponding 

discharges for the same period at a station with a long record, where 

the mean annual flood is known, would permit estimation of the mean 

annual flood at the short-term station.
44 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Partial-duration series, 1946-4?

Order

l___--__-

2

3-----_~

4---- 

5-_----_.

Discharge in cubic feet per second

Short-term station (Olentangy 

River near Delaware, Ohio, 

1946-47)

9, 160 

8,640 

6,060 

5,840 

5,730

Long-term station (Scoito 

River near Columbus, Ohio, 

1913, 1921-48)

31, 600 

20, 500 

19, 600 

19, 600 

14, 000

Order

6

7._ -_ 

8-_-_----

>9__--___.

Discharge in cubic feet per second

Short-term station (Olentangy 

River near Delaware, Ohio, 

1946-47)

5, 190 

5, 190 

4,200 

3,840

Long-term station (Scoito 

River near Columbus, Ohio, 

1913, 1921 '48)

13, 600 

11, 700 

11,700 

11,000

The data for corresponding order numbers are plotted against each 

other in a logarithmic chart, as shown in figure 15. A straight line 

was drawn to average the plotted points. The mean annual flood for 

the long-term record is 27,500 cfs. The estimate for the short-term 

station is 10,000 cfs. The record on Olentangy Kiver near Delaware 

Ohio, is in fact 36 years long, and the mean annual flood is 9,000 

cfs, a 10 percent difference from the foregoing estimate.

10

2 3

DISCHARGE, IN TEN THOUSANDS OF CUBIC FEET PER SECOND

Scioto River at Columbus, Ohio 

FIGURE 15. Determination of mean annual flood from a short record.
FLOOD-FREQUENCY ANALYSES 45

DEFINITION OF A FREQUENCY CURVE

To define a flood-frequency curve at any place within the study 

region, the mean annual flood is found from the curve relating this 

factor to the drainage area (fig. 14). The mean annual flood is multiplied by the flood ratios previously computed (fig. 13) to obtain 

the discharge corresponding to a selected frequency. A complete 

frequency curve can be produced by plotting discharges for several 

frequencies and drawing the curve they define.

An example of a flood-frequency curve defined for any 100-squaremile drainage basin in the Youghiogheny or Kiskiminetas Kiver 

basins in Pennsylvania is presented as figure 16.

SUMMARY OF PROCEDURE

The step-by-step procedure for making a regional flood-frequency 

analysis follows:

1. Tabulate flood data for all gaging stations in the region having 

a record of 5 years or more. List the maximum annual floods 

only. Leave space for tabulating adjusted data.

2. Prepare a bar graph and select the base period for study; generally this will be the period of the longest record.

3. Adjust all the records to the base period.

4. Number the floods for each station in the order of magnitude, 

the greatest flood is number 1.

5. Compute the recurrence intervals.

6. Plot the discharge against the recurrence intervals and draw 

frequency curves, one for each station.

7. Test for homogeneity.

8. Compute the median flood ratios.

1AUIAJ

10,000 

8000

4000 

2000

0

1

 - -

,.- 

,* ,»*

^

^"

^

'

^

s

s/

/

^

s

'-'

01 1.1 1.5 2 5 10 20 50 10

RECURRENCE INTERVAL, IN YEARS 

FIGURE 16. Flood-frequency curve for a 100-square-miIe drainage basin.
46 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

9. Plot the median flood ratios and draw a regional frequency 

curve.

10. Plot the mean annual floods against the drainage areas. Draw 

a curve, or curves, to show tne relation applicable for the 

region.

11. Determine flood frequency for any place in the region from the 

curves of items 9 and 10.

SPECIAL TREATMENT FOB LARGE STREAMS

Large streams that traverse more than one flood-frequency region 

or hydrologic area may not fit into the general pattern, and special 

treatment is indicated. The frequency relation may follow one of 

the regional curves plotted for tributary streams, where the regional 

curve would be used. If it follows an independent pattern, a frequency curve may be plotted for the one stream. This is illustrated 

by the curve defined for the Mississippi River by Searcy (1955, 

fig. 12).

The mean annual floods for large streams may require separate 

curves derived from records for stations along those streams. The 

curves smooth the individual station records and are a means of 

interpolation between the stations, on the basis of either size of drainage area or river mileage. An example of such a curve for the Mississippi River, taken from Searcy (1955, fig. 11) is presented as 

figure 17.

600

500

400

300

200

100

Thebes, Illino

it. Louis, Missouri

Alton, Illinoi

__|.

£

Mississippi f?/\er

0 100 200 300

DISTANCE, IN MILES, ABOVE OHIO RIVER 

FIGURE 17. Variation of mean annual flood with distance along stream.

400
FLOOD-FREQUENCY ANALYSES 47

350.000

300,000

a: 250,000

200,000

3 150,000

100,000

50,000

52,000 54,000 56,000 58,000 60,000 62,000 64,000 66,000 68,000

DRAINAGE AREA. IN SQUARE MILES 

FIGURE 18. Relation of flood frequencies to drainage area.

Some large streams show a change in both the flood-frequency 

relation and the mean annual flood at gaging stations along the 

stream. Separate curves for selected floods may be defined and 

used for interpolation along the stream. An example of this treatment has been made for the Red River in Louisiana by Cragwall 

(1952, fig. 22) and is presented as figure 18.

Regional flood-frequency reports have been prepared by methods 

described in this report and have been published by the Geological 

Survey or cooperating agencies. Knowledge of techniques may be 

gained from a study of these reports. All are listed in "Selected 

references"; they are listed below in brief, for ready reference:

Alabama. __________ (Peirce, 1954)

Columbia River (Rantz and

basin. Riggs, 1949).

Connecticut ________ (Bigwood and

	Thomas, 1955) 

Delaware River (Tice, 1958)

basin.

Florida,.___________ (Pride, 1957)

Georgia.___________ (Carter, 1951)

Illinois---..._______ (Mitchell, 1954)

Iowa_____________ (Schwob, 1953)

Kentucky. _________ (McCabe, 1958)

Louisiana._________ (Cragwall, 1952)

Minnesota________ (Prior, 1949)

Missouri. __________ (Searcy, 1955)

Montana, eastern

part.

Nebraska____---- 

North Carolina. 

North and South

Dakota.

Washington, western 

part.

Youghiogheny and 

Kiskiminetas River 

basins, Pennsylvania and 

Maryland.

(Berwick, 1958)

(Furness, 1955) 

(Riggs, 1955) 

(McCabe, 1957)

(Cross, 1946) 

(Bodhaine and

Robinson,

1952) 

(Noecker, 1952)
48 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

PLOTTING POSITIONS IN FREQUENCY ANALYSIS

By W. B. LANGBEIN

w-4-1 

The formula m for recurrence intervals has become nearlyJ

standard in Survey reports on flood and drought frequency. The 

first use of this formula is ascribed to Kimball (1946), but except 

for Survey reports and a few papers in statistical literature, this 

formula has been little used. It, therefore, may be unfamiliar to 

many of the readers of Survey reports, and some explanation is desirable concerning the nature of the formula in relation to others 

mentioned in flood literature.

For simplicity this explanation will treat only the highest flood in 

an annual series for which the several formulas for plotting positions 

differ most.

Consider first, for example, the question: What is the chance that 

a 50-year flood will be the maximum in a 10-year period ? A 50-year 

flood has a probability of 0.02 of being equaled or exceeded in any 

year and a probability of 0.98 or not being equaled or exceeded in 

any year. A flood of this magnitude can be the maximum in a 10- 

year annual series only if all 10 values are equal to or less than the 

50-year flood. Hence 0.9810 represents the probability that a 50-year 

flood will be the greatest in a 10-year period. This logic is general 

and is independent of the nature of the flood-frequency graph. The 

general formula therefore is:

z=(i-Py (i)

2>=^ (2)

in which z represents the probability (percent chance) that a flood of 

recurrence interval T or probability p will be the maximum in an 

w-year period.

Assume that several records of floods are available, each of n years. 

The top floods in each list would vary greatly in their magnitude; 

some would be of 5-year size, some of 50-year size, and so on. The 

ratio s given by equations 1 and 2 may also be viewed as representing the relative proportion of floods less than the 77-year flood among 

the top floods in the groups of n floods. If there were several lists 

of n floods, one could verify equation 1, but there is only one list 

of floods of n years and only one maximum of this list. The problem 

is to assign a representative plotting position to this single experience.
FLOOD-FREQUENCY ANALYSES 49

A single value offers no opportunity for comparison or for any 

other measure of central tendency. The measures of central tendency 

generally are the median, the average, or the mode, as defined in any 

elementary statistics text. The only possible course is to accept one 

of these measures of central tendency; the median, the arithmetic 

mean, or the mode. The selection of one of these is the basis for 

the several formulas for "plotting positions."

MEDIAN

This assumption was introduced by Beard (1943). The value of

z is set equal to 0.50. Plotting positions are therefore as follows:

i^

7)=1 0.50W or the reciprocal where recurrence intervals are desired. 

For recurrence intervals the solution converges toward 0.5-f-L443" 

as an approximate answer.

MEAN

In discussing plotting positions Gumbel (1945b, p. 70) says, "We 

may conclude from the distribution of the largest value that the mean 

surpasses the median, and the occurrence-interval surpasses 1.433". 

Therefore, the mean of the largest value cannot be used for the determination of the occurrence-interval." In his report, Gumbel does 

not consider the mean of the probabilities of the largest values. 

However, in a later article Gumbel and von Schelling (1950) discuss 

the distribution of exceedences from which Chow (1953) show the

mean number of exceedences in N future trials to be N= nr > or n-\-l '

simply - for the probability of the mean number of occurrences.

This is the formula for plotting position used by the Survey. This 

formula can be derived for the special case of the highest value, 

m=l, simply by direct integration of equation 1 between the limits 

of 0=0, and z=\. The integration is as follows:

I (\ zn )dz -Jo V / =

p f]

I dz

Jo

for the plotting position of the highest value in terms of its probability 

and n+1 in teims of recurrence interval.

This formula for plotting position can also be stated as follows: 

A list of n events contains (n+1) class intervals. A future event 

has equal probability of falling within any of these intervals. Because
50 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

the highest value represents the bottom of the top class interval, it 

represents an event which has a probability of ., of being equaled

or exceeded.

MODE

This is the basis of the plotting positions employed by Gumbel in 

his original treatment of the problem. Equation 1 has no mode, 

hence a modal position can only be considered in relation to a distribution of the flood magnitudes. Combining equation 1 with the 

equation for the distribution of extreme values l p=e~e~v , in which 

y is a linear function of the discharges, leads to the equation

d?z -

z=(e~e ")*. Setting ^ 2 equal to zero for the mode gives: p=l en 

ay

for the probability of the mode. For values of n greater than 10,

1 I

the value of l e n equals very closely. Hence, where there is a

fi.

long record, the modal plotting position in terms of recurrence intervals equals n the number of years in the record. This is the procedure in the California method and is implied in duration-curve 

work.

How does the Hazen method (see p. 15) fit into the scheme of plotting position? Although it is not defined by reference to equation 1, 

it is possible to determine where it fits. According to the Hazen 

formula, the highest flood has a plotting position that corresponds to

2n in terms of recurrence intervals and ^- in terms of probability p. ZiTl

/ 1V Therefore z=( 1 J whereby 2=0.60 for most values of n.

Comparison of the results of the preceding plotting positions for the highest flood

Method

Hazen _ __ __

Beard.... '________

GumbeL

Designation

Median

IS^ean

Plotting position

1

2n

1-0.50"

1

n+1i

i-i"

e

Percentage of

future floods

that will be less 

than plotting

position

[100- (1-f)]

60

i 50

40

37

1 By definition.
FLOOD-FREQUENCY ANALYSES 51

The final column lists the percentage of the floods in the future 

that will be less than the given plotting position. The Hazen method, 

which gives the highest position, will be fallen short of by 60 percent 

of future floods, whereas the Gumbel modal position will be less than 

37 percent of the future floods.

Also note that the "mean" position lies in between those of the 

median and the mode, which may seem anomalous, because in most 

examples it is the median that lies between the mean and the mode. 

However, the mode in this table applies to a somewhat different 

situation than the median and mean.

The selection of a plotting position for the top flood of a list is 

like choosing a stand on a political question; there is a wide range 

of reasonable positions. But uniformity is essential if reports on 

flood frequencies are to be comparable. The term (n+1) for the 

recurrence interval of the highest flood of record was adopted for 

use in flood-frequency reports of the Geological Survey because it 

corresponds to a mean probability and because it is simple to apply.

CHARACTERISTICS OP FREQUENCY CURVES BASED ON 

A THEORETICAL 1,000-YEAR RECORD

By M. A. BEN SON

This report contains a summary of the procedures followed and 

the results obtained in analyzing a "perfect 1,000-year record" of 

maximum annual peak floods.

The object of the investigation was to study possible variations in 

frequency curves computed from short periods of record, each taken 

from a long-term record whose frequency characteristics were exactly 

determinable. There is no "theoretically correct" frequency distribution of flood events, although claims have been made for one or 

another system either based on empirical evidence or (with necessary 

assumptions) on statistical theory. For this reason, we are not sure 

of the "true" shapes of the frequency curves of even the longest flood 

records.

It was therefore decided to start from an assumed list of 1,000 

peak floods, defining exactly a simple-frequency curve. Separate 

parts of the array could be analyzed independently and the results 

compared with the known characteristics of the base curve. This 

could demonstrate the variations, due to chance alone, in frequency 

curves from short records.

BASE DATA

The data forming the basis for this study are a list of 1,000 figures 

of varying magnitudes between the arbitrary limits of 9,910 and 380. 

A partial listing of the base data is shown in the following table:
Flood frequency data, 1,000-year record (par

Order No. 

(1-1,000)

(TO)

1

I .

2........................

3 .

4...-.-..-...-...-.-.-...

5 ............... ......

6.. ...................

7... .

8

9

10 .

11 .

12.......................

13.-.--... .... ....

14.......................

15 

T

(years) 

1,000+1

TO

2

1,001 

500 

334 

250 

200 

167 

143 

125 

111 

100 

91.0 

83.4 

77.0 

71.5 

66.7

Discharge

(cfs)

3

9,910 

9,180 

8,730 

8,410 

8,190 

7,990 

7,820 

7,670 

7,540 

7,420 

7,310 

7,210 

7,120 

7,030 

6,960

Year 

(1-1,000)

(TO)

4

957 

456 

167 

348 

75 

237 

977 

276 

175 

868 

146 

597 

410 

137 

305

Discharge

(cfs)

5

1,220 

2,980 

4,280 

3,370 

5,180 

3,840 

1,040 

3,660 

4,220 

1,690 

4,430 

2,550 

3,130 

4,500 

3,540

10-year period

Order 

No.

6

9 

7 

2 

6 

1 

4 

10 

5 

3 

8

2 

8 

6 

1 

5

T

(years)

7

1.22 

1.57 

5.50 

1.83 

11.0 

2.75 

1.10 

2.20 

3.67 

1 38

5.50 

1.38 

1.83 

11.0 

2.20

25

Orde

No.

8
FLOOD-FREQUENCY ANALYSES 53

The data used were obtained as follows:

An arbitrary straight line was drawn on the frequency graph of

figure 19, with an arbitrary discharge (ordinate) scale. 

One thousand values of discharge were read from this graph for

n+1 abscissa values of T -

m

where n equals 1,000, and m, the

order number, ranges from 1 to 1,000. The computed values of 

T and the corresponding discharges are shown in columns 2 and 3 

of the table.

3. Numbers ranging from 1 to 1,000, representing m, were written on 

slips of papers, and these were thoroughly shuffled, then with7 

drawn one at a time in random order, as shown in column 4 of the 

table. The corresponding discharges are shown in column 5.

The random variation from an "infinite" population could have 

been approximated by returning each slip to the pack and reshuffling 

before drawing another. However, the infinite range in discharge 

existing in an infinite population could not be duplicated, so that the 

resultant arrray would have uncertain properties. The objective 

was to obtain a random distribution of floods in a postulated 1,000- 

year record.

The list of 1,000 peak floods is similar to what might happen in 

nature, assuming that in a 1,000-year period a "perfect" distribution 

can occur. In this distribution one 1,000-year flood; two 500-year

10,000

345 10 20 50 

RECURRENCE INTERVAL, IN YEARS

FIGURE 19. Theoretical frequency curve.
54 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

floods; three 333-year floods; and so on occur throughout the entire 

range. They are in random order, and short consecutive periods of 

this array are similar to what might be found in short periods of 

record in nature. Analysis of the frequency curves computed from 

short periods may be compared with the known characteristics of the 

basic long-term frequency curve. Since the order of the items was 

established by random drawing, any differences from the basic curve 

are due to chance alone.

MEAN ANNUAL FLOOD

According to the theory of largest values (Gumbel, 1945a), the 

mean of all the annual floods has a value corresponding to the flood 

of 2.33-year recurrence interval. The arithmetic mean of the 1,000 

items shown in column 5 of the preceding table is 3,064, which is 

the same as the value at 2.33-years on the base curve of figure 19.

SHORT PERIODS OF RECORD 

PERIODS USED

The base data were divided into "records", of various lengths for 

purposes of analysis; for example, in constituting twenty 50-year 

"records" the first 50 on the random list was the first "record" and 

so on. Division was as follows:

1. One hundred consecutive 10-year periods: 

Items 1-10, 11-20, 21-30 991-1,000.

2. Forty consecutive 25-year periods: 

Items 1-25, 26-50 976-1,000.

3. Twenty consecutive 50-year periods: 

Items 1-50, 51-100 951-1,000.

4. Ten consecutive 100-year periods: 

Items 1-100, 101-200 901-1,000.

Each of the 170 periods was analyzed separately, as though it were 

an independent record of 10, 25, 50, or 100 years. Recurrence intervals were computed and plotted and individual frequency curves were 

drawn for each.

DRAWING OF FREQUENCY CURVES

In drawing the frequency curves, the same procedures were followed as when actual station records are being studied:

1. The curves were drawn by eye as the graphical curves of best fit.

2. They were drawn as straight lines or curves depending on the trend of the 

points.

3. The curves were drawn to average rather than to follow individual points, 

the object being to avoid sharp breaks or bends.

4. The curves were kept as close as was reasonably possible to the plotted 

points in the vicinity of the 2.33-year line.
FLOOD-FREQUENCY ANALYSES 55

5. Floods of high recurrence interval are likely to occur within a short period 

of record. For this reason, the highest points were not given much weight 

in defining the curves, if they were to the left and above the trend line 

through the remaining points. If the higher points lined up consistently 

with each other and with the lower points, they were followed although 

the resulting curve bent upward to the left.

It is not possible to avoid the personal factor in drawing curves 

fitted visually as described here. However, this is believed the most 

practical method and gives the best results. A general confirmation 

of the method was furnished by the following means. Values of the 

2.33 (mean annual), 10-, 25-, 50-., and 100-year floods were taken 

from the individual curves (except that the 50-year flood was the 

highest taken from the 10-year curves). The averages of these values 

compared with the known true values (from the base curve, fig. 19) 

are shown in the following table:

Comparison of true values with averages of computed values of mean annual flood

Length of record 

(years)

10- 

25__-_-_ __

50..- 

10(L-_-____-

True value. __

Mean 

annual 

flood (cfs)

3, 100

3, 110

3, 100

3,080

3,064

10-year 

flood (cfs)

5, 060

4, 940

4,900

4,860

4,860

25-year 

flood (cfs)

6, 160

5,960

5,900

5,940

5,860

50-year 

flood (cfs)

6,970

6,720

6,660

6,770

6,680

100-year 

flood (cfs)

7, 440

7,390

7,570

7,420

Most of these averages are within small percentages of the known 

values. The average values of the 10-, 25-, and 50-year floods as derived from 10-year records are between 4 and 5 percent higher than 

the true values. This may seem excessive and might demonstrate 

that curves are being drawn too high. However, a review of the 

individual frequency curves shows that this is not a general tendency.

There is always some uncertainty as to whether to draw the curve 

to follow the upper points, if it is necessary to bend the curve upward 

to do so. This study can give no guidance for any single example 

which may be met in practice, but some frequency curves have such a 

tendency. The only way to determine whether such a graph represents a physical actuality or is merely chance, is to compare it with 

curves for other places in the region.

Some investigators prefer to fit frequency curves as straight lines 

by computations as outlined by Powell (1943). A serious objection 

to this procedure is that if a short period of record contains one or 

more long-term floods, the line is unduly influenced by these and may 

be drawn to the left of most of the other floods. To avoid this, some
56 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

investigators advocate drawing a straight line through the lower 

points, giving little weight to the upper points. Where all points 

indicate definite curvature throughout the range, this procedure becomes wholly arbitrary and subjective. The greatest objection to the 

drawing of straight-line graphs, whether computed or by eye, is that 

it is first necessary to support the belief that they should be straight 

lines. There is no firm basis for rigid adherent to straight lines.

The individual frequency curves for 10-, 25-, 50-, and 100-year 

periods are shown as figures 20 to 23, respectively. All those based 

on a common length of period have been plotted together to show 

graphically the extent of variation obtained.

ARITHMETIC VERSUS GRAPHICAL MEAN ANNUAL FLOOD

The mean of the annual floods, or "mean annual flood," is an extremely important factor, used in correlation studies and in regional 

flood-frequency compilations. It is desirable to determine it from 

fairly short-term records.

In short-term records particularly, the arithmetic mean is affected 

considerably by the chance inclusion of one or more major floods. 

The Geological Survey uses the graphical mean annual flood to avoid 

this adverse condition. The graphical mean is the value determined 

by the intersection of the visually best fitting frequency curve with 

the mean line (the line corresponding to the 2.33-year recurrence interval). The graphical mean is more stable and dependable than 

the arithmetic mean. This method of determining the mean gives 

greater weight to the medium floods than to the extreme floods with 

large sampling errors. The resulting figure is no longer the mean 

of the annual floods, but is the "mean annual flood" by definition.

This investigation makes possible a practical comparison between 

values of arithmetic and graphical means, and a demonstration of 

the variation in the value of the mean with the length of the record.

The average of the arithmetic means of all the short-term periods 

is 3,064. The averages of the graphical means, as obtained from the 

individual short periods, vary as follows:

Length of

record

(years)

10

25

50

100

Number of

records

100

40

20

10

Average of

graphical

means

3, 100

3, 110

3, 100

3,080

These are within small percentages (less than 1.5 percent) of the 

true values, so that no gross errors are involved in using the graphical mean.
FLOOD-FREQUENCY ANALYSES 57

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58 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

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FLOOD-FREQUENCY ANALYSES 59

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60 MANUAL OF HYDROLOGY! FLOOD-FLOW TECHNIQUES

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FLOOD-FREQUENCY ANALYSES 61 

RELIABILITY OF MEAN ANNUAL-FLOOD VALUES

In statistical studies, the most favorable expectancy during 95 percent of the time is commonly used as the criterion for dependable 

results.

The extent of variation from the true mean of the mean annual 

floods (determined graphically) in this study, during 95 percent of 

the time, was 28 percent for 10-year periods, 14 percent for 25-year 

periods, 12 percent for 50-year periods, and about 5 percent for 100- 

year periods.

These percentages cannot be assumed to apply generally, because 

they will vary with the slope of the base-frequency curve. They are 

a general indication of what may be expected from short- and longterm records. The 10-year records will give a much wider range 

in determining the mean.

It is desirable to have some means of allowing for the range in 

value due to chance, and a method is hereby outlined which gives 

consideration to the length of the record from which the mean annual 

flood has been determined.

"WORKING RANGE" OF MEAN ANNUAL FLOOI>S

Based on the theory of extreme values, it is possible to compute, 

for conditions existing 95 percent of the time, and for any length of 

record, the range in recurrence intervals which would be found for 

the 2.33-year flood. The theoretical limiting curves for these values 

are shown as figure 24.

These curves are generally applicable. The extent to which results 

of this 1,000-year frequency study conform with the theoretical 

curves is shown by the plotted points on figure 24. These points represent the apparent recurrence intervals corresponding to the known 

mean annual flood of 3,064, as determined from the 170 individual 

frequency curves developed in the study. Theoretically, 5 percent 

of these points, or 8.5 points, should be outside the limits; actually, 

7 of the points lie outside.

In regional flood-frequency studies, the relation between the mean 

annual flood and the drainage area (or other basin characteristics) 

is required and can be determined graphically by plotting one against 

the other. Instead of plotting a point, a range line may be plotted 

using the chance range in the mean annual flood based on the length 

of record. An average curve passing through all the range lines 

may then be considered as the best determination of the relation 

curve.

The method of determining the chance or working range of the 

mean annual flood is illustrated in figure 25. Curve A is an assumed 

frequency curve, based on a presumed record of 10 years. The curves
62 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

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FLOOD-FREQUENCY ANALYSES 63

6000

5000

4000

03000

UJ CD 2000

1000

1.1 1.5 2 5 10 20 30 

RECURRENCE INTERVAL, IN YEARS

FIGURE 25. Computing range for mean annual flood.

of figure 25 show that the recurrence interval for a mean annual flood 

based on 10 years of record might have an apparent recurrence interval of between 1.30 and 5.20 years (95 percent of the time). The 

mean annual flood, determined from curve J., is 3,000 cfs. Curves 

B and C are drawn parallel to curve A passing through 3,000 cfs at 

recurrence intervals of 1.30 and 5.20 years. These represent the extreme positions which curve A might take due to the chance variation in 10 years of record. (It is assumed that the slope would be 

parallel within these short segments.) These curves intersect the 

2.33-year line at 2,100 and 3,800 cfs; this range is interpreted as 

showing that the actual value of the mean annual flood might, due 

to chance variation, lie anywhere between 2,100 and 3,800 cfs. The 

mean annual flood for this station record is plotted as a range line 

between these limits, against any other desired factor.

RELIABILITY OF FLOOD MAGNITUDES

A study was made of the maximum and minimum values of the 

10-, 25-, 50-, and 100-year floods as determined graphically from records of various lengths, for 80, 95, and 100 percent of the time.

These results show, for example, that (19 out of 20 times) for 

these data, any 39-year record can define the 50-year flood within 25 

percent of the true, long-term value.
64 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Lengths of record necessary to come within 25 percent of the correct value 95 or 80

percent of the lime

Magnitude of flood 

(Tin years)

2.33_--_ --__-

10

25______- ________

50_-__- _ ___

100- _

Length of record 

in years

95 percent 

of the time

12 

18 

31 

39 

48

80 percent 

of the time

8 

12 

15

Lengths of record necessary to come within 10 percent of the correct value 95 or 80

percent of the time

Magnitude of flood

Length of record 

in years

( T in years)

2.33

10______ ____

25-_ _--

50__--__

100_--_-

95 percent 

of the time

40

90

105

110

115

80 percent 

of the time

25

38

75

90

100

In recent years, many crest-stage gaging stations have been established for the purpose of defining the flood potential of a region. 

The mean annual flood can be determined within 25 percent (95 percent of the time) by a 12-year record of such stations.

The results of this study are a general qualitative indication of 

long- and short-term records. They show, for example, that if we 

do not demand too great a degree of accuracy, the 50- and 100-year 

floods may be determined from the 40- or 50-year records which are 

commonly available.

It seems that the individual short-term station record is perhaps 

less reliable than we have generally considered it, but the individual 

long-term record is surprisingly dependable.

For less than the 100-year flood, a longer record than the period 

of the desired flood is necessary, for the result to be within 10 percent 

of the correct answer. This is increasingly true for the short-term 

floods. The correct answer for short-term floods probably would be 

10 percent or less. In general, we should feel content if we are reasonably sure of predicting a given flood within 25 percent.

The figures shown cannot be expressed as percentages of the correct values and applied generally. The percentages are dependent 

on the slope of the individual frequency curve in question.
FLOOD-FREQUENCY ANALYSES 65

Using the theory of extreme values, theoretical curves can be derived showing, for 95 percent expectancy, the range of recurrence 

interval which might be assigned to an actual flood of known recurrence interval. This set of curves for the 10-year flood is shown as 

figure 26. (This is the same set of curves used in the homogeneity 

test.) On figure 26 are plotted the recurrence intervals corresponding to the actual 10-year flood (4,860 cfs), as determined from the 

170 individual frequency curves of this study. Theoretically, 5 percent of these points, or 8.5 points should lie outside the limits; actually, 8 lie outside.

COMPOSITE FLOOD-FREQUENCY CURVES

The preceding section gives actual figures on the variability of 

results from different periods of record. Figures 20 to 23 show the 

same record graphically. Figure 20 illustrates the variability among 

individual short-term records. Many of the records are short, but 

in spite of their unreliability each one is valuable in adding to the 

accuracy of the basic relations to be established.

The Geological Survey makes predictions of frequency on regional 

flood-frequency studies where possible. For methods of regional 

analysis see Dalrymple (1950), Carter (1951), and other published 

reports. Evidence indicates that a flood-frequency graph that is 

based on the combined experience of a group of stations has firmer 

support than one drawn to fit the data at a single station. The process of combining the individual records tends to minimize the effect 

of the erratic samples.

In combining a group of station records, the individual records 

are not combined end-to-end to produce a long-term record. This 

procedure, (the "station-year method") is not considered justifiable 

in the use of discharge records. Combining ten 10-year records does 

not give a 100-year record but can provide a more dependable 10- 

year record, if records for homogeneous regions are being combined.

The data for this study afford an opportunity for a general check 

on these methods. Individual station records in the same general 

area show about as much variation as appears in the curves of figures 

20, 21, 22, or 23. Therefore it was justifiable to treat them as though 

they were individual records in producing composite curves, and the 

results probably are comparable with those found in combining separate station records.

PERIODS COMBINED

Using procedures as outlined according to Dalrymple (1950) and 

Carter (1951), the short periods of record were combined as follows: 

1. The first 10, second 10, and so forth of the 100 10-year periods 

were combined, giving a total of 10 composite graphs.
66 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

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FLOOD-FREQUENCY ANALYSES 67

2. The first 10, second 10, and so forth, of the forty 25-year periods 

were combined, giving 4 composite graphs.

3. The first 10 and second 10 of the twenty 50-year periods were 

combined, giving 2 composite graphs.

For each single record, the separate items were expressed as ratios 

to the individual graphic means. Homogeneity tests were applied 

and in a few examples a record was discarded and not combined with 

the group. This practice was followed although the true mean was 

known and the separate parts all came from a single and therefore 

homogeneous record. The process was the same as is followed when 

combining actual station records.

The composite graphs derived in this manner are shown in figures 

27 to 29 for combined 10-, 25-, and 50-year periods, respectively.

RESULTS FROM COMPOSITE GRAPHS

Analysis of these curves shows that the composites of the 10-, 25-, 

and 50-year periods define the 100-year flood within 22.7, 18.2 and 

4.5 percent respectively. These are the outside limits represented 

by the curves most of the values are much closer. These results 

cannot be applied quantitatively, however they are near the correct 

value. By means of any one of these combinations of 10-year periods, 

the 100-year flood could be predicted within 22.7 percent.

EFFECT OF NUMBER OF STATIONS COMBINED

An investigation was made of the accuracy of results as affected 

by the number of stations used to define a composite curve. Five 10- 

year periods were chosen at random, and a composite curve derived 

from them. Five more periods were chosen at random, and a composite curve based on these 10. The process was repeated up to the 

final combining of 20 records.

Results from the first five periods were so near the true values 

that there was not much opportunity for improvement with an increasing number of records. A second trial was therefore made repeating the entire process. For this trial, the values jumped erratically and showed no improvement with number in fact, the results 

of combining 20 records were less accurate than for 5 records.

After a large number of records have been combined (possibly 

50 or 100), there should be a definite trend toward the correct answer. Apparently with a small number of records, and both 5 and 

20 are small numbers statistically in this sense, the results of chance 

may be very erratic. There may be available 5 records whose composite answer is near the true values, or there may be 20 records 

whose composite result is less accurate. However, the results of using 

composite frequency curves seems to be good enough even with a 

small number of stations available. Using medians and combining
68 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

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FLOOD-FREQUENCY ANALYSES 69

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3.0

2.6

§2.2

1.8

o 

o

1.4

1.0

0.6

0.21

1.01 1.1 1.5 2 5 10 

RECURRENCE INTERVAL, IN 

FIGURE 29. Composite frequency curves based on 
FLOOD-FREQUENCY ANALYSES 71

from five to twenty 10-year records, the 10-year flood could be predicted with 5.7 percent, the 25-year flood within 7.3 percent, the 50- 

year flood within 7.8 percent, and the 100-year flood within 8.7 percent.

Statistical theory gives some indication of what to expect from 

increasing samples. The results of composite curves may be considered as mean values and probably the results have somewhat the 

same characteristics as means. Statistics state that the standard deviation of the mean of samples drawn from a single population varies 

inversely as the square root of the sample size. The standard deviation is a measure of the expected departure from the true value, and 

samples of 20 should have one-half the standard deviations of samples of 5, on a long-time average.

MEDIAN VERSUS AVERAGE RATIOS

In combining the dimensionless flood ratios in the process of obtaining a composite graph, the median value of the ratios for any 

specific order number has ordinarily been used (Dalrymple 1950, 

p. 15, col. 9). There is some question as to whether it might be preferable to use the arithmetic average instead of the median.

A comparison of the two has been made for all the composite 

graphs developed in this study. Because the correct values of any 

frequency of flood are known, the departures from the correct values 

are an indication of the validity of any procedure.

The ranges in value of the 10-, 25-, 50-year floods, based on records 

of various lengths, show that use of the average rather than the 

median flood ratios practically does away with the apparent inconsistency, previously mentioned, shown by 10- and 25-year records. 

However, both the minimum and maximum values are increased.

Studies were made of the values of 10-, 25-, 50-, and 100-year floods 

based on the 2 trials in which the number of combined stations was 

increased from 5 to 20. The values based on averages were generally 

less accurate than those based on medians. Also, those based on 

averages were predominantly too high.

On the basis of these two sets of comparisons, the medians have 

an advantage on the averages, and it seems best to continue the use 

of medians as recommended. The use of medians rather than averages agrees with the use of graphical rather than arithmetic means. 

Both procedures tend to nullify the effect of abnormally large floods 

in sKort-term records.

THEORETICAL DISTRIBUTION OF COMPOSITE CURVES

The results of using composite curves were so favorable that a 

theoretical stuHy was made to confirm these specific results.
72 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

The following discussion refers to figure 30. Lines A and F represent the extreme positions of a group of frequency curves based on 

the same period of record such as the group of curves in figure 20.

Lines B, C, Z>, and E divide the total range into 5 areas 1, 2, 3, 4, 

5), each containing 20 percent of the entire number of curves. Because the curves do not necessarily parallel these lines, but may cross 

them, the lines, more precisely, have such a position that for any 

recurrence interval one-fifth of the curve intersections would lie 

within each area.

All points within each of the areas 1, 2, 3, 4, and 5 have been represented by average numerical values of 1.0, 2.0, 3.0, 4.0, and 5.0. The 

positions of the averages of several points are represented by the 

numerical average of their individual values. Where the analysis 

applies equally at any recurrence interval, applies to the composite 

frequency curve.

Consider a curve in area 1 and a curve in area 5; if they are combined they will have a position somewhere in area 3. This is repre10,000

9000

8000

7000

6000

5000

-.4000

3000

2000

1000

(5.0)

1.01 1.1 1.5 2 3 4 5678910 

RECURRENCE INTERVAL, IN YEARS

20 30 40 50

FIGURE 30. 'fhqperetical distribution of composite curves.
FLOOD-FREQUENCY ANALYSES 73

sented numerically by the average of their numerical values, 1.0 and 

5.0, is 3.0. The distribution is slightly unbalanced, but in actual 

practice, medians are used rather than averages, and the effect of 

medians is to lower the result somewhat.

The effect of combining any five curves can be represented by the 

average of their numerical values. If all 5 curves were in area 1, 

the numerical value of their average would be 1.0; if all 5 curves were 

in area 5, their average would be 5.0. There are 3,125 (that is 55 ) 

possible variations within the 5 areas of 5 curves chosen at random. 

These may be considered as all the possible combinations of 5 numbers, each of which may have the value 1, 2, 3, 4, or 5. The average 

of each combination is then computed. All averages lying between 

1.0 and 1.5 are considered as lying in area 1, averages between 1.5 

and 2.5 as lying in area 2, and so forth. The distribution of all the 

3,125 means is shown below in percentage.

The number of stations combined in regional frequency studies 

generally is 10 or more. If 10 records are combined, there are 

9,765,625 (that is, 510 ) possible variations. By statistical theory, 

(using the binomial theorem), the variation of the means has been 

computed and the percentage of distribution is as follows:

Percentage of distribution

Area

1

2______ ___________

3-

4-----------------

5-_._ ___________

Total_______

5 records 

combined

0.7

21. 3

56.0

21.3

.7

100.0

10 records 

combined

0.04

13. 1

73.7

13. 1

.04

100.0

These results indicate that the composites of either 5 or 10 stations 

have only a negligible chance of lying in either areas 1 or 5. With 

5 records combined, chances are better than 50-50 that the composite 

curve will lie within the narrow band of area 3, and with 10 combined records, chances are 3 out of 4 that it will do so.

CONCLUSIONS

1. A confirmation by actual computation is obtained of the theoretically derived value of 2.33 years as the recurrence interval for 

the mean annual flood, for a straight-line distribution on a Gumbel 

graph.

2. A general confirmation is obtained of the method of drawing 

frequency graphs as visually best-fitting curves.
74 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

3. The graphical mean gives good results in general and is to be 

preferred to the arithmetic mean since it avoids large errors due to 

the occurrence of large floods within a short period.

4. The reliability of the mean annual flood increases with length 

of record. Short periods of record do not define it closely enough 

for some purposes. A method is available whereby a "working 

range" can be computed and used in correlation studies.

5. The reliability of flood magnitudes other than the mean also 

increases with length of record. Short periods of record (up to 

about 25 years) cannot define satisfactorily even short-term floods. 

Long-term records (40 to 50 years or longer) can define flood magnitudes up to the length of record or longer, depending upon the accuracy required. The individual short-term record is perhaps less reliable than generally considered, but the individual long-term record 

is surprisingly dependable.

6. Accuracy within 10 percent from individual station records is 

rarely attainable. Accuracy within 25 percent seems attainable with 

a fair length of record.

7. The results of combining even short-term records, similar to 

procedures in regional flood studies, are good.

8. The number of station records generally combined is always 

statistically small, and, a combination of 5 records may give as good 

results as a combination of 20 records. However, even a combination 

of five records can give reliable results.

9. Accuracy of 10 percent seems indicated from composite curves 

for frequencies ordinarily desired (50 years or less) based on records 

of the number and length generally available.

10. Theoretical consideration of the results of combining frequency 

curves shows a high degree of probability for favorable results.

SELECTED REFERENCES

American Society Civil Engineers, 1951, Review of flood frequency methods: 

Proc. Separate 110; 1953, Trans., v. 118, p. 1220-1230.

Anderson, H. W., 1949, Flood frequencies and sedimentation from forest watersheds : Am. Geophys. Union Trans., v. 30, p. 567-584.

Arkin, Herbert, and Colton, Raymond, 1939, An outline of statistical methods: 

New York, Barnes & Noble, 220 p.

Beard, L. R., 1943, Statistical analysis in hydrology: Am. Soc. Civil Engineers 

Trans., v. 108, p. 1110-1160.

 1952, Statistical methods in hydrology: U.S. Army, Corps of Engineers, 

Civil Works Engineer Bull. 52-24.

1954, Estimation of flood probabilities: Am. Soc. Civil Engineers Proc.

Separate 438. 

 1956, Statistical evaluation of runoff volume frequencies : Internat. Assoc.

Hydrology (de 1' U.G.G.I.), Symposia Darcy, pub. no. 42, p. 171-183. 

Benson, M. A., 1950, Use of historical data in flood-frequency analysis: Am. 

Geophys. Union Trans., v. 31, p. 419-424.
FLOOD-FREQUENCY ANALYSES 75

Benson, M. A., 1952, Characteristics of frequency curves based on a theoretical

1,000-year record: U.S. Geol. Survey open-file report 

Bernard, M. M., 1936, Determination of flood flow by unit-hydrograph method,

in Jarvis, C. S., and others, Floods in the United States, magnitude and frequency : U.S. Geol. Survey Water-Supply Paper 771, p. 451-^461. 

Berwick, V. K., 1958, Floods in eastern Montana, magnitude and frequency:

U.S. Geol. Survey open-file report. 

Bigwood, B. L., and Thomas, M. P., 1955, A flood-flow formula for Connecticut:

U.S. Geol. Survey Circ. 365, 16 p. 

Bodhaine, G. L., and Robinson, W. H., 1952, Floods in western Washington,

frequency and magnitude in relation to drainage-basin characteristics: U.S.

Geol. Survey Circ. 191, 124 p. 

California Department of Public Works, 1923, Flow of California streams:

Div. Engineering and Irrigation, Bull. 5, chap. 5. 

Carter, R. W., 1951, Floods in Georgia, frequency and magnitude: U.S. Geol.

Survey Circ. 100, 127 p. 

Chow, Ven Te, 1950, Discussion of Annual floods and the partial duration flood

series, by W. B. Langbein: Am. Geophys. Union Trans., v. 30, p. 939-941.

 1951, A general formula for hydrologic frequency analysis: Am. Geophys. Union Trans., v. 32, p. 231-237.

 1953, Frequency analysis of hydrologic data with special application to

rainfall intensities: Illinois Univ. Bng. Bxpt. Sta. Bull. 414. 

 1954, The log-probability law and its engineering applications: Am. Soc.

Civil Engineers Proc. Separate 536.

1956, Hydrologic studies of floods in the United States: Internat. Assoc.

Hydrology (de 1' U.G.G.I.), Symposia Darcy, pub. no. 42, p. 134-170.

Cragwall, J. S., Jr., 1952, Floods in Louisiana, magnitude and frequency: Louisiana Dept. Highways, 281 p.

Creager, W. P., Justin, J. D., and Hinds, J., 1944, Engineering for dams: New 

York, John Wiley & Sons, v. 1, 245 p.

Cross, W. P., 1946, Floods in Ohio, magnitude and frequency: Ohio Dept. Public 

Works, Water Resources Board Bull. 7, 154 p.

Dalrymple, Tate, 1950, Regional flood frequency: Highway Research Board, 

Research Rept. no. 11-B, p. 4-20.

 1957, Flood frequency relations for gaged and ungaged streams : Internat. 

Assoc. Hydrology (de 1' U.G.G.I.), Gen. Assembly Toronto, pub. no. 45, p. 

268-279.

Ezekiel, Mordecai, 1945, Methods of correlation analysis: New York, John Wiley 

& Sons, 522 p.

Fisher, R. A., 1946, Statistical methods for research workers: London, Oliver 

& Boyd, 354 p.

Fisher, R. A., and Tippett, L. H. C., 1928, Limiting forms of the frequency distribution of the largest or smallest member of a sample: Proc. Cambridge 

Philos. Soc., v. 24, p. 180-190.

Foster, E. E., 1948, Rainfall and runoff: New York, Macmillan Co., 487 p.

Foster, H. A., 1924, Theoretical frequency curves and their application to engineering problems: Am. Soc. Civil Engineers Trans., v. 87, p. 142-203.

Furness, L. W., 1955, Floods in Nebraska, magnitude and frequency: Nebraska 

Dept. Roads and Irrigation, 103 p.

Grover, N. C., 1937, The Floods of March, 1936, part 1, New England rivers: 

U.S. Geol. Survey Water-Supply Paper 798, 466 p.
76 MANUAL OF HYDROLOGY: FLOOD-FLOW TECHNIQUES

Grover, N. C., 1937, The Floods of March, 1936, part 2, Hudson River to Susquehanna Biver region: U.S. Geol. Survey Water-Supply Paper 799, 380 p.

 1937, The Floods of March, 1936, part 3, Potomac, James, and upper 

Ohio Rivers: U.S. Geol. Survey Water-Supply Paper 800, 351 p.

Gumbel, E. J., 1941, The return period of flood flows: Annals Math. Statistics, 

v. 12, no. 2, p. 163-190.

 1942, Statistical control-curves for flood-discharges: Am. Geophys. Union 

Trans., pt. 2, p. 489-509.

 1945a, Floods estimated by probability method: Bng. News-Rec., v. 134,

no. 24, p. 833-837. 

 1945b, Simplified plotting of statistical observations: Am. Geophys. Union

Trans., v. 26, p. 69-82. 

 1948, The statistical forecast of floods: Ohio Water Resources Board

pamphlet.

1952, Discussion of A general formula for hydrologic frequence analysis,

by Ven Te Chow: Am. Geophys. Union Trans., v. 33, p. 277-282.

1954, Statistical theory of extreme values and some practical applications : U.S. Bur. Standards Appl. Mathematics, ser. 33, 51 p. 

 1958, Statistics of extremes: New York, Columbia Univ. Press, 371 p.

Gumbel, B. J., and Von Schelling, H., 1950, The distribution of the number 

of exceedances: Annals Math. Statistics, v. 21, no. 2, p. 247-262.

Hazen, Alien, 1930, Flood flows: New York, John Wiley & Sons, 199 p.

Hoel, P. G., 1947, Introduction to mathematical statistics: New York, John 

Wiley & Sons, 258 p.

Horton, R. E., 1936, Hydrologic conditions as affecting the results of the application of methods of frequency analysis to flood records, in Jarvis, C. S., and 

others, Floods in the United States, magnitude and frequency: U.S. Geol. 

Survey Water-Supply Paper 771, p. 433-450.

Hoyt, W. G., and others, 1936, Studies of relations of rainfall and run-off in 

the United States: U.S. Geol. Survey Water-Supply Paper 772, 301 p.

Jarvis, C. S., and others, 1936, Floods in the United States, magnitude and frequency : U.S. Geol. Survey Water-Supply Paper 771, 497 p.

Johnstone, Don, and Cross, W. P., 1949, Elements of applied hydrology: New 

York, Ronald Press, 276 p.

Kimball, B. F., 1946, Assignment of frequencies to a completely ordered set of 

sample data: Am. Geophys. Union Trans., v. 27, p. 843-846.

Kinnison, H. B., and Colby, B. R., 1945, Flood formulas based on drainage 

basin characteristics: Am. Soc. Civil Engineers Trans., v. 110, p. 849-904.

Langbein, W. B., 1949, Annual floods and the partial-duration series: Am. 

Geophys. Union Trans., v. 30, p. 879-881.

Langbein, W. B., and others, 1947, Topographic characteristics of drainage 

basins: U.S. Geol. Survey Water-Supply Paper 968-C, p. 125-157.

Lieblein, Julius, 1954, A new method of analysing extreme-value data: U.S. 

Bur. Standards, Nat. Advisory Comm. Aeronautics Tech. Note 3053, 88 p.

Linsley, R. K., Kohler, M. A., Paulhus, J. L. H., 1949, Applied hydrology: New 

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 1958, Floods in Kentucky, magnitude and frequency: U.S. Geol. Survey 

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FLOOD-FREQUENCY ANALYSES 77

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INDEX

Page

Acknowledgments..___-___-.____.. 1 

Adjustment from short to long period...-__ 40,42 

Annual flood, definition.___.-._____ 5 

Annual-flood series__..__ _____ 5,6,8 

Annual reports, preparation 11

Bankfull stage . . . - 17 

Bar graph_________..... _____ 34,45

Base discharge, selection... _. 11,12,13 

Base period, adjustment 34,45 

selection .. 32 

Basic-frequency curve 26 

Basin lag-...--. . .. 30 

Beard method-_.......__.....__... . 16

Big Piney Run near Salisbury, Pa..-.____ 36,37

"California" method.________ __ 15,16,50 

Channel storage, effect on mean annual flood.. 41,43 

Combining of short periods. ______ 65 

Combining records within a region. - __ __ 25,73 

Composite curves, theoretical distribution- 71 

Composite factors..____ _____ 30 

Correlation with physical factors.- 43 

Cyclic changes in weather.. ... . 32

Data, listing . 8-13

Degrees of freedom 31 

Dimensionless frequency curves 26,27 

Discharge-frequency curve - 5 

Discharge records, length- .......... 3,18,20

Drought years, effect on discharge . 43

Envelope curve__ . 21 

Estimation of years of no record- . .... 35 

Examples of regional reports - 47 

Exceedences, distribution_ .- _-_ 49 

Extrapolation.. ...... . -. 20,21

Fitting curves to data, graphically.-. 3,

20,42,43,54,56,73

mathematically- ------------- 3,20,37,56

Flexible skew-distribution curve ----- 2

Flood-control storage - 8 

Flood-frequency curves, for short-term periods-. ..... 21,24,25,40,51,54,64,74

types . -- -- 5,7,8

Flood-frequency method, application... 2,

3,5,20,21,25 

Flood-frequency relation, regional.... . 24,

25,26,27,39,43,45-46,61,65,73 

Flood-frequency, relation to drainage-basin

characteristics - 4,26,29-31,41,43,61 

Flood-frequency reports, Statewide or regional-. ..... 1,3,4,24.47

Flood insurance_ __ . _. 2 

Flood magnitudes, reliability.. . - - 63 

Flood potential of a region.-. 64

Flood stages, significance___ ____ . 10 

Flood zoning._______-----__-___ 2

Floods, considered in structure design . - 2

magnitude and frequency- --... 2,3

Frequency curve, definition.. . .. 45

Page

Gaging-station data, in water-supply papers .. 11 

Geological Survey method...-- .... 3,4,6,16,49 

Q umbel method. ........... 3,15,16,28,49,50,73

Handbook for Hydrologists._________ 2 

Hazen method . .... . 3,15,51 

Historical flood data.. 8,13,17,18,37 

Homogeneity test..______ 27-29,38,39,45,65,67 

Hurricane conditions, effect on flood peaks..... 15 

Hydrologic areas... __-__-_____ 31

Ice-affected streams . . . 7,10 

Interpolation____________.__.__ 20

Licking River at Toboso, Ohio. _______ 13,14 

Logarithmic discharge scales . 19,21,37 

Logarithmic skew-frequency curve.__ __ 3

Mean annual flood, definition__. 4,26,29,31,37,54

estimation . . 41,55-63

Mean annual rainfall_______ _ _. 30

Mean annual runoff.._ .. .. .... 30

Median flood ratio..___._______ 39,45,46

Meteorologic factors _ 30

Mississippi River, selection of stations on. 31

special treatment for. . .... 46

Multiple correlation techniques 43

Newspaper sources for historical information. 13

Nomographs...- 43

Nueces River below Uvalde, Tex ....... 13

Number of stations, effect on results .... 67

Olentangy River near Delaware, Ohio . 44

Order numbers, adjustment. 17,18

computation_.._ . . 36

Partial-duration series_ . 5,6,13,16,19,21,43,44 

Physiographic factors.. . 29 

Plotting paper, preparation___.__. . 19,21 

Plotting positions, computation_ 5,15,17,18,49-51 

Preliminary frequency curve.. 37,39; fig. 11

Rainfall-intensity frequencies 25 

Range line . 42,61 

Rectangular discharge scales 19,21 

Recurrence interval, computation 4,

15, 17, 18,19, 20, 28, 29, 31, 37, 38, 45,

47, 49, 54, 63.

definition. 5,6,15,24,72 

Red River in Louisiana.. . 47 

Regional frequency curve 27,39,40,41,43,46,65 

Relation between annual-flood and partialduration series.._ 6 

Reliability of mean annual-flood values 61 

Rules for publishing peaks 11

Sampling errors..._____-_ 2,3,24,25,37,56 

San Antonio River at San Antonio, Tex 13 

Single-station curve.. 24 

Skewness of the data.. 3,15 

Snowmelt, effect on peak discharge 12,15 

Stage-discharge relation 7

79
80 INDEX

Page

Stage-frequency curve__ _ 5 

Standard deviation, definition.. . 71 

Station-year method of analysis 25 

Straight-line graphs_ ____ __ 55,56 

Stream channels, shifting_______ ___ 7 

Streams, manmade regulation.. 11 

Susquehanna River at Harrisburg, Pa..__.. 13,18 

Suwanee River at White Springs, Fla..__.. 10

Theoretical 1,000-year record...... 24,28,27,28,51,53

Time graduation, equation___ . 19 

Total usable storage capacity_______.. 31

Use of short-term records.

Page 

. 43

Volume-frequency curveWorking range of mean annual floods.. 61

Work sheets, listing of data on______ 34

Works of man, effect on floodflow______ 4,31

Youghiogheny and Klskiminetas River basins,

flood-frequency curve defined 45 

stations used in analysis _____.... 32,33

o

U. S. GOVERNMENT PRINTING OFFICE : 1960 O - 540570