FREQUENCY CURVES - USGS

[Pages:25]Techniques of Water-Resources Investigations of the United States Geological Survey

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Chapter A2

FREQUENCYCURVES

By H. C. Riggs

Book 4 HYDROLOGIC ANALYSIS AND INTERPRETATION

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DEPARTMENT OF THE INTERIOR DONALD PAUL HODEL, Secretary U.S. GEOLOGICAL SURVEY Dallas L. Peck, Director

First printing 1968 Second printing 1969 Third printing 1978 Fourth printing 1989

UNITED STATES GOVERNMENT PRINTING OFFICE : 1968

For sale by the Books and Open-File Reports Section U.S. Geological Survey Federal Center, Box 25425 Denver, CO 80225

PREFACE

The series of manuals on techniques describes procedures for planning and executing specialized work in water-resources investigations. The material is grouped under major headings called books and further subdivided into sections and chapters; section A of Book 4 is on statistical analysis.

The unit of publication, the chapter, is limited to a narrow field of subject matter. This format permits flexibility in revision and publication as the need arises.

Provisional drafts of chapters are distributed to field offices of the U.S. Geological Survey for their use. These drafts are subject to revision because of experience in use or because of advancement in knowledge, techniques, or equipment. After the technique described in a chapter is sufficiently developed, the chapter is published and is sold by the U.S. Geological Survey Books and Open-File Reports Section, Federal Center, Box 25425, Denver, CO 80225 (authorized agent,of Superintendent of Documents, Government Printing Office).

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TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS OF THE

UNITED STATES GEOLOGICAL SURVEY

a

The U.S. Geological Survey publishes a series of manuals describing procedures for planning and conducting specialized work in water-resources investigations. The manuals published to date are listed below and may be ordered by mail from the U.S. Geological Survey, Books and Open-File Reports, Federal Center, Box 25425, Denver, Colorado 80225 (an authorized agent of the Superintendent of Documents, Government Printing Office).

Prepayment is required. Remittance should be sent by check or money order payable to U.S. Geological Survey. Prices are not included in the listing below as they are subject to change. Current prices can be obtained by writing to the USGS, Books and Open File Reports. Prices include cost of domestic surface transportation. For transmittal outside the U.S.A. (except to Canada and Mexico) a surcharge of 25 percent of the net bill should be included to cover surface transpo.rtation. When ordering any of these publications, please give the title, book number, chapter number, and "U.S. Geological Survey Techniques of Water-Resources Investigations."

TWI l-D1

TWI l-D2.

TWX 2-Dl. TWI 2-D2. TWI 2-El. TWI 3-Al. TWI 3-A2. TWI 3-A3. TWI 3-A4. TWI 3-As. TWI 3-A6. TWI 3-A7. TWI 3-A8. TWI 3-A9.

TWI 3-AlO. TWI 3-All. TWI 3-A12.

TWI 3-A13. TWI 3-A14. TWI 3-A15. TWI 3-A16. TWI 3-Al?. TWI 3-Bl. TWI 3-B2.' TWI 3-B3. TWI 3-BS.

TWI 3-B6.

TWI 3-Cl. TWI 3X2. TWI 3X3. TWI 4-Al. TWI 4-A2. TWI 4-Bl. TWI 4-B2. TWI 4-B3. TWI 4-Dl. TWI 5-Al.

Water temperature-influential factors, field measurement, and data presentation, by H.H. Stevens, Jr., J.F. Ficke, and G.F. Smoot. 1975. 65 pages.

Guidelines for collection and field analysis of ground-water samples for selected unstable constituents, by W.W. Wood. 1976. 24 pages.

Application of surface geophysics to ground water investigations, by A.A.R. Zohdy, G.P. Eaton, and D.R. Mabey. 1974. 116 pages. Application of seismic-refraction techniques to hydrologic studies, by F.P. Haeni. 1988. 86 p. Application of borehole geophysics to water- resources investigations, by W.S. Keys and L.M. MacCary. 1971. 126 pages. General field and office procedures for indirect discharge measurement, by M.A. Benson and Tate Dahymple. 1967. 30 pages. Measurement of peak discharge by the slope-area method, by Tate Dahymple and M.A. Benson. 1967. 12 pages. Measurement of peak discharge at culverts by indirect methods, by G.L. Bodhaine. 1968. 60 pages. Measurement of peak discharge at width contractions by indirect methods, by H.F. Matthai. 1967. 44 pages. Measurement of peak discharge at dams by indirect methods, by Harxy Hulsing. 1967. 29 pages. General procedure for gaging streams, by R.W. Carter and Jacob Davidian. 1968. 13 pages. Stage measurements at gaging stations, by T.J. Buchanan and W.P. Somers. 1968. 28 pages. Discharge measurements at gaging stations, by T.J. Buchanan and W.P. Somers. 1969. 65 pages. Measurement of time of travel and dispersion in streams by dye tracing, by E.F. Hubbard, F.A. Kilpatrick, L.A. Martens, and

J.F. Wilson, Jr. 1982. 44 pages. Discharge ratings at gaging stations, by E.J. Kennedy. 1984. 59 pages. Measurement of discharge by moving-boat method, by G.F. Smoot and C.C. Novak. 1969. 22 pages. Fluorometric procedures for dye tracing, Revised, by James F. Wilson, Jr., Ernest D. Cobb, and Frederick A. Kilpatrick. 1986. 41

pages. Computation of continuous records of streamflow, by Edward J. Kennedy. 1983. 53 pages. Use of flumes in measuring discharge, by F.A. Kilpatrick, and V.R. Schneider. 1983. 46 pages. Computation of water-surface profiles in open channels, by Jacob Davidian. 1984. 48 pages. Measurement of discharge using tracers, by F.A. Kilpatrick and E.D. Cobb. 1985. 52 pages. Acoustic velocity meter systems, by Antonius Laenen. 1985. 38 pages. Aquifer-test design, observation, and data analysis, by R.W. Stallman. 1971. 26 pages. Introduction to ground-water hydraulics, a programmed text for self-instruction, by G.D. Bennett. 1976. 172 pages. Type curves for selected problems of flow to wells in confined aquifers, by J.E. Reed. 1980. 106 pages.

Definition of boundary and initial conditions in the analysis of saturated ground-water flow systems-an introduction, by 0. Lehn Franke, Thomas E. Reilly, and Gordon D. Bennett. 1987. 15 pages.

The principle of superposition and its application in ground-water hydraulics, by Thomas E. Reilly, 0. Lehn Franke, and Gordon D. Bennett. 1987. 28 pages.

Fluvial sediment concepts, by H.P. Guy. 1970. 55 pages. Field methods of measurement of tluvial sediment, by H.P. Guy and V.W. Norman. 1970. 59 pages. Computation of fluvial-sediment discharge, by George Porterfield. 1972. 66 pages. Some statistical tools in hydrology, by H.C. Riggs. 1968. 39 pages. Frequency curves, by H.C. Riggs, 1968. 15 pages. Low-flow investigations, by H.C. Riggs. 1972. 18 pages. Storage analyses for water supply, by H.C. Riggs and C.H. Hardison. 1973. 20 pages. Regional analyses of streamflow characteristics, by H.C. Riggs. 1973. 15 pages. Computation of rate and volume of stream depletion by wells, by C.T. Jenkins. 1970. 17 pages. Methods for determination of inorganic substances in water and fluvial sediments, by M.W. Skougstad and others, editors. 1979.626

pages.

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`Spanish tl .anslation also available.

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TWI 5-A2. TWI 5-A3.

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TWI %Al. TWI 8-A2. TWI S-B2.

Determination of minor elements in water by emission spectroscopy,by P.R. Bamett and E.C. Mallory, Jr. 1971. 31 pages. Methods for the determination of organic substancesin water and fluvial sediments, edited by R.L. Wershaw, M.J. Fishman, RR.

Grabbe, and L.E. Lowe. 1987. 80 pages.This manual is a revision of "Methods for Analysis of Organic Substancesin Water" by Donald F. Goerlitz and Eugene Brown, Book 5, Chapter A3, published in 1972. Methods for collection and analysis of aquatic biological and microbiological samples, edited by P.E. Greeson, T.A. Ehlke, G.A. Inuin, B.W. Lium, and K.V. Slack. 1977. 332 pages. Methods for determination of radioactive substancesin water and fluvial sediments, by L.L. Thatcher, V.J. Janzer, and K.W. Edwards. 1977. 95 pages.

Quality assurancepractices for the chemical and biological analysesof water and fluvial sediments, by L.C. Friedman and D.E. Erdmann. 1982. 181 pages.

Laboratory theory and methods for sediment analysis,by H.P. Guy. 1969. 58 pages. A modular three-dimensional finite-difference ground-water flow model, by Michael G. McDonald and Arlen W. Harbaugh. 1988.

586 pages. Finite difference model for aquifer simulation in two dimensions with results of numerical experiments, by P.C. Trescott, G.F.

Pinder, and S.P. Larson. 1976. 116 pages. Computer model of two-dimensional solute transport and dispersion in ground water, by L.F. Konikow and J.D. Bredehoeft. 1978.

90 pages. A model for simulation of flow in singular and interconnected channels, by R.W. Schaffranek, R.A. Baltzer, and D.E. Goldberg.

1981. 110 pages. Methods of measuring water levels in deep wells, by M.S. Garber and F.C. Koopman. 1968.23 pages. Installation and service manual for U.S. Geological Survey monometers, by J.D. Craig. 1983. 57 pages. Calibration and maintenance of vertical-axis type current meters, by G.F. Smoot and C.E. Novak. 1968. 1.5pages.

V

CONTENTS

Preface_-.-----------------------------------

III

Abstract__---------------.---.------------.--

1

Introduction _____ - _____ - _______ -_- ___..______ 1

Cumulative distributions- _ _ ________ _-___- _____ 1

Distributions used in hydrology- - __- - __________ 3

Normal distribution _____. _______ ___-__ -___ 3

Lognormal distribution- __- _- - __- - _________ 4

Type I extreme-value distribution (Gumbel) _ 4 Type III extreme-value distribution--- ______ 4

Pearson Type III distribution_ ___________- _ 4

Graphically defined distributions ______- _____ 4

Page

Mathematical curve fitting ___________________ -_ 4 Normal distribution _______________________ 4 Three-parameter distributions..-. ___________ 5 Type I extreme-value distribution (Gumbel) _ 6 Type III extreme-value distribution--- ______ 7

Graphical fitting--- ____________- - _- ___________ 7 Example of graphical fitting ______- ____- ____ 8

Use of historical data ______.___.___ -- ____ __ 11 Comparison of mathematical and graphical fitting- 11 Describing frequency curves ____________. _______ 12 Interpretation of frequency curves _____________ _ 12 Special cumulative frequency curves _____________ 13 References cited- _____. _____________. _. _______ 15

FIGURES

Page

1. Graphs showing two normal distributions and their cumulative forms- _______- - -

2

2. Graphs showing normal and skewed distributions and their cumulative forms on

a normal-probability plot ____________ - ___________________________________

2

3. Graphs showing relative positions of the mean, median, and mode for right-

skewed (upper) and left-skewed (lower) distributions ____-_ ____________- - ----

3

4. Frequency curves showing effect of direction of skew and direction of cumulation

on position of the mean with respect to the median _________________________

3

5. Gumbel frequency curve of annual floods on Columbia River near The Dalles,

Oreg., showing agreement with the plotted points- ______ ___- _____- _____ __--

7

6. Frequency curve based on data from table 3, assuming that data are annual

maxirnurns_____----~~---~-------------.~-----~--~-~-~--~~-~--~---~----

9

7. Frequency curve based on data from table 3, assuming that data are annual

minimurns__~-----~--~---------------~----~---~------.------~----------

10

8. Design-probability curves (lower graphs) and the frequency curve on which

they are based (upper graph)- _____- ______________.___ - __________ ______--

13

9. Frequency curve of annual minimum flows and plot showing serial correlation, South Fork Obion River near Greenfield, Term_---_ - - _________- - ___-__ ___. _ 14

TABLES

Page

1. Frequency factors for Pearson Type III distribution- ______. . ___. - - _. - - - - - - - - -

5

2. Means and standard deviations of reduced extremes- ___________- - - _- - - - - - - - - -

6

3. Computation of plotting position ________________________________________---

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VI

FREQUENCY CURVES

By H. C. Riggs

Abstract

Cumulative Distributions

This manual describes graphical and mathematical

procedures for preparing frequency curves from samples of hydrologic data. It also discusses the theory of frequency curves, compares advantages of graphical and mathematical fitting, suggests methods of describing graphically defined frequency curves analytically,

Book 4, chapter Al of the seriesof Techniques of Water-Resources Investigations (Riggs, 1967) describes the relation of a frequency distribution or probability density curve to its cumu-

and emphasizes the correct interpretations of a fre- lative form. A more detailed examination of

quency curve.

this relation helps in understanding the cumu-

lative distribution, or frequency curve. We

Introduction

begin with the two normal distributions shown in figure 1. Their cumulative forms can be ex-

A frequency curve relates magnitude of a variable to frequency of occurrence. The curve is an estimate of the cumulative distribution of the population of that variable and is prepared from a sample of data.

Frequency curves have many uses in hydrology. Flood-frequency curves are widely used in the design of bridge openings, channel capacities, and roadbed elevations; for flood-plain zoning; and in studies of economics of floodprotection works. Frequency curves of annual low flows are used in design of industrial and domestic water-supply systems, classification of streams as to their potential for waste dilution, definition of the probable amount of water available for supplemental irrigation, and maintenance of certain channel discharges as required by agreement or by law. Frequency curves of annual mean flows are sometimes used in studies of the carryover of annual storage (Beard, 1964).

Frequency curves also provide a means of classifying data for use in subsequent analyses. For example, Benson (1962a) used intensity of rainfall for a given frequency in his regional flood-frequency analysis for New England, and Riggs (1953) used a frequency curve of runoff

pressed as straight lines by use of the special abscissascale which is derived from the characteristics of the normal distribution. Both distributions have the same median value, 20, and these medians plot at 0.5 probability on the cumulative graph. The variability of a distribution is indicated by the slope of the cumulative distribution; that is, the greater the variability, the greater the slope. The standard deviation is half the difference between magnitudes at probabilities of 0.16 and 0.84 (Dixon and Massey, 1957, table A-4).

Many frequency distributions are nonsymmetrical. For such distributions, the mean, median, and mode have different values which consequently correspond to different probabilities on the cumulative graph. A nonsymmetrical distribution is classified as skewed. Skewness may be shown graphically as right or left; it may be described mathematically by a number, either positive or negative. Two skewed distributions and a symmetrical distribution are shown in figure 2, which also shows the corresponding cumulative distributions (frequency curves).

For a normal, or any symmetrical, distribu-

tion the mean and median are the same value.

in excess of assured flow in a forecasting prob- Thus, the value corresponding to the proba-

lem. Many other applications have been and bility of 0.5 on the cumulative frequency curve

B can be made.

is the mean as well as the median for such

1

TECHNIQUES

OF WATER-RESOURCES

INVESTIGATIONS

26-

MAGNITUDE

8

I I

I

0.9

0.8 0.7

05

I I

I

0.3 0.2

0.1

PROBABILITY THAT A RANDOMLY DRAWN INDIVIDUAL

WILL EXCEED THE INDICATED MAGNITUDE

Figure l.- Two normal distributions and their cumulative forms.

10

15

20

MAGNITUDE A. Right skewed 8 Normal C. Left skewed

0

I

I

I

I

I

30

0.9

0.8 0.7

0.5

03 02

0.1

PROBABILITY THAT A RANDOMLY DRAWN INDIVIDUAL WILL EXCEED THE INDICATED MAGNITUDE

Figure I.-Normal

and skewed distributions ond their cumulative forms on a normal-probability

plot.

distributions. The relative positions of the

mean, median, and mode for skewed distributions are shown in figure 3. Only the median value can be determined from the cumulative plot. The position of the mean with respect to the median on the cumulative plot depends on the degree of skewness, the direction of skewness, and the direction in which the frequency distribution is cumulated. For example, the mean of a particular right (positive)-skewed distribution will be exceeded 43 percent of the time; but 57 percent of the time it will not be exceeded. Thus, if the distribution is cumulated

from the high end, the mean is to the right of

the median; if cumulated from the low end, the mean is to the left of the median. These relations are reversed for a left-skewed distribution. Figure 4 illustrates the relations. The probability scales of the two plots of figure 4 are different. Each is designed for the particular distribution plotted.

Frequency curves of a time series commonly relate magnitude to recurrence interval or return period instead of to probability of exceedence or nonexceedence. Recurrence interval is the average length of time between exceeden-

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