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S t a t is t ic a l M e t h o d s in

HYDROLOG

Statistical Methods in

HY Second

ROLOGY dition

CHARLES T HAAN

owa State

ress

lackwellll Publishing Company lackwe

C H A R L E S T H A A N is Regents Professor and S Sarkeys arkeys Distinguished Professor, Emeritus, from

the Department of Biosystems Biosystem s and A,gicul A,gicultura turall Engineering, Engine ering, Oklahoma Okla homa State University, Stillwater.

1974 Iowa Iowa State University Press 2002 Iowa State Press A Blackwell Publishing Company All rights reserved Iowa State Press 2121 State S tate Avenu Avenue, e, Ames, Iowa 50014 5 0014 Orders:

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Authorizatio Authori zation n to photocopy iteins for internal or personal use, or the internal or personal use of specific clients, clients , is granted by Iowa State Press, provided that the base fee of of . lo per copy is paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For those organizations that have have been granted granted a photocopy ph otocopy license by CCC, a separate system system of payments has been arranged. The fee code for users of the Transactional Reporting Service is 0-8 138-1503712002 . 10. @Printed @Print ed on on acid-free paper in the United States of America America First edition, 1974 Second edition, 2002 Library Librar y of Congress Cataloging-in-Publication Data Haan, C. T. (Charles Thomas) Statistical Statist ical methods in hydrology hydrol ogy / Charles T. Haan Haan..-2n 2nd d p. cm. Includes bibliographical references and index. ISBN 0-8 138-1503-7 138-1503-7 (acid-free paper) 1. Hydrology-Stat Hydrology-Statistica isticall methods. metho ds. I. Title. GB656.2.S7 H3 2002 551.48 07'27-4~21

2002000060

The last digit is the print number: 9 8 7 6 5 4 3 2 1

ed.

I dedicate this book onc once e again t o my wife Janice who has been been

m y constant companion friend helpmate an d sourc source e of encouragement for the past 3 4 years. Secondly I dedica dedicate te the book to m y tw o daught daughters ers Patti an d Pam an d t o m y son Chris his wz wzye Rie and their two children Katrina and Daniel. Daniel. n i r d l y I dedicate the book to m y parents Charl Charles es an d Doroth Dorothy y who gaue gaue me a start in life and taught me m an y of of the values I hold dear: Finall Fina lly y the book book is dedicated t o the ma ny gradu ate students that I have worked with. They have been a constant source of renew al challenge inspiration inspira tion an d jo joy. y.

ontents REFACE TO SECOND EDITION EDIT ION

xv

REFA RE FACE CE TO FIRST FIRST EDITION

xv~i

CKNOWLEDGMENTS FOR THE SECOND EDITION

xix

CKNOWLEDGMENTS FOR THE FIRST EDITION EDIT ION

1

xx

INTRODUCTION

9

ydrologic data PROBABILITY AND PROBABILITY DISTRIBUTI DISTRIBUTIONS ONS BASIC robability otal probability theorem Bayestheorem Counting raphical presentation presentation Randomvariables

CONCEPTS

16 17 24 5 26 9

1

nivariate probability distributions

3

ivariate ivaria te distributions

40

arginal distributions

41

onditional distributions

41

Independence

43

Deriveddistributions

44

ixed ixe d distributions

48

xercises

49

PROPERTIES PROPER TIES OF RANDOM VARIABLES oments and expectation expectation univariate univariate

distributions

easures of central tendency

53 55

Arithmeticmean

55

Geometricinean

56

Median

56

Mode

56

Weightedmean

57

easures of dispersion dispe rsion

57

Range

57

Variance

57

easures of symmetry

58

Measuresofpeakedness

59

oments and expectation expectation jointly

distributed random variables

Covariance orrelation coefficient urther properties of moments

4

52

60 62 62 65

ample moments

66

robability weighted moments and L moments

68

arameter estimation

70

Unbiasedness

70

Consistency

70

Efficiency

71

Sufficiency

71

ethod of moments moment s

72

aximum likelihood

74

hebyshevinequality

76

Lawoflargenumbers xercises

77 78

SOME DISCRETE PROBABILITY DISTRIBUTIONS DISTRIBUTIONS AND THEIR APPLICATIONS

1

ypergeometric distribution

81

ernoulli processes

84

inomial distribution

84

eometric distribution

89

egative binomial distribution

90

ummary of Bernoulli process

90

CONTENTS

ix

oissonprocess oisson distribution Exponential distribution

93 93

ummary of Poisson process

-94

Exercises

NORMALDISTRIBUTION NORMALDISTRIBUTION eneral normal distribution Reproductiveproperties tandard normal normal distribution

95 96

100 100 101 -102

pproximations for standard normal distribution

104

entral limit theorem

106

onstructing onstructi ng pdf pdf curves for data

107

ormall approximations for other distributions orma

109

inomial distribution distribut ion

109

egative binomial distribution distribut ion

110

oisson distribution

111

ontinuous distributions Exercises

CONTINUOUS PROBABILITY DISTRIBUTIONS

111 111

114

niform distribution

114

riangular distribution

116

xponential xponent ial distribution

117

Gammadistribution ognormal distribution xtreme value distribut distributions ions xtreme xtre me Value Type Extr eme Value Extreme Value Type III Minimum Weibull) Discussion eneralized eneralize d extreme value distribut distribution ion Betadistribution

120 -126 129 -132 -13 2 34 138 139 140

earson distributions

141

ome important distributions distribut ions of sample statistics

142

hi-square distribution

142

he distribution

143

TheFdistribution

44

ransformations Exercises

91

amma distribution ultinomial distribution

5

91

145 46

7

FREQUENCYANALYSIS FREQUENCYA NALYSIS

49

Probability plotting plottin g

151

Historicaldata

56

Outliers

58

Analytical hydrologic frequency analysis Normal distribution

159

Lognormal distribution Log Pearson Pearso n type I11 distribution distrib ution xtreme value type I distribution distribution Gumbel distribution) Other distributions

160 164 165

Confidenceintervals

167

Treatmentofzeros

168

Truncation Trunc ation of low flows

76

se of paleohydrolo paleoh ydrologic gic data

177

robable maximum flood flood

177

iscussion of flood frequency determinations

178

egionalfrequencyanalysis

180

Delineation of homogeneous homog eneous regions region s

80

Historical Histor ical development

181

tatistical methods Frequencydistributions

182 82

egression-based procedures Index-floodmethod

183 186

Regional index-flood relationship relation ship

86

egionalization using L-moments and the GEV distribution

187

egionalization using modeling

189

Frequency Frequen cy analysis of precipitation data requency requen cy analysis of other hydrologic variables Exercises

-189 191 92

CONFIDENCE INTERVAL INTERVALS S AND HYPOTHESIS TESTING TESTI NG Confidence intervals Confidence Mean of a normal distribution Variance Varianc e of a normal distribution distrib ution One-sided confidence intervals Parameters of probability distributions Hypothesis testing

94 96 197 199 00 201 -201

H, p

=

p l . Ha p

, p

=

p . Ha: p

p2.normal distribution. unknown variance

206

, p

=

po. Ha: p

po. normal distribution. known variance

207

H, p

=

po. Ha: p

po.normal distribution. unknown variance

-207

=

p2. normal distribution. known variance variance

est for differences in means of two normal distributions

60

165

eneralconsiderations

8

58

06

208

CONTENTS

xi

Test of H : u populati on a; versus Ha: a a : normal population Test of H a: = a; versus Ha a: a; for two normal populations Test for equality of variances from several normal distributions distribu tions esting the goodness good ness of fit of data to probability distributions hi square goodness of fit test Distributional tests based on cumulative distributions omparing two empirical distributions eneral comments on goodness of fit tests xercises 9

SIMPLE LINEAR REGRESSION Simple regression Evaluating the regression Confidence intervals intervals and tests of hypotheses Inferences on regression coefficients Confidence intervals intervals on regression line Confidence intervals on standard error er ror xtrapolation eneral considerations considerations xercises

221

.

0 MULTIPLE LINEAR REGRESSION Notation Generallinearmodel onfidence intervals and tests of hypotheses onfidence intervals on standard error err or nferences on the regression coefficients onfidence intervals on the regression line ther inferences in regression Whichlineisbest Extrapolation utocorrelated errors esting for serial correlation orrective action Multicolinearity etection of multicolinearity n application of multiple regression ransforming linear models ndicator variables in regression Generalcomments ogistic regression Exercises

209 09 09 210 211 13 219 221

242 242 242 249 249 249 251 251 254 56 257 259 260 60 262 262 266 268 272 272 78

11

CORRELATI CORRELATION ON nferences nferenc es about population c correlation orrelation coefficient coefficientss Serialcorrelation

12

Correlation and cause and effect

-291

Spurious correl correlation ation

91

Exercises

93

MULT MULTIVARI IVARIATE ATE ANAL ANALYSIS YSIS

97

Notation

97

Principalcomponents

98 -307

Multivariate Multiv ariate multiple regression

11

Canonical corre correlation lation

12

Exercises DATAGENER DATAGENERATION ATION Univariate Univari ate data genera generation tion Multivariate Multiv ariate data generation Multivariate. corre correlated. lated. normal random variables Multivariate. Multiv ariate. correlated. nornorma nornormall random variab variables les Applications of data genera generation tion xercises ANA ANALYS LYSIS IS O F HYDROLOGIC TIME SERIES Definitions

313 18 321 -321 27 327 28 -331 334 -336 336

Trendanalysis

40

Jumps

46

Autocorrelation Periodicity Autoregressive integra integrated ted movin moving g avera average ge models ARIMA) Moving Average Processes MA) Autoregressive processes utoregressive Moving Average Models ARMA ARMA p, q) Autoregressive Integra Integrated ted Moving Average ARIMA p. d. q)

87 290

Cluster analysi analysiss

14

282

Correlation and regiona regionall analysis

Regression on principal compone components nts

13

81

348 50 355 56 58 362 363

~ s t i m a t ef ef noise variance o

-364

Parameter Paramet er estimation via least squares

364

ARmodels

64

MAmodels

64

Parameter Paramet er estimation via maxim maximum um likelihood

366

Exercises

367

CONTENTS 15

xu1

SOME STOCHASTIC HYDROLOGIC MODELS Purely random stochastic models First order Markov process Higher order autoregressive models

379 80 388

PROBABILISTIC METHODS FOR UNCERT UNCERTAIN AINTY. TY. RISK. AND RELIABILITY RE LIABILITY ANALYSIS Sensitivity analysis

90 91

Traditional or local sensitivity analysis

391

Global sensitivity analysis

392

Uncertainty analysis Reliability and risk analysis Uncertainty. Uncert ainty. risk. risk. and reliability analysis methods

96 96 98

First order approximation method

398

Simplified Simpli fied FOA FOA estimates for some functional forms

399

Monte Carlo simulation

404

Corrected Correct ed FOA FOA method

406

Correcting FOA mean and variance estimates of an individual function Second order approximation method method

06 411

First order reliability method

412

Generic expectation functions

18

Othermethods Second order reliability methods Point estimation methods Transform methods

GEOSTATISTICS

423 23 424 24

425

Descriptive statistics

26

Semivariogrammodels Combination semivariogra semivariogram m models

30 32

stimation

433

nexample

438

nisotropy

443

okriging

445

Local and global estimation Polygon declustering elldeclustering Pointkriging lockkriging

75 378

xercises

7

374

First order Markov process with periodicity Markovchainmodels

16

370

446 446 447 447 447

xiv

CONTENTS stimation of cumu lative distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .447 ncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 Modeling using geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 APPENDIXES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 A . Comm on distr distributi ibutions ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Hydrologicdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 A.2.. M onthly runoff in.), Cave Creek near Fort Spring, Kentucky . . . . . . . 454 A.2 A.3.. Peak discharge cfs), Cum berland River at Cumberland Falls, A.3 Falls, entucky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 A.4. Peak discharge cfs), Pisc ataqu is River, Dover-Foxcroft, M aine . . . . . . 5 7 A.5. A.5. Total Precipitation Precipitation in.) for week of March 1 to March 7, Ashland, Kentucky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8 A.6.. Flow and sedim ent load, Green River at M unfordville, A.6 unfordville, Kentucky . . . . . . 58 A.7.. Streamflow in.) A.7 in.),, Walnut Gulch near Tom bstone, Arizona . . . . . . . . . . . . 5 9 A.8. Monthly Rainfall in.), Walnut G ulch near Tom bstone, Arizona . . . . . . . 6 0 A.9. Ann ual discharge cfs , Spray River, Banff Canada . . . . . . . . . . . . . . . . 6 1 A.lO. Ann ual discharge cfs), Pisc ataqu is River, Dover-Foxcroft, M aine . . . . 6 1 A .l l . Annual discharge discharge cfs), Llano River, Junction, Junction, Texas . . . . . . . . . . . . . . 6 1 Statistical tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 A . 2 Standard normal distributi distribution on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 .13 Percentile values for the t distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 464 A.14. Percentile values for the c hi squar e dist distribution ribution . . . . . . . . . . . . . . . . . . . 6 5 A.15. Percentile values for the F distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 A 6. Critical values for the Kolmogorov-Smirnov test statistic . . . . . . . . . . . 6 9 .17 Durban-Watson test bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 NDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 8 3

Preface to the Second dition SINCE THE publication of the first edition of this book statistics has come to play an increasingly important role in hydrology. The advancements in computing technology and data management have made the application applicati on o off statistical techniques that were previously previously known but difficult to implement allnost routine. User friendly software for personal computers has made powerful statistical routines available to nearly nearly all hydrologists. Generally this software comes with user manuals or help files that lead a ne new w user through the steps needed to use the programs. program s. Unfortunately Unfortunate ly these aids rarely indicate the assumptions inherent in in the techniques the limitations of the techniques and the situations in which the techniques should or should not not be used. They are generally weak in instructing one on the interpretation interpret ation of of the results of the analysis as well. This software is a tool that is available for use in hydrology but does not replace sound hydrologic understanding of the problem problem at hand nor does it rreplace eplace a basic understanding of of the statistical technique being used. This current edition should serve as a companion to many of the software programs availabl ava ilable-no e-nott

to explain how to use the softwar software e but to provide guidance guid ance as to the proper rou-

tines to use for a particular problem and the interpretation of the results of the analysis. The basic philosophy of of the current edition is the same as that of the first edition. Enough detail on particular statistical methods is presented to gain a working understanding of the technique. Certainly the treatment on any particular statistical technique is not exhaustive. Much theory and derivation are omitted and left to more in-depth treatments found in books dealing specifically specifica lly with the various topics. Two chapters have been added to the book. One of these chapters deals with uncertainty analysis and the other with geostatistics. Both of these topics have received great emphasis in

xvi

PREFACE PREF ACE TO THE SECOND EDITION

the past decade. Uncertainty analysis is a growing concern as it is increasingly recognized that both statistical and deterministic analyses result in estimates that are far from absolute answers. Increasingly. attempts are made to evaluate how much uncertainty should be associated with various types of analyses. Rather than providing a point estimate of some quantity, confidence limits are sought, such that one can assert with various degrees of confidence bounds within which the sought after quantity is thought to be. Geostatistics has become of increasing importance as geographically geographically referenced information becomes availabl available e and is used in g geographical eographical information systems GISs) to produce hydrologic estimates. The chapter on uncertainty was written by Aditya Tyagi, a former PhD candidate at Oklahoma State University and currently a water resources engineer with CH2M Hill. Jason Vogel, Vog el, a research research engineer and PhD P hD candidate at Oklahoma Stat State e University, was was a coauthor of the chapter on geostatistics.

Preface t th Firs Fi rstt Ed iti on THE RANDOM variability variabil ity o off such hydrologic variables as streamflow and precipitation has been recognized for centuries. The general field field of hydrol hydrology ogy was one of of the first areas of of science and engineering to use statistical st atistical concepts in an effort eff ort to anal analyze yze natural phenomena. Many papapers have been published that amply demonstrate the value of statistical tools in analyzing and solving hydrologic problems. In spite of the long history and proven utility of statistical techniques in hydrology, relatively relatively few comprehensi comprehensive ve and basic ttreatments reatments of statistical stati stical methods in hydrology have been published. This book has been prepared to assist engineers and hydrologists develop an elementary knowledge of some statistical stati stical tools that have been successfully applied to hydrologic problems. The intent of the book is to familiarize the reader with various statistical techniques, point out their strengths and weaknesses and demonstrate their usefulness. The serious reader will want to supplement the material with formal f ormal courses or independent study of those individu individual al topics tha thatt are major interests. No single topic has been developed completely. Books have been written covering many of the topics discussed as single chapters in this presentation.

gain the purpose here

is to develop understanding and illustrate the usefulness of the ttechniques. echniques. Most of the techniques are discussed in sufficient sufficient detail for a thorough thor ough understanding and and application appli cation to problem situations. The philosophy of of the presentation has been that one does not have to understand hydrodynamics to swim even though it could help one to become becom e a more proficient proficient swimmer. The book has not been written for statisticians stat isticians or for those primarily interested in statistical theory. Rather it has been prepared for hydrologists and engineers interested in learning how statistical models and methods can be valuable tools in the analysis and solution of many hydrologic and engineering problems. The basic premise has b been een take taken n and justifiably justifiably so) that

PREFACE PREFAC E TO THE FIRST EDITION

xviii

statisticians are competent so that many statistical results are presented without developing a rigorous proof proof of their validit validity. y. Proofs for most results can be found in mathematical statistics books many of which are listed liste d in the bibliography. No prior knowledge of of statistics sta tistics is required if one starts with Chapter 2 Those with varying degrees of statistical knowledge knowledge may may choose to start with later chapters. A knowledge o off calculus is required throughout and some familiarity with matrices is needed for material in later chapters. Appendix D is a review of the basic matrix manipulation used in the book (not in this new edition). This is is not a statistical cookbook cookbook for hydrologists. It does non non contain step-by-step calculation procedures procedures for sta nda rd hydrologic problems. Basic st statistical atistical concepts are discussed and illustrated in enough detail so that one can develop his own computational procedures or methods. Most of the computations in actual work situations would be done on digital computers. Computer programs have not been included because it is felt that most computer centers will have programs or programmers available. Likewise computational techniques are not emphasized. For example, in the chapter on multiple regression, efficient efficient techniques for matrix inversion are not presented presented as it is felt that these techniques are readil readily y available at most computer t e n ters. The emphasis is thus retained on the statistical technique being used and not on the computational aspects of of the problem. Some liberties have been taken in that many terms are not precisely defined in a mathematical sense unless such a definition is warranted. Where terms are loosely defined, it is hoped that the meticulous reader will accept the general connotation of the terms for purposes of simplicity and to avoid placing emphasis on terms rather than concepts. Many of of the problems require sets of data. Those dat data a may may be supplied by the reader or selected from the data in Appendix C. I am grateful gra teful to the Literary Lit erary Executor Execu tor of the late Si Sirr Ronald A. Fisher, F F.. R S., to Dr. Frank Yates, F. R. S. and to London Group Ltd., London, for permission to reprint Table E.5 from their thei r book Statistical Tables or Biological Agricultural arzd Medical Research 6~ Edition 1 974) (not in this new edition).

cknowledgmentsfor the Second Edition IT HAS been nearly a quarter century since I wrote the first edition of this book. During that time I have become indebted to many people. I have spent nearly this entire period with the Biosystems and Agricultural Engineering Department at Oklahoma State University. This Department has provided a wonderful atmosphere for intellectual growth and accomplishment. The faculty faculty staff and students students that I have been associated with have helped to create a working environment environme nt that was challenging challengi ng friendly and one in which my my only limitation was myself. myself.

I

m

grateful to many individuals. Bill Barfield has continued to be a valued friend and

coworker. Dan Storm Bruce Wilson Wilson and many graduate students studen ts have have been especial especially ly instrumental in much of my research and teaching in the field of statistical hydrology. My daughter Dr. Dr. Patricia Haan assistant professor in the Biological Biological and Agricultural Engineering Department at Texas A

M University has been very helpful in clarifyin clarifying g some

points in the text and correcting errors. Certainly my wife of 34 years Jan has been most most supportive and forgiving as I have devoted far too much time to work. As is true of all of us I owe whatever I have accomplished to my Creator without Whom

I could accomplish nothing.

cknow cknowlledgm edgments ents for the First Edition MUCH OF the material material presented in this book was developed for a course cou rse taught to students in the Agricultural Engineering and Civil Engineerin Engin eering g Departments Departmen ts at the University of Kentucky. The suggestions and clarificat tucky. clari fications ions made by the students stude nts in this course over the past have been a great aid in attempting to to make this book more understandable. un derstandable.

years

Special acknowledgment must be given to Dan Carey for his careful readings of the entire manuscript. These readings resulted in several corrections and clarifications. Several individuals have read parts of the book and made valuable suggestions for its improvement. Among those reviewing parts of the manuscript were Donn DeCoursey David Allen David Culver and personnel of the U.S U.S.. Soil Conservation Servic S ervice e under the direction of Neil Bogner. Several individuals individuals in the Agricultural Agricu ltural Engineering Departm Department ent at the University University of Kent Kentucky ucky offered valuable valuable suggestions suggestio ns and considerable conside rable encouragement. encour agement. Deserving special mention mention are Billy Barfield Blaine Blain e Parker. Parker. and John Walker. Walker. This undertaking has has required sacrifice on the part of my family and especially my wife Janice. She not only typed the early drafts of the book but offered continued continu ed encouragement over the years as work work and revisions were done d one on the book. boo k. This manuscript was reproduced from photo-ready copy. The excellent typing involved in preparing this this final draft as well as an earlier draft was done by Pat Owens. Buren Plaster drafted all of the figures.

Of course any failings and shortcomings of this book must be credited to me. me. My hope is that it will be found useful in at least partially meeting the need for an elementary treatment o off statistical methods in hydrology. hydrology. Whatever Whatever is accomplished along alon g these lines I owe to our Father for giving givin g me the will to s

this project through and the ability to withstand the setbacks setback s experienced experienced along the way.

Finally I express my my appreciation to all of the members of the Agricultural Engineering Departmentt at the University Departmen University of Kentucky for their understanding understanding during the preparation of this manuscript.

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