Stochastic Hydrology

275 Pages·2021·21.367 MB·English

by  P. Jayarami Reddy

STOCHASTICHYDROLOGY
StochaStic hydrology
By
dr. P. Jayarami reddy
Principal
G. Pulla Reddy Engineering College
Kurnool-518002
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StochaStic hydrology
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First Published : 1987, Second Edition : 2016, Edition : 2021
ISBN : 978-81-318-0983-9
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To the MemoryofmyFather
BUGGA RAMI REDDY

PREFACETOTHE SECONDEDITION
Inbringingoutthesecondeditionofthebook,theopportunity
has been utilised to thoroughly revise and enlarge the subject
matter taking into account the suggestions offered by many. Few
mistakes which crept into the first edition have been rectified.
Solvedexampleshave beenincreasedmanifold.Andinthisedition
all chapters are provided with Exercises at the end to help the
readers to testtheircomprehensionofthe subject.
Suggestions for further improvement of the book will be
thankfullyreceived.
P. JAYARAMI REDDY
Kurnool
January, 1997
PREFACETOTHEFIRSTEDITION
The importance ofstochastic modellingofhydrological vari
ables like runoffand precipitation in the design ofwater resources
systems is too well recognised to need any further emphasis. Most
of the Indian Universities offering Postgraduate programmes in
Hydraulics and Water Resources Engineering have included
"Stochastic Hydrology" as one ofthe subjects in their syllabi. The
material presented inthis text-bookis the outcomeofteachingthe
subject to postgraduate students at Regional EngineeringCollege,
Warangal, for over five years and the extensive study and use of
stochastic models in mydoctoralwork.
After a briefintroduction in Chapter 1, the fundamentals of
probabilitytheoryaredevelopedinChapter2.Mostoftheimportant
probabilitydistributionsofbothdiscrete and continuousvariables,
which are frequently used to model the hydrologicphenomena are
described in Chapter 3. Chapter 4 concentrates on methods of
estimation and fitting appropriate distributions to the observed
data,whileChapter5coversthetopicsofcorrelationandregression.
Allthesechaptersform the basichackgroundtotheunderstanding
and the appreciation ofthe principles ofstochastic processes and
theirapplicationsinhydrologythatarediscussedinthesubsequent
chapters. Chapter6dealswiththedefinitionsandclassificationsof
stochastic processes and time series. Chapters 7and8 are, respec-
tively, devoted to the two important analysis techniques, namely,
the autocorrelation analysis and the spectral analysis. Synthetic
flow generation models are elaborately discussed in Chapter 9.
Finally Chapter 10 contains a briefdescription ofthe methods of
generationofrandom numbers.
Readers already familiar with probability theory can per
haps, skipthe Chapters 2through5. Butthematerialpresentedin
thistext-bookisapplicationorientedand isquitedifferentfromthe
presentations found in the other text-books on probabilitywritten
by mathematicians. Most ofthe numerical examples are solved on
a pocket calculator. The data (or the correlogram and spectral
densityofthe examples in Chapters 7 and 8 are obtained through
computer,analysis. When the generation of synthetic flows is re
quired for longer periods, one has to necessarily resort to the
computer. As the book stresses more on applications, several
results,especiallyinautocorrelationanalysisandspectralanalysis
havebeentakenforgranted, andmoreorlessthroughoutthebook,
a compromise has been struck between mathematical rigour and
methodology.
Iexpressmygratitudetotheauthorsofthevariousbooksand
technical papers listed inthebibliography, which I consulted. I am
indebted to all my colleagues and postgraduate students of the
Division ofWater Resources ~t the Regional EngineeringCollege,
Warangalfortheirencouragementandhelpinwritingthisbook,In
particular, I would like to thank professor R. Raghavendran who
hasbeena sourceofinspirationfor me.
Iexpressmygratefulnessto Sri R.K. Guptafor hiseffortsin
bringingoutthisbook.
Suggestions for improvement shall be gratefully acknow
ledged.
P. JAYARAMI REDDY
Vidyanagar
September, 1986.
CONTENTS
Pages
1. INTRODUCTION 1-4
Exercises 3
2. PROBABILITY 5-40
2.1. Introduction 5
2.2. SetTheory 6
Events andSets 7
Algebraofsets 9
2.3. AxiomsofProbability 14
2.4. InterpretationofProbability 15
TheclassicalinterpretatiQn 15
Thefrequencyinterpretation 15
Thesubjectiveinterpretation 15
2.5. ElementaryPropertiesofProbability 16
2.6. ConditionalProbability 18
2.7. TotalProbabilityandBayes".Theorem 22
2.8. RandomVariable 23
2.9. ProbabilityFunctionandDistribution
FunctionofaDiscreteRandom
Variable 24
2.10. ProbabilityDensityFunctionand
DistributionF\lnctionofa Continuous
RandomVariable 26
2.11. DistributionFunctionofaMixed
RandomVariable 28
2.12. BivariateandMultivariate
Distributions 30
2.13. MarginalDistributions 32
2.14. ConditionalDistributions 33
2.15. DerivedDistributions 35
Exercises 37
3. SOMECHARACTERISTICSOFDISTRmUTIONS 41-54
3.1. Introduction 41
3.2. Mean 41
3.3. Variance 43
3.4. Chebyshev'sInequality 44
3.5. Covariance 46
(viii)
Pages
3.6. Moments 48
Skewnesscoefficient 49
Kurtosiscoefficient 50
3.7. MODlentGeneratingFunction 52
Exercises 53
4. CONTINUOUSDISTRmUTIONS 55-73
4.1. Introduction 55
4.2. NormalDistribution 55
ApproximationstoF(z) 58
Centrallimittheorem 59
4.3. Log-NormalDistribution 61
4.4. GammaDistribution 64
Special CasesofGammaDistribution 66
4.5. BetaDistribution 67
4.6. UniformDistribution 67
4.7. ExtremeValueDistribution 68
ExtremeValueTypeIDistributionfor
LargestValue(Gumbel) 68
ExtremeValueTypeIIIDistribution
forSmallestValue(Weibull) 70
4.8. PearsonDistributions 71
PearsonTypeIIIDistribution 71
Log-PearsonTypeIIIDistribution 71
4.9. OtherContinuousDistributions 72
Exercises '72
5. DISCRETEDISTRmUTIONS 74-85
5.1. BinomialDistribution 74
5.2. NegativeBinomialDistribution 79
5.3. GeometricDistribution 80
5.4. PoissonDistribution 81
5.5. HypergeometricDistribution 83
5.6. OtherDiscreteDistributions 84
Exercises 84
6. ESTIMATIONANDPROBABILITYPL01TING 86-114
6.1. SampIeStatistics 86
6.2. Estimationofthe parameters
ofthe Distribution 88
MethodofMoments 89
MethodofMaximumLikelihood 89
6.3. SelectionofDistribution 91