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Stochastic Hydrology PDF

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STOCHASTICHYDROLOGY StochaStic hydrology By dr. P. Jayarami reddy Principal G. Pulla Reddy Engineering College Kurnool-518002 laxmi Publications (P) ltd (An iso 9001:2015 company) bengaluru • chennai • guwahati • hyderabad • jalandhar Kochi • KolKata • lucKnow • mumbai • ranchi new delhi StochaStic hydrology Copyright © by Laxmi Publications Pvt., Ltd. All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012, no part of this publication may be reproduced, stored in a retrieval system, translated into any other languages or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. Any such act or scanning, uploading, and or electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder’s intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained from the publishers. Printed and bound in India First Published : 1987, Second Edition : 2016, Edition : 2021 ISBN : 978-81-318-0983-9 limits of liability/disclaimer of warranty: The publisher and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activities contained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, common sense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publisher nor the author shall be liable or assumes any responsibility for any injuries or damages arising here from. The fact that an organization or Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers must be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. All trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmi appearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries or affiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks or service marks of their respective owners. & Bengaluru 080-26 75 69 30 & Chennai 044-24 34 47 26 & Guwahati 0361-254 36 69 es& Hyderabad 040-27 55 53 83 h nc& Jalandhar 0181-222 12 72 a br& Kochi 0484-405 13 03 Published in india by & Kolkata 033-40 04 77 79 & Lucknow 0522-430 36 13 Laxmi Publications (P) Ltd. & Ranchi 0651-224 24 64 (An ISO 9001:2015 Company) 113, GOLDEN HOUSE, GURUDWARA ROAD, DARYAGANJ, NEW DELHI - 110002, INDIA Telephone : 91-11-4353 2500, 4353 2501 www.laxmipublications.com C—R/021/10 [email protected] Printed at : Ajit Printing Press, Delhi. 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

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