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Parameter Estimation and Inverse Problems PDF

316 Pages·2005·1.607 MB·English
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Parameter Estimation and Inverse Problems ThisisVolume90inthe INTERNATIONALGEOPHYSICSSERIES Aseriesofmonographsandtextbooks EditedbyRENATADMOWSKA,JAMESR.HOLTON,andH.THOMASROSSBY Acompletelistofbooksinthisseriesappearsattheendofthisvolume. Parameter Estimation and Inverse Problems Richard C.Aster, Brian Borchers, and Clifford H. Thurber Amsterdam • Boston (cid:127) Heidelberg (cid:127) London (cid:127) NewYork (cid:127) Oxford Paris (cid:127) SanDiego (cid:127) SanFrancisco (cid:127) Singapore (cid:127) Sydney (cid:127) Tokyo AcquisitionEditor FrankCynar ProjectManager KyleSarofeen EditorialCoordinator JenniferHelé MarketingManager LindaBeattie CoverDesign SuzanneRogers Composition CephaImagingPrivateLtd. CoverPrinter PhoenixColorCorporation InteriorPrinter Maple-VailBookManufacturingGroup ElsevierAcademicPress 30CorporateDrive,Suite400,Burlington,MA01803,USA 525BStreet,Suite1900,SanDiego,CA92101-4495,USA 84Theobald’sRoad,LondonWC1X8RR,UK Thisbookisprintedonacid-freepaper. Copyright©2005,ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopy,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartment inOxford,UK:phone:(+44)1865843830,fax:(+44)1865853333,e-mail: permissions@elsevier.com.uk.YoumayalsocompleteyourrequestonlineviatheElsevier homepage(http://elsevier.com),byselecting“CustomerSupport”andthen“ObtainingPermissions.” LibraryofCongressCataloging-in-PublicationData Applicationsubmitted BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN: 0-12-065604-3 ForinformationonallElsevierAcademicPressPublications visitourWebsiteatwww.books.elsevier.com PrintedintheUnitedStatesofAmerica 04 05 06 07 08 09 9 8 7 6 5 4 3 2 1 Contents Preface xi 1 INTRODUCTION 1 1.1 ClassificationofInverseProblems 1 1.2 ExamplesofParameterEstimationProblems 4 1.3 ExamplesofInverseProblems 7 1.4 WhyInverseProblemsAreHard 11 1.5 Exercises 14 1.6 NotesandFurtherReading 14 2 LINEAR REGRESSION 15 2.1 IntroductiontoLinearRegression 15 2.2 StatisticalAspectsofLeastSquares 17 2.3 UnknownMeasurementStandardDeviations 26 2.4 L Regression 30 1 2.5 MonteCarloErrorPropagation 35 2.6 Exercises 36 2.7 NotesandFurtherReading 40 3 DISCRETIZING CONTINUOUS INVERSE PROBLEMS 41 3.1 IntegralEquations 41 3.2 QuadratureMethods 41 v vi Contents 3.3 ExpansioninTermsofRepresenters 46 3.4 ExpansioninTermsofOrthonormalBasisFunctions 47 3.5 TheMethodofBackusandGilbert 48 3.6 Exercises 52 3.7 NotesandFurtherReading 54 4 RANK DEFICIENCYAND ILL-CONDITIONING 55 4.1 TheSVDandtheGeneralizedInverse 55 4.2 CovarianceandResolutionoftheGeneralizedInverseSolution 62 4.3 InstabilityoftheGeneralizedInverseSolution 64 4.4 AnExampleofaRank-DeficientProblem 67 4.5 DiscreteIll-PosedProblems 73 4.6 Exercises 85 4.7 NotesandFurtherReading 87 5 TIKHONOV REGULARIZATION 89 5.1 SelectingaGoodSolution 89 5.2 SVDImplementationofTikhonovRegularization 91 5.3 Resolution,Bias,andUncertaintyintheTikhonovSolution 95 5.4 Higher-OrderTikhonovRegularization 98 5.5 ResolutioninHigher-OrderTikhonovRegularization 103 5.6 TheTGSVDMethod 105 5.7 GeneralizedCrossValidation 106 5.8 ErrorBounds 109 5.9 Exercises 114 5.10 NotesandFurtherReading 117 6 ITERATIVE METHODS 119 6.1 Introduction 119 6.2 IterativeMethodsforTomographyProblems 120 6.3 TheConjugateGradientMethod 126 6.4 TheCGLSMethod 131 Contents vii 6.5 Exercises 135 6.6 NotesandFurtherReading 136 7 ADDITIONALREGULARIZATIONTECHNIQUES 139 7.1 UsingBoundsasConstraints 139 7.2 MaximumEntropyRegularization 143 7.3 TotalVariation 146 7.4 Exercises 151 7.5 NotesandFurtherReading 152 8 FOURIER TECHNIQUES 153 8.1 LinearSystemsintheTimeandFrequencyDomains 153 8.2 DeconvolutionfromaFourierPerspective 158 8.3 LinearSystemsinDiscreteTime 161 8.4 WaterLevelRegularization 164 8.5 Exercises 168 8.6 NotesandFurtherReading 170 9 NONLINEAR REGRESSION 171 9.1 Newton’sMethod 171 9.2 TheGauss–NewtonandLevenberg–MarquardtMethods 174 9.3 StatisticalAspects 177 9.4 ImplementationIssues 181 9.5 Exercises 186 9.6 NotesandFurtherReading 189 10 NONLINEAR INVERSE PROBLEMS 191 10.1 RegularizingNonlinearLeastSquaresProblems 191 10.2 Occam’sInversion 195 10.3 Exercises 199 10.4 NotesandFurtherReading 199 viii Contents 11 BAYESIAN METHODS 201 11.1 ReviewoftheClassicalApproach 201 11.2 TheBayesianApproach 202 11.3 TheMultivariateNormalCase 207 11.4 MaximumEntropyMethods 212 11.5 Epilogue 214 11.6 Exercises 216 11.7 NotesandFurtherReading 217 A REVIEW OF LINEARALGEBRA 219 A.1 SystemsofLinearEquations 219 A.2 MatrixandVectorAlgebra 222 A.3 LinearIndependence 228 A.4 SubspacesofRn 229 A.5 OrthogonalityandtheDotProduct 233 A.6 EigenvaluesandEigenvectors 237 A.7 VectorandMatrixNorms 240 A.8 TheConditionNumberofaLinearSystem 242 A.9 TheQRFactorization 244 A.10 LinearAlgebrainSpacesofFunctions 245 A.11 Exercises 247 A.12 NotesandFurtherReading 249 B REVIEW OF PROBABILITYAND STATISTICS 251 B.1 ProbabilityandRandomVariables 251 B.2 ExpectedValueandVariance 257 B.3 JointDistributions 258 B.4 ConditionalProbability 262 B.5 TheMultivariateNormalDistribution 264 B.6 TheCentralLimitTheorem 265 B.7 TestingforNormality 265 Contents ix B.8 EstimatingMeansandConfidenceIntervals 267 B.9 HypothesisTests 269 B.10 Exercises 271 B.11 NotesandFurtherReading 272 C REVIEW OF VECTOR CALCULUS 273 C.1 TheGradient,Hessian,andJacobian 273 C.2 Taylor’sTheorem 275 C.3 LagrangeMultipliers 276 C.4 Exercises 278 C.5 NotesandFurtherReading 280 D GLOSSARYOF NOTATION 281 283 Bibliography 291 Index

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