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- A. Tarantola.pdfContents
Preface xi
1 The General Discrete Inverse Problem 1
1.1 ModelSpaceandDataSpace ...................... 1 1.2 StatesofInformation .......................... 6 1.3 ForwardProblem ............................ 20 1.4 MeasurementsandAPrioriInformation . . . . . . . . . . . . . . . . 24 1.5 DefiningtheSolutionoftheInverseProblem . . . . . . . . . . . . . . 32 1.6 UsingtheSolutionoftheInverseProblem . . . . . . . . . . . . . . . 37
2 Monte Carlo Methods 41
2.1 Introduction ............................... 41 2.2 TheMovieStrategyforInverseProblems . . . . . . . . . . . . . . . . 44 2.3 SamplingMethods............................ 48 2.4 MonteCarloSolutiontoInverseProblems . . . . . . . . . . . . . . . 51 2.5 SimulatedAnnealing .......................... 54
3 The Least-Squares Criterion 57
3.1 Preamble: TheMathematicsofLinearSpaces . . . . . . . . . . . . . 57 3.2 TheLeast-SquaresProblem....................... 62 3.3 EstimatingPosteriorUncertainties ................... 70 3.4 Least-SquaresGradientandHessian .................. 75
4 Least-Absolute-Values Criterion and Minimax Criterion 81
4.1 Introduction ............................... 81 4.2 Preamble:lp-Norms........................... 82 4.3 Thelp-NormProblem.......................... 86 4.4 Thel1-NormCriterionforInverseProblems . . . . . . . . . . . . . . 89 4.5 Thel∞-NormCriterionforInverseProblems. . . . . . . . . . . . . . 96
5 Functional Inverse Problems 101
5.1 RandomFunctions............................101 5.2 SolutionofGeneralInverseProblems. . . . . . . . . . . . . . . . . .108 5.3 IntroductiontoFunctionalLeastSquares . . . . . . . . . . . . . . . . 108 5.4 Derivative and Transpose Operators in Functional Spaces . . . . . . . 119
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7
Appendices
6.1 Volumetric Probability and Probability Density . . . . 6.2 Homogeneous Probability Distributions . . . . . . . . 6.3 Homogeneous Distribution for Elastic Parameters . . 6.4 Homogeneous Distribution for Second-Rank Tensors 6.5 Central Estimators and Estimators of Dispersion . . . 6.6 GeneralizedGaussian ..........................174 6.7 Log-NormalProbabilityDensity ....................175 6.8 Chi-SquaredProbabilityDensity ....................177 6.9 MonteCarloMethodofNumericalIntegration . . . . . . . . . . . . . 179 6.10 SequentialRandomRealization.....................181 6.11 CascadedMetropolisAlgorithm.....................182 6.12 DistanceandNorm ...........................183 6.13 TheDifferentMeaningsoftheWordKernel . . . . . . . . . . . . . . 183 6.14 TransposeandAdjointofaDifferentialOperator . . . . . . . . . . . . 184 6.15 TheBayesianViewpointofBackus(1970) . . . . . . . . . . . . . . . 190 6.16 TheMethodofBackusandGilbert ...................191 6.17 DisjunctionandConjunctionofProbabilities . . . . . . . . . . . . . . 195 6.18 PartitionofDataintoSubsets ......................197 6.19 MarginalizinginLinearLeastSquares . . . . . . . . . . . . . . . . .200 6.20 RelativeInformationofTwoGaussians . . . . . . . . . . . . . . . . .201 6.21 ConvolutionofTwoGaussians .....................202 6.22 Gradient-BasedOptimizationAlgorithms. . . . . . . . . . . . . . . .203 6.23 ElementsofLinearProgramming....................223 6.24 SpacesandOperators ..........................230 6.25 UsualFunctionalSpaces.........................242 6.26 MaximumEntropyProbabilityDensity . . . . . . . . . . . . . . . . .245 6.27 TwoPropertiesoflp-Norms.......................246 6.28 DiscreteDerivativeOperator ......................247 6.29 LagrangeParameters ..........................249 6.30 MatrixIdentities.............................249 6.31 InverseofaPartitionedMatrix .....................250 6.32 NormoftheGeneralizedGaussian ...................250
Problems 253
5.5 5.6 5.7 5.8
GeneralLeast-SquaresInversion ....................133
Example: X-Ray Tomography as an Inverse Problem Example: Travel-Time Tomography . . . . . . . . . Example: Nonlinear Inversion of Elastic Waveforms .
......... 140 . . . . . . . . . 143 . . . . . . . . . 144
159
7.1 Estimation of the Epicentral Coordinates of a Seismic Event 7.2 MeasuringtheAccelerationofGravity . . . . . . . . . . . 7.3 ElementaryApproachtoTomography. . . . . . . . . . . . 7.4 Linear Regression with Rounding Errors . . . . . . . . . . 7.5 UsualLeast-SquaresRegression. . . . . . . . . . . . . . . 7.6 Least-Squares Regression with Uncertainties in Both Axes
......253 ......256 ......259 ......266 ......269 ......273
......... 159 ......... 160 ......... 164 ......... 170 ......... 170
Contents ix
7.7 LinearRegressionwithanOutlier....................275 7.8 Condition Number and A Posteriori Uncertainties . . . . . . . . . . . 279 7.9 ConjunctionofTwoProbabilityDistributions. . . . . . . . . . . . . . 285
7.10 Adjoint of a Covariance Operator . 7.11 Problem7.1Revisited . . . . . . . 7.12 Problem7.3Revisited . . . . . . . 7.13 An Example of Partial Derivatives 7.14 Shapesofthelp-NormMisfitFunctions . . . . . . . . . . . . . . . .290 7.15 UsingtheSimplexMethod .......................293 7.16 Problem7.7Revisited..........................295 7.17 GeodeticAdjustmentwithOutliers ...................296
7.18 InversionofAcousticWaveforms.............. 7.19 UsingtheBackusandGilbertMethod. . . . . . . . . . . . 7.20 The Coefficients in the Backus and Gilbert Method . . . . . 7.21 The Norm Associated with the 1D Exponential Covariance 7.22 The Norm Associated with the 1D Random Walk . . . . . 7.23 The Norm Associated with the 3D Exponential Covariance
References and References for General Reading Index
......297 ......304 ...... 308 ...... 308 ...... 311 ...... 313
317 333
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