Inverse Problem Theory and Methods for Model Parameter Estimation
- 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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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
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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
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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