Inverse Problem Theory and Methods for Model Parameter Estimation

happytiger1966

Contents
Preface xi
1 The General Discrete Inverse Problem 1
1.1 Model Space and Data Space . . . . . . . . . . . . . . . . . . . . . . 1
1.2 States of Information . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Measurements and A Priori Information . . . . . . . . . . . . . . . . 24
1.5 Defining the Solution of the Inverse Problem . . . . . . . . . . . . . . 32
1.6 Using the Solution of the Inverse Problem . . . . . . . . . . . . . . . 37
2 Monte Carlo Methods 41
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 The Movie Strategy for Inverse Problems . . . . . . . . . . . . . . . . 44
2.3 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Monte Carlo Solution to Inverse Problems . . . . . . . . . . . . . . . 51
2.5 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 The Least-Squares Criterion 57
3.1 Preamble: The Mathematics of Linear Spaces . . . . . . . . . . . . . 57
3.2 The Least-Squares Problem . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Estimating Posterior Uncertainties . . . . . . . . . . . . . . . . . . . 70
3.4 Least-Squares Gradient and Hessian . . . . . . . . . . . . . . . . . . 75
4 Least-Absolute-Values Criterion and Minimax Criterion 81
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Preamble:
p-Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 The
p-Norm Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 The
1-Norm Criterion for Inverse Problems . . . . . . . . . . . . . . 89
4.5 The
∞-Norm Criterion for Inverse Problems . . . . . . . . . . . . . . 96
5 Functional Inverse Problems 101
5.1 Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Solution of General Inverse Problems . . . . . . . . . . . . . . . . . . 108
5.3 Introduction to Functional Least Squares . . . . . . . . . . . . . . . . 108
5.4 Derivative and Transpose Operators in Functional Spaces . . . . . . . 119
vii
viii Contents
5.5 General Least-Squares Inversion . . . . . . . . . . . . . . . . . . . . 133
5.6 Example: X-Ray Tomography as an Inverse Problem . . . . . . . . . 140
5.7 Example: Travel-Time Tomography . . . . . . . . . . . . . . . . . . 143
5.8 Example: Nonlinear Inversion of Elastic Waveforms . . . . . . . . . . 144
6 Appendices 159
6.1 Volumetric Probability and Probability Density . . . . . . . . . . . . . 159
6.2 Homogeneous Probability Distributions . . . . . . . . . . . . . . . . . 160
6.3 Homogeneous Distribution for Elastic Parameters . . . . . . . . . . . 164
6.4 Homogeneous Distribution for Second-Rank Tensors . . . . . . . . . 170
6.5 Central Estimators and Estimators of Dispersion . . . . . . . . . . . . 170
6.6 Generalized Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.7 Log-Normal Probability Density . . . . . . . . . . . . . . . . . . . . 175
6.8 Chi-Squared Probability Density . . . . . . . . . . . . . . . . . . . . 177
6.9 Monte Carlo Method of Numerical Integration . . . . . . . . . . . . . 179
6.10 Sequential Random Realization . . . . . . . . . . . . . . . . . . . . . 181
6.11 Cascaded Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . 182
6.12 Distance and Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.13 The Different Meanings of the Word Kernel . . . . . . . . . . . . . . 183
6.14 Transpose and Adjoint of a Differential Operator . . . . . . . . . . . . 184
6.15 The Bayesian Viewpoint of Backus (1970) . . . . . . . . . . . . . . . 190
6.16 The Method of Backus and Gilbert . . . . . . . . . . . . . . . . . . . 191
6.17 Disjunction and Conjunction of Probabilities . . . . . . . . . . . . . . 195
6.18 Partition of Data into Subsets . . . . . . . . . . . . . . . . . . . . . . 197
6.19 Marginalizing in Linear Least Squares . . . . . . . . . . . . . . . . . 200
6.20 Relative Information of Two Gaussians . . . . . . . . . . . . . . . . . 201
6.21 Convolution of Two Gaussians . . . . . . . . . . . . . . . . . . . . . 202
6.22 Gradient-Based Optimization Algorithms . . . . . . . . . . . . . . . . 203
6.23 Elements of Linear Programming . . . . . . . . . . . . . . . . . . . . 223
6.24 Spaces and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.25 Usual Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.26 Maximum Entropy Probability Density . . . . . . . . . . . . . . . . . 245
6.27 Two Properties of
p-Norms . . . . . . . . . . . . . . . . . . . . . . . 246
6.28 Discrete Derivative Operator . . . . . . . . . . . . . . . . . . . . . . 247
6.29 Lagrange Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.30 Matrix Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.31 Inverse of a Partitioned Matrix . . . . . . . . . . . . . . . . . . . . . 250
6.32 Norm of the Generalized Gaussian . . . . . . . . . . . . . . . . . . . 250
7 Problems 253
7.1 Estimation of the Epicentral Coordinates of a Seismic Event . . . . . . 253
7.2 Measuring the Acceleration of Gravity . . . . . . . . . . . . . . . . . 256
7.3 Elementary Approach to Tomography . . . . . . . . . . . . . . . . . . 259
7.4 Linear Regression with Rounding Errors . . . . . . . . . . . . . . . . 266
7.5 Usual Least-Squares Regression . . . . . . . . . . . . . . . . . . . . . 269
7.6 Least-Squares Regression with Uncertainties in Both Axes . . . . . . 273
Contents ix
7.7 Linear Regression with an Outlier . . . . . . . . . . . . . . . . . . . . 275
7.8 Condition Number and A Posteriori Uncertainties . . . . . . . . . . . 279
7.9 Conjunction of Two Probability Distributions . . . . . . . . . . . . . . 285
7.10 Adjoint of a Covariance Operator . . . . . . . . . . . . . . . . . . . . 288
7.11 Problem 7.1 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 289
7.12 Problem 7.3 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 289
7.13 An Example of Partial Derivatives . . . . . . . . . . . . . . . . . . . 290
7.14 Shapes of the
p-Norm Misfit Functions . . . . . . . . . . . . . . . . 290
7.15 Using the Simplex Method . . . . . . . . . . . . . . . . . . . . . . . 293
7.16 Problem 7.7 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 295
7.17 Geodetic Adjustment with Outliers . . . . . . . . . . . . . . . . . . . 296
7.18 Inversion of Acoustic Waveforms . . . . . . . . . . . . . . . . . . . . 297
7.19 Using the Backus and Gilbert Method . . . . . . . . . . . . . . . . . . 304
7.20 The Coefficients in the Backus and Gilbert Method . . . . . . . . . . . 308
7.21 The Norm Associated with the 1D Exponential Covariance . . . . . . 308
7.22 The Norm Associated with the 1D Random Walk . . . . . . . . . . . 311
7.23 The Norm Associated with the 3D Exponential Covariance . . . . . . 313
References and References for General Reading 317
Index 333

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【书    名】inverse problem theory and methods for model parameter estimation
【作    者】 Albert Tarantola
【出版日期】2004
【出 版 社】SIAM
【文件格式】 PDF
【页    数】358
【内容简介】the general discrete inverse problem, monte carlo methods, the least-squares criterion, least-absolute-values criterion and minimax criterion, functional inverse problems,
【附件个数】6
【威望要求】0