Information-Based Inversion and Processing with Applications (eBook)

Cover 1
Contents 6
Some Basic Concepts 32
Introduction 32
Probability Distributions, Stationarity & Ensemble Statistics
Essentials of Probability Distributions 33
Ensembles, Expectations etc 36
The Ergodic Hypothesis 39
The Chebychev Inequality 39
Time Averages and Ergodidty 40
Properties of Estimators 40
Bias of an Estimator 41
An Example 41
Variance of an Estimator 42
An Example 42
Mean Square Error of an Estimator 42
Orthogonality 43
Orthogonal Functions and Vectors 43
Orthogonal Vector Space 44
Gram-Schmidt Orthogonalization 45
Remarks 47
Orthogonality and Correlation 47
Orthogonality and Eigenvectors 48
Fourier Analysis 51
Introduction 51
Orthogonal Functions 51
Fourier Series 53
The Fourier Transform 53
Properties of the Fourier Transform 55
The FT of Some Functions 56
Truncation in Time 59
Symmetries 60
Living in a Discrete World 63
Aliasing and the Poisson Sum Formula 64
Some Theoretical Details 66
Limits of Infinite Scries 67
Remarks 68
The z Transform 68
Relationship Between z and Fourier Transforms 69
Discrete Fourier Transform 71
Inverse DFT 72
Zero Padding 73
The Fast Fourier Transform (FFT) 74
Linearity and Time Invariance 76
Causal Systems 78
Discrete Convolution 79
Convolution and the z Transform 80
Dcconvolution 80
Dipole Filters 82
Invertibility of Dipole Filters 83
Properties of Polynomial Filters 84
Some Toy Examples for Clarity 85
Least Squares Inversion of Minimum Phase Dipoles 91
Inversion of Minimum Phase Sequences 95
Inversion of Nonminimum Phase Wavelets: Optimum Lag SpikingFilters 98
Discrete Convolution and Circulant Matrices 98
Discrete and Circular Convolution 98
Matrix Notation for Circular Convolution 100
Diagonalization of the Circulant Matrix 100
Applications of the Circulant 102
Convolution 102
Deconvolution 102
Efficient Computation of Large Problems 104
Polynomial and FT Wavelet Inversion 105
Expectations etc., 107
The Covariance Matrix 109
Lagrange Multipliers 109
Linear Time Series Modelling 112
Introduction 112
The Wold Decomposition Theorem 112
The Moving Average. MA, Model 113
Determining the Coefficients of the MA Model 114
Computing the Minimum Phase Wavelet via the FFT 115
The Autoregressive, AR, Model 117
Autocovariance of the AR Process 118
Estimating the AR Parameters 119
The Levinson Recursion 121
Initialization 123
The Prediction Error Operator, PEO 123
Phase Properties of the PEO 126
Proof of the Minimum Delay Property of the PEO 126
The Autoregressive Moving Average, ARMA, Model 127
A Very Special ARMA Process 128
MA, AR and ARMA Models in Seismic Modelling and Processing 131
Extended AR Models and Applications 133
A Little Predictive Deconvolution Theory 134
The Output of Predictive Deconvolution 135
Remarks 137
Summary 138
A Few Words About Nonlinear Time Series 139
The Principle of Embedding 140
Summary 143
Levinson's Recursion and Reflection Coefficients 144
Theoretical Summary 144
Summary and Remarks 147
Minimum Phase Property of the PEO 149
PROOF I 149
Eigenvectors of Doubly Symmetric Matrices 149
Spectral decomposition 150
Minimum phase property 152
PROOF II 152
Discussion 154
Information Theory and Relevant Issues 156
Introduction 156
Entropy in Time Series Analysis 156
Some Basic Considerations 156
Entropy and Things 157
Differential (or Relative) Entropy 158
Multiplicities 159
The Kullback-Lciblcr Information Measure 160
The Kullback-Leibler Measure and Entropy 160
The Kullback-Leibler Measure and Likelihood 161
Jaynes' Principle of Maximum Entropy 161
The Jaynes Entropy Concentration Theorem, ECT 162
The Jaynes Entropy Concentration Theorem, ECT 162
Example 1. The Famous Die Problem 163
Example 2. The Gull and Newton Problem 165
Shannon Entropy Solution 166
Least Squares Solution 166
Burg Entropy Solution 166
The General MaxEnt Solution 168
Entropic justification of Gaussianity 170
MaxEnt and the Spectral Problem 172
John Burg's Maximum Entropy Spectrum 172
Remarks 175
The Akaike Information Criterion, AIC 177
Relationship of the AIC to the FPE 180
Mutual Information and Conditional Entropy 181
Mutual Information 182
Entropy and Aperture 185
Discussion 186
The Inverse Problem 188
Introduction 188
The Linear (or Linearized) Inverse Formulation 188
The Lagrange Approach 189
The Hyperparameter Approach 190
A Hybrid Approach 191
A Toy Example 192
Total Least Squares 194
The TLS Solution 195
Computing the Weight Matrix 197
Parameter Covariancc Matrix 198
Simple Examples 199
The General TLS Problem 200
SVD for TLS 201
SVD Solution for TLS - Overdetermiiied Case (M > TV)
An Illustration 204
Extensions of TLS 206
Discussion 211
Probabilistic Inversion 212
Minimum Relative Entropy Inversion 213
Introduction to MRE 213
The Bayesian Approach 214
MRE Theoretical Details 215
Determining the Lagrange Multipliers 218
Confidence Intervals 219
The Algorithm 220
Taking Noise Into Account 220
Generalized Inverse Approach 221
Applications of MRE 222
Bandlimited Extrapolation 222
Hydrological Plume Source Reconstruction 224
Discussion 226
Bayesian Inference 227
A Little About Priors 229
A Simple Example or Two 231
Likelihood and Things 232
Non Random Model Vector 233
The Controversy 234
Inversion via Baycs 235
Determining the Hyperparameters 237
Parameter Errors: Confidence and Credibility Intervals 238
A Bit More About Prior Information 238
Parameter Uncertainties 239
A Little About Marginals 240
Parameter Credibility Intervals 241
Computational Tractability and Minimum Relative Entropy 242
More About Priors 242
Bayes, MaxErit and Priors 243
The MaxEnt pdf 243
Incorporating Sample Size via Baycs 244
Summary 248
Bayesian Objective Functions 249
Zero Order Quadratic Regularization 250
Regularization by the Cauchy-Gauss Model 251
Summary and Discussion 253
Hierarchical Issues 254
Empirical Issues 255
Singular Value Decomposition, SVD 257
Signal to Noise Enhancement 260
Introduction 260
f - x Filters 260
The Signal Model 261
AR f - x Filters 262
The Convolution Matrix 264
Some Examples 264
Nonlinear Events: Chirps in / — x 265
Gap Filling and Recovery of Near Offset Traces 267
f -x Projection Filters 268
Wavenuniber Domain Formulation 268
Space Domain Formulation 271
A Wrong Formulation of the Problem 272
ARMA Formulation of Projection Filters 273
Estimation of the ARMA Prediction Error Filter 273
Noise Estimation 274
ARMA and Projection Filters 275
Discussion 279
Principal Components, Eigenimages and the KL Transform 281
Introduction 281
PC A and a Probabilistic Formulation 281
Eigenimages 283
Eigenimages and the KL Transformation 285
Eigenimages and Entropy 288
KL Transformation in Multivariatc Statistics 289
KL and Image Processing 290
Eigenimages and the Fourier Transform 290
Computing the Filtered Image 291
Applications 292
Signal to Noise Enhancement 292
Eigcnimagc Analysis of Common Offset Sections 293
Eigenimages and Velocity Analysis 297
Residual Static Correction 299
3D PCA - Eigensections 302
Introducing Eigensections 302
Eigenfaces 302
Computing the Eigensections 304
SVD in 3D 305
Detail Extraction 306
Remarks 307
Discussion 309
Radon Transforms 310
The Linear Radon Transform (LRT) 310
The Inverse Slant Stack Operator 312
The Sampling Theorem for Slant Stacks 314
Discrete Slant Stacks 315
Least Squares Inverse Slant Stacks 316
Parabolic Radon Transform (PRT) 317
High Resolution Radon Transforms 318
Computational Aspects 320
Least Squares Radon Transform 320
High Resolution Parabolic Radon Transform 325
Non-iterative High Resolution Radon Transform 326
Time variant Radon Transforms 327
Discussion 334
Deconvolution with Applications to Seismology 336
Introduction 336
Layered Earth Model 336
Normal Incidence Formulation 337
Impulse Response of a Layered Earth 339
Deconvolution of the Reflectivity Series 341
The Autocovariancc Sequence and the White Reflectivity Assumption 342
Deconvolution of Noisy Seismograms 343
Deconvolution in the Frequency Domain 344
Sparse Deconvolution and Bayesian Analysis 348
Norms for Sparse Deconvolution 349
Modifying J 350
ID Impedance Inversion 356
Acoustic Impedance 357
Bayesian Inversion of Impedance 359
Linear Programming Impedance Inversion 362
Autoregressive Recovery of the Acoustic Impedance 363
AR Gap Prediction 366
Gap Prediction with Impedance Constraints 367
Minimum Entropy Extension of the High Frequencies 368
Nonminimum Phase Wavelet Estimation 368
Nonminimum Phase System Identification 368
The Bicepstrum 370
The Tricepstrum 371
Computing the Bicepstrum and Tricepstrum 372
Some Examples 373
Algorithm Performance 373
Blind, Full Band Deconvolution 381
Minimum Entropy Deconvolution, MED 381
Minimum Entropy Estimators 382
Entropy Norms and Simplicity 384
Wiggins Algorithm 385
Frequency Domain Algorithm 386
Blind Deconvolution via Independent Component Analysis 387
Introduction 387
Blind Processing 388
Independence 388
Definition of ICA 391
Specifying Independence 391
Finally, the Reason to "Why Independence" ? 393
Blind Deconvolution 394
The ICA Algorithm 394
ICA, BD and Noise 397
A Synthetic Example 397
Remarks 399
Discussion 399
A Potpourri of Some Favorite Techniques 400
Introduction 400
Physical Wavelet Frame Dcnoising 400
Frames and Wavelet Frames 401
Prcstack Seismic Frames 403
Noise Suppression 403
Synthetic and Real Data Examples 405
Discussion 405
Stein Processing 407
Principles of stacking 408
Trimmed Means 408
Weighted stack 409
The Stein Estimator 409
The Bootstrap and the EIC 412
The Bootstrap Method 413
The Extended Information Criterion 414
The Expected Log Likelihood and the EIC 415
Extended Information Criterion, EIC 416
Application of the EIC to Harmonic Retrieval 417
Discussion 418
Summary 419