Information-Based Inversion and Processing With Applications : Information-Based Inversion and Processing with Applications - T. J. Ulrych

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Information-Based Inversion and Processing With Applications

Information-Based Inversion and Processing with Applications

By: T. J. Ulrych, M. D. Sacchi

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Hardcover | 18 January 2006

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Hardcover
436 Pages

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23.5 x 16.51 x 1.91

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Information-Based Inversion and Processing with Applications examines different classical and modern aspects of geophysical data processing and inversion with emphasis on the processing of seismic records in applied seismology.

Chapter 1 introduces basic concepts including: probability theory (expectation operator and ensemble statistics), elementary principles of parameter estimation, Fourier and z-transform essentials, and issues of orthogonality. In Chapter 2, the linear treatment of time series is provided. Particular attention is paid to Wold decomposition theorem and time series models (AR, MA, and ARMA) and their connection to seismic data analysis problems. Chapter 3 introduces concepts of Information theory and contains a synopsis of those topics that are used throughout the book. Examples are entropy, conditional entropy, Burg's maximum entropy spectral estimator, and mutual information. Chapter 4 provides a description of inverse problems first from a deterministic point of view, then from a probabilistic one. Chapter 5 deals with methods to improve the signal-to-noise ratio of seismic records. Concepts from previous chapters are put in practice for designing prediction error filters for noise attenuation and high-resolution Radon operators. Chapter 6 deals with the topic of deconvolution and the inversion of acoustic impedance. The first part discusses band-limited extrapolation assuming a known wavelet and considers the issue of wavelet estimation. The second part deals with sparse deconvolution using various 'entropy' type norms. Finally, Chapter 7 introduces recent topics of interest to the authors.

The emphasis of this book is on applied seismology but researchers in the area of global seismology, and geophysical signal processing and inversion will find material that is relevant to the ubiquitous problem of estimating complex models from a limited number of noisy observations.

Non-conventional approaches to data processing and inversion are presented
Important problems in the area of seismic resolution enhancement are discussed
Contains research material that could inspire graduate students and their supervisors to undertake new research directions in applied seismology and geophysical signal processing

Industry Reviews

"The book is beautifully produced and very well referenced...[it] should be available in all technical libraries. Some readers will feel that they must have the book. All geophysicists should have a look..." -Ronald Green for THE LEADING EDGE

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Non-Fiction Science Physics Applied Physics Geophysics Earth Sciences, Geography, Environment, Planning Earth Sciences Volcanology Seismology Computing & I.T.Computer Science

Some Basic Concepts	p. 1
Introduction	p. 1
Probability Distributions, Stationarity & Ensemble Statistics	p. 1
Essentials of Probability Distributions	p. 2
Ensembles, Expectations etc.	p. 5
The Ergodic Hypothesis	p. 8
The Chebychev Inequality	p. 8
Time Averages and Ergodicity	p. 9
Properties of Estimators	p. 9
Bias of an Estimator	p. 10
An Example	p. 10
Variance of an Estimator	p. 11
An Example	p. 11
Mean Square Error of an Estimator	p. 11
Orthogonality	p. 12
Orthogonal Functions and Vectors	p. 12
Orthogonal Vector Space	p. 13
Gram-Schmidt Orthogonalization	p. 14
Remarks	p. 16
Orthogonality and Correlation	p. 16
Orthogonality and Eigenvectors	p. 17
Fourier Analysis	p. 20
Introduction	p. 20
Orthogonal Functions	p. 20
Fourier Series	p. 22
The Fourier Transform	p. 22
Properties of the Fourier Transform	p. 24
The FT of Some Functions	p. 25
Truncation in Time	p. 28
Symmetries	p. 29
Living in a Discrete World	p. 32
Aliasing and the Poisson Sum Formula	p. 33
Some Theoretical Details	p. 35
Limits of Infinite Series	p. 36
Remarks	p. 37
The z Transform	p. 37
Relationship Between z and Fourier Transforms	p. 38
Discrete Fourier Transform	p. 40
Inverse DFT	p. 41
Zero Padding	p. 42
The Fast Fourier Transform (FFT)	p. 43
Linearity and Time Invariance	p. 45
Causal Systems	p. 47
Discrete Convolution	p. 48
Convolution and the z Transform	p. 49
Deconvolution	p. 49
Dipole Filters	p. 51
Invertibility of Dipole Filters	p. 52
Properties of Polynomial Filters	p. 53
Some Toy Examples for Clarity	p. 54
Least Squares Inversion of Minimum Phase Dipoles	p. 60
Inversion of Minimum Phase Sequences	p. 64
Inversion of Nonminimum Phase Wavelets: Optimum Lag Spiking Filters	p. 67
Discrete Convolution and Circulant Matrices	p. 67
Discrete and Circular Convolution	p. 67
Matrix Notation for Circular Convolution	p. 69
Diagonalization of the Circulant Matrix	p. 69
Applications of the Circulant	p. 71
Convolution	p. 71
Deconvolution	p. 71
Efficient Computation of Large Problems	p. 73
Polynomial and FT Wavelet Inversion	p. 74
Expectations etc.	p. 76
The Covariance Matrix	p. 78
Lagrange Multipliers	p. 78
Linear Time Series Modelling	p. 81
Introduction	p. 81
The Wold Decomposition Theorem	p. 81
The Moving Average, MA, Model	p. 82
Determining the Coefficients of the MA Model	p. 83
Computing the Minimum Phase Wavelet via the FFT	p. 84
The Autoregressive, AR, Model	p. 86
Autocovariance of the AR Process	p. 87
Estimating the AR Parameters	p. 88
The Levinson Recursion	p. 90
Initialization	p. 92
The Prediction Error Operator, PEO	p. 92
Phase Properties of the PEO	p. 95
Proof of the Minimum Delay Property of the PEO	p. 95
The Autoregressive Moving Average, ARMA, Model	p. 96
A Very Special ARMA Process	p. 97
MA, AR and ARMA Models in Seismic Modelling and Processing	p. 100
Extended AR Models and Applications	p. 102
A Little Predictive Deconvolution Theory	p. 103
The Output of Predictive Deconvolution	p. 104
Remarks	p. 106
Summary	p. 107
A Few Words About Nonlinear Time Series	p. 108
The Principle of Embedding	p. 109
Summary	p. 112
Levinson's Recursion and Reflection Coefficients	p. 113
Theoretical Summary	p. 113
Summary and Remarks	p. 116
Minimum Phase Property of the PEO	p. 118
PROOF I	p. 118
Eigenvectors of Doubly Symmetric Matrices	p. 118
Spectral decomposition	p. 119
Minimum phase property	p. 121
PROOF II	p. 121
Discussion	p. 123
Information Theory and Relevant Issues	p. 125
Introduction	p. 125
Entropy in Time Series Analysis	p. 125
Some Basic Considerations	p. 125
Entropy and Things	p. 126
Differential (or Relative) Entropy	p. 127
Multiplicities	p. 128
The Kullback-Leibler Information Measure	p. 129
The Kullback-Leibler Measure and Entropy	p. 129
The Kullback-Leibler Measure and Likelihood	p. 130
Jaynes' Principle of Maximum Entropy	p. 130
The Jaynes Entropy Concentration Theorem, ECT	p. 131
The Jaynes Entropy Concentration Theorem, ECT	p. 131
Example 1. The Famous Die Problem	p. 132
Example 2. The Gull and Newton Problem	p. 134
Shannon Entropy Solution	p. 135
Least Squares Solution	p. 135
Burg Entropy Solution	p. 135
The General MaxEnt Solution	p. 137
Entropic justification of Gaussianity	p. 139
MaxEnt and the Spectral Problem	p. 141
John Burg's Maximum Entropy Spectrum	p. 141
Remarks	p. 144
The Akaike Information Criterion, AIC	p. 146
Relationship of the AIC to the FPE	p. 149
Mutual Information and Conditional Entropy	p. 150
Mutual Information	p. 151
Entropy and Aperture	p. 154
Discussion	p. 155
The Inverse Problem	p. 157
Introduction	p. 157
The Linear (or Linearized) Inverse Formulation	p. 157
The Lagrange Approach	p. 158
The Hyperparameter Approach	p. 159
A Hybrid Approach	p. 160
A Toy Example	p. 161
Total Least Squares	p. 163
The TLS Solution	p. 164
Computing the Weight Matrix	p. 166
Parameter Covariance Matrix	p. 167
Simple Examples	p. 168
The General TLS Problem	p. 169
SVD for TLS	p. 170
SVD Solution for TLS - Overdetermined Case (M > N)	p. 171
An Illustration	p. 173
Extensions of TLS	p. 175
Discussion	p. 180
Probabilistic Inversion	p. 181
Minimum Relative Entropy Inversion	p. 182
Introduction to MRE	p. 182
The Bayesian Approach	p. 183
MRE Theoretical Details	p. 184
Determining the Lagrange Multipliers	p. 187
Confidence Intervals	p. 188
The Algorithm	p. 189
Taking Noise Into Account	p. 189
Generalized Inverse Approach	p. 190
Applications of MRE	p. 191
Bandlimited Extrapolation	p. 191
Hydrological Plume Source Reconstruction	p. 193
Discussion	p. 195
Bayesian Inference	p. 196
A Little About Priors	p. 198
A Simple Example or Two	p. 200
Likelihood and Things	p. 201
Non Random Model Vector	p. 202
The Controversy	p. 203
Inversion via Bayes	p. 204
Determining the Hyperparameters	p. 206
Parameter Errors: Confidence and Credibility Intervals	p. 207
A Bit More About Prior Information	p. 207
Parameter Uncertainties	p. 208
A Little About Marginals	p. 209
Parameter Credibility Intervals	p. 210
Computational Tractability and Minimum Relative Entropy	p. 211
More About Priors	p. 211
Bayes, MaxEnt and Priors	p. 212
The MaxEnt pdf	p. 212
Incorporating Sample Size via Bayes	p. 213
Summary	p. 217
Bayesian Objective Functions	p. 218
Zero Order Quadratic Regularization	p. 219
Regularization by the Cauchy-Gauss Model	p. 220
Summary and Discussion	p. 222
Hierarchical Issues	p. 223
Empirical Issues	p. 224
Singular Value Decomposition, SVD	p. 226
Signal to Noise Enhancement	p. 229
Introduction	p. 229
f - x Filters	p. 229
The Signal Model	p. 230
AR f - x Filters	p. 231
The Convolution Matrix	p. 233
Some Examples	p. 233
Nonlinear Events: Chirps in f - x	p. 234
Gap Filling and Recovery of Near Offset Traces	p. 236
f - x Projection Filters	p. 237
Wavenumber Domain Formulation	p. 237
Space Domain Formulation	p. 240
A Wrong Formulation of the Problem	p. 241
ARMA Formulation of Projection Filters	p. 242
Estimation of the ARMA Prediction Error Filter	p. 242
Noise Estimation	p. 243
ARMA and Projection Filters	p. 244
Discussion	p. 248
Principal Components, Eigenimages and the KL Transform	p. 250
Introduction	p. 250
PCA and a Probabilistic Formulation	p. 250
Eigenimages	p. 252
Eigenimages and the KL Transformation	p. 254
Eigenimages and Entropy	p. 257
KL Transformation in Multivariate Statistics	p. 258
KL and Image Processing	p. 259
Eigenimages and the Fourier Transform	p. 259
Computing the Filtered Image	p. 260
Applications	p. 261
Signal to Noise Enhancement	p. 261
Eigenimage Analysis of Common Offset Sections	p. 262
Eigenimages and Velocity Analysis	p. 266
Residual Static Correction	p. 268
3D PCA - Eigensections	p. 271
Introducing Eigensections	p. 271
Eigenfaces	p. 271
Computing the Eigensections	p. 273
SVD in 3D	p. 274
Detail Extraction	p. 275
Remarks	p. 276
Discussion	p. 278
Radon Transforms	p. 279
The Linear Radon Transform (LRT)	p. 279
The Inverse Slant Stack Operator	p. 281
The Sampling Theorem for Slant Stacks	p. 283
Discrete Slant Stacks	p. 284
Least Squares Inverse Slant Stacks	p. 285
Parabolic Radon Transform (PRT)	p. 286
High Resolution Radon Transforms	p. 287
Computational Aspects	p. 289
Least Squares Radon Transform	p. 289
High Resolution Oarabolic Radon Transform	p. 294
Non-iterative High Resolution Radon Transform	p. 295
Time variant Radon Transforms	p. 296
Discussion	p. 303
Deconvolution with Applications to Seismology	p. 305
Introduction	p. 305
Layered Earth Model	p. 305
Normal Incidence Formulation	p. 306
Impulse Response of a Layered Earth	p. 308
Deconvolution of the Reflectivity Series	p. 310
The Autocovariance Sequence and the White Reflectivity Assumption	p. 311
Deconvolution of Noisy Seismograms	p. 312
Deconvolution in the Frequency Domain	p. 313
Sparse Deconvolution and Bayesian Analysis	p. 317
Norms for Sparse Deconvolution	p. 318
Modifying J	p. 319
1D Impedance Inversion	p. 325
Acoustic Impedance	p. 326
Bayesian Inversion of Impedance	p. 328
Linear Programming Impedance Inversion	p. 331
Autoregressive Recovery of the Acoustic Impedance	p. 332
AR Gap Prediction	p. 335
Gap Prediction with Impedance Constraints	p. 336
Minimum Entropy Extension of the High Frequencies	p. 337
Nonminimum Phase Wavelet Estimation	p. 337
Nonminimum Phase System Identification	p. 337
The Bicepstrum	p. 339
The Tricepstrum	p. 340
Computing the Bicepstrum and Tricepstrum	p. 341
Some Examples	p. 342
Algorithm Performance	p. 342
Blind, Full Band Deconvolution	p. 350
Minimum Entropy Deconvolution, MED	p. 350
Minimum Entropy Estimators	p. 351
Entropy Norms and Simplicity	p. 353
Wiggins Algorithm	p. 354
Frequency Domain Algorithm	p. 355
Blind Deconvolution via Independent Component Analysis	p. 356
Introduction	p. 356
Blind Processing	p. 357
Independence	p. 357
Definition of ICA	p. 360
Specifying Independence	p. 360
Finally, the Reason to "Why Independence"?	p. 362
Blind Deconvolution	p. 363
The ICA Algorithm	p. 363
ICA, BD and Noise	p. 366
A Synthetic Example	p. 366
Remarks	p. 368
Discussion	p. 368
A Potpourri of Some Favorite Techniques	p. 369
Introduction	p. 369
Physical Wavelet Frame Denoising	p. 369
Frames and Wavelet Frames	p. 370
Prestack Seismic Frames	p. 372
Noise Suppression	p. 372
Synthetic and Real Data Examples	p. 374
Discussion	p. 374
Stein Processing	p. 376
Principles of stacking	p. 377
Trimmed Means	p. 377
Weighted stack	p. 378
The Stein Estimator	p. 378
The Bootstrap and the EIC	p. 381
The Bootstrap Method	p. 382
The Extended Information Criterion	p. 383
The Expected Log Likelihood and the EIC	p. 384
Extended Information Criterion, EIC	p. 385
Application of the EIC to Harmonic Retrieval	p. 386
Discussion	p. 387
Summary	p. 388
Table of Contents provided by Ingram. All Rights Reserved.

Information-Based Inversion and Processing With Applications

Information-Based Inversion and Processing with Applications

At a Glance

Hardcover

Industry Reviews

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