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Ulrych, T.J.; Sacchi, M.D.

Information-Based Inversion and Processing with Applications

Autor: Ulrych, T.J.; Sacchi, M.D.
Verlag: Elsevier Science, Oxford
Reihentitel: Handbook of geophysical exploration
Band: Band 36
Zusatzinfo: 437 Seiten.
ISBN13: 9780080461342
ISBN10: 0080461344
Erschienen: 16.12.2005
Medientyp: E-Book
Einbandart: ebook
Land: Großbritannien
Sprache: Englisch
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This book 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
  • Cover
  • Contents
  • Some Basic Concepts
  • Introduction
  • Probability Distributions, Stationarity & Ensemble Statistics
  • Essentials of Probability Distributions
  • Ensembles, Expectations etc
  • The Ergodic Hypothesis
  • The Chebychev Inequality
  • Time Averages and Ergodidty
  • Properties of Estimators
  • Bias of an Estimator
  • An Example
  • Variance of an Estimator
  • An Example
  • Mean Square Error of an Estimator
  • Orthogonality
  • Orthogonal Functions and Vectors
  • Orthogonal Vector Space
  • Gram-Schmidt Orthogonalization
  • Remarks
  • Orthogonality and Correlation
  • Orthogonality and Eigenvectors
  • Fourier Analysis
  • Introduction
  • Orthogonal Functions
  • Fourier Series
  • The Fourier Transform
  • Properties of the Fourier Transform
  • The FT of Some Functions
  • Truncation in Time
  • Symmetries
  • Living in a Discrete World
  • Aliasing and the Poisson Sum Formula
  • Some Theoretical Details
  • Limits of Infinite Scries
  • Remarks
  • The z Transform
  • Relationship Between z and Fourier Transforms
  • Discrete Fourier Transform
  • Inverse DFT
  • Zero Padding
  • The Fast Fourier Transform (FFT)
  • Linearity and Time Invariance
  • Causal Systems
  • Discrete Convolution
  • Convolution and the z Transform
  • Dcconvolution
  • Dipole Filters
  • Invertibility of Dipole Filters
  • Properties of Polynomial Filters
  • Some Toy Examples for Clarity
  • Least Squares Inversion of Minimum Phase Dipoles
  • Inversion of Minimum Phase Sequences
  • Inversion of Nonminimum Phase Wavelets: Optimum Lag SpikingFilters
  • Discrete Convolution and Circulant Matrices
  • Discrete and Circular Convolution
  • Matrix Notation for Circular Convolution
  • Diagonalization of the Circulant Matrix
  • Applications of the Circulant
  • Convolution
  • Deconvolution
  • Efficient Computation of Large Problems
  • Polynomial and FT Wavelet Inversion
  • Expectations etc.,
  • The Covariance Matrix
  • Lagrange Multipliers
  • Linear Time Series Modelling
  • Introduction
  • The Wold Decomposition Theorem
  • The Moving Average. MA, Model
  • Determining the Coefficients of the MA Model
  • Computing the Minimum Phase Wavelet via the FFT
  • The Autoregressive, AR, Model
  • Autocovariance of the AR Process
  • Estimating the AR Parameters
  • The Levinson Recursion
  • Initialization
  • The Prediction Error Operator, PEO
  • Phase Properties of the PEO
  • Proof of the Minimum Delay Property of the PEO
  • The Autoregressive Moving Average, ARMA, Model
  • A Very Special ARMA Process
  • MA, AR and ARMA Models in Seismic Modelling and Processing
  • Extended AR Models and Applications
  • A Little Predictive Deconvolution Theory
  • The Output of Predictive Deconvolution
  • Remarks
  • Summary
  • A Few Words About Nonlinear Time Series
  • The Principle of Embedding
  • Summary
  • Levinson's Recursion and Reflection Coefficients
  • Theoretical Summary
  • Summary and Remarks
  • Minimum Phase Property of the PEO
  • PROOF I
  • Eigenvectors of Doubly Symmetric Matrices
  • Spectral decomposition
  • Minimum phase property
  • PROOF II
  • Discussion
  • Information Theory and Relevant Issues
  • Introduction
  • Entropy in Time Series Analysis
  • Some Basic Considerations
  • Entropy and Things
  • Differential (or Relative) Entropy
  • Multiplicities
  • The Kullback-Lciblcr Information Measure
  • The Kullback-Leibler Measure and Entropy
  • The Kullback-Leibler Measure and Likelihood
  • Jaynes' Principle of Maximum Entropy
  • The Jaynes Entropy Concentration Theorem, ECT
  • The Jaynes Entropy Concentration Theorem, ECT
  • Example 1. The Famous Die Problem
  • Example 2. The Gull and Newton Problem
  • Shannon Entropy Solution
  • Least Squares Solution
  • Burg Entropy Solution
  • The General MaxEnt Solution
  • Entropic justification of Gaussianity
  • MaxEnt and the Spectral Problem
  • John Burg's Maximum Entropy Spectrum
  • Remarks
  • The Akaike Information Criterion, AIC
  • Relationship of the AIC to the FPE
  • Mutual Information and Conditional Entropy
  • Mutual Information
  • Entropy and Aperture
  • Discussion
  • The Inverse Problem
  • Introduction
  • The Linear (or Linearized) Inverse Formulation
  • The Lagrange Approach
  • The Hyperparameter Approach
  • A Hybrid Approach
  • A Toy Example
  • Total Least Squares
  • The TLS Solution
  • Computing the Weight Matrix
  • Parameter Covariancc Matrix
  • Simple Examples
  • The General TLS Problem
  • SVD for TLS
  • SVD Solution for TLS - Overdetermiiied Case (M >TV)
  • An Illustration
  • Extensions of TLS
  • Discussion
  • Probabilistic Inversion
  • Minimum Relative Entropy Inversion
  • Introduction to MRE
  • The Bayesian Approach
  • MRE Theoretical Details
  • Determining the Lagrange Multipliers
  • Confidence Intervals
  • The Algorithm
  • Taking Noise Into Account
  • Generalized Inverse Approach
  • Applications of MRE
  • Bandlimited Extrapolation
  • Hydrological Plume Source Reconstruction
  • Discussion
  • Bayesian Inference
  • A Little About Priors
  • A Simple Example or Two
  • Likelihood and Things
  • Non Random Model Vector
  • The Controversy
  • Inversion via Baycs
  • Determining the Hyperparameters
  • Parameter Errors: Confidence and Credibility Intervals
  • A Bit More About Prior Information
  • Parameter Uncertainties
  • A Little About Marginals
  • Parameter Credibility Intervals
  • Computational Tractability and Minimum Relative Entropy
  • More About Priors
  • Bayes, MaxErit and Priors
  • The MaxEnt pdf
  • Incorporating Sample Size via Baycs
  • Summary
  • Bayesian Objective Functions
  • Zero Order Quadratic Regularization
  • Regularization by the Cauchy-Gauss Model
  • Summary and Discussion
  • Hierarchical Issues
  • Empirical Issues
  • Singular Value Decomposition, SVD
  • Signal to Noise Enhancement
  • Introduction
  • f - x Filters
  • The Signal Model
  • AR f - x Filters
  • The Convolution Matrix
  • Some Examples
  • Nonlinear Events: Chirps in / - x
  • Gap Filling and Recovery of Near Offset Traces
  • f -x Projection Filters
  • Wavenuniber Domain Formulation
  • Space Domain Formulation
  • A Wrong Formulation of the Problem
  • ARMA Formulation of Projection Filters
  • Estimation of the ARMA Prediction Error Filter
  • Noise Estimation
  • ARMA and Projection Filters
  • Discussion
  • Principal Components, Eigenimages and the KL Transform
  • Introduction
  • PC A and a Probabilistic Formulation
  • Eigenimages
  • Eigenimages and the KL Transformation
  • Eigenimages and Entropy
  • KL Transformation in Multivariatc Statistics
  • KL and Image Processing
  • Eigenimages and the Fourier Transform
  • Computing the Filtered Image
  • Applications
  • Signal to Noise Enhancement
  • Eigcnimagc Analysis of Common Offset Sections
  • Eigenimages and Velocity Analysis
  • Residual Static Correction
  • 3D PCA - Eigensections
  • Introducing Eigensections
  • Eigenfaces
  • Computing the Eigensections
  • SVD in 3D
  • Detail Extraction
  • Remarks
  • Discussion
  • Radon Transforms
  • The Linear Radon Transform (LRT)
  • The Inverse Slant Stack Operator
  • The Sampling Theorem for Slant Stacks
  • Discrete Slant Stacks
  • Least Squares Inverse Slant Stacks
  • Parabolic Radon Transform (PRT)
  • High Resolution Radon Transforms
  • Computational Aspects
  • Least Squares Radon Transform
  • High Resolution Parabolic Radon Transform
  • Non-iterative High Resolution Radon Transform
  • Time variant Radon Transforms
  • Discussion
  • Deconvolution with Applications to Seismology
  • Introduction
  • Layered Earth Model
  • Normal Incidence Formulation
  • Impulse Response of a Layered Earth
  • Deconvolution of the Reflectivity Series
  • The Autocovariancc Sequence and the White Reflectivity Assumption
  • Deconvolution of Noisy Seismograms
  • Deconvolution in the Frequency Domain
  • Sparse Deconvolution and Bayesian Analysis
  • Norms for Sparse Deconvolution
  • Modifying J
  • ID Impedance Inversion
  • Acoustic Impedance
  • Bayesian Inversion of Impedance
  • Linear Programming Impedance Inversion
  • Autoregressive Recovery of the Acoustic Impedance
  • AR Gap Prediction
  • Gap Prediction with Impedance Constraints
  • Minimum Entropy Extension of the High Frequencies
  • Nonminimum Phase Wavelet Estimation
  • Nonminimum Phase System Identification
  • The Bicepstrum
  • The Tricepstrum
  • Computing the Bicepstrum and Tricepstrum
  • Some Examples
  • Algorithm Performance
  • Blind, Full Band Deconvolution
  • Minimum Entropy Deconvolution, MED
  • Minimum Entropy Estimators
  • Entropy Norms and Simplicity
  • Wiggins Algorithm
  • Frequency Domain Algorithm
  • Blind Deconvolution via Independent Component Analysis
  • Introduction
  • Blind Processing
  • Independence
  • Definition of ICA
  • Specifying Independence
  • Finally, the Reason to "Why Independence" ?
  • Blind Deconvolution
  • The ICA Algorithm
  • ICA, BD and Noise
  • A Synthetic Example
  • Remarks
  • Discussion
  • A Potpourri of Some Favorite Techniques
  • Introduction
  • Physical Wavelet Frame Dcnoising
  • Frames and Wavelet Frames
  • Prcstack Seismic Frames
  • Noise Suppression
  • Synthetic and Real Data Examples
  • Discussion
  • Stein Processing
  • Principles of stacking
  • Trimmed Means
  • Weighted stack
  • The Stein Estimator
  • The Bootstrap and the EIC
  • The Bootstrap Method
  • The Extended Information Criterion
  • The Expected Log Likelihood and the EIC
  • Extended Information Criterion, EIC
  • Application of the EIC to Harmonic Retrieval
  • Discussion
  • Summary
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