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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2016
Seismic inversion aims to reconstruct a quantitative model of the Earth subsurface, by solving an inverse problem based on seismic measurements. There are at least three fundamental issues to be solved simultaneously: non-linearity, non-uniqueness, and instability. This book covers the basic theory and techniques used in seismic inversion, corresponding to these three issues, emphasising the physical interpretation of theoretical concepts and practical solutions. This book is written for master and doctoral students who need to understand the mathematical tools and the engineering aspects of the inverse problem needed to obtain geophysically meaningful solutions. Building on the basic theory of linear inverse problems, the methodologies of seismic inversion are explained in detail, including ray-impedance inversion and waveform tomography etc. The application methodologies are categorised into convolutional and wave-equation based groups. This systematic presentation simplifies the subject and enables an in-depth understanding of seismic inversion. This book also provides a practical guide to reservoir geophysicists who are attempting quantitative reservoir characterisation based on seismic data. Philosophically, the seismic inverse problem allows for a range of possible solutions, but the techniques described herein enable geophysicists to exclude models that cannot satisfy the available data. This book summarises the author's extensive experience in both industry and academia and includes innovative techniques not previously published.
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Table of Contents
Cover
Dedication
Title Page
Copyright
Preface
Chapter 1: Basics of seismic inversion
1.1 The linear inverse problem
1.2 Data, model and mapping
1.3 General solutions
1.4 Regularisation
Chapter 2: Linear systems for inversion
2.1 The governing equation and its solution
2.2 Seismic scattering
2.3 Seismic imaging
2.4 Seismic downward continuation
2.5 Seismic data processing
Chapter 3: Least-squares solutions
3.1 Determinant and rank
3.2 The inverse of a square matrix
3.3 LU decomposition and Cholesky factorisation
3.4 Least-squares solutions of linear systems
3.5 Least-squares solution for a nonlinear system
3.6 Least-squares solution by QR decomposition
Chapter 4: Singular value analysis
4.1 Eigenvalues and eigenvectors
4.2 Singular value concept
4.3 Generalised inverse solution by SVD
4.4 SVD applications
Chapter 5: Gradient-based methods
5.1 The step length
5.2 The steepest descent method
5.3 Conjugate gradient method
5.4 Biconjugate gradient method
5.5 The subspace gradient method
Chapter 6: Regularisation
6.1 Regularisation versus conditional probability
6.2 The L
p
norm constraint
6.3 The maximum entropy constraint
6.4 The Cauchy constraint
6.5 Comparison of various regularisations
Chapter 7: Localised average solutions
7.1 The average solution
7.2 The deltaness
7.3 The spread criterion
7.4 The Backus-Gilbert stable solution
Chapter 8: Seismic wavelet estimation
8.1 Wavelet extraction from seismic-to-well correlation
8.2 Generalised wavelet constructed from power spectrum
8.3 Kurtosis matching for a constant-phase wavelet
8.4 Cumulant matching for a mixed-phase wavelet
Chapter 9: Seismic reflectivity inversion
9.1 The least-squares problem with a Gaussian constraint
9.2 Reflectivity inversion with an L
p
norm constraint
9.3 Reflectivity inversion with the Cauchy constraint
9.4 Multichannel reflectivity inversion
9.5 Multichannel conjugate gradient method
Chapter 10: Seismic ray-impedance inversion
10.1 Acoustic and elastic impedances
10.2 Ray impedance
10.3 Workflow of ray-impedance inversion
10.4 Reflectivity inversion in the ray-parameter domain
10.5 Ray-impedance inversion with a model constraint
Chapter 11: Seismic tomography based on ray theory
11.1 Seismic tomography
11.2 Velocity-depth ambiguity in reflection tomography
11.3 Ray tracing by a path bending method
11.4 Geometrical spreading of curved interfaces
11.5 Joint inversion of traveltime and amplitude data
Chapter 12: Waveform tomography for the velocity model
12.1 Inversion theory for waveform tomography
12.2 The optimal step length
12.3 Strategy for reflection seismic tomography
12.4 Multiple attenuation and partial compensation
12.5 Reflection waveform tomography
Chapter 13: Waveform tomography with irregular topography
13.1 Body-fitted grids for finite-difference modelling
13.2 Modification of boundary points
13.3 Pseudo-orthogonality and smoothness
13.4 Wave equation and absorbing boundary condition
13.5 Waveform tomography with irregular topography
Chapter 14: Waveform tomography for seismic impedance
14.1 Wave equation and model parameterisation
14.2 The impedance inversion method
14.3 Inversion strategies and the inversion flow
14.4 Application to field seismic data
14.5 Conclusions
Appendices
Appendix A: Householder transform for QR decomposition
Appendix B: Singular value decomposition algorithm
Appendix C: Biconjugate gradient method for complex systems
Appendix D: Gradient calculation in waveform tomography
Exercises and solutions
References
Author index
Subject index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Tables
Chapter 2: Linear systems for inversion
Table 2.1 The physical meaning of parameters in the governing equation
This book is dedicated to my wife Guo-ling, and my two children Brian and Claire.
Seismic Inversion
Theory and Applications
Yanghua Wang
Professor of Geophysics Imperial College London, UK
This edition first published 2017 © 2017 by John Wiley & Sons, Ltd
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Library of Congress Cataloging-in-Publication Data
Names: Wang, Yanghua.
Title: Seismic inversion : theory and applications / by Yanghua Wang, professor of geophysics, Imperial College London, UK.
Description: Malden, MA : Wiley-Blackwell Publishing, Ltd., 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016024723 (print) | LCCN 2016025497 (ebook) | ISBN 9781119257981 (cloth) | ISBN 9781119258049 (pdf) | ISBN 9781119258025(epub)
Subjects: LCSH: Seismic traveltime inversion. | Seismic reflection method-Deconvolution. | Seismic tomography. | Seismology\endash Mathematics.
Classification: LCC QE539.2.S43 W37 2016 (print) | LCC QE539.2.S43 (ebook) | DDC 551.22028/7-dc23
LC record available at https://lccn.loc.gov/2016024723
A catalogue record for this book is available from the British Library.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Cover image: © Yanghua Wang
Preface
Seismic inversion aims to reconstruct an Earth subsurface model based on seismic measurements. Such a subsurface model is quantitatively represented by spatially variable physical parameters, and is extracted from seismic data by solving an inverse problem. For seismic inversion, we need to resolve at least three fundamental issues simultaneously: (1) non-linearity, because the solving procedure is dependent upon the solution, that is, seismic wave propagation involved in the inversion is a function of the current model estimate, (2) non-uniqueness due to data incompleteness, and (3) instability, as a small amount of data errors may cause huge perturbations in the model estimate. The last two complicated issues are due to the inverse problem being ill-posed mathematically.
This book introduces the basic theory and solutions of the inverse problems, in correspondence to the above three issues related to seismic inversion. Practically, we must understand the following how-to’s: to solve a nonlinear problem by iterative linearisation, to solve an underdetermined problem with model constraints, and to solve an ill-posed problem by regularisations. This book also introduces some applications with which to extract meaningful information from seismic data for reservoir characterisation, in order to stimulate readers’ interest for pursuing advanced research in seismic inversion.
This textbook is based on lecture notes of Seismic Inversion and Quantitative Analysis, which have been presented to master and doctoral students in geophysics at Imperial College London. The syllabuses are
1.
Linear inverse problem
2.
Matrix analysis
3.
Least-squares method
4.
Iterative method
5.
Quadratic minimisation
6.
Steepest descent method
7.
Conjugate gradient method
8.
Subspace gradient method
9.
Eigenvalues and eigenvectors
10.
Singular value analysis
11.
Generalised inverse by SVD
12.
Maximum entropy method
13.
Maximum likelihood method
14.
The Cauchy inversion method
15.
General L
p
norm method
16.
Localised average solution
17.
Wavelet estimation
18.
Reflectivity inversion
19.
Ray-impedance inversion
20.
Traveltime tomography
21.
Waveform tomography for velocity
22.
Waveform tomography for impedance
Many of these syllabuses are named as mathematical terminologies. The focus of this book will be their physical meanings. This book is divided into two parts. The first part, consisting of seven chapters, is the fundamentals of linear inverse problems. The second part is methodologies of seismic inversion. The essence of seismic inversion is regularisation. Regularisation can be defined as a model constraint, used additively in an objective function of the inverse problem. Regularisation can also be an action applied directly to the geophysical operator.
Building on the basic theory of linear inverse problems, the methodologies of seismic inversion are explained in detail, including ray-impedance inversion and waveform tomography etc. The application methodologies are categorised into convolutional and wave-equation based groups. This systematic presentation simplifies the subject and enables an in-depth understanding of seismic inversion.
This book summarises the author’s extensive experience in both industry and academia and includes innovative techniques not previously published. Conventionally, the convolutional model is used for seismic reflectivity and impedance inversion, and wave equation-based waveform tomography, or full waveform inversion, is inverting for velocity variation. This book presents for the first time the use of the wave equation-based inversion method for the reconstruction of subsurface impedance images.
This book provides a practical guide to reservoir geophysicists who are attempting quantitative reservoir characterisation based on seismic data. Philosophically, the seismic inverse problem allows for a range of possible solutions, but the techniques described herein enable geophysicists to exclude models that cannot satisfy the available data. This book deals with the engineering aspects of the inverse problem, for understanding the mathematical tools and in turn to generate geophysically meaningful solutions.
Yanghua Wang21 February 2016
