Seismic Inversion - Yanghua Wang - E-Book

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Yanghua Wang

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Beschreibung

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