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Global optimization methods in Geophysical inversion

s e c o n d e d i t i o n

making inferences about systems in the earth’s subsurface from remotely sensed, sparse measurements is a challenging task. Geophysical inversion aims to find models that explain geophysical observations – a model-based inversion method attempts to infer model parameters by iteratively fitting observations with theoretical predictions from trial models. Global optimization often enables the solution of non-linear models, employing a global search approach to find the absolute minimum of an objective function so that predicted data best fit the observations.

this new edition provides an up-to-date overview of the most popular global optimization methods, including a detailed description of the theoretical development underlying each method, and a thorough explanation of the design, implementation, and limitations of algorithms.

a new chapter provides details of recently developed methods, such as the neighborhood algorithm and particle swarm optimization. an expanded chapter on uncertainty estimation includes a succinct description on how to use optimization methods for model space exploration to characterize uncertainty and now discusses other new methods such as hybrid monte carlo and multi-chain mcmc methods. other chapters include new examples of applications, from uncertainty in climate modeling to whole-earth studies. several different examples of geophysical inversion, including joint inversion of disparate geophysical datasets, are provided to help readers design algorithms for their own applications.

this is an authoritative and valuable text for researchers and graduate students in geophysics, inverse theory, and exploration geoscience, and an important resource for professionals working in engineering and petroleum exploration.

mrinal K. sen is a Jackson chair professor in applied seismology at the University of texas, austin, and in January 2012, while on leave, became director of the national Geophysical research institute at hyderabad, india. his research areas include seismic wave propagation, inverse problems, seismic imaging, reservoir characterization, and computational geophysics, and he is an expert on seismic wave propagation, including anisotropy and fractures, developing analytic and numerical techniques for forward and inverse modeling. professor sen is the principal author of two books on geophysical inversion and co-author of over 130 papers in peer-reviewed journals and has been an instructor for several industry short courses. he serves on several national and international committees and is an associate editor of leading journals such as Geophysics, and Journal of Seismic Exploration.

paUl l. stoffa has held the shell distinguished chair in Geophysics at the department of Geological sciences, University of texas, austin, since 1997 and was director of the institute for Geophysics at the same institution from 1994 to 2008. his areas of expertise are in multichannel seismic acquisition, signal processing, acoustic and elastic wave propagation, modeling, imaging, and inversion of geophysical data, and employing his knowledge of parallel computers, he works on developing new seismic data acquisition and processing methods that can be used to address complex geologic problems. dr stoffa has published over 100 research articles in refereed journals. a member of the society of exploration Geophysicists (seG), the american Geophysical Union (aGU), and the european association of Geoscientists and engineers (eaGe), he is also a recipient of the society of brazilian Geophysicists’ foreign Geophysicist recognition award.

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-1-107-01190-8 - Global Optimization Methods in Geophysical Inversion: Second EditionMrinal K. Sen and Paul L. StoffaFrontmatterMore information

http://www.cambridge.org/9781107011908

http://www.cambridge.org







Global optimization methods in Geophysical

inversion

s e c o n d e d i t i o n

mrinal K. sen University of Texas, Austin, USA

CSIR-National Geophysical Research Institute, India

and

paUl l. stoffaUniversity of Texas, Austin, USA






cambridge university presscambridge, new york, melbourne, madrid, cape town,

singapore, são paulo, delhi, mexico city

cambridge University pressthe edinburgh building, cambridge cb2 8rU, UK

published in the United states of america by cambridge University press, new york

www.cambridge.orginformation on this title: www.cambridge.org/9781107011908

first edition published by elsevier science b.v., 1995, as Global Optimization Methods in Geophysical Inversion

first edition © elsevier science b.v. 1995.second edition © mrinal K. sen and paul l. stoffa 2013

this publication is in copyright. subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written

permission of cambridge University press.

first published by cambridge University press 2013

printed in the United Kingdom at the University press, cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication datasen, mrinal K.

Global optimization methods in geophysical inversion / mrinal K. sen, paul l. stoffa, the University of texas at austin, institute for Geophysics,

J.J. pickle research campus. – second edition.pages cm

includes bibliographical references and index.isbn 978-1-107-01190-8 (hardback)

1. Geological modeling. 2. Geophysics–mathematical models. 3. inverse problems (differential equations) 4. mathematical optimization. i. stoffa, paul l., 1948– ii. title.

Qe43.s46 2013550.1′515357–dc23 2012033212

isbn 978-1-107-01190-8 hardback

additional resources for this publication at www.cambridge.org/sen_stoffa

cambridge University press has no responsibility for the persistence or accuracy of Urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is,

or will remain, accurate or appropriate.






v

Preface to the first edition (1995) page ixPreface to the second edition (2013) xii

1 preliminary statistics 11.1 random variables 11.2 random numbers 11.3 probability 31.4 probability distribution, distribution function, and density

function 41.5 Joint and marginal probability distributions 81.6 mathematical expectation, moments, variances, and covariances 91.7 conditional probability and bayes’ rule 111.8 monte carlo integration 131.9 importance sampling 141.10 stochastic processes 151.11 markov chains 171.12 homogeneous, inhomogeneous, irreducible, and aperiodic

markov chains 201.13 the limiting probability 21

2 direct, linear, and iterative-linear inverse methods 232.1 direct inversion methods 242.2 model-based inversion methods 292.3 linear/linearized inverse methods 322.4 solution of linear inverse problems 352.5 iterative methods for non-linear problems: local optimization 532.6 solution using probabilistic formulation 672.7 summary 69

contents






Contentsvi

3 monte carlo methods 723.1 enumerative or grid-search techniques 723.2 monte carlo inversion 743.3 hybrid monte carlo–linear inversion 793.4 directed monte carlo methods 80

4 simulated annealing methods 814.1 metropolis algorithm 854.2 heat bath algorithm 914.3 simulated annealing without rejected moves 1004.4 fast simulated annealing (fsa) 1014.5 very fast simulated reannealing 1024.6 mean field annealing 1064.7 Using sa in geophysical inversion 1164.8 summary 117

5 Genetic algorithms 1195.1 a classical Ga 1195.2 schemata and the fundamental theorem of genetic algorithms 1285.3 problems 1315.4 combining elements of sa into a new Ga 1325.5 a mathematical model of a Ga 1345.6 multimodal fitness functions, genetic drift, Ga with sharing,

and repeat (parallel) Ga 1425.7 Uncertainty estimates 1445.8 evolutionary programming – a variant of Ga 1455.9 summary 146

6 other stochastic optimization methods 1486.1 the neighborhood algorithm (na) 1486.2 particle swarm optimization (pso) 1536.3 simultaneous perturbation stochastic approximation (spsa) 154

7 Geophysical applications of simulated annealing and genetic algorithms 1577.1 1d seismic waveform inversion 1597.2 prestack migration velocity estimation 1847.3 inversion of resistivity sounding data for 1d earth models 2047.4 inversion of resistivity profiling data for 2d earth models 2087.5 inversion of magnetotelluric sounding data for 1d earth models 2117.6 stochastic reservoir modeling 217






Contents vii

7.7 seismic deconvolution by mean field annealing (mfa) and hopfield network 221

7.8 Joint inversion 229

8 Uncertainty estimation 2418.1 methods of numerical integration 2438.2 simulated annealing: the Gibbs sampler 2468.3 Genetic algorithm: the parallel Gibbs sampler 2498.4 numerical examples 2508.5 hybrid monte carlo 2628.6 summary 265

Bibliography 268Index 279











ix

preface to the first edition (1995)

one of the major goals of geophysical inversion is to find earth models that explain geophysical observations. thus the branch of mathematics known as optimization has found significant use in many geophysical applications. Geophysical inversion in this context involves finding an optimal value of a function of several vari-ables. the function that we want to minimize (or maximize) is a misfit (or fitness) function that characterizes the differences (or similarities) between observed and synthetic data calculated by using an assumed earth model. the earth model is described by physical parameters that characterize the properties of rock layers, such as the compressional-wave velocity, shear-wave velocity, resistivity, etc.

both local and global optimization methods are used in the estimation of mater-ial properties from geophysical data. as the title of this book suggests, our goal is to describe the application of several recently developed global optimization methods to geophysical problems. although we emphasize the application aspects of these algorithms, we describe several parts of the theory in sufficient detail for readers to understand the underlying fundamental principles on which these algo-rithms are based. at this stage we take the opportunity to define some commonly used terms.

for many geophysical applications, the misfit surface as a function of the model parameters that are described by the mismatch between the predicted and observed geophysical data may be highly complicated and characterized by multiple hills and valleys. thus such a function will have several minima and maxima; the min-imum of all the minima is called the global minimum, and all other minima are called local minima. note that the global minimum is one of the local minima, but the converse is not true, and it is also possible to have several minima of nearly the same depth. local optimization or search algorithms such as gradient-descent methods typically attempt to find a local minimum in the close neighborhood of the starting solution. almost all the local search methods are deterministic algorithms.






Preface to the first edition (1995)x

they use local properties of the misfit function to calculate an update to the current answer and search in the downhill direction. thus these algorithms will miss the global minimum if the starting solution is nearer to one of the local minima than the global minimum. the local minimum syndrome has plagued geophysicists for over a decade now.

recently (owing to the advent of powerful and relatively inexpensive comput-ers), global optimization methods have been applied to several geophysical prob-lems. Unlike local optimization methods, these methods attempt to find the global minimum of the misfit function. most of the global optimization algorithms are stochastic in nature and use more global information about the misfit surface to update their current position. the convergence of these methods to the globally optimal solution is not guaranteed for all the algorithms. only for some of the simulated annealing algorithms under certain conditions is convergence to the glo-bally optimal solution statistically guaranteed. also, with real observational data it is never possible to know whether the derived solution corresponds to the global minimum or not. however, our experience indicates that we are able to find many good solutions starting with only poor initial models using global optimization methods.

these global optimization methods are computationally intensive, but with the advent of vector computers, parallel computers, and powerful desktop workstations, use of these methods is becoming increasingly practical. While finding the optimal solution will always be a goal, and the global optimization methods described here are well suited for this purpose, they can also be used to obtain additional infor-mation about the nature of the solution. in particular, the description of a solution is not complete without assigning uncertainties to the derived answer. With noisy data it may not even be advisable to search for the so-called global minimum. in these situations, a statistical formulation of the inverse problem is often appealing. consequently, we also describe how global optimization methods can be applied in a statistical framework to estimate the uncertainties in the derived result.

this is not a book on inverse theory per se; several excellent texts already exist (e.g., menke 1984; tarantola 1987). our goal is to describe in sufficient detail the fundamentals of several optimization methods with application to geophysical inversion such that students, researchers, and practitioners will be able to design practical algorithms to solve their specific geophysical inversion problems. We attempted to make this book virtually self-contained so that there are no prereq-uisites, except for a fundamental mathematical background that includes a basic understanding of linear algebra and calculus. the material presented in this book can easily be covered in a one-semester graduate-level course on geophysical inver-sion. We believe that after reviewing the materials presented in this book, readers will be able to develop specific algorithms for their own applications. We will be






Preface to the first edition (1995) xi

happy to mail sample fortran codes of prototype metropolis simulated annealing (sa), heat bath sa, very fast simulated annealing (vfsa), and a basic genetic algorithm to those interested.

much of our work on non-linear inversion has been supported by grants from the national science foundation, the office of naval research, cray research, inc., and the texas higher education coordinating board. We acknowledge the University of texas system center for high performance computing for their sup-port and computational resources. adre duijndam, Jacob fokkema, cliff frohlich, John Goff, lester ingber, tad Ulrych, and lian-she zhao reviewed the manuscript and offered valuable suggestions. milton porsani, who worked with us for one year as a visiting scientist, along with several of our graduate students, including faruq akbar, carlos calderon, raghu chunduru, mike Jervis, vik sen, mehmet tanis, and carlos varela, participated in the research and reviewed the manuscript. their contribution to this project has been extremely valuable. We thank milo backus for his many critical comments during the early stages of the work that helped tremendously in shaping our ideas on inverse theory in general. charlene palmer receives our thanks for painstakingly typesetting the manuscript. We also thank Gus berkhout and Jacob fokkema for inviting us to write the book for the series Advances in Exploration Geophysics.

several figures and parts of the text in chapter 8 are based on a paper presented at the 1994 eaeG 56th annual meeting and exposition in vienna, austria, and have been printed by permission of the copyright holders. the copyright of the paper belongs to the european association of Geoscientists and engineers.

mrinal K. sen wishes to thank his wife, alo, and children, amrita and ayon, for their sacrifice and encouragement. he also thanks neil frazer for his suggestion following the 1990 society of exploration Geophysicists meeting to write a book on this subject.

paul l. stoffa wishes to thank his wife, donna, for her constant support and Gus berkhout, who motivated the writing of this book.

Mrinal K. Sen ([email protected]) Paul L. Stoffa ([email protected])

Institute for Geophysics The University of Texas at Austin






xii

the first edition of the book went out of print a few years ago. We received many requests for a copy of the book, and therefore, an invitation from cambridge University press to publish a revised version of this book was a welcome mes-sage. since the publication of the first edition, a couple of review articles on monte carlo methods have been published in the geophysical literature. a large num-ber of applications have also been reported. this book, however, still remains as the only publication providing a comprehensive overview of global optimiza-tion methods for geophysical applications. Global optimization methods, which were at the time of the first edition considered too slow to be of practical use, are now being used routinely in many applications. in this revised version, we have expanded several sections, including one on local optimization; we include recent global optimization methods such as the neighborhood algorithm (na) and par-ticle swarm optimization (pso), and we add several new applications of global optimization methods, including the joint inversion of diverse data. although no major new algorithmic developments of genetic algorithms (Gas) and simulated annealing (sa) have been reported, their application to solving complex problems has increased significantly since the first edition. We apologize for not being able to include all these successful applications in our reference list.

We remain indebted to our graduate students and postdoctoral fellows for col-laboration. We thank our families for their unconditional support.

the University of texas institute for Geophysics contribution number 2355.

Mrinal K. Sen ([email protected]) Paul L. Stoffa ([email protected])

Institute for Geophysics The University of Texas at Austin

preface to the second edition (2013)






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Global optimization methods in Geophysical inversion › MediaFiles › Texts › 8 › ... · 2013-02-19 · Global optimization methods in Geophysical inversion second edition making