Mathematics for Seismic Data Processing and Interpretation

Mathematics for Seismic Data Processing and Interpretation

A. R. Camina and G. J. Janacek

School of Mathematics and Physics University of East Anglia

Foreword by

R. L. French Racal Geophysics Limited

Introduction by

M. Bacon Shell

Graham ~ Trotman

First published in 1984 by

Graham & Trotman Limited Sterling House 66 Wilton Road London SWlV IDE

© A. R. Camina and G. J. Janacek, 1984

Softcover reprint of the hardcover 1st edition 1984

British Library Cataloguing in Publication Data Camina, A. R.

Mathematics for seismic data. I. Engineering mathematics I. Title II. Janacek, G. J. 510'.2462 TA330

ISBN 978-0-86010-576-3 ISBN 978-94-011-7767-2 (eBook) DOI 10.1007/978-94-011-7767-2

This publication is protected by international copyright law. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanica~ photo-copying, recording or otherwise, without the prior permission of the publishers.

Typeset in Great Britain by J. W. Arrowsmith Ltd, Bristol

CONTENTS

FOREWORD ix

PREFACE xiii

INTRODUCTION xv

Chapter 1 SPECIAL FUNCTIONS

1 Functions 1 2 Polynomials and Step Functions 2 3 Trigonometric Functions 7 4 Power and Exponential Functions 14 5 Inverse Functions 16 6 New Functions from Old 20 7 Numbers 20

Chapter 2 CALCULUS: DIFFERENTIATION 27

1 Introduction 27 2 Higher Derivatives 35 3 Maxima and Minima 37 4 Taylor Series and Approximations 41 5 Partial Derivatives 43 6 Higher Order Partial Derivatives 46 7 Optimisation 48

Chapter 3 INTEGRATION 51

1 Introduction and Definition 51 2 The Relationship between Integration and Differentiation 54 3 Numerical Integration (Quadrature) 61

v

vi Contents

4 Double Integration 63 5 Line Integrals 66 6 Differential Equations 73

Chapter 4 COMPLEX NUMBERS 79

1 Introduction 79 2 The Beginning 79 3 Functions of Complex Variables 83 4 Differentiation and Integration 86

Chapter 5 MATRICES 89

1 Introduction 89 2 Definitions and Elementary Properties 89 3 Matrices 91 4 Multiplication of Matrices 93 5 Special Types of Matrices 96 6 Matrices as Functions 98 7 Linear Equations 100 8 Eigenvalues and Quadratic Forms 108

Chapter 6 STOCHASTIC PROCESSES, PROBABILITY AND STATISTICS 112

1 Introduction 112 2 Probability 115 3 Permutations and Combinations 119 4 Probability Distributions 121 5 Joint Distributions 128 6 Expected Values and Moments 129 7 Real Data Samples 134 8 Two Variables 138 9 Simulation and Monte Carlo Methods 141

10 Confidence Intervals 144 11 Stochastic Processes 147

Chapter 7 FOURIER ANALYSIS 149

1 Introduction 149 2 Fourier Series 149 3 Some Examples of Fourier Analysis 152 4 The Phase, Amplitude and Exponential Formulation 154 5 Fourier Transform 158 6 The z-Transform 162

Contents vii

7 The Discrete Fourier Transform 163 8 Fast Fourier Transform 168 9 Frequency Domain 171

Chapter 8 TIME SERIES 174

1 Stationary and Related Series 175 2 Aliasing and Sampling 183 3 Filters and Convolutions 188

Chapter 9 APPLICATIONS 197

1 Wavelets 197 2 Predictive Deconvolution 204

Appendix 1 REFERENCES TO APPLICATIONS 211

Appendix 2 SOME USEFUL FORMULAE FOR READY REFERENCE 215

Appendix 3 PROGRAMS 218

INDEX 251

FOREWORD

With the growth of modern computing power it has become possible to apply far more mathematics to real problems. This has led to the difficulty that many people who have been working in various jobs suddenly find themselves not understanding the modern processing which is being applied to their own professional field. It also means that the people presently being trained in these subjects need to understand a much wider range of mathe­matics than in the past. It is to both of these groups that this book is addressed.

The major objective is to present the reader with the basic mathematical understanding to follow the new developments in their own field. The mathematics in this book is based on the need to understand signal process­ing. The modern work in this area is mathematically very sophisticated and our purpose is not to train professional mathematicians but to make far more of the literature accessible. Since this book is based on courses devised for Racal Geophysics there is clearly going to be a bias towards the applications in that area, as the title implies. It is also true that the bibliogra­phy has been chosen in order to aid the reader in that field by pointing them in the direction of recent applications in geophysics.

Whilst every attempt has been made to make the material comprehensible to the non-mathematically inclined, it is important to remember that this is a mathematical textbook. This has the implication that it must be read very carefully and not like a novel. One of the great advantages of mathe­matics is its very conciseness and precision. This has the disadvantage of making it difficult to read. So have patience and take your time. There is a collection of computer programs (American spelling) which we hope will be useful, not for commercial use, but to enable the reader to work through some of these ideas for themselves, especially to foster understanding of the mathematics.

This work arose out of the realisation at Racal Geophysics that numeracy had become a problem as we moved from our traditional business under the Decca Survey flag of single-channel analogue seismics into newer fields of engineering hazard surveys and multi-channel exploration. It became clear that, whether our personnel's background was classical geology, engineering geology or geophysics, each had problems to some degree with the mathematical concepts involved with current seismic data processing techniques.

ix

x Foreword

It was also clear that there were no really suitable courses in the UK to teach the relevant mathematics (and the related physics and computing). This is not meant as a criticism of the various courses currently available, it is purely a comment on their usefulness to our specific needs. Most of the instruction offered in seismics is of two varieties, either of an acquaint­ance type or of an advanced nature to already practising experts neither of which include the fundamentals which were required. The length and size of these courses usually also preclude a high level of direct one-to-one instructor-student contact. None of this was what we at Racal required and consequently, in December 1981, in conjunction with the Racal College at Brixham, we decided to take the bull by the horns and set up a course tailored to our specific needs. After consultation with colleagues at the University of East Anglia, University College of Wales, Aberstwyth, Merlin Geophysical Company Limited and Racal College, we produced a proposed curriculum for a three-part course.

Part One, to be taught by lecturers of Racal College at Brixham, would involve a review of electronics, physics, acoustics and the use of a suite of the analogue equipment employed by Racal Geophysics and would be of three weeks' duration. Part Two, to be provided by staff from both Racal Geophysics Limited and Merlin Geophysical Company Limited, would include multi-channel digital acquistion processing, by computer, terminating i~a period of hands-on experience at Merlin Geophysical Company Limited at Woking. This would be a two week course. Part Three, to be provided by the School of Mathematics of the J)niversity of East Anglia, would be a series of correspondence notes in mathematics to be supplied to the student over a period of six months. The idea being that they would be in stages and lead into the other parts of the course. The text of this book is an expansion of these notes.

The course has been inclined specifically towards digital seismic recording and processing, although it would obviously provide a good general back­ground to any time-series, digital analysis. The starting point for this course has been at the end of high school. The early chapters are a review of the A-level syllabus and a review of first year university additional mathematics in areas of specific relevance. The subjects covered are special functions, trigonometric functions for waveforms and calculus. The content and the examples, although of an easy high schoolj first year nature, are all orientated towards seismics and were designed to assist with Part I of the course. In that the objective was to prepare a mathematics book for use by geophysicists all instr~ction is of a purely mathematical nature and no attempt is made to apply the text to physical theory. Therefore, in the first three chapters the mathematics used with alternating current theory, operational amplifiers, Huygens' principle and the Rayleigh-Willis curve will be explained but no attempt to apply it will be made. One departure from this practice is the section on number systems where a description of binary, octal and hexadecimal bases could be equally at home in a text book on physics or electronics. However, as this is so crucial to the further development of digital time series analysis it was decided to include this.

Foreword xi

The later chapters are much more complex and ultimately go beyond what is usually taught in a first degree subsidiary mathematics course.

The middle section covers complex numbers which introduces the idea of vector transform, a section that is further developed in the chapter on Fourier analysis; matrices which are the main number system used in computers and is the direct form for de-multiplexing. The next chapter is devoted to stochastic processes and probability and a development of the concepts of mean, median, root mean square and standard deviation. The final chapters deal with Fourier Analysis and Transforms and Time Series Analysis. Covered in these chapters are the mathematical concepts involved in Fourier synthesis and decomposition, predictive deconvolution and Weiner Filtering.

The actual processing concepts and details were covered by the course of lectures produced by Merlin Geophysical Company. The point of the mathematics course was that the Merlin lecturers would not have to be interrupted by definitions of triple integral signs and the convolution and de-convolution signs. Short sections are also included on the z-transform, and the wave equation and the frequency-wave number domain.

To expand the last point, this text is not meant to be a mathematical discussion of predictive deconvolution and the like, neither is it meant to be a rigorous and complete mathematical text. The idea is that, those who have not had a heavy University mathematical training, will be able to use this book as a reader and background to the classic processing papers by Robinson, Backus, Claerbout, Weiner, Berkhout, etc.

One of the problems encountered in designing this course in general, which was also highlighted with the mathematics content, was the bringing together of a number of advanced level ideas from a variety of subjects and disciplines. This is evident in the number of starting points, almost with the beginning of each chapter, and the way in which the whole text only really comes together in the final chapters. Consequently, the text does not flow from one chapter to next as would be the case in a conventional text book. In other words, in order to carry out seismic data processing it is necessary to have an understanding of binary number systems, discrete wavelet sampling, digital filtering and Fourier analysis as well as the work­ings of digital recording and large mainframe computing techniques. Incidentally, an understanding of the geology, and geophysics of the earth is quite helpful as it gives one an idea after all the processing as to how well you have done.

It cannot be emphasised too much that this is a textbook, covering many of the mathematical techniques employed in modern day seismic data processing and as such most of the examples and references are linked to specific problems. Indeed, an attempt has been made to correlate the examples through to final processing ideas and to only consider that which is rei event to seismics; consequently, this is not a complete mathematical course. But like all mathematics text books, it cannot be read from cover to cover in an evening like a novel. Time has to be taken over each chapter and the examples worked in order to achieve a reasonable understanding.

Although written essentially for geophysical work, most of the techniques involved are equally relevant to the field of data transfer, whether it be by

xii Foreword

telecommunication, both radio and wire, fibre optics, or acoustics. The only difference being that remarked on by one of the authors towards the end of the final drafting, that in seismics we were looking for and at the noise, rather than at the signal which is usually the case. The problem is that there is noise arid noise. Hopefully this book will remove some of the noise from the processing.

R. L. French Racal Geophysics Limited

PREFACE

In this book we have tried to present the mathematical foundations for understanding signal processing in seismic analysis. It is not meant to be a mathematics text book in the traditional sense but rather a mathematics text to give the reader an understanding of the concepts involved. The object is not to turn the reader into a mathematician but rather to enable him or her to be mathematically literate.

The chapters of the book could be divided into three sections, the first three chapters making up the first section. In this section, most of the material is essentially" A-level material" (in the English context) or first calculus course (in the American context). There are various sections especially relevant to geophysics. It could be said that most of this is essential to the understanding of continuous processes.

Chapters 4 and 5 are not always covered in such basic courses but nevertheless underlie a lot of the work done in the last three chapters. It is difficult to imagine a modern engineer or applied scientist who has not a need to know about the contents of these chapters. It is perhaps in these two chapters that the reader will first find material which is not necessarily familiar. It is important at this point to recall that reading mathematics is a slow procedure. By its nature the writing is condensed, many ideas can be compressed into a short formula. So read slowly and carefully, but don't be afraid to go on and then return. Often the way a topic develops leads to greater understanding of its roots.

The last three chapters are the meat of the book. In one sense the justification of the first six chapters is to enable the reader to understand these last three chapters. One could almost say that this is presented as a precursor to the work of Enders Robinson.

Throughout the main text the reader will occasionally find a "P" in the margin. This means that there is a program (written in BASIC) which is relevant to the material in the text at that point. These are not meant to be the latest, fastest, most efficient programs. Their purpose is to enable the reader to use them on most micros and to get some feeling for the mathe­matics involved.

We would like to thank the Chairman and Directors of Racal Electronics plc and the Directors of their subsidiaries Racal Training Services Limited and Racal Geophysics Limited, for permission to write and publish this book.

xiii

xiv Preface

In particular, thanks to Phillip Holden (Oxon.) C. Eng. M.I.E.E. of the Racal Training College at Brixham for checking the original manuscript and for his constant encouragement throughout the project. Thanks are also due to Robert Whittington BSc., MSc., PhD, F.G.S. of University College of Wales, Aberystwyth, members of the School of Environmental Studies at the University of East Anglia, the Directors and staff of Merlin Geophy­sical Company Limited of Woking and of course colleagues at Racal Geophysics.

Finally, we would like to thank Carol Haines who did a magnificent job typing the manuscript, correcting our inadequate English and spelling as she went along. We would also like to thank Nick Bartlett for the figures and many colleagues for comment and criticism. However, like all authors, we must make it quite clear that all errors, mistakes and inaccuracies are due entirely to us.

A. R. Camina and G. J. Janacek University of East Anglia

INTRODUCTION

One of the most important practical applications of geophysics is in the search for accumulations of oil and gas. This is largely a matter of looking for suitable rock structures that might have trapped the accumulations. Typically, useful reserves of oil and gas are found at depths of several thousand metres below the Earth's surface. It is only at such depths that the conditions of pressure and temperature are right to cook the organic remains in a source rock, leading to the formation of oil and gas. These petroleum products are expelled into the surrounding rocks, displacing the water which is otherwise present in the pore spaces. Being less dense than the water, the oil and gas tend to rise up through the prorous rock until their path is blocked by a layer of rock with no pore spaces, through which they cannot pass. Thus, a petroleum accumulation will tend to form at the top of a "buried hill" (anticline) of porous rock, which is capped by an impermeable layer. Therefore, petroleum exploration is largely a matter of looking for such buried structures.

Sometimes, it is fairly easy to guess the strucure at a depth of several thousand metres by extrapolating the structures visible at the surface. In such a case, the hypothesis that deep structures are geometrically similar to the shallow ones can be easily tested by drilling a few wells. However, such easy cases have usually been drilled long ago. If we want to look for new oil-fields today, we must search for deeply buried structures that have no expression at all at the surface. In principle, this search could be carried out by drilling a large number of exploration wells. However, wells are expensive to drill, and in many cases, especially offshore, trying to find fields by more-or-Iess blind drilling would be hopelessly uneconomic.

It is at this point that the geophysicist can offer a great deal of help. Geophysical methods permit us to build up a picture of the sub-surface structure to depths of several thousand metres. With this knowledge, a small number of wells can be precisely situated so as to test the most propsective structures; furthermore, the geological information gained from the well, which in itself tells us only about a zone within a few feet of the borehole, can be extrapolated laterally with some confidence, perhaps for many kilometres in favourable cases. Various geophysical methods (gravity, mag­netics, seismic refraction) can be used to delineate sub-surface structure, but they mostly have very poor vertical and horizontal resolution, giving

xv

xvi I ntrod uction

us a very generalised picture of the sub-surface. One method, however, can give us rather precise knowledge of sub-surface structure, with a resolution down to a few tens of metres in many cases. This is the seismic reflection technique, which in the last twenty years has become a basic tool of petroleum exploration.

In essence, the idea behind this technique is an extension of that of the ship's echo-sounder. Sound waves are generated at the Earth's surface, using a rather strong source such as a controlled explosion. The sound travels down into the Earth, and is reflected back from sub-surface layers where a change in rock physical properties occurs. These echoes from sub-surface discontinuities can be detected by receivers, analogous to micro­phones, on the ground surface and recorded on magnetic tape. It is then possible to measure the time taken for the sound to travel from the surface down to the reflecting layer and back again; this travel time is a measure of the depth of the reflecting interface. Because of the great depths to the reflectors, the echoes are rather faint, and considerable skill is needed in the design of field equipment that will record them satisfactorily. It is usually necessary to use an array of receivers and add together their signals, so as to achieve partial cancellation of random noise, such as might for example be generated by road traffic or by the wind blowing in the trees. For the same reason, it is often necessary to repeat the observation several times at each location, and add together the resulting echo signals from each separate explosion.

Even when the echoes have been satisfactorily recorded, our problems are not at an end. It is fairly easy to interpret the echo travel-times from a ship's echo-sounder in terms of depth to the sea-bed, because the sound has travelled through a (nearly) homogeneous sea-water layer. In seismic reflection exploration, the sound has travelled through what may be very complex structures below the surface, and this gives rise to a number of complications. By way of illustration, let us single out just two of the problems that arise.

Firstly, we should ideally like to put a very sharp pulse of sound into the ground, and record the echoes as a series of sharp pulses also. In this way, it would be possible to measure the travd-times very accurately. Unfortunately, this ideal is unattainable, for two reasons. It is not possible to generate an ideally sharp pulse with any practical course, even an explosion. Even if this were possible, however, it is a property of the real Earth that, over a path of several thousand metres, it will smear out an initially sharp pulse, perhaps turning it into quite a complex waveform. This will limit the vertical resolution we can attain; if we have a series of closely-spaced reflectors, the echoes from them will overlap in time, and the resulting jumble will be difficult to sort out into the echoes from the individual layers.

Secondly, if we have a number of reflecting layers, sound energy can be partly trapped, bouncing backwards and forwards between them. Some of this energy eventually arrives back at the surface, but later than expected because of the time spent in the part of the path where it is bouncing up and down. If the reflecting layers involved are close together, the extra time taken will be small, and the effect contributes to the pulse-broadening

Introduction xvii

mentioned above. If, however, the layers are several thousand metres apart, the delayed arrival will mimic a reflection from a much deeper layer. Clearly, we would not want to drill an apparent deep structure which is actually such an artefact of the method.

Great progress has been made over the last twenty years in dealing with these problems. Perhaps the greatest single advance has been the use of digital recording of the receiver signals. This has meant that the data can be fed into a computer, where a wide variety of ingenious signal-processing techniques can be applied. In this way, the problems outlined above can be partially solved. Thus, it is possible to estimate the shape of the smeared­out pulse that actually generated the data, and then process the records to approximate what they would have been with an ideally sharp pulse. It is also possible to predict the arrival times of energy that has spent some time trapped between sub-surface layers, and then subtract this signal from the record, so that we are no longer fooled into believing that these echoes come from genuine deep reflectors.

These signal-processing techniques were first applied, of necessity, in the search for deep oil accumulations, but in recent years they have been increasingly used in the rather simpler problem of shallow sub-surface investigation. This is usually carried out for engineering purposes (predic­tion of drilling hazards, and selection of sites for offshore production platforms or pipeline routes). In this application, the depth of investigation required is usually only a few hundred metres, but a relatively high resolution may be needed.

To apply these signal-processing techniques with confidence, it is impor­tant to understand their nature and limitations. A cookerybook approach is not enough; what is ideally needed is a thorough understanding of what happens to the seismic signal as it propagates through the earth, and the effects of the source and receiver parameters on our picture of the sub­surface. It is not only the geophysicist directly concerned with processing the data who needs this appreciation; anyone who interprets the data in geological terms needs a clear understanding of the distortions introduced into his picture of the sub-surface by the imperfections of the seismic reflection technique.

Any useful and detailed account of the seismic reflection method inevi­tably involves a good deal of mathematics. Many texts cover this ground, but they all assume that the reader has a good grasp of the mathematical background, and make no real attempt to explain the mathematics (as opposed to the physics) of the development of the subject. However, it is a dubious assumption that the user of such a book will have enough mathematics to be able to follow the arguments in detail. With the increased use of the seismic method, a wide variety of people need to learn about it, and their pre-existing mathematical knowledge will range from a fairly elementary school level in some cases to University level in others. To fill gaps in his knowledge, it would be possible for the student to read parts of various traditional texts, assembling a "course" directly related to his needs; this type of study, however, requires careful guidance from an experienced teacher. This book brings together these scattered topics, to give a coherent account of the background mathematics needed to

xviii Introduction

understand the seismic method, and thus covers ground which no other single textbook does. It begins with a section on basic concepts, which will be useful as a reference source even to those familiar with them, and goes on to develop the subject to a level at which the student can read for himself the (sometimes rather abstruse) literature on seismic processing. The final section of the book provides a bridge to the more advanced material to be found elsewhere.

An important feature of the text is the provision of numerous examples, and some illustrative computer programs. By working through these, the student can acquire the detailed familiarity with mathematical manipulation which he will need if he is to understand the basis of modern seismic techniques.

M. Bacon Shell UK Exploration and Production Ltd