Table Of ContentOptimal
Seismic Deconvolution
AN ESTIMATION-BASED APPROACH
JERRY M. MENDEL
Department of Electrical Engineering
University of Southern California
Los Angeles, California
With a Foreword by Enders A. Robinson
1983
ACADEMIC PRESS
A Subsidiary of Harcourt Brace Jovanovich, Publishers
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Quotations on pp. 125-126 and Figure 6.2-1 are from Melsa and Cohn (1978), "Decision
and Estimation Theory," © 1978 McGraw-Hill, New York. Reprinted with permission of
the McGraw-Hill Book Company.
Figures 10.1-1 —10.1-8 are reproduced by arrangement with the editors of Geophysics.
COPYRIGHT © 1983, BY ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Mendel, Jerry M., Date
Optimal seismic deconvolut ion.
Includes index.
1. Seismology--Data processing. 2. Seismic
reflection method--Deconvolut ion. 3. Time-series
analys i s. I. Title.
QE539.MA 1983 82-22739
ISBN 0-12-^90780-6
PRINTED IN THE UNITED STATES OF AMERICA
83 84 85 86 9 8 7 6 5 4 3 2 1
To my students, past, present, and future.
Foreword
Today we are in the midst of a universal digital revolution of epic
proportions. The silicon chip has made possible computer power un
dreamed of as recently as a decade ago. Tremendous potential now
exists to extend our scientific computations both quantitatively and
qualitatively. We are faced not only with more computations but also
with more difficult ones.
The deconvolution of seismic data has a long history, going back to
the first of the digital computers in the early 1950s. Deconvolution has
been used as a routine computational procedure on virtually all ex
ploration seismic data since the early 1960s. The tremendous discover
ies in the North Sea, Alaska, Africa, the Far East, the Persian Gulf,
and the many continental shelf regions of the world are testaments to
the efficacy and power of deconvolution in exploration for new re
serves of petroleum and natural gas. Deconvolution lifted the veil that
hung over seismic data, and its success depends to a large measure on
achieving the proper balance between what the theory requires and
what the computer can do. Theory and practice have gone hand in
hand in its history.
Today we are faced with a new level of exploration problems. We
must explore for oil in regions we would have passed over just a few
years ago as being too difficult. The computer power is available and
waiting. But, by and large, the same deconvolution procedures used in
the past are in use today. Now is the time to develop more sophisti
cated and powerful methods of deconvolution. New theoretical devel-
xi
xii Foreword
opment will have a profound influence, not only in geophysics, but in
many other scientific disciplines as well.
This book contains the new deconvolution methods that will fulfill
this requirement. It is an important book and will revolutionize the
whole approach to deconvolution, not only in petroleum exploration
but in most other physical sciences. The material in the book is diffi
cult, but Professor Mendel has written it with clarity and in a form that
is accessible to all serious scientists; it will yield great rewards to those
who study it well. We are indebted to Professor Mendel for the theo
retical developments and practical insights in his path-breaking work.
His is an impressive accomplishment.
ENDERS A. ROBINSON
Tulsa, Oklahoma
Preface
During the past seven years my students and I have approached the
problem of seismic deconvolution from many points of view. This book
brings together those seven years of work. It is addressed to two differ
ent audiences—practitioners of recursive estimation theory and geo
physical signal processors—who, it is hoped, will be drawn closer to
gether by it.
Estimation practitioners will find that seismic deconvolution is a
beautiful and rich area of application for many familiar techniques,
such as Kaiman filtering, optimal smoothing, maximum-likelihood pa
rameter estimation, and maximum-likelihood detection. To these people
this book will be a potpourri of applied estimation and detection the
ory, one that treats a variety of problems for the following models: (1)
a linear time-varying system excited by Gaussian white noise, (2) a lin
ear system excited by Bernoulli-Gaussian white noise, (3) a linear sys
tem excited by multiplicative noise, and (4) a linear system excited by
a mixture of Bernoulli-Gaussian and Gaussian white noises.
Geophysical signal processors will find that seismic deconvolution,
one of their bread-and-butter processing procedures, can be treated
from what will (most likely) be to them a new point of view, namely,
state-variable models and recursive estimation and detection theory.
This approach allows a wide variety of deconvolution problems to be
treated from a unified viewpoint. Effects such as spherical spreading,
backscatter, recording equipment, and prefiltering are easily incor
porated into a state-variable model for seismic deconvolution; thus, no
Xlll
xiv Preface
substantially new theory is needed each time the model is modified to
account for a different effect.
Processors who are used to viewing a multitude of real data results
may be disappointed with this book in that the number of real data re
sults is quite low. Such results are by no means ignored (see, e.g.,
Chapter 9) but most of the techniques described herein can be, and are,
illustrated quite well using synthetic data. We leave it to the potential
user of the book's results to add all the bells and whistles that are inev
itably necessary to make them run efficiently and economically on large
amounts of real data.
To make this book readable to both practitioners of recursive estima
tion theory and geophysical signal processors, I have made it self-con
tained. Appendix A to Chapter 1 presents a brief introduction to state-
variable models and methods. Chapters on minimum-variance estima
tion theory and maximum-likelihood and maximum a posteriori
methods (i.e., Chapters 2 and 4) are included for readers who are unfa-
milar with these optimal estimation procedures. Having used this mon
ograph a number of times in short courses, I suggest that geophysical
signal processors read the entire book, whereas estimation practitioners
can (most likely) omit Sections 1.3.2, 1.3.3, 1.3.4, and Appendix A to
Chapter 1, as well as Chapters 2 and 4.
I do not claim originality for all of the material in this book. Most of
it, however, was generated either by myself, by my students John Kor-
mylo, Faroukh Habibi-Ashrafi, and Mostafa Shiva, or by all of us to
gether. Without their dedicated efforts nothing would have happened. I
am pleased to acknowledge their contributions. I also wish to thank
John Goutsias and A. C. Hsueh, graduate students at USC, and Fred
Aminzadeh and John Kormylo for the assistance in reviewing the
book's manuscript. Finally, I wish to thank the numerous sponsors of
our research, without whose support none of this would have been pos
sible.
CHAPTER 1
Deconvolutìon
1.1 Introduction
Reflection seismology, by far the most widely used geophysical tech
nique for oil exploration, generates a picture of the subsurface lithology
from surface measurements. A vibrational source of seismic energy, such
as an explosive, is located at the surface near an array of sensors, as
shown in Figs. 1.1-1 and 1.1-2. The waves thus generated are reflected
and transmitted at interfaces owing to the impedance mismatches
between different geological layers, and these reflected waves are then
transmitted back to the surface, where they are recorded by the sensors.
By repeating this procedure at many source and sensor locations, one can
produce an image of the subsurface reflectors, as shown in Fig. 1.1-3
[Anstey (1970) and Dobrin (1976), for example].
To generate this reflector image, many levels of signal processing are
performed that attempt to remove various undesired aspects of the raw
data. One such undesired aspect is the duration of the wavelet produced
by the seismic source, because reflected signals from two contiguous re
flectors may overlap and interfere. Because the transmission of seismic
waves is a relatively linear process, we can regard the observed signal z(f)
as a convolution of the source wavelet (the input signal from the seismic
source) V{t) with a reflection signal (the impulse response of the Earth)
μ(0, or
zit) = f μ{τ)ν{ί -τ)άτ + n(t), (1.1-1)
Jo
1
2 1. Deconvolution
Fig. 1.1-1 Land seismic data gathering (from Kormylo, 1979).
where n(f) is an error or noise term. The object of de convolution is to re
move the effects of the source wavelet and the noise term from the ob
served signal, so that one is left with the desired reflection signal, or at
least an estimate thereof [Ricker (1940, 1953) and Robinson (1967b), for
example]. Deconvolution is the signal processing subject of this book.
With the increasing use of digital computers for signal processing it has
become popular to use discrete-time models and signal processing tech
niques. By appropriate discretization methods we can rewrite convolu
tion model (1.1-1) as (see Fig. 1.1-4)
z(k) = (k) + n(k) = £ μϋ)ν« - j) + n(k), (1.1-2)
yi
where k = 1, 2, . . . , N. In this model yi(k) is the ideal seismic trace
(i.e., the noise-free signal); n(k) is "measurement" noise, which accounts
Fig. 1.1-2 Marine seismic data gathering (from Kormylo, 1979).
1.1. Introduction 3
for physical effects not explained by yi(k), as well as sensor noise; V(i),
i — 0, 1, . . . , /, is a sequence associated with the seismic source
wavelet (i.e., the signal distorting system) that is usually much shorter
in duration than N and is band limited in frequency; and μ,Ο'), j =
1, 2, . . . , is the reflectivity sequence (i.e., the desired signal) (see
Appendix A to Chapter 5 for a careful discussion of continuous- and
discrete-time models for μ,). This convolution summation model can be
derived from physical principles and some simplifying assumptions, such
as: normal incidence, each layer is homogeneous and isotropie, small
strains, and pressure and velocity satisfy a one-dimensional wave equa
tion. Signal yi(k) which is recorded as z(k) by a seismic sensor, is a super
position of wavelets reflected from the interfaces of subsurface layers.
The μθ') are related to interface reflection coefficients.
In much of our work we shall assume that V(0) = 0, in which case we
simply change the upper limit of the summation in Eq. (1.1-2) from k to
k - 1. Note also that we do not include a direct reflection term μ,(Ο)V(k) in
the summation. If such a term exists, we assume that it has been absorbed
into z(k), on the left-hand side of Eq. (1.1-2).
Although we have described the model of Eq. (1.1-2) in the context of
reflection seismology, it occurs in many other fields, such as astronomy
(Scargle, 1979) and communication systems (Belfiore and Parks, Jr., 1979;
Lucky, 1965, 1966, 1973). In the latter field, μ,(&) is di message transmitted
over channel V(k), which distorts it. Equalization, the counterpart to de-
convolution, removes the effects of the channel and provides an estimate
of the message.
In this book we assume that μ(β is a white non-Gaussian sequence.
This is a commonly made assumption in seismic deconvolution. Addition
ally, we assume that μθ') and n(J) are statistically independent, and that
n(j) is zero mean.
From linear system theory (Chen, 1970; Kailath, 1980; Schwartz and
Friedland, 1965) we know that the output y(k) of a linear, discrete-time,
time-invariant, causal system whose input m(k) is zero prior to and in
cluding time zero is
y(k) = % m(j)h(k - j), (1.1-3)
where h(i) is the unit response of the system. When comparing Eqs.
(1.1-2) and (1.1-3), we are led to the following important system interpre
tation for the seismic trace model (Mendel, 1977; Mendel and Kormylo,
1978): signal y^k) can be thought of as the output of a linear time-invariant
system whose unit response is V(0 and whose input sequence is the re
flectivity sequence μ{ϊ) (see Fig. 1.1-5).
