Table Of ContentQUANTITATIVE
SEISMOLOGY
SECOND EDITION
Keiiti Aki
Formerly with Observatoire Volcanologique du Piton de la Fournaise
Paul G. Richards
Lamont-Doherty Earth Observatory of Columbia University
University Science Books
Mill Valley, California
University Science Books
www.uscibooks.c orn
Editor: Jane Ellis
Production Manager: Ann Knight
Manuscript Editor: Lee A. Young
Designer: Robert Ishi
Compositor: Windfall Sojiiare
Illustrators: John and Judy Waller
Printer and Binder: Integrated Book Technology
This book is printed on acid-free paper.
Copyright 02 009 by University Science Books, First Paperback Impression,
Corrected Printing
Copyright 0 2002 by University Science Books
Reproduction or translation of any part of this work beyond that permitted
by Section 107 or 108 of the 1976 United States Copyright Act without
the permission of the copyright owner is unlawful. Requests for permission
or further information should be addressed to the Permissions Department,
University Science Books.
Library of Congress Cataloging-in-Publication Data
Aki, Keiiti, 1930-
Quantitative Seismology / Keiiti Aki, Paul G. Richards.-2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-891389-63-4, Paperback
1. Seismology-Mathematics. I. Richards, Paul G., 1943- 11. Title.
QE539.2.M37 A45 2002
551.22-dc21 2002071360
Printed in the United States of America
10 9 8 7 6 5 4 3
Preface
TO THE FIRST EDITION
In the past decade, seismology has matured as a quantitative science through an exten-
sive interplay between theoretical and experimental workers. Several specialized journals
recorded this progress in thousands of pages of research papers, yet such a forum does
not bring out key concepts systematically. Because many graduate students have expressed
their need for a textbook on this subject and because many methods of seismogram analysis
now used almost routinely by small groups of seismologists have never been adequately
explained to the wider audience of scientists and engineers who work in the peripheral ar-
eas of seismology, we have here attempted to give a unified treatment of those methods of
seismology which are currently used in interpreting actual data.
We develop the theory of seismic wave propagation in realistic Earth models. We study
specialized theories of fracture and rupture propagation as models of an earthquake, and we
supplement these theoretical subjects with practical descriptions of how seismographs work
and how data are analyzed and inverted.
Our text is arranged in two volumes. Volume I gives a systematic development of the
theory of seismic-wave propagation in classical Earth models, in which material properties
vary only with depth. It concludes with a chapter on seismometry. This volume is intended
to be used as a textbook in basic courses for advanced students of seismology. Volume I1
summarizes progress made in the major frontiers of seismology during the past decade. It
covers a wide range of special subjects, including chapters on data analysis and inversion,
on successful methods for quantifying wave propagation in media varying laterally (as well
as with depth), and on the kinematic and dynamic aspects of motions near a fault plane
undergoing rupture. The second volume may be used as a textbook in graduate courses on
tectonophysics, earthquake mechanics, inverse problems in geophysics, and geophysical
data processing.
Many people have helped us. Armando Cisternas worked on the original plan for the
book, suggesting part of the sequence of subjects we eventually adopted. Frank Press’s
encouragement was a major factor in getting this project started. Chapter 12, on inverse
problems, grew out of a course given at MIT by one of the authors and Theodore R.
Madden, to whom we are grateful for many helpful discussions. Our students’ Ph.D. theses
have taught us much of what we know, and have been freely raided. We drew upon the
explicit ideas and results of several hundred people, many of them colleagues, and hope xvii
xviii PREFACE TO THE FIRST EDITION
their contributions are correctly acknowledged in the text. Here, we express our sincere
thanks.
Critical readings of all or part of the manuscript were undertaken by Roger Bilham, Jack
Boatwright, David Boore, Roger Borcherdt, Michel Bouchon, Arthur Cheng, Tom Chen,
Wang-Ping Chen, Bernard Chouet, George Choy, Vernon Cormier, Allan Cox, Shamita Das,
Jim Dewey, Bill Ellsworth, Mike Fehler, Neil Frazer, Freeman Gilbert, Neal Goins, Anton
Hales, David Harkrider, Lane Johnson, Bruce Julian, Colum Keith, Gerry LaTorraca, Wook
Lee, Dale Morgan, Bill Menke, Gerhard Miiller, Albert Ng, Howard Patton, Steve Roecker,
Tony Shakal, Euan Smith, Teng-Fong Wong, Mai Yang, and George Zandt. We appreciate
their attention, their advice, and their encouragement.
About fifteen different secretaries typed for us over the four years during which we
prepared this text. Linda Murphy at Lamont-Doherty carried the major burden, helping us
to salvage some self-respect in the way we handled deadlines. We also thank our manuscript
editor, Dick Johnson, for his sustained efforts and skill in clarifying the original typescript.
We acknowledge support from the Alfred P. Sloan Foundation and the John Simon
Guggenheim Memorial Foundation (P.G.R.). This book could not have been written without
the support given to our research projects over the years by several funding agencies: The
U.S. Geological Survey and the Department of Energy (K.A.); the Advanced Research
Projects Agency, monitored through the Air Force Office of Scientific Research (K.A. and
P.G.R.); and the National Science Foundation (K.A. and P.G.R.).
Keiiti Aki
Paul G. Richards
June 1979
Preface
TO THE PAPERBACK VERSION OF THE SECOND EDITION
In 1975 I received a surprising letter from Kei Aki, beginning “I wonder if you would be
interested in coauthoring a text book on theoretical seismology with me . . .
”
We had both taught advanced seismology courses at our respective institutions, the
Massachusetts Institute of Technology and Columbia University. But his were informed
by many years as a leading researcher. He had worked on everything from the practical
study of noise to theoretical frameworks for interpreting free oscillation signals. I was in
my fourth year as an assistant professor, knew nothing about vast areas of seismology, and
had focussed on some details in source theory and wave propagation that he knew a great
deal more about than I did. But I said yes to his invitation, and thus began a wonderful four-
year period of being forced to learn about seismology well enough to write explanations of
the underlying theory.
At that time, in the mid to late 1970s, the concept of quantifying seismic sources
with a moment tensor had just begun to take hold. Film chips of analog data from the
Worldwide Standardized Seismographic Network were the best source of seismograms for
most geophysicists in academia, but their narrow band and limited dynamic range were
problematic (the seismograms, not the geophysicists). Broadband instruments and digital
methods of recording were only beginning to show their potential.
Kei sent me 300 pages of his teaching notes. We quickly drafted a sequence of chapter
titles and began writing. In 1978 we sent our first draft to the publisher. It was promptly
rejected as being three times longer than planned, and unmarketable. I was devastated. But
Kei calmly responded with the suggestion that we could do a little re-organization and offer
the material as two volumes. The original publishers agreed, and after two more years of
editing and figure preparation the first edition appeared in February 1980.
The IRIS Consortium and the Federation of Digital Seismographic Networks emerged
in the 1980s to meet growing needs for high-quality broadband seismic data. Global,
national, regional, and local networks of broadband seismometers have since been deployed
at thousands of locations, and quantitative seismology is conducted today on a scale that
could hardly be imagined in the 1970s and 1980s. Every generation of seismologists
correctly knows that it is working at new levels of excellence. As always, the rationale for
support to seismology is multi-faceted: to study the Earth’s internal structure, to conduct
...
research in the physics of earthquakes, to quantify and mitigate earthquake hazard, and Xlll
xiv PREFACE TO THE PAPERBACK VERSION OF THE SECOND EDITION
to monitor explosions both to evaluate the weapons development programs of a potential
adversary and to support initiatives in nuclear arms control.
These different applications of seismology are illustrated by our own careers. In 1984
Kei Aki moved from MIT to the University of Southern California, and promoted inte-
gration of scientific information about earthquakes and its public transfer as the founding
science director of the Southern California Earthquake Center. At the Center, for example,
input from earthquake geologists was used together with the fault model of quantitative
seismology, to generate output useful for earthquake engineers. In this work, the concept of
seismic moment was central to unifying information from plate tectonics, geology, geodesy,
and historical and instrumental seismology. The public transfer of the integrated informa-
tion was made in the form of probabilistic estimates of earthquake hazards. The Center is
still alive and well, long after Kei left for an on-site prediction of volcanic eruptions using
seismic signals from an active volcano (RCunion) in the Indian Ocean. In the mid-l980s,
I also changed my interests to applied aspects of seismology and began work on practical
problems of monitoring compliance with nuclear test ban treaties. At first the main issue
was estimating the size of the largest underground nuclear explosions, in the context of
assessing compliance with the 150 kiloton limit of the Threshold Test Ban Treaty. Later
the focus changed to a series of technical issues in detection, location, and identification
of small explosions, in the context of verification of the Comprehensive Nuclear-Test-Ban
Treaty. This latter treaty became a reality in 1996, and is now associated with an Interna-
tional Data Centre in Vienna and an International Monitoring System currently being built
with stations at hundreds of new sites around the world. In early 1996, Xiaodong Song and
I working at Columbia’s Lamont-Doherty Earth Observatory discovered small changes in
the travel time with which seismic waves traverse the Earth’s inner core-evidence that we
interpreted as due to inner core motion with respect to the rest of the solid Earth.
These developments in understanding earthquake hazard, explosion monitoring, and
Earth’s internal structure and processes, directly show that people do seismology for utterly
different reasons. The common thread is interpretation of seismograms. The quality of
data and ease of data access have greatly improved since 1980, but the fundamentals of
seismogram interpretation are little changed. Progress in applications of seismology relies
upon sophisticated methods of analysis, often incorporated into software that students must
learn to use soon after beginning graduate school. The purpose of this book is to provide
students and other researchers with the underlying theory essential to understanding these
methods-and their pitfalls, and possibilities for improvement.
We received numerous requests to keep the 1980e dition in print, and it would have been
easy to accept invitations simply to republish. But I decided in late 1994 to rewrite rather
than republish, because the emergence of new methods for detecting and recording seismic
motions meant that much of the instrumentation chapter would have to be completely
reworked, and rewriting could accommodate new problems, up-to-date references, and
thousands of small changes as well as major revision of some sections. The new publisher,
University Science Books, working with Windfall Software, enabled this second edition
with modern methods of design and typesetting. Numerous typographical errors have been
corrected for this paperback edition. I thank many people for finding them, especially Koji
Uenishi, who worked on the Japanese edition.
Dropped from the first edition are chapters on inverse theory, methods of data analysis,
and seismic wave propagation in media with general heterogeneity. (Note that whole books
PREFACE TO THE PAPERBACK VERSION OF THE SECOND EDITION xv
have been published since 1980 on these subjects.) Parts of our discarded chapters have
been reworked into the chapters that remain. Numerous sections elsewhere are brought
up-to-date (for example, an explanation of the centroid moment tensor). The revised and
rewritten material emphasizes basic methods that have turned out to be most important in
practice.
In May 2005 we received the news of Kei Aki’s untimely death in his home town
on RCunion Island. In the pages of Seismological Review Letters (76, 551-553, issue of
Sept/Oct 2005) I have described some of his many accomplishments. I am so fortunate to
have been among the more than one hundred people who worked as co-authors with him.
He was a gentle leader, informal and approachable, who provided the quantitative methods
that now guide the work of thousands of Earth scientists around the world.
Books like this are more than scaled-up versions of research papers-teams of people
have to work together for years to turn concepts into reality. I thank Jane Ellis, my editor
at University Science Books, for encouragement, tact, patience, help, and stamina since
we began this project in 1994. The help of the first edition publisher, W. H. Freeman and
Co., in allowing us to use original figures where possible, is gratefully acknowledged. I
thank Paul Anagnostopoulos of Windfall Software who introduced me to ZzTS and solved
electronic design and typesetting problems on this second edition over more than ten years;
Kathy Falato and Violeta Tomsa who took care of my office at Lamont; and Kathy Falato,
Elizabeth Jackson, MaryEllen Oliver, and Gillian Richards for entering text and equations
to recreate something like the original edition electronically, thus giving me an entity that
could be revised. (How else could piles of notes for revision be merged with a text generated
in the 1970s with IBM Selectrics?)
I received support during the rewriting from Los Alamos National Laboratory in 1997,
and from several federal agencies back at Lamont. Many people helped with comments on
the first edition, with suggestions for new material, critical reading, supplying references
and figures, and checking the new problems. It is a pleasure here to acknowledge such
contributions to the second edition from Duncan Agnew, Joe Andrews, Yehuda Ben-Zion,
Phil Cummins, Steve Day, Tony Dahlen, Wen-xuan Du, Goran Ekstrom, Karen Fischer,
Steve Grand, John Granville, David Harkrider, Klaus Jacob, Bruce Julian, Richard Katz,
Vitaly Khalturin, Debi Kilb, Won-Young Kim (who selected the broadband seismogram
shown in red on the cover, and the filtered versions with all their different character as shown
also in Figure 12.1), Boris Kostrov, Anyi Li, Wenyi Li, Gerhard Miiller, Jeffrey Park, Mike
Ritzwoller, Peter Shearer, Jinghua Shi, Bob Smith, Stan Whitcomb, Riidi Widmer, Bob
Woodward, and Jian Zhang.
To facilitate commentary on the second edition (both hard copy and this paperback),
and to provide supplementary material as it may accumulate in future years, a website is
maintained at http://www.LDEO.columbia.edu/-richards/Aki-Richards.html
Jody Richards has stayed with all this, and with me, since the very beginning. I owe
her more than thanks, and am so glad we can still dance together.
Paul G. Richards
February 2009
Contents
Preface to the Paperback Version of the Second Edition xiii
Preface to the First Edition xvii
1. INTRODUCTION 1
Suggestions for Further Reading 6
2. BASIC THEOREMS IN DYNAMIC ELASTICITY 11
BOX 2.1 Examples of representation theorems 12
2.1 Formulation 12
BOX2.2 Notation 14
BOX 2.3 Euler or Lagrange? 19
2.2 Stress-Strain Relations and the Strain-Energy Function 20
2.3 Theorems of Uniqueness and Reciprocity 24
2.3.1 Uniqueness theorem 24
2.3.2 Reciprocity theorems 24
BOX 2.4 Use of the term “homogeneous” as applied to equations and boundary
conditions 25
BOX2.5 Parallels 26
2.4 Introducing Green’s Function for Elastodynamics 27
2.5 Representation Theorems 28
2.6 Strain-Displacement Relations and Displacement-Stress Relations in General
Orthogonal Curvilinear Coordinates 30
BOX 2.6 General properties of orthogonal curvilinear coordinates 31
Suggestions for Further Reading 34
Problems 35
3. REPRESENTATION OF SEISMIC SOURCES 37
3.1 Representation Theorems for an Internal Surface; Body-Force Equivalents for
Discontinuities in Traction and Displacement 38
3.1.1 Body-force equivalents 39 V
vi
CONTENTS
BOX 3.1 On the use of effective slip and effective elastic moduli in
the source region 41
3.2 A Simple Example of Slip on a Buried Fault 42
3.3 General Analysis of Displacement Discontinuities across
an Internal Surface X 49
BOX 3.2 On uses of the word “moment” in seismic source theory 53
3.4 Volume Sources: Outline of the Theory and Some Simple Examples 53
BOX 3.3 Body-force equivalents and the seismic moment tensor 54
BOX 3.4 The strain energy released by earthquake faulting 55
Suggestions for Further Reading 58
Problems 59
4. ELASTIC WAVES FROM A POINT DISLOCATION SOURCE 63
4.1 Formulation: Introduction of Potentials 63
+
BOX4.1 On the outgoing solution of g = 6(x)6(t) c2V2g with zero initial
conditions 65
4.1.1 Lame’s theorem 67
4.2 Solution for the Elastodynamic Green Function in a Homogeneous, Isotropic,
Unbounded Medium 68
BOX 4.2 On potentials 69
BOX 4.3 Evaluation of a sui$ace integral 71
4.2.1 Properties of the far-field P -wave 73
4.2.2 Properties of the far-field S-wave 73
4.2.3 Properties of the near-field term 74
4.3 The Double-Couple Solution in an Infinite Homogeneous Medium 76
4.4 Ray Theory for Far-Field P-waves and S-waves from a Point Source 82
4.4.1 Properties of the travel-time function T (x) associated with
velocity field C(X) 87
4.4.2 Ray coordinates 90
4.4.3 The geometrical solution for P-waves in spherically symmetric
media 92
4.4.4 The geometrical solution for S-waves in spherically symmetric media:
Introduction of the components 95
4.4.5 The geometrical ray solutions in general inhomogeneous media 96
4.5 The Radiation Pattern of Body Waves in the Far Field for a Point Shear
Dislocation of Arbitrary Orientation in a Spherically Symmetric Medium 101
4.5.1 A method for obtaining the fault-plane orientation of an earthquake and
the direction of slip using teleseismic body-wave observations 102
4.5.2 Arbitrary orientation of the double couple in a homogeneous
medium 106
4.5.3 Adapting the radiation pattern to the case of a spherically symmetric
medium 110
BOX 4.4 Cartesian components of the moment tensor for a shear dislocation of
arbitrary orientation 11 2
Suggestions for Further Reading 113
Problems 114
CONTENTS vii
5. PLANE WAVES IN HOMOGENEOUS MEDIA AND THEIR REFLECTION
AND TRANSMISSION AT A PLANE BOUNDARY 1 19
5.1 Basic Properties of Plane Waves in Elastic Media 120
BOX 5.1 Notation 121
5.1.1 Potentials for plane waves 123
5.1.2 Separation of variables; steady-state plane waves 124
BOX 5.2 The sign convention for Fourier transforms used in solving
wave-propagation problems 125
5.2 Elementary Formulas for ReflectionlConversion/TransmissionC oefficients 128
5.2.1 Boundary conditions 128
BOX 5.3 The distinction between kinematics and dynamics 129
5.2.2 Reflection ofplane P-waves and SV-waves at a free surface 130
BOX 5.4 Impedance 132
5.2.3 Reflection and transmission of SH-waves 136
5.2.4 Reflection and transmission of P-SV across a solid-solid interface 139
5.2.5 Energy flux 145
5.2.6 A useful approximation for reflection/transmissionc oefficients between
two similar half-spaces 147
5.2.7 Frequency independence of plane-wave reflection/transmission
coefficients 149
5.3 lnhomogeneous Waves, Phase Shifts, and Interface Waves 149
BOX 5.5 Phase shifts: phase delay andphase advance 151
BOX 5.6 The Hilbert transform and the frequency-independentphase advance 152
5.4 A Matrix Method for Analyzing Plane Waves in Homogeneous Media 157
5.5 Wave Propagation in an Attenuating Medium: Basic Theory for
Plane Waves 161
BOX 5.7 Different dejnitions of Q 162
5.5.1 The necessity for material dispersion in an attenuating medium 163
5.5.2 Some suggested values for material dispersion in an attenuating
medium 165
BOX 5.8 Relations between the amplitude spectrum and phase spectrum of a causal
propagating pulse shape 167
5.6 Wave Propagation in an Elastic Anisotropic Medium: Basic Theory for
Plane Waves 177
BOX 5.9 Shear-wave splitting due to anisotropy 181
Suggestions for Further Reading 183
Problems 183
6. REFLECTION AND REFRACTION OF SPHERICAL WAVES; LAMB'S
PROBLEM 189
6.1 Spherical Waves as a Superposition of Plane Waves and Conical Waves 190
BOX 6.1 Fundamental signi3cance of Wql and Sommerfeld integrals 193
6.2 Reflection of Spherical Waves at a Plane Boundary: Acoustic Waves 195
BOX 6.2 Determining the branch cuts of q'w = 5 in the complex p-plane,
so that Im 5 > 0 for a whole plane. 19 7
