Seismic Waves and Sources

1126 Pages·1981·22.044 MB·English

by  Ari Ben-Menahem, Sarva Jit Singh (auth.)

Seismic Waves
and Sources
Seismic Waves
and Sources
by
Ari Ben-Menahem
Professor of Geophysics
Weizmann Institute of Science
Rehovot, ISRAEL
and
Sarva Jit Singh
Professor of Mathematics
Maharshi Dayanand University
Rohtak, INDIA
Springer-Verlag
New York Heidelberg Berlin
Ari Ben-Menahem
Professor of Geophysics
Weizmann Institute of Science
Rehovot 76100
ISRAEL
Sarva Jit Singh
Professor of Mathematics
Maharshi Dayanand University
Rohtak 124001
INDIA
With 307 Illustrations
Library of Congress Cataloging in Publication Data
Ben-Menahem, Ari.
Seismic waves and sources.
Includes bibliographies and indexes.
1. Seismic waves. 2. Seismology-Mathematics.
I. Singh, Sarva Jit, joint author. II. Title.
QE538.5.B46 551.2'2 80-12298
Cover: View ofa fault scarp, 3.5 meters high, near Quiches in the Peruvian Andes; associ
ated with the earthquake of November 10, 1946. (photograph by E. Silgado. 1946)
All rights reserved.
No part of this book may be translated or reproduced in any form without written
permission from Springer-Verlag.
The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such
names, as understood by the Trade Marks and Merchandise Marks Act, may
accordingly be used freely by anyone.
© 1981 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1981
9 8 765 432 I
ISBN-13: 978-1-4612-5858-2 e-ISBN-13: 978-1-4612-5856-8
001: 10.1007/978-1-4612-5856-8
In memory of my father, MOSHE BEN-MENAHEM
(A.B-M.)
To my parents
(S.J.S.)
Preface
Earthquakes come and go as they please, leaving behind them trails of destruc
tion and casualties. Although their occurrence is little affected by what we do
or think, it is the task of earth scientists to keep studying them from all possible
angles until ways and means are found to divert, forecast, and eventually
control them.
In ancient times people were awestruck by singular geophysical events,
which were attributed to supernatural powers. It was recognized only in 1760
that earthquakes originated within the earth. A hundred years later, first
systematic attempts were made to apply physical principles to study them.
During the next century scientists accumulated knowledge about the effects
of earthquakes, their geographic patterns, the waves emitted by them, and the
internal constitution of the earth.
During the past 20 years, seismology has made a tremendous progress, mainly
because of the advent of modern computers and improvements in data acquisi
tion systems, which are now capable of digital and analog recording of ground
motion over a frequency range of five orders of magnitude. These technologic
developments have enabled seismologists to make measurements with far
greater precision and sophistication than was previously possible. Advanced
computational analyses have been applied to high-quality data and elaborate
theoretical models have been devised to interpret them. As a result, far
reaching advances in our knowledge of the earth's structure and the nature of
earthquake sources have occurred.
The primary objective of this book is to give the reader a comprehensive
account of the generation of elastic waves by realistic earthquake sources and
their propagation through realistic earth models. There has been a wide gap
between the levels of the textbooks available to seismologists and the work
appearing in research journals. No previous modern work has bridged this
gap, even partially. We hope that our treatise indeed fulfills this objective.
We seek to represent earthquake seismology as a science that stands firm
on its own theoretical foundations and is able to render a satisfactory quanti
tative account of seismic observations over the entire spectral range of recorded
wave phenomena.
The hard core and the general framework of the material presented here
are based on lectures given by one of us (A.B-M.) to students of the Feinberg
vi Preface
Graduate School of the Weizmann Institute of Science through the years 1966-
1975. The uniform representation of sources and fields, which is the theore
tical backbone of the treatise, was developed mostly by the authors in their
joint work during the period 1967-1974. This unifying formalism has enabled
us to develop the mathematical theory of seismic fields from first principles
and to take the reader to the latest developments in the subject. Functional
both as a textbook and a handbook, this work should prove useful to university
students and research workers in the various branches of earth sciences, applied
mathematics, theoretical physics, and engineering.
The material covers some 160 years in the history of seismology, starting
with the equations of motion derived by Louis Navier in 1821. In the pre
seismograph era (1821-1891), intensive theoretical work was done by mathe
maticians and physicists, who laid the foundations to the mathematical theories
of infinitesimal elasticity and seismic fields. The precomputer era (1892-1950)
was characterized by the availability of instrumental data, which motivated
theoretical research. Simple models for earth structure and seismic sources
were established and tested against the data. The fundamental properties of
seismic body and surface waves were discovered. Concurrently, mathematicians
and theoretical physicists discovered new methods for tackling problems of wave
propagation. Among these were the method of steepest descents, integral
relations among plane, spherical and cylindrical waves, operational methods,
variational techniques, and asymptotic solutions of differential equations.
These methods were applied by seismologists to solve problems ofw ave propaga
tion in the earth. The third era (1951 onward) has been marked by two out
standing features: the development of sensitive long-period seismographs and
the increasing influence of the computer, both on the choice of problems and
on the methods of attack.
In selecting material out of this vast literature, we were guided by two
principles: First, only well-established theoretical results were reported,
ones that had been verified by repeated observations and careful data analyses.
Second, the scope of the book forced us to concentrate our efforts on the topics
that belong to the mainstream of contemporary seismology. For example, a
discussion of dynamic theories of fracture was deleted for the first reason and a
discussion of scattering and diffraction of seismic waves, as well as the theory of
leaking modes, were excluded for the second reason.
The book gives an up-to-date, comprehensive, rigorous, and lucid account
of the propagation of elastic waves in the earth. Although the main emphasis
is on earthquake waves, the theories of gravity waves in water and acoustic
gravity waves in the atmosphere are also included. The book is well illustrated
with figures, tables, and solved examples and is made self-sufficient with the
addition of several appendices, which introduce most of the mathematical
tools employed. The dyadic approach is used for elegance and brevity. The
material is fully indexed and a fairly comprehensive list of references is given at
the end of each chapter and at the end of the appendices. We make no preten
sions as to completeness or historical balance. The references are not included
to give credit for results but to help the reader find more complete discussions
of various topics.
Preface
VB
Chapter 1 is concerned with a brief but lucid account of those results of
classical continuum dynamics which are essential to seismology. The fun
damental concepts of stress and strain are presented and the basic field equa
tions are derived. It includes the Lagrangian formulation, which is found to be
a useful tool in the amplitude theory of surface waves in elastic and anelastic
media.
In Chapter 2 we deal mainly with the eigenvector solutions of the Navier
equation for infinite space and the plane-wave solutions of this equation in
Cartesian, circular cylinder, and spherical coordinates. Emphasis is laid on
the integral interrelations between plane, cylindrical, and spherical waves.
The concepts of primary (P), vertical shear (SV), and horizontal shear (SH)
waves are introduced.
Chapter 3 concerns the interaction of vector plane waves with planar dis
continuities. The basic laws of reflection and refraction are derived. The
properties of surface waves are exhibited, first for simple structures and
eventually for a multilayered half-space with the aid of the matrix propagator
algorithm. Later in the chapter, we introduce numerical methods that enable
one to derive crustal and upper-mantle structure of selected geological provinces
from observed dispersion data of seismic surface waves over that region.
Chapter 4 renders a comprehensive and systematic study of seismic sources
and the application of elasticity theory of dislocations to seismology. Starting
from the fundamental Stokes-Love solution of the inhomogeneous Navier
equation, the theory of dipolar point sources is developed and is finally
linked with the key concept of displacement dislocation via the Volterra rela
tion. The discussion goes on to incorporate other relevant aspects of modern
seismology which bear on the theory of earthquake sources, such as fault
plane geometry, theoretical seismograms in an infinite medium, displacements
in the near and the far fields, explosions in pre-stressed media and finally the
theory of radiation of body waves from finite moving sources.
Chapter 5 contains a detailed account of one of the greatest triumphs of
modern seismology, namely the ability to account for the observed amplitudes
of surface waves from earthquakes in terms of the kinematic parameters of the
source. The new field of "terrestrial interferometry" by which the fault
length and rupture velocity can be routinely calCulated from the seismic
"Doppler effect" via the directivity function, is explained and illustrated with
several examples. The material includes details of numerical algorithms and
data reduction routines that are widely used in the analysis of earthquake
signals.
One of the noteworthy achievements of seismology since the late fifties is
the discovery that earthquakes can excite the earth's free oscillations to a level
that can be measured by most long-period seismographs. This new field is
known as "terrestrial spectroscopy." Chapter 6 gives a detailed account of the
theory of excitation of the free oscillations of the earth by earthquakes. It
includes numerical methods for the calculation of the eigenfrequencies and
spectral amplitudes for realistic structural models. The effects of the rotation
of the earth and the source's finiteness and motion on the line spectra are also
given.
viii Preface
Chapter 7 is about rays. While the free oscillation data renders important
information concerning the gross features of the earth and surface waves are
useful in deriving average crustal structures and the parameters of seismic
sources, our main tool for the investigation of the core and deep mantle comes
chiefly from the study of body waves that travel everywhere inside the earth
in the form of rays. Starting again from the equation of motion, the amplitude
theory of body waves in a radially inhomogeneous earth is developed. The
useful concept of a generalized ray is introduced and numerical methods are
developed by which ground motion can be represented as a sum of generalized
rays. The theory of initial motions and the phenomenon of caustics are dis
cussed.
The duality of the normal modes and rays is explored in Chapter 8. Here,
the asymptotic theory of earth's normal modes is studied in great detail. It is
shown how the exact normal-mode solution yields the partial fields of body
waves, diffracted waves in the shadow zone and the field on a caustic. Finally,
the "rainbow expansion" is used to generate generalized rays in spherical
earth models and to derive amplitudes of isolated body wave signals. Topics
such as Fresnel diffraction, tunnelling, and earth-flattening transformation are
also presented.
Atmospheric and water waves associated with earthquakes and explosions
are discussed in Chapter 9. The chapter opens with a brief summary of the basic
principles of hydrodynamics, including the equations of sound waves and long
gravity waves in liquids. The theoretical results are then applied to waves and
oscillations excited by earthquakes such as tsunamis, seiches, air waves, pressure
induced surface waves, and coupled air-sea waves. Rayleigh waves and acoustic
gravity waves excited by nuclear explosions are also discussed. It is demonstrated
that many of the concepts and numerical algorithms developed earlier for
layered elastic media are valid also for waves in fluid media.
In Chapter lOwe return to seismic waves and examine their propagation in
low-loss anelastic substances of which the real earth is made. After introducing
the basic concepts of the theory of viscoelastic solids, we discuss the propaga
tion of seismic pulses in unbounded anelastic media and apply the results to
the attenuation of seismic waves in the earth. The causal dispersion of attenu
ated surface waves is explained.
The appendices furnish ample information on the various mathematical
techniques used in the book. There should be no need for the reader to look
for the material in other sources, where his comprehension may be unneces
sarily hampered by alien notation and applications to fields foreign to seismol
ogy. The information given in the appendices is brief, concentrated and with
seismological applications in mind.
Parts of this book were written while one of us (A.B-M.) was a visiting
professor at the department of Geophysics of Stanford University (1975-1977)
and the Institute of Geophysics and Planetary Physics, UCLA (1979). The
hospitality and technical assistance afforded there are gratefully acknowledged.
Thanks are due especially to Kathleen Hart, Linda and Bob Kovach, and
Prof. George A. Thompson of Stanford University, and Dr. A. K. Chatterjee
Preface ix
of UCLA. Among those who made useful suggestions we must single out Dane
Brooke of our Geophysical Laboratory and Dr. Shahar Ben-Menahem of the
Stanford Linear Accelerator Center. We are thankful to the Department of
Applied Mathematics of the Weizmann Institute for grants which enabled one
of the authors (S.J.S.) to travel several times to Rehovot and work at the In
stitute. We are much indebted to Professor Robert L. Kovach of Stanford
University for his help in writing Sections 3.9, 6.5, and 6.7.1. The technical
production of the book has been accomplished with the devoted assistance of
Ms. Sara Fligelman, who prepared the typescript. Mr. Yehuda Barbut drew
most of the figures and Mr. A. Silberberg of the Wix Library assisted us in the
photographic reproductions. Special acknowledgement must be made to
Adolpho Bloch for his generous support of seismological research at the Weiz
mann Institute of Science. Finally, we would like to thank the staff of Springer
Verlag for their unfailing efforts in producing this book.
Rehovot Ari Ben-Menahem
November 4, 1980
Seismology-Milestones
of Progress
Science is the knowledge of many,
orderly and methodically digested and arranged,
so as to become attainable by one.
(j. F. W. Herschel)
1660 Robert Hooke (England) stated his law: "Vt tensio sic vis."
1760 John Michell (England) recognized that earthquakes originate within
the earth and send out elastic waves through the earth's interior.
1821 Louis Navier (France) derived the differential equations of the
theory of elasticity.
1828 Simeon-Denis Poisson (France) predicted theoretically the existence
of longitudinal and transverse elastic waves.
1849 George Gabriel Stokes (England) conceived the first mathematical
model of an earthquake source.
1857 First systematic attempt to apply physical principles to earthquake
effects by Robert Mallet (Ireland).
1883 Rossi-Forel scale for earthquake effects published.
1885 C. Somigliana (Italy) produced formal solutions to Navier equations
for a wide class of sources and boundary conditions.
Lord Rayleigh (England) predicted the existence of elastic surface
waves.
1892 John Milne (England) constructed in Japan a seismograph suitable for
world-wide use. Seismological observatories were set up on global
basis to measure ground movements.
1897 Emil Wiechert (Germany) conjectured on the presence of a central
fluid core in the earth.
R. D. Oldham (England) identified in seismograms the three main
types of seismic waves.
1899 C. G. Knott (England) derived the general equations for the reflection
and refraction of plane seismic waves at plane boundaries.
1901 First Geophysical Institute founded in Gottingen, Germany.