Seismology: Body Waves and Sources PDF
Preview Seismology: Body Waves and Sources
Methods in Computational Physics Advances in Research and Applications 1 STATISTICAL PHYSICS 2 QUANTUM MECHANICS 3 FUNDAMENTAL METHODS IN HYDRODYNAMICS 4 APPLICATIONS IN HYDRODYNAMICS 5 NUCLEAR PARTICLE KINEMATICS 6 NUCLEAR PHYSICS 7 ASTROPHYSICS 8 ENERGY BANDS OF SOLIDS 9 PLASMA PHYSICS 10 ATOMIC AND MOLECULAR SCATTERING 11 SEISMOLOGY: SURFACE WAVES AND EARTH OSCILLATIONS 12 SEISMOLOGY: BODY WAVES AND SOURCES METHODS IN COMPUTATIONAL PHYSICS Advances in Research and Applications Series Editors BERNI ALDER Lawrence Livermore Laboratory Livermore, California SIDNEY FERNBACH MANUEL ROTENBERG Lawrence Livermore Laboratory University of California Livermore, California La Jolla, California Volume 12 Seismology: Body Waves and Sources Volume Editor BRUCE A. BOLT Seismographie Station Department of Geology and Geophysics University of California Berkeley, California AN ACADEMIC PRESS REPLICA REPRINT ACADEMIC P R E SS A Subsiciiury of Harcourt Bruce* Jovanovich. Publinliers New York London Toronto Sydney San Francisco This is an Academic Press Replica Reprint reproduced directly from the pages of a title for which type, plates, or film no longer exist. Although not up to the standards of the original, this method of reproduction makes it possible to provide copies of books which otherwise would be out of print. COPYRIGHT © 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 LIBRARY OF CONGRESS CATALOG CARD NUMBER: 63-18406 PRINTED IN THE UNITED STATES OF AMERICA 82 9 8 7 6 5 4 3 2 Contributors Numbers in parentheses indicate the pages on which the authors* contributions begin. Z. ALTERMAN, Department of Environmental Science, Tel Aviv University, Ramat-Aviv, Israel (33) ARI BEN-MENAHEM, Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel (299) C. H. CHAPMAN, Institute of Earth and Planetary Physics, Department of Physics, University of Alberta, Edmonton, Alberta, Canada (165) FREEMAN GILBERT, Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California (231) F. HRON, Institute of Earth and Planetary Physics, Department of Physics, The University of Alberta, Edmonton, Alberta, Canada (1) LEONARD E. JOHNSON,* Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California (231) D. LOEWENTHAL, Department of Environmental Science, Tel Aviv University, Ramat-Aviv, Israel (35) R. A. PHINNEY, Department of Geological and Geophysical Sciences, Princeton University, Princeton, New Jersey (165) M. J. RANDALL, Geophysics Division, Department of Scientific and Industrial Research, Wellington, New Zealand (267) SARVA JIT SINGH,f Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel (299) * Present address: Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado. t On leave from the Department of Mathematics, Kurukshetra University, Kuruk- shetra, India. vii Preface THIS VOLUME, LIKE ITS companion Volume 11, deals with recent seismological research in which advanced computational analysis has played a critical role. In this volume the numerical solution of some central problems on seismic body wave propagation and generation in the Earth are covered. A major aim in seismology is to infer the minimum set of properties of the earthquake source and of the Earth, which will explain in detail the recorded wave trains. By the end of the 1950s this geophysical interpretation of earthquake observations was limited severely by the laborious arithmetic needed to solve the theoretical models, yet great technical improvement occurred in the worldwide network of seismographic observatories. The impasse was broken with the application of the high speed computer. A clear example is given by F. Hron in his discussion of the power of the computer to trace out detailed seismic response by means of ray theory. Hron brings together for the first time some important but rather inaccessible work from the Soviet Union and the Continent. A different numerical procedure of generating synthetic seismograms to allow comparison with observations is set out in detail by Z. Alterman and D. Loewenthal. Correspondence with the ray theoretical results is pointed out. Chapman and Phinney have made an important contribution to the problem of the diffraction of seismic waves by the sharp boundary of the Earth's liquid core, a phenomenon which has been known for about sixty years. However, no numerical solutions for the diffraction of elastic spherical waves propagating through an inhomogeneous medium were available until their work. The basic seismological input to knowledge of the structure of the Earth is to provide times of travel of seismic body waves from one point to another on the surface. Given the empirical times, numerical integration is required to infer a seismic velocity distribution with depth which will satisfy the times within the measurement uncertainties. This optimal constraint formulation is typical of a general class of inverse geophysical problems. Johnson and Gilbert describe a method for the travel-time inversion problem including an assessment of uniqueness. The final two chapters describe the latest methods of analysis of earth quake mechanisms. Randall shows how discrimination between source mechanisms is possible using the amplitudes of seismic waves as well as the direction of initial motion, which has traditionally been used since its development by P. Byerly of the University of California. The major review ix X PREFACE by Ben-Menahem and Sarva Jit Singh is designed to acquaint physicists and geoscientists with the application of elastic dislocation theory to investi gations of earthquake sources. Important computer applications to the study of faulting dynamics and the line spectra of terrestrial eigenvibrations are also considered. Numerical Methods of Ray Generation in Multilayered Media F. HRON INSTITUTE OF EARTH AND PLANETARY PHYSICS DEPARTMENT OF PHYSICS THE UNIVERSITY OF ALBERTA, EDMONTON, ALBERTA, CANADA I. Introduction 1 II. Notation and Identification of Phases 2 III. Groups of Kinematic and Dynamic Analogs 5 A. Unconverted Waves 5 B. Kinematic and Dynamic Analogs of Simply Converted Waves . . .. 11 C. Kinematic and Dynamic Analogs of Head Waves 14 IV. Flow Charts of Subroutines 19 A. Flow Chart of Subroutine Creating Symbols for Kinematic and Dynamic Analogs of Unconverted Waves 19 B. Flow Chart of Subroutine Creating Codes of Representative Waves for Groups of Dynamic Analogs of Unconverted Waves 26 V. Numerical Examples 28 Appendix A. Number of Arrangements of n Identical Balls into «i Pockets, 2 Some of Which Can Be Empty 32 Appendix B. Number of Arrangements of n Identical Balls into tii Pockets, 2 None of Which Can Be Empty 33 References 33 I. Introduction ALTHOUGH THE INCREASING NUMBER of publications dealing with the computa tion of synthetic (theoretical) seismograms for layered media has been due mainly to the rapidly developing computing facilities of universities and geo physical research centers, the first few quite acceptable agreements between synthetic seismograms and field records (Helmberger, 1968; Helmberger and Morris, 1969) suggest that synthetic seismograms could play an important role in interpreting field materials in the near future. After evaluating results, it becomes obvious that a ray theory, which decomposes the displacement field into contributions attributed to the individual rays, is one of the most efficient methods that could be used for computation of synthetic seismograms. To date, two different approaches within ray theory can be distinguished. 1 2 F. HRON The first one, often called a generalized ray theory (or exact ray theory, method of generalized reflection and refraction coefficients), was used for the calculation of synthetic seismograms, for example, by Spencer (1965), Cerveny (1965, 1966), Helmberger (1968), Müller (1968a,b, 1970), and Helm- berger and Morris (1969). Special attention should be given to Müller's last paper, where a list of all important publications dealing with generalized ray theory is given. The second approach, called here an asymptotic ray theory (or a ray method in most of the Russian literature), has been used quite recently for the calculation of synthetic seismograms by Hron and Kanasewich (1971), who also give an outline of this theory. As both generalized and asymptotic versions of ray theory furnish well established methods of evaluating the displacement contribution attributed to the individual rays, the main problem which was encountered during the computation of synthetic seismograms lay in the selection of rays used for their construction. The crucial importance of the selection of rays becomes clear if one realizes that the "quality" or, in other words, the "reliability" of synthetic seismograms depends very strongly on the family of rays which are selected from the infinite number of rays existing between source and receiver for the calculation of synthetic seismograms. Until recently such selections were performed mostly manually according to human experience and that is why interpretations which were drawn from synthetic seismograms could only be inconclusive. It is obvious that any practical solution of this problem cannot be achieved without using computers which would replace human judgment by an appropriate algorithm based on strictly defined criteria. In order to do this, some rules, by which significant rays could be selected, cataloged, and their contribution to a synthetic seismogram calculated, have to be determined. One possible approach to the solution of this task is outlined in Sections II and III of this article, which are based on a recently published paper of the author (Hron, 1971). Then, flow charts of two subroutines, which perform automatic ray generation and create codes of socalled " representative waves," are given in Section IV while the corresponding numerical examples can be found in Section V. II. Notation and Identification of Phases Suppose the medium consists of / + 1 layers including a half-space. Then, according to ray theory, an infinite number of rays propagate from an arbi trarily placed source to a receiver at any other location. In this article, both the source and receiver will be restricted to the surface as this corresponds to a commonly used geometry in deep seismic sounding or refraction studies. For RAY GENERATION IN MULTILAYERED MEDIA 3 simplicity, the medium will be considered to consist of homogeneous layers separated by plane interfaces parallel to the earth's surface. The object of this article is to describe rules by which phases with significant energy are selected, cataloged, and their contribution to a synthetic seismogram determined. Those waves which travel from the source to the receiver along different paths but with identical travel time curves are kinematically equivalent and are called "kinematic analogs." The groups of kinematic analogs may be further divided into subgroups of waves whose amplitude curves are identical. The members belonging to each of these subgroups of phases may be called "dynamic analogs" The advantage of the distinction between dynamic and kinematic analogs is evident. For example, if we know the number of all dynamic analogs in the subgroup and multiply that number by the amplitude of one of them, we obtain the total dynamic effect of the subgroup. This classification is only efficient in media with plane layers whose properties do not change in the direction parallel to interfaces. Another factor which can influence the efficiency of this distinction is the number of con versions from P- to S-waves and vice versa. The method is effective only for waves with few conversions. The surface of the layered media will be called the zeroth interface while the top of the half-space will be the /th interface. The properties of the layers are described by the elastic wave velocities, density, and thickness as follows: (Xi—compressional or P-wave density, ßt—shear or S-wave velocity, p—density, t h—thickness, t / = 1, 2, ...,/,/+ 1, —layer index. Every ray travels in a straight line through each layer and the ray may be divided into k = 1 to K segments which always connect points where the ray changes its direction after reflection or refraction. Compressional and shear waves are identified in each segment by a code C as follows k C = 1, compressional or P-wave in the &th segment, k C = 2, shear or S-wave in the /cth segment. k A ray is completely specified by K contiguous pairs of integers as follows {C i} *=!,...,*. (1) k9k9 The computing code used for the ray in Fig. 1 is 1, 1;2, 2; 2, 2; 1,2; 1, 3; 2, 3; 2, 2; 2, 1. (2) In more conventional notation the ray would be specified by (3) P\» «3 2 ' ^2 ■> *2 ·> * 3 » ^3 » ^2 * ^1 · F. HRON \ 7 β1· &\>P\ ^ Λ / 2 β2·02·/>2 \ / V 3 «2'03>P3 βΙ-Ι·^Μ·^Ι-Ι I-» ί αΙ· Ρ\· *>1 FIG. 1. A ray for the wave with the code Λ, S , S, Pi, P$, S, S, Si in a medium 2 2 3 2 with / plane interfaces parallel to the surface. The segments of the ray are numbered sequentially from 1 to 8. If a ray has no phase conversions a sequence of K integers may be used in place of the code (1): {/»},*=!,...,*. (4) Such a wave is described as an " unconverted " wave. Two waves are different if their code as given by Eq. (1) or (4) is different. When the source and receiver are on the surface the number of segments in each layer is even. There will be n downgoing waves and n upgoing waves t ( in each layer /. A " coupled segment" is defined as consisting of one down- going and the next nearest upgoing segment in a layer. There are n coupled t segments in each layer, in Fig. 1 the second and third segments constitute the first couple in the second layer while the fourth and seventh segments con stitute the second couple. An ''element of (he jth class" is defined to be the continuous chain of segments commenced and terminated by a coupled segment in theyth layer. The symbol J will be given to the deepest layer traversed by the ray. Elements of the /th class do not overlap and are numbered 1, ..., n along the ray in the } direction of propagation. For example, in Fig. 1 the first element of the second class consists of segments 2 and 3 while the second element is composed of segments 4, 5, 6 and 7. If an element consists of only 2 segments, as in the first case, it is called a "trivial element" The second case is an example of a "normal element" which consists of at least one coupled segment in the (7 + l)th layer. Head waves are diffractions from a surface and have segments of rays that are parallel to an interface. The symbol " H " will be given to these waves
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