ADVANCES IN GEOPHYSICS
CONTENTS CHAPTER SEISMIC RAY METHOD: RECENT DEVELOPMENTS 1 INTRODUCTION 2 SEISMIC RAY THEORIES FOR ISOTROPIC AND ANISOTROPIC MEDIA 2 1 SEISMIC RAYS AND TRAVEL TIMES INITIAL-VALUE RAY TRACING 2 2 RAY HISTORIES, TWO-POINT RAY TRACING, WAVEFRONT TRACING 2 3 DYNAMIC RAY TRACING PARAXIAL RAY METHODS 2 4 RAY-THEORY AMPLITUDE 2 5 EFFECTS OF INTERFACES 2 6 RAY-THEORY GREEN FUNCTION 2 7 RAY-THEORY SEISMOGRAMS 3 RAY-THEORY PERTURBATIONS 3 1 PERTURBATION PARAMETERS 3 2 PERTURBATION OF TRAVEL TIME 3 3 OPTIMIZING MODEL UPDATES DURING LINEARIZED INVERSION OF TRAVEL TIMES 4 COUPLING RAY THEORY FOR S WAVES 4 1 COUPLING EQUATION FOR S WAVES 4 2 COUPLING-RAY-THEORY S-WAVE PROPAGATOR MATRIX 4 3 QUASI-ISOTROPIC APPROXIMATIONS OF THE COUPLING RAY THEORY 4 4 NUMERICAL EXAMPLES 5 SUMMATION OF GAUSSIAN BEAMS AND PACKETS 5 1 GAUSSIAN BEAMS 5 2 GAUSSIAN PACKETS 5 3 OPTIMIZATION OF THE SHAPE OF GAUSSIAN BEAMS OR PACKETS 5 4 ASYMPTOTIC SUMMATION OF GAUSSIAN BEAMS AND PACKETS 5 5 DECOMPOSITION OF A GENERAL WAVE?ELD INTO GAUSSIAN PACKETS OR BEAMS 5 6 SENSITIVITY OF WAVES TO HETEROGENEITIES 5 7 MIGRATIONS 6 RAY CHAOS, LYAPUNOV EXPONENTS, MODELS SUITABLE FOR RAY TRACING 6 1 LYAPUNOV EXPONENTS AND ROTATION NUMBERS 6 2 MODELS SUITABLE FOR RAY TRACING 7 OTHER TOPICS RELATED TO THE RAY METHOD 7 1 HIGHER-ORDER RAY APPROXIMATIONS 7 2 DIRECT COMPUTATION OF FIRST-ARRIVAL TRAVEL TIMES 7 3 RAY METHOD WITH COMPLEX EIKONAL 7 4 HYBRID METHODS 7 5 SEVERAL OTHER EXTENSIONS OF THE RAY METHOD ACKNOWLEDGEMENTS REFERENCES CHAPTER INTRODUCTION TO MODE COUPLING METHODS FOR SURFACE WAVES 1 INTRODUCTION 2 MODES AND MODE COUPLING 3 LOCAL MODES 3 1 A SIMPLE EXAMPLE OF LOCAL MODE COUPLING: WELDED SPACES 3 2 LOCAL MODE COUPLING: THE GENERAL CASE 3 3 SURFACE WAVE RAY TRACING 4 REFERENCE MODES 4 1 TWO-DIMENSIONAL METHOD OF KENNETT 4 2 MULTIPLE-SCATTERING 4 3 SINGLE-SCATTERING APPROXIMATION 5 CONCLUSION AND PERSPECTIVES BOUNDARY INTEGRAL EQUATIONS AND BOUNDARY ELEMENTS METHODS IN ELASTODYNAMICS 1 INTRODUCTION 2 THE DIRECT BIE 3 THE INDIRECT BIE 4 THE CALCULATION OF THE GREEN’S FUNCTIONS 4 1 APPROXIMATE REPRESENTATIONS 4 2 2-D GREEN’S FUNCTIONS 4 3 2 5-D GREEN’S FUNCTIONS 4 4 3-D GREEN’S FUNCTIONS 4 5 NON-HOMOGENEOUS MEDIA GREEN’S FUNCTIONS 5 DISCRETIZATION AND INVERSION 6 TIME DOMAIN IMPLEMENTATION 7 OTHER BOUNDARY METHODS 8 HYBRID METHODS 9 DOMAINS OF APPLICATION 10 CONCLUDING REMARKS GENERATION AND PROPAGATION OF SEISMIC SH WAVES IN MULTI-LAYERED MEDIA WITH IRREGULAR INTERFACES 1 INTRODUCTION 2 SEISMIC SH WAVE GENERATION AND PROPAGATION IN MULTI-LAYERED MEDIA WITH IRREGULAR INTERFACES 2 1 INTEGRAL EQUATION AND ITS DISCRETE WAVE NUMBER REPRESENTATION 2 2 SIMULTANEOUS MATRIX EQUATIONS 2 3 GLOBAL GENERALIZED RE?ECTION/TRANSMISSION MATRICES 2 4 SOLUTION SYNTHESIS 2 5 NUMERICAL VALIDATION BY THE ANALYTICAL SOLUTION 2 6 COMPARISONS WITH OTHER EXISTING METHODS 2 7 MORE NUMERICAL EXAMPLES 3 LOVE WAVE IN MULTI-LAYERED MEDIA WITH IRREGULAR INTERFACES 3 1 MODAL SOLUTION OF LOVE WAVE IN AN ARBITRARILY IRREGULAR MULTI- LAYERED MEDIUM 3 2 EXCITATION FORMULA OF LOVE WAVE IN IRREGULAR MULTI-LAYERED ME- DIA 3 3 COMPARISONS WITH CLASSIC THEORY OF LOVE WAVE IN LATERALLY HO- MOGENEOUS MULTI-LAYERED MEDIA 4 CONCLUDING REMARKS APPENDIX A APPENDIX B APPENDIX C APPENDIX D ONE-WAY AND ONE-RETURN APPROXIMATIONS (DE WOLF APPROXIMATION) FOR FAST ELASTIC WAVE MODELING IN COMPLEX MEDIA 1 INTRODUCTION 2 BORN, RYTOV, DE WOLF APPROXIMATIONS AND MULTIPLE SCATTERING SERIES 2 1 BORN APPROXIMATION AND RYTOV APPROXIMATION: THEIR STRONG AND WEAK POINTS 2 2 DE WOLF APPROXIMATION 2 3 THE DE WOLF SERIES (DWS) OF MULTIPLE SCATTERING 3 A DUAL-DOMAIN THIN-SLAB FORMULATION FOR ONE-RETURN (MFSB) SYN- THETICS 3 1 THE CASE OF SCALAR MEDIA 3 2 THE CASE OF ACOUSTIC MEDIA 3 3 THE CASE OF ELASTIC MEDIA 3 4 IMPLEMENTATION PROCEDURE OF THE ONE-RETURN SIMULATION 4 FAST ALGORITHM OF THE ELASTIC THIN-SLAB PROPAGATOR AND SOME PRACTICAL ISSUES 4 1 FAST IMPLEMENTATION IN DUAL DOMAINS 4 2 INCORPORATION OF BOUNDARY TRANSMISSION/RE?ECTION INTO THE ONE-RETURN METHOD 4 3 TREATMENT OF ANELASTICITY: THE Q-FACTOR 4 4 NUMERICAL TESTS FOR THE ELASTIC THIN-SLAB METHOD 5 THE SCREEN APPROXIMATION 5 1 SCREEN PROPAGATORS FOR ACOUSTIC MEDIA 5 2 SCREEN APPROXIMATION FOR ELASTIC MEDIA: ELASTIC COMPLEX SCREEN METHOD 5 3 COMPLEX SCREEN METHOD WITH FIRST-ORDER CORRECTIONS 6 RE?ECTED WAVE FIELD MODELING USING THIN-SLAB METHOD 6 1 RE?ECTION COEF?CIENTS OF SEDIMENTARY INTERFACES 6 2 RE?ECTIONS FROM A DIPPING SANDSTONE RESERVOIR 6 3 RESERVOIR RE?ECTIONS WITH SCATTERING AND ATTENUATION FROM A HETEROGENEOUS OVERBURDEN 6 4 THIN LAYER AVO RESPONSE 6 5 AVO RESPONSE IN LATERALLY VARYING MEDIA 7 CONCLUSIONS SIMULATION OF HIGH-FREQUENCY WAVE PROPAGATION IN COMPLEX CRUSTAL WAVEGUIDES USING GENERALIZED SCREEN PROPAGATORS RU-SHAN WU, XIAN-YUN WU AND XIAO-BI XIE 1 INTRODUCTION 2 A BRIEF DESCRIPTION OF THE GENERALIZED SCREEN PROPAGATOR FOR GUIDED WAVES 3 SH WAVE CASE 3 1 HALF-SPACE SCREEN PROPAGATOR 3 2 TREATMENT OF THE MOHO DISCONTINUITY 3 3 NUMERICAL VERI?CATIONS AND SIMULATION EXAMPLES 3 4 APPLICATION TO ENERGY PARTITION AND ATTENUATION IN CRUSTAL WAVEGUIDE WITH RANDOM HETEROGENEITIES 3 5 SH-WAVES IN CRUSTAL WAVEGUIDES WITH IRREGULAR SURFACE TOPOG- RAPHY 4 P-SV CASE 5 CONCLUSION SPECTRAL-ELEMENT ANALYSIS IN SEISMOLOGY EMMANUEL CHALJUB, DIMITRI KOMATITSCH, JEAN-PIERRE VILOTTE, YANN CAPDEVILLE, BERNARD VALETTE AND GAETANO FESTA 1 INTRODUCTION 2 THE WAVE EQUATION IN REGIONAL AND GLOBAL SEISMOLOGY 2 1 GENERAL FRAMEWORK 2 2 REFERENCE CON?GURATION 2 3 THE GRAVITO-ELASTODYNAMIC EQUATIONS 2 4 POTENTIAL FORMULATION IN FLUID REGIONS 2 5 INITIAL AND BOUNDARY CONDITIONS 2 6 COUPLING BOUNDARIES 2 7 WEAK FORMULATION 3 SPECTRAL-ELEMENT APPROXIMATION 3 1 SPATIAL DISCRETIZATION 4 TIME DISCRETIZATION 4 1 NEWMARK TIME STEPPING METHOD 4 2 ROTATION 4 3 SELF-GRAVITATION 4 4 ATTENUATION 4 5 IMPLEMENTATION OF A PERFECTLY MATCHED LAYER 5 IMPLEMENTATION 5 1 CHOICE OF PARAMETERS FOR ACCURATE CALCULATIONS 5 2 PARALLEL IMPLEMENTATION 6 APPLICATIONS 6 1 3-D GROUND MOTION IN ALPINE VALLEYS 6 2 3-D SPECTRAL-ELEMENT REFERENCE SOLUTION 6 3 3-D MODELING IN THE D?? REGION 7 CONCLUSIONS AND PERSPECTIVES APPENDIX A MODI?ED HOOKE’S LAW A 1 INTRINSIC ATTENUATION A 2 FICTITIOUS ATTENUATION IN ABSORBING LAYERS APPENDIX B DTN OPERATORS B 1 DYNAMIC COUPLING B 2 STATIC COUPLING B 3 OCEAN LOAD APPROXIMATION THE FINITE-DIFFERENCE TIME-DOMAIN METHOD FOR MODELING OF SEISMIC WAVE PROPAGATION PETER MOCZO, JOHAN O A ROBERTSSON AND LEO EISNER 1 INTRODUCTION 1 1 THE GRID 1 2 THE FD APPROXIMATIONS TO DERIVATIVES 1 3 BASIC PROPERTIES OF THE FD EQUATIONS AND THEIR SOLUTION 1 4 EXPLICIT AND IMPLICIT FD SCHEMES 1 5 HOMOGENEOUS AND HETEROGENEOUS FD SCHEMES 1 6 THE EQUATION OF MOTION, HOOKE’S LAW, AND FD SCHEMES 1 7 ALGORITHMS FOR ENHANCING COMPUTATIONAL PERFORMANCE 2 OPTIMALLY ACCURATE FD OPERATORS 2 1 GENERAL CRITERION FOR OPTIMALLY ACCURATE FD OPERATORS 2 2 OPTIMALLY ACCURATE FD OPERATORS FOR 1-D PROBLEM IN A HOMO- GENEOUS MEDIUM 3 COMPLEX VISCOELASTIC MEDIA WITH MATERIAL DISCONTINUITIES 3 1 A MATERIAL DISCONTINUITY IN THE ELASTIC MEDIUM 3 2 INCORPORATION OF THE REALISTIC ATTENUATION 4 ANISOTROPIC MEDIA 5 FREE SURFACE 5 1 TRACTION-FREE BOUNDARY CONDITION 5 2 PLANAR FREE SURFACE 5 3 FREE-SURFACE TOPOGRAPHY 6 WAVE?ELD EXCITATION 6 1 DIRECT MODELING OF THE POINT SOURCES 6 2 INTRODUCING THE SOURCE WAVE?ELD ALONG A BOUNDARY INTERNAL TO THE GRID 6 3 DYNAMIC MODELING OF EARTHQUAKE RUPTURE 6 4 NON-RE?ECTING BOUNDARIES 7 MEMORY OPTIMIZATION AND PARALLELIZATION 7 1 MEMORY OPTIMIZATION 7 2 PARALLELIZATION A LATTICE BOLTZMANN APPROACH TO ACOUSTIC-WAVE PROPAGATION LIANJIE HUANG 1 INTRODUCTION 2 THE HISTORY OF THE USE OF DISCRETIZATION 3 LATTICE GAS APPROACH 3 1 PARTICLE MOVEMENT 3 2 PARTICLE COLLISION RULES 3 3 MICRODYNAMICAL EQUATION 3 4 CONSERVATION RELATIONS 4 LATTICE BOLTZMANN APPROACHES 4 1 I: A FERMI–DIRAC-TYPE APPROACH 4 2 II: A BOSE-LIKE APPROACH 5 A LATTICE BOLTZMANN METHOD FOR MODELING ACOUSTIC WAVES: PHONONIC LATTICE SOLID BY INTERPOLATION 5 1 TRANSPORTATION STEP 5 2 TRANSMISSION AND RE?ECTION STEPS 5 3 COLLISION STEP 5 4 THE FIRST-ORDER CHAPMAN–ENSKOG EXPANSION OF THE QUASI- PARTICLE NUMBER DENSITY N? 5 5 THE ACOUSTIC-WAVE EQUATION 6 ABSORBING BOUNDARY CONDITION 6 1 ZERO-VALUED RE?ECTION COEF?CIENTS AT ABSORBING BOUNDARIES 6 2 VISCOUS ABSORBING BOUNDARY LAYERS 6 3 INCIDENCE ANGLE DEPENDENCE OF THE ABSORBING BOUNDARY CONDI- TION 7 FREE-SURFACE RE?ECTIONS 7 1 NUMERICAL SIMULATIONS OF FREE-SURFACE RE?ECTIONS 7 2 NUMERICAL SIMULATIONS OF RE?ECTIONS FROM FREE-SURFACE TOPOG- RAPHY 8 ACOUSTIC WAVES IN STRONGLY HETEROGENEOUS MEDIA 8 1 WAVE PROPAGATION IN A FINELY LAYERED MEDIUM 8 2 WAVE PROPAGATION IN A MEDIUM WITH ALIGNED HETEROGENEITIES 8 3 WAVE PROPAGATION IN A MEDIUM WITH A CIRCULAR EMPTY BUBBLE 8 4 WAVE PROPAGATION IN A MEDIUM WITH RANDOM EMPTY BUBBLES 9 CONCLUSIONS SYNTHESIS OF SEISMOGRAM ENVELOPES IN HETEROGENEOUS MEDIA HARUO SATO AND MICHAEL C FEHLER 1 INTRODUCTION 2 SCALAR WAVE EQUATION IN 2-D RANDOM MEDIA 3 FINITE DIFFERENCE SIMULATION IN GAUSSIAN-TYPE RANDOM MEDIA 3 1 REALIZATION OF RANDOM MEDIA 3 2 FINITE DIFFERENCE SIMULATION 3 3 FD ENVELOPES AS REFERENCE ENVELOPES 4 MARKOV APPROXIMATION FOR GAUSSIAN-TYPE RANDOM MEDIA 4 1 COHERENT WAVE FIELD 4 2 TWO-FREQUENCY MUTUAL COHERENCE FUNCTION 4 3 QUASI-MONOCHROMATIC WAVES 4 4 ANALYTIC SOLUTION FOR THE CASE OF GAUSSIAN ACF 4 5 MARKOV ENVELOPES IN GAUSSIAN-TYPE RANDOM MEDIA 5 MARKOV APPROXIMATION FOR VON KÁRMÁN-TYPE RANDOM MEDIA 5 1 FD ENVELOPES AS REFERENCE ENVELOPES 5 2 MARKOV ENVELOPES 5 3 COMPARISON OF MARKOV ENVELOPES WITH REFERENCE ENVELOPES 6 RADIATIVE TRANSFER THEORY FOR VON KÁRMÁN-TYPE RANDOM MEDIA OF ? = 0 1 6 1 MOMENTUM TRANSFER SCATTERING COEF?CIENT 6 2 RADIATIVE TRANSFER INTEGRAL EQUATION FOR ISOTROPIC SCATTERING PROCESS 8 OVERVIEW OF ENVELOPE STUDIES 8 1 ENVELOPE BROADENING STUDIES 8 2 DIFFUSION STUDIES 8 3 THREE-COMPONENT ENVELOPE STUDIES 8 4 ENVELOPES OF TELESEISMIC SEISMOGRAMS 9 CONCLUSION AND DISCUSSION