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This page intentionally left blank Fundamentals of Seismic Wave Propagation presents a comprehensive introduction to the propagation of high-frequency body waves in elastodynamics. The theory of seismic wave propagation in acoustic, elastic and anisotropic media is developed to allow seismic waves to be modelled in complex realistic three-dimensional Earth models. This book provides a consistent and thorough development of modelling methods widely used in elastic wave propagation ranging from the whole Earth through regional and crustal seismology exploration seismics to borehole seismics sonics and ultrasonics. Methods developed include ray theory for acoustic isotropic and anisotropic media transform techniques including spectral and slowness methods such as the Cagniard and WKBJ seismogram methods and extensions such as the Maslov seismogram method quasi-isotropic ray theory Born scattering theory and the Kirchhoff surface integral method. Particular emphasis is placed on developing a consistent notation and approach throughout which highlights similarities and allows more complicated methods and extensions to be developed without difficulty. Although this book does not cover seismic interpretation the types of signals caused by different model features are comprehensively described. Where possible these canonical signals are described by simple standard time-domain functions as well as by classical spectral results. These results will be invaluable to seismologists interpreting seismic data and even understanding numerical modelling results. Fundamentals of Seismic Wave Propagation is intended as a text for graduate courses in theoretical seismology and a reference for all seismologists using numerical modelling methods. It will also be valuable to researchers in academic and industrial seismology. Exercises and suggestions for further reading are included in each chapter and solutions to the exercises and computer programs are available on the Internet at http://publishing.cambridge.org/resources/052181538X. Chris H. Chapman is a Scientific Advisor at Schlumberger Cambridge Research, Cambridge England. Professor Chapman’s research interests are in theoretical seismology with applications ranging from exploration to earthquake seismology. He is interested in all aspects of seismic modelling but in particular extensions of ray theory and anisotropy and scattering with applications in high-frequency seismology. He has developed new methods for efficiently modelling seismic body waves and used them in interpretation and inverse problems. He held academic positions as an Associate Professor of Physics at the University of Alberta Professor of Physics at the University of Toronto and Professor of Geophysics at Cambridge University before joining Schlumberger in 1991. He was a Killam Research Fellow at Toronto a Cecil H. and Ida Green Scholar at the University of California San Diego twice. He is a Fellow of the American Geophysical Union and the Royal Astronomical Society and an Active Member of the Society of Exploration Geophysicists. Professor Chapman has been an (associate) editor of various journals – Geophysical Journal of the Royal Astronomical Society Journal of Computational Physics Inverse Problems Annales Geophysics and Wave Motion – and is author of more than 100 research papers. Contents Preface Preliminaries Nomenclature Symbols Special functions Canonical signals Introduction Basic wave propagation Plane waves A point source Travel-time function in layered media Types of ray and travel-time results Calculation of travel-time functions Transforms Temporal Fourier transform Spatial Fourier transform Fourier–Bessel transform Tau-p transform Review of continuum mechanics and elastic waves Infinite stress tensor and traction Infinite strain tensor Boundary conditions Constitutive relations Navier wave equation and Green functions Stress glut source Asymptotic ray theory Acoustic kinematic ray theory Acoustic dynamic ray theory Anisotropic kinematic ray theory Anisotropic dynamic ray theory Isotropic kinematic ray theory Isotropic dynamic ray theory One and two-dimensional media Rays at an interface Boundary conditions Continuity of the ray equations Refection transmission coefficients Free surface refection coefficients Fluid–solid refection transmission coefficients Interface polarization conversions Linearized coefficients Geometrical Green dyadic with interfaces Differential systems for strati?ed media One-dimensional differential systems Solutions of one-dimensional systems Inverse transforms for strati?ed media Cagniard method in two dimensions Cagniard method in three dimensions Cagniard method in strati?ed media Real slowness methods Spectral methods Canonical signals First-motion approximations using the Cagniard method First-motion approximations for WKBJ seismograms Spectral methods Generalizations of ray theory Maslov asymptotic ray theory Quasi-isotropic ray theory Born scattering theory Kirchhoff surface integral method Appendices Useful integrals Useful Fourier transforms Ordinary differential equations Saddle-point methods Bibliography Author index Subject index Ключевые слова: jk xk, model, arrival, zr, ? xr, astr, derivative, xs ps, ray, vector, x xs, slowness, dimensional, phase, smirnovs lemma, surface, p p, wkbj, pv, dx, propagation, series, interface, ha, anisotropic, asymptotic, caustic, ut, nm g, condition, xtj, method, order, result, differential, tensor, travel, volume, term, ray theory, expansion, re?ection, real, parameter, receiver, pkc, tc, fourier, ln, pi dt, tn xr, solution, ar br, turning, axis, cagniard, depth, isotropic, simple, br, ds, problem, pc, zs zr, note, xs xr, zs, inverse, w w, singularity, xr, ti, general, small, pdj, sin, theory, lr, dt dt, jlk, jk g, amplitude, velocity, transforms, high-velocity lid, green function, zn wn, notation, iv, dv, approximation, re?ection transmission, pka, za zb, case, transform, zv, chapman, v t, frequency, hn ap, signal, coef?cients, travel time, scattering, unit, green, point, expression, zr t, transmission, g gt, de?ned, homogeneous, acoustic, soc, px, amer, component, jk il, ss xr, ai, source, xr zs, z dz, ei, tik, dt, xp, polarization, sp, numerical, contour, begat, xs ds, range, zr ms, xs, q p, tn, wave, form, wavefront, grow exponentially, xray, response, jk tk, integral, dx dp, time, skm, coef?cient, perturbation, xr ln, geometrical, equation, medium, riemann sheet, elastic, illustrated, ta, layer, xr xs, direction, ls, chapter, matrix, four-leafed clover, sv, seism, geophys, gradient, aa ab, plane, dimension, jkl, function, dt dx, symmetry, xr t, ?rst, normal, erveny, eit, jm pk, pr, ps, dy dt, tick mark, jk ek, sign