Applied Seismic Wave Theory

Introduction. I. Capita Selecta from Vector Analysis. Scalar product, gradient, curl and divergence. Theorem of Stokes, theorem of Gauss and Green's theorem. II. One-Dimensional Discrete Spectral Analysis. The delta pulse and discrete functions. Fourier series of periodic time functions. Fourier integral of transients. Relationship between the discrete property and periodicity. Sampling and aliasing in time and frequency. Matrix formulations. Decomposition of a broad band experiment into monochromatic simulations. III. Multi-Dimensional Discrete Spectral Analysis. Basic theory. Spatial aliasing. Two-dimensional spectral analysis and plane wave decomposition. Extensions to three dimensions. IV. Vibrations. Basic concepts. Free vibrations. Forced vibrations. Coupled systems. From vibrations to waves. V. Acoustic Waves. Derivation of the acoustic wave equation. One-way versions of the acoustic wave equation. Plane waves. Spherical waves. Cylindrical waves. Principle of numerical modeling with the acoustic wave equation. VI. Elastic Waves. Compressional waves in homogeneous isotropic solids. Shear waves in homogeneous isotropic solids. Derivation of the elastic wave equation. One-way versions of the elastic wave equation. Principle of numerical modeling with the elastic wave equation. VII. Boundary Conditions. Reflection and transmission coefficients for acoustic boundaries. Reflection in terms of convolution. The fluid-solid boundary. Reflection and transmission coefficients for elastic boundaries. Summary. VIII. Kirchhoff and Rayleigh Integrals. Derivation of the Kirchhoff integral for homogeneous media. Derivation of the Rayleigh integrals for homogeneous media. Rayleigh integrals in terms of convolution. Transformation of Rayleigh integrals to the wave number domain. Kirchhoff and Rayleigh integrals for inhomogeneous fluid-like media. Rayleigh integrals as one-way versions of the Kirchhoff integral. Discrete version of the Kirchhoff integral. Discrete versions of the Rayleigh integrals. Summary. IX. Directivity Properties of Wave Fields. Fraunhofer approximations in terms of the Fourier integral. Directivity patterns. Far field expressions of scattered wave fields. Summary. X. Forward and Inverse Problems. Principle of one-way forward wave field extrapolation. Principle of one-way inverse wave field extrapolation. Principle of two-way techniques. Example. Summary. (Each chapter includes an introduction and references). Appendices. Subject Index.