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A class of plane inhomogeneous waves which propagate in linear viscoelastic media is considered. A theoretical description of the physical properties and energy associated with these waves is given. Attention is restricted to harmonic Pand SV-waves.

The theory of Love-and Rayleigh-wave dispersion for plane-layered earth models has undergone a number of developments since the initial work of Thomson and Haskell. Most of these were concerned with computational difficulties associated with numerical overflow and loss of precision at high frequencies in the original Thomson-Haskell formalism. Several seemingly distinct approaches have been followed, including the delta matrix, reduced delta matrix, Schwab-Knopoff, fast SchwabKnopoff, Kennett's Reflection-Transmission Matrix and Abo-Zena methods. This paper analyses all these methods in detail and finds explicit transformations connecting them. It is shown that they are essentially equivalent and, contrary to some claims made, each solves the loss of precision problem equally well. This is demonstrated both theoretically and computationally. By extracting the best computational features of the various methods, we develop a new algorithm (see Appendix A5), called the fast delta matrix algorithm. To date, this is the simplest and most efficient algorithm for surface-wave dispersion computations (see Fig. 4).The theory given in this paper provides a complete review of the principal methods developed for Love-and Rayleigh-wave dispersion of free modes in plane-layered perfectly elastic, isotropic earth models and puts to rest controversies that have arisen with regard to computational stability.

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Kirchhoff's time-dependent surface integral representation of a scalar wavefield is applied to the problem of computing synthetic seismograms for P-waves in the Earth. By means of an appropriate parameterization, the Kirchhoff integral is transformed into a convolution of a weight function with the derivative of the source function in the time domain. The weight function is calculated using simple ray theory. The method extends the applicability of simple ray theory to caustics and other diffraction phenomena and allows certain kinds of departures from spherical symmetry to be taken into account. The method is illustrated in detail by application to the PKPwavefield in the Earth.