INTEGRAL REPRESENTATION IN GEOMETRICAL SEISMICS

Abstract

The paper presents an analysis of seismic-wave propagation expressed geometrically in terms of the ray method. The objective was to describe the wave field at regular points of a ray set in the vicinity of its initial point (on the ray-orthogonal plane). A new approach is proposed to compute the terms of ray series confining to the discontinuous part of the wave field. Describing wave propagation in terms of discontinuities is describing propagation of ideal waves. Such a description is as exact as are the laws of mechanics applied to propagation of material points. Discontinueties give an exact description of wave field functionals rather than of the field itself. However, these functionals exhaustively describe the geometrical aspect of wave propagation phenomenon. Another novel approach consists in using the standard integral field presentations involving fundamental Green's tensor where "integrand functions" are replaced by the corresponding discontinuities. This permits a straightforward first-approximation computation of all components of compressional and shear waves in homogeneous media by reducing the problem to integration of δ-function. Explicit formulas have been obtained that describe the amplitude of a seismic wave along the ray (the novelty is in the use of two leading terms of the ray series). The method proposed does not claim to be an improvement in the computation techniques but is thought as a tool for qualitative investigation of elastic-wave propagation. The authors analyze a regular case of wave propagation, considered as a test for the new approache to be further applied to caustic fileds and inhomogeneous media