A ray-Kirchhoff method for body-wave calculations in inhomogeneous media: theory

Abstract

SUMMARY A ray-Kirchhoff method is developed for body-wave calculations which extends previous ray methods to rapidly varying media. It is based on a newly derived integral solution to wave equations which indicates that the wave field at a receiver point is given by a superposition of ray solutions determined by the transport and extended eikonal equations. The latter is in turn solved by an asymptotic series. In a slowly varying medium, only the leading term of this series needs to be considered, and the extended eikonal equation reduces to the well-known eikonal equation. Wave fields in this case can be calculated using asymptotic ray theory. For a rapidly varying medium where velocity gradients are no longer small, the higher-order terms of the series must not be disregarded. These frequency-dependent higher-order terms represent the scattering effect of velocity gradients and provide a basis for avoiding caustics. The new method also includes a procedure for estimating the errors introduced by truncating higher-order terms from the asymptotic series. In particular, validity conditions for the ray-Kirchhoff method in elastic media are formulated which indicate that the new method is less restrictive than some previous ray methods such as the Gaussian-beam technique. For implementation, a perturbation scheme is developed for solving the ray and transport equations. In addition to computing the higher-order terms of the asymptotic series, this scheme avoids most of the ray tracing required for computing wave fields in median with weak lateral variations. Using this scheme, the ray-Kirchhoff method is extended to anelastic media. Approaches for removing singularities on an integral surface used in the ray-Kirchhoff method are proposed which not only prevent the infinite amplitude at a caustic, but also predict the phase shift caused by this singularity.