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

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.