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

Abstract

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.

Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media

“…It can be shown that J 6 ¼ J 0 as x j j < 1 because this condition is not satisfied (ZHU, 1988). Particularly, it can be readily verified that J 6 ¼ 0 as J 0 !…”

Elastic Kirchhoff-Helmholtz Synthetic Seismograms

“…where is the circular frequency, U + p are amplitude functions, and~p are corresponding phase functions determined formally from the differential equations of exact ray theory (ZHU, 1988;FOREMAN, 1989;DRUZHININ, 1994). The index p= 1, 2, .…”

“…In the ART approximation (Č ERVENÝ et al, 1977;DRUZHININ, 1993), the amplitude G (vw) and phase~( vw) functions in (8) can be determined from the corresponding equations for the main and additional components of the wave field. According to the exact ray theory (ZHU, 1988;FOREMAN, 1989;DRUZHININ, 1994), these functions can be frequencydependent and may give rise to a very useful frequency-dependent ray diagram near the edges, caustics, etc. In the regular domains of the ray field, the diagonal (w= v) components (8) account for the energy transport along the ray tube of the considered wave mode.…”

“…This problem can be solved by using a more accurate approximation of the KH integrand, based on the exact ray theory (ZHU, 1988;FOREMAN, 1989;DRUZHININ, 1994) or the complex continued ART Green's function (FELSEN, 1984;WU, 1985). Their extension was proposed in the Method of the Interference Integral (MII) (KRAVTSOV and ORLOV, 1993).…”

Study of Focusing of Field Refracted by the Quartic Wood Lens Into an Uniaxial Crystal

“…A number of theoretical techniques have been developed to circumvent this problem, including Maslov's theory [4], coherent state approximations [5], direct use of the caustic classification theorem [6], and Kirchhoff's approximation [7]. However, each of these methods has its own difficulties.…”