Vicinity ray tracing: an alternative to dynamic ray tracing

Abstract

SUMMARY A method for computing seismic wavefields in a high-frequency approximation is proposed based on the integration of the kinematic ray tracing equations and a new set of differential equations for the dynamic properties of the wavefront, which we call the vicinity ray tracing (VRT) equations. These equations are directly obtained from the Hamiltonian in ray centred coordinates, using no paraxial approximations. This system is comparable to the standard dynamic ray tracing (DRT) system, but it is specified by fewer equations (four versus eight in 3-D) and only requires the specification of velocity and its first spatial derivative along a ray. The VRT equations describe the trajectory of a ray in the ray centred coordinates of a reference ray. Quantities obtained from vicinity ray tracing can be used to determine wavefront curvature, geometric spreading, traveltime to a receiver near the reference ray, and the KMAH index of the reference ray with greater numerical precision than is possible by differencing kinematically traced rays. Since second spatial derivatives of velocity are not required by the new technique, parametrization of the medium is simplified, and reflection and transmission of beams can be calculated by applying Snell's law to both vicinity rays and central rays. Conversion relations between VRT and DRT can be used to determine the paraxial vicinity of DRT, in which the errors of the paraxial approximations of DRT remain small. In either DRT or VRT, the width of Gaussian beams can be physically defined from the width of the Fresnel volume surrounding the central ray. Because no paraxial approximations are made, the superposition of the Gaussian beams defined from vicinity rays should exhibit a much slower breakdown in accuracy as the scale length of the medium given by ν/Δν/ approaches the beamwidth.