Modeling, Migration and Inversion with Gaussian Beams, Revisited

Abstract

Gaussian beams have been shown in some cases to provide smoother leading order asymptotic expansions of the solution of wave equations than classical ray theory provides. They do this through complexvalued traveltime functions that describe the propagating wave in a tube around some central ray. Quadratic exponential decay orthogonal to the central ray arising from the imaginary part of the traveltime provides a "skin depth" to the ray tube. Gaussian beams also have complex-valued amplitudes that avoid the vanishing ray Jacobians of classical ray theory in the neighborhood of caustics. By contrast, classical ray theory requires an extension to more general waveforms (Airy functions, for example) to describe the wavefield in the neighborhood of a caustic. Gaussian beam representations are influenced by the neighborhood of the ray through the sum over wavefields on nearby rays, whereas classical ray-theory produces a solution that depends only on the medium heterogeneity along a single ray, up to second derivatives of those functions. Further, no special tracking of a KMAH index is necessary for the phase shift through caustics to be effected by the sum/integral over Gaussian beams. This note presents a hierarchy of ray methods starting from classical ray theory, followed by dynamic ray tracing and modeling in ray-centered coordinates with real traveltime and amplitude and then with complex traveltime and amplitude leading to Gaussian beams. We introduce a sample application of the representations of wavefields-the Green's function-for source point and beam fan initial point coincident and non-coincident. Thereafter, we show how these wavefield representations can be used in seismic migration/inversion (M/I) for the common-shot configuration