Understanding finite-frequency wave phenomena: phase-ray formulation and inhomogeneity scattering

Abstract

SUMMARY A straight line connecting two points in a homogeneous elastic medium defines a body-wave ray path. A single wavefront can be said to be uniquely associated with a body-wave signal propagating along this path. Ray path and wavefront are concepts at once straightforward and useful in such media, where exact solutions may be attainable. In inhomogeneous media, signal distortions accompany the physical processes of focusingfdefocusing and scattering. The ray path depends on the signal frequency; the very concept of a single wavefront for each propagating body wave is blurred. To assume otherwise is tantamount to neglecting a host of wave phenomena intrinsically connected with the scattering effects due to medium inhomogeneities. An example of such a scattering-related wave phenomenon is frequency-dependent partial reflection of a body wave propagating in a zone of high-velocity gradient. These phenomena are not modelled by the published ray methods for seismic wavefield computation, limiting their applicability to homogeneous and weakly inhomogeneous media. The scattering-induced wave phenomena become increasingly more difficult to ignore as the wave-medium heterogeneity strengthens. To achieve a higher order of accuracy than the published ray methods, it is necessary to incorporate the scattering effects. The principal objective of this study is to extend our wavefield modelling capability to rapidly varying media by proper accounting for the scattering effects embodying, among others, frequency-dependent ray paths and scatteringinduced wave attenuation and dispersion. We demonstrate that the new formulation based on a phase eikonal equation can now model frequency-dependent partial reflections from a gradient zone. It is shown that the scattering process introduces signal smoothing and its incorporation in the wavefield calculation results in natural removal of the singularity of asymptotic ray theory near a caustic.