Refraction across an angular unconformity between nonparallel TI media

Abstract

Assuming elliptical wavefronts, we reformulate refraction theory for transversely isotropic (TI) media based on the use of the auxiliary angle, α, which is intermediate between the phase angle, θ, and the group angle, ϕ. When considering the application of stretching to transform elliptically anisotropic media into isotropic media, the auxiliary angle is a natural one to use because both θ and ϕ → α under such stretching. Our present formulation for TI media makes the assumption of elliptical anisotropy, which is valid generally for SH-waves but only as a special case for P-and SV-waves, where, in the SV case, the only possible ellipses are circles. Nevertheless, the theory has useful applications for P-waves over limited ranges of propagation direction (e.g., in the short-spread approximation). Our formulation provides explicit results for all angles of incidence and for what we term an angular unconformity between two TI media, that is, for all orientations of the axes of symmetry for each of the media, and for all orientations of the interface, assuming these two axes and the interface normal to lie in the same vertical plane. Our conclusions have been verified by showing that the phase angles and phase velocities of the incident and refracted waves obey Snell's law across the interface. We also demonstrate, using auxiliary angles, that the description of refraction between elliptically anisotropic media by stretching the media to make them isotropic, then applying isotropic refraction, is also valid for our general angular-unconformity case. However, both stretching (1D) and either scaling (2D) or shearing must be applied correctly and separately to the two media. The refraction algorithm developed from this theory and another developed by Byun in terms of phase-velocity theory are currently the only published noniterative algorithms known to us for refraction across an angular unconformity where the axes of anisotropy are parallel neither to each other nor to the interface. Based on this theory, we have developed a demonstration program, AUXDEMOC, that computes the refracted-ray angles for any combination of parameters by the two equivalent methods: (1) anisotropic refraction and (2) stretching plus isotropic refraction. This program can be downloaded from http://www.crewes.org/under Free Software.