Free oscillations in an anisotropic earth: path-integral asymptotics

We derive expressions for the influence of seismic anisotropy on the coupling of free oscillations using the Born approximation. By rotating to epicentral coordinates, the angular part of the coupling integrals can be approximated, in the limit of smooth structure, by an integral over the source-receiver great circle. To illustrate asymptotic coupling relationships, we examine a simplified case, in which the fourth-order elastic tensor Λαβγδ has an axis of ŵ orientation along the source-receiver great circle. In the path-integral formulation, mode-mode interaction satisfies a set of selection rules based on the angular degrees l1, l2 of the modes and the wavenumber s of elastic perturbation along the path. The familiar odd/even-parity selection rules for isotropic structure are a subset of these rules. If the elastic tensor is rotated to generalized spherical coordinates in an epicentral frame of reference, its components become complex-valued. Spheroidal-spheroidal and toroidal-toroidal coupling is dominated by ℛℯαβγδ Spheroidal-toroidal coupling is dominated by ImΛαβγδ For simple cos 2η azimuthal velocity variations with a horizontal axis of symmetry ŵ, mixed-type coupling via ImΛαβγδ is maximal if ŵ is 45° to the propagation path. This agrees with full coupled-mode computations of long-period Love-Rayleigh scattering. Several terms in the elastic tensor Λαβγδ are non-zero only for a tilted axis of symmetry, and have distinct selection rules. Using the leading-order asymptotics for Legendre functions, coupling for smoothly varying structure is predicted to be small unless s=|l1–l2|. For coupling between like-type modes along a single dispersion branch, tomographic ‘source shifts’, focusing, and refraction effects can be identified.