Xiongwei Liu, Michael H Ritzwoller

It is well known that for a weakly anisotropic medium, at angular frequency $\omega$ and propagation azimuth $\psi$ Rayleigh and Love wave phase speeds are given approximately by $V(\omega,\psi)
= A_0 + A_{2c} \cos 2 \psi + A_{2s} \sin 2 \psi + A_{4c} \cos 4 \psi + A_{4s} \sin 4 \psi$. Previous theories of the propagation of surface waves in anisotropic media based on non-degenerate perturbation theory predict that the dominant components are expected to be $2\psi$ for Rayleigh waves and $4\psi$ for Love waves. This paper is motivated by recent observations of previously unexpected anisotropy: the 2$\psi$ component for Love waves and 4$\psi$ for Rayleigh waves.
To illuminate this phenomenon, we present a quasi-degenerate theory of Rayleigh-Love coupling based on the application of Hamilton's Principle to Rayleigh and Love waves propagating in a weakly anisotropic medium. We show that the unexpected components are actually expected in the presence of strong Rayleigh-Love coupling and recent observations of Rayleigh and Love wave 2$\psi$ and 4$\psi$ anisotropy can be fit successfully with physically plausible models of a depth-dependent tilted transversely isotropic (TTI) medium. In addition, the ellipticity parameter $\eta_X$, introduced here, is better constrained and we present evidence that the mantle should be modeled as a tilted orthorhombic medium rather than a TTI medium. We also provide information about the polarization of the quasi-Love waves, coupling between fundamental mode Love and overtone Rayleigh waves in both continental and oceanic settings, and practical suggestions for observers. For comparison, we present a theory of SV-SH coupling for horizontally propagating body waves, with particular emphasis on results for a TTI medium.