The Effect of Rayleigh-Love Coupling in an Anisotropic Medium

For a weakly anisotropic medium, Rayleigh and Love wave phase speeds at angular frequency $\omega$ and propagation azimuth $\psi$ are given approximately by $V(\omega,\psi) = A_0 + A_{2c} \cos 2 \psi + A_{2s} \sin 2 \psi + A_{4c} \cos 4 \psi + A_{4s} \sin 4 \psi$. Earlier 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 the the 2$\psi$ component for Love waves and 4$\psi$ for Rayleigh waves, referred to here as ``unexpected anisotropy". To explain these observations, we present a quasi-degenerate theory of Rayleigh-Love coupling in a weakly anisotropic medium based on Hamilton's Principle in Cartesian coordinates, benchmarking this theory with numerical results based on SPECFEM3D. We show that unexpected anisotropy is expected to be present when Rayleigh-Love coupling is strong 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, when observations of the 2$\psi$ and 4$\psi$ components of Rayleigh and Love anisotropy are used in the inversion, the ellipticity parameter $\eta_X$, introduced here, is better constrained, we can constrain the absolute dip direction based on polarization measurements, and we provide evidence that the mantle should be modeled as a tilted orthorhombic medium rather than a TTI medium. Ignoring observations of unexpected anisotropy may bias the estimated seismic model significantly. We also provide information about the polarization of the quasi-Love waves and coupling between fundamental mode Love and overtone Rayleigh waves in both continental and oceanic settings. The theory of SV-SH coupling for horizontally propagating body waves is presented for comparison with the surface wave theory, with emphasis on results for a TTI medium.