A new formalism for the effect of lateral heterogeneity on normal modes and surface waves-II. General anisotropic perturbation

Abstract

Using an operator formalism and simple geometrical relations on the sphere, we have derived the interaction terms for the free oscillations of an aspherical anisotropic Earth, as needed for seismogram calculations in the framework of first-order perturbation theory, without needing to expand the model perturbations into spherical harmonics. The expressions obtained are compact and free of summation over azimuthal orders. In the case of transverse isotropy they are particularly simple. Their application may be very useful in the case of localized heterogeneities which would require expansion in spherical harmonics up to a large angular order. We then give several straightforward applications in the high-frequency limit, which generalize results previously obtained by different authors in the isotropic case. By going to a propagating-wave formalism, we generalize the first-order scattering formulation of Snieder & Nolet (1987) as well as Snieder & Romanowicz (1988). In the asymptotic limit of propagation along the great circle, we readily obtain the extension of the results of Smith & Dahlen (1973) for surface-wave phase velocities in a flat-layered anisotropic Earth. Finally, to order 1/l where l is the angular order of the mode, we generalize the results obtained by Romanowicz (1987), for the perturbation in amplitude of modes and surface waves in the framework of the great-circle approximation, to the case of transverse isotropy. In the general anisotropic case, additional terms depending on odd and even multiples of the azimuth of the great circle must be taken into account.