A new approach to Gaussian beams on a sphere: theory and application to long-period surface wave propagation

The subject of this paper is the application of the Gaussian beam method to the propagation of surface waves on a laterally heterogeneous, spherical earth taking fully into account the curvature of the Earth’s surface. This is achieved by constructing a ray centred coordinate system on general curved surfaces. The application of this method to a spherical shell leads to a ray centred coordinate system adjusted to the spherical geometry. Solving the elastodynamic equation in this coordinate system by a perturbation ansatz yields two interesting results. (1) The well-known parabolic differential equation for the zeroth order displacement follows directly from a solvability condition imposed on the right-hand side of an inhomogeneous differential equation arising from the perturbation ansatz. (2) The dynamic ray-tracing system which is derived from the parabolic equation is a special case of a general dynamic ray-tracing system valid for arbitrary curved surfaces which can be derived by perturbing the ray equations. It is shown that the geometry of the surface enters only by its intrinsic curvature into the dynamic ray-tracing system. Numerical computations of rays and synthetic seismograms based on this new approach for the model M84C of Woodhouse & Dziewonski show that there can occur considerable deviations of the ray-paths from great circles leading to visible amplitude anomalies in the synthetic seismograms. Comparison of synthetic seismograms with selected data recorded at the Black Forest Observatory (BFO) indicate that the model M84C is still too smooth to yield satisfactory agreement of the synthetics with the data.