Ray Tracing on a Heterogeneous Sphere by Lie Series

We propose a method for fast analytical ray tracing on a heterogeneous sphere for surface waves. We first select the specific coordinates of orbital motion which have action/angle properties. We then apply the Lie perturbation approach and, when the square of the slowness is expanded in spherical harmonics, we obtain an analytical formula for the perturbed parameters of the ray. These expressions are sensitive to both the odd and even parts of the expansion. Traveltimes are computed by perturbation, while geometrical spreading is estimated numerically between two nearby perturbed rays. For the 'Gulf of Alaska' earthquake of November 1987, the analytical ray follows the same deviations with respect to the great circle as the numerical one, when we use the phase velocity model of Montagner & Tanimoto (1990) at period of 167s. The agreement is excellent for traveltime computations. When the numerical ray tracing predicts a focus/defocusing effect, the perturbed ray tracing gives the same trend. Moreover, variations of the shooting angles between trains can be as high as 20-degrees which might modify the radiation pattern seen by the station for different trains. When the perturbed ray deviates too strongly, a reinitialization technique will guarantee a given accuracy. This reinitialization, which is not required for long periods (> 150 s), is probably necessary at shorter periods.