Coupling the spectral element method with a modal solution for elastic wave propagation in global earth models

We present a new method for wave propagation in global earth models based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust. In the outer shell, the solution is sought in terms of the spectral element method, which stem from a high order variational formulation in space and a second-order explicit scheme in time. In the inner sphere, the solution is sought in terms of a modal solution in frequency after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of finite element methods with the exponential convergence rate of spectral methods. It avoids the pole problems and allows for local mesh refinement, using a non-conforming discretization, for the resolution of sharp variations and topography along interfaces. The modal solution allows for an accurate isotropic representation in the inner sphere. The coupling is introduced within the spectral element method via a Dirichlet-to-Neumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized spherical harmonics. The inverse transform in space and time requires special attention and an asymptotic regularization. The coupled method allows a significant speed-up in the simulation of the wave propagation in earth models. For spherically symmetric earth model, the method is shown to have the accuracy of spectral transform methods and allow the resolution of wavefield propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis.