Modeling 3D Wave Propagation in Global Earth Models Using a Spectral/Mortar Element Method

The Spectral Element Method (SEM) has been shown in recent years to provide an efficient, accurate solution to wave propagation in the Earth at both regional and global scales. Based on a high order polynomial approximation of the weak form of the wave equation, the SEM naturally takes into account the free surface condition (which is crucial for surface wave propagation) and allows for considering localized density/velocity heterogeneities. For global wave propagation the method requires to tile the 3-sphere with hexaedra. This is achieved by paving the Earth's major interfaces with quadrangles and connecting them radially, with a cube being set inside the inner core to avoid any singularity at the center of the mesh. However, such a process yields a non-uniform grid-points distribution that causes the seismic wavelengths to be largely over-sampled for increasing depth. This drawback is overcome by coarsening the mesh through some special ``non-conforming'' interfaces, which do not necessarily coincide with composition or phase changes. The matching between the fine and the coarse grids is achieved by the mortar element method which consists in relaxing the continuity conditions across the non-conforming interfaces. Examples of the method will be presented for some radial Earth models where the local effects of gravity are taken into account during wave propagation. The approximation is based on a formulation in displacement for the solid parts and in the velocity potential for the fluid regions, and the non-conforming interfaces can be set either in the solid or in the fluid. For realistic applications of the method to 3D Earth models, the possibility of refining the spectral element mesh laterally must also be considered. A striking example illustrating this need is the discretization of the Earth's crustal structure, for which thickness variations of a factor up to 10 can be encountered. Adapting the grid-points sampling to these lateral contrasts is a key ingredient to study the propagation of surfaces waves in presence of topography on both the surface of the Earth and the Mohorovicic discontinuity. We will present how the mortar element method must be adapted in order to deal with such local refinements, and show some examples for models of increasing complexity.