Using the pseudospectral method on curved grids for 2D elastic forward modelling

When applying the conventional Fourier pseudospectral method (FSM) on a Cartesian grid that has a sufficient size to propagate a pulse, spurious diffractions from the staircase representation of the curved interfaces appear in the wavefield. It is demonstrated that these non-physical diffractions can be eliminated by using curved grids that conform to all the interfaces of the subsurface. Methods for solving the 2D acoustic wave equation using such curved grids have been published previously by the authors. Here the extensions to the full 2D elastic wave equations are presented. The curved grids are generated by using the so-called multiblock strategy which is a well-known concept in computational fluid dynamics. In principle the subsurface is divided into a number of contiguous subdomains. A separate grid is generated for each subdomain patching the grid lines across domain boundaries to obtain a globally continuous grid. Using this approach, even configurations with pinch outs can be handled. The curved grid is taken to constitute a generalized curvilinear coordinate system. Thus, the elastic equations have to be written in a curvilinear frame before applying the numerical scheme. The method implies that twice the number of spatial derivatives have to be evaluated compared to the conventional FSM on a Cartesian grid. However, it is demonstrated that the extra terms are more than compensated for by the fewer grid points needed in the curved approach