Three-dimensional elastic wave numerical modelling in the presence of surface topography by a collocated-grid finite-difference method on curvilinear grids

Surface topography has been considered a difficult task for seismic wave numerical modelling by the finite-difference method (FDM) because the most popular staggered finite-difference scheme requires a rectilinear grid. Even though there are numerous collocated grid schemes in other computational fields that could be used to solve the first-order hyperbolic equations, the lack of a stable free-surface boundary condition implementation for curvilinear grids also obstructs the adoption of curvilinear grids in seismic wave FDM modelling. In this study, we use generalized curvilinear grids that can fit the surface topography to discretize the computational domain and describe the implementation of a collocated grid finite-difference scheme, a higher order MacCormack scheme, to solve the first-order hyperbolic velocity-stress equations on the curvilinear grid. To achieve a sufficiently accurate and stable free-surface boundary condition implementation on the curvilinear grids, we propose the traction image method that antisymmetrically images the traction components instead of the stress components to the ghost points above the free surface. Since the velocity derivatives at the free surface are provided by the free-surface condition, we use a compact scheme to compute the velocity derivatives near the free surface and avoid the use of velocity values on the ghost points. Numerical tests verify that using the curvilinear grid, the collocated finite-difference scheme and the traction image technique can simulate seismic wave propagation in the presence of surface topography with sufficient accuracy.