Discontinuous Staggered Finite Difference Method for Anelastic Wave Simulations

Abstract:

Simulating seismic waves on uniform grids in realistic heterogeneous crustal media typically requires a relatively small grid spacing determined by the minimum velocity, which leads to a large number of grid points and small time step. Discontinuous grid (DG) methods, on the other hand, using finer grids in the lower-velocity region and coarser grids in the higher-velocity part of the model, are much more efficient for geological models including a wide range of seismic velocities. Here, we have developed a 3-D DG fourth-order staggered-grid visco-elastic finite difference (FD) method with a ratio of 3 between coarse and fine grid partitions, and we explore its numerical properties. In particular, we focus on the stability of the scheme for many timesteps (100,000+), often needed for regional and/or high-frequency simulations.

FD DG implementations suffer inherently from stability problems due to the nature of exchange of wavefield information between the fine and coarse grids. In particular, staggered grids, where analytical stability conditions are less tractable, provide a challenge. The cause of instability is likely related to down-sampling of the wavefield from the fine grid into the coarse grid, and possibly the interpolation to obtain the wave field when transferring the wave field from coarse to fine grids. We have tested the accuracy and stability of several techniques to deal with these issues. For example, we have explored bilinear, bicubic and wavenumber domain interpolation approaches, which all tend to provide good accuracy. For the downsampling we have tested both Lanczo’s and Gaussian filters as well as a new approach, the Weakly Enforced Discontinuous Grid Interface Elements (WEDGIE) method. While these down sampling methods show a significant improvement of stability, our preliminary analysis suggests that stability is affected by several factors, including media properties, spatial dimension, the presence of absorbing boundaries, and anelastic attenuation. Specifically, we find that smaller visco-elastic models with little or no velocity variation near the fine and coarse grid interface and including absorbing boundary conditions tend to be more stable.