Free-surface Boundary Condition in 3D Fully Staggered Grid Finite Difference Seismic Simulation with Topography

Abstract

We extend the Robertson’s image method (Robertsson, 1996) to a 3D finite-difference time-domain (FDTD) wave propagation in the presence of surface topography. Robertson method is accurate for a horizontal free surface and easily implemented on a standard staggered grid (SSG), but has some artefact for a topographic surface if it is staircase represented. We use a surface-conforming mapping to transfer the physical topographic domain (x,y,z) to a regular Cartesian domain (ξ,κ,η), which is equivalent to a curvilinear coordinate system. We take the velocity-stress first-order equation form for the (anisotropic) elastic wave equation, and discretize the transferred Cartesian domain on a fully staggered grid (FSG). Following the Robertsson procedure, we put all the particle velocity zero above the free surface, the velocity derivative with respect to η at the free surface is replaced with velocity derivatives with respect to ξ and κ, and anti-symmetrize the traction component normal to the curvilinear surface. We present the numerical results to show the efficiency of this extension.