Free-surface boundary condition in 3D fully staggered grid finite difference seismic simulation with topography

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 (ζ,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 ζ 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.