3−D Surface Topography Boundary Conditions in Seismic Wave Modelling

Abstract

New alternative formulations of exact boundary conditions for arbitrary three{dimensional (3−D) free surface topographies on seismic media have been derived. They are shown to be equivalent with previously published formulations, thereby serving as a verification of the validity of each set of formulations. The top of a curved grid represents the free surface topography while the grid's interior represents the physical medium. We assume the velocity{stress version of the viscoelastic wave equations to be valid in this grid before transforming the equations to a rectangular grid. In order to do the numerical discretization we apply the latter version of equations for seismic wave propagation simulation in the interior of the medium. The numerical discretization of the free surface topography boundary conditions by second−order finite−differences (F−Ds) is shown in detail, as well as spatially unconditional stability of the resulting system of equations. The F−D order is increased by two for each point away from the free surface up to eight, which is the order used in the interior. We use staggered grids both in space and time and the second-order leap-frog and Crank-Nicholson methods for wave field time propagation. We simulate point sources at the surface of a homogeneous medium, with a plane surface containing a hill and a trench, respectively. The main features of these general cases are outlined. Then, we present results using parameters typical of teleseismic earthquakes and explosions with a 200 × 100 km[superscript 2] area of real topography from southwestern Norway over a homogeneous medium. A dipping plane wave simulates a teleseismic P−wave incident on the surface topography. Results show clear conversion from P− to Rg− (short period fundamental mode Rayleigh) waves in the steepest and/or roughest topography, as well as attenuated waves in valleys and fjords. The codes are parallellized for simulation on fast supercomputers to model higher frequencies and/or larger areas than before.