3-D finite‐difference elastic wave modeling including surface topography

Abstract

Three‐dimensional finite‐difference (FD) modeling of seismic scattering from free surface topography has been pursued. We have developed exact 3-D free surface topography boundary conditions for the particle velocities. A velocity‐stress formulation of the full elastic wave equations together with the boundary conditions has been numerically modeled by an eighth‐order FD method on a staggered grid. We give a numerical stability criterion for combining the boundary conditions with curved‐grid wave equations, where a curved grid represents the physical medium with topography. Implementation of this stability criterion stops instabilities from arising in areas of steep and rough topographies. We have simulated scattering from teleseismic P-waves using a plane, vertically incident wavefront and real topography from a 40 × 40 km area centered at the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P-waves to Rg (short period fundamental mode Rayleigh) waves in an area of rough topography approximately 10 km east of NORESS. This result is consistent with numerous observations.

Three Dimensional Finite-Difference Seismic Signal Propagation

“…Greater detail can be found in Hestholm and Ruud (1998). We implement a 3D FD-TD elastic wave equation model using a staggered grid, central difference scheme.…”

“…Details of this rotation and transformation are given by Hestholm and Ruud (1998). The result is a system of linear equations, which are solved by direct matrix inversion using 2 nd order vertical and horizontal particle velocity derivatives from the grid nodes immediately below the free-surface.…”

“…To consider the effects of topography, we incorporate Hestholm and Ruud's (1998) curvilinear coordinate system transformation. The approach begins by specifying the surface of the earth as an arbitrary single-valued elevation function z o (x,y).…”

Seismic Waves - Research and Analysis

“…The stress is forced to vanish at the free surface by using a virtual plane located half a grid interval above the free surface where the stress is forced to have opposite values to that located just below the free surface. In case of more complex topographies, one strategy is to adapt the topography to the grid structure at the expense of numerical dispersion effect (Robertsson, 1996) or to deform the underlying meshing used in the numerical method to the topography (Hestholm, 1999;Hestholm & Ruud, 1998;Tessmer et al, 1992). In the first case, because of stair-case approximation, a local fine sampling is required (Hayashi et al, 2001).…”

Accurate seabed modeling using finite difference methods

“…To achieve this, the continuous model is first sampled along any problematic interfaces to form a curved grid; the curved grid is then mapped to an orthogonal grid via coordinate transform. There are two widely used methods: the curvilinear coordinate method [16,17] and the body-fitted grid method [22,38,55]. Both these methods are commonly used to model irregular topography of the free surface.…”