Free‐boundary conditions of arbitrary polygonal topography in a two‐dimensional explicit elastic finite‐difference scheme

Abstract

We present a simple method for simulating 2-D elastic waves in a model with free-surface topography of polygonal shape, i.e., a continuous but irregular surface composed of line segments. Our method requires special treatment for each of the six specific cases involving line segments of various slopes as well as transition points between the sloping segments. For brevity, only nonnegatively sloping segments are specifically included.On an inclined free surface, vanishing stress conditions are implemented using a rotated coordinate system parallel to the inclined boundary. At transition points on the topography between line segments, we use a first-order approximate boundary condition in a locally rotated coordinate system aligned with the bisector of the corner. As in the classical one-sided explicit approximation scheme widely used for the flat freesurface case, these extrapolation formulas are accurate to first order in spatial increment. Numerical tests indicate that the present scheme is stable over a range of Poisson's ratios of practical interest (v > 0.3) for fairly complicated geometric shapes consisting of ridges and valleys with both steep and gentle slopes. Stability for complicated shapes enables us to study realistic problems for which the topography plays a significant role in shaping the wave field and for which analytical solutions are not generally available.

Numerical Models of Quarry Blast Sources: The Effects of the Bench

“…The seismic signals from this source result from a linear component, the signals due to ;, dilatational source in this structure; and a nonlinear component, the nonlinear motion (spall and ballistic motion) of the material broken from the quarry face. To calculate the linear component to the quarry blast source, we use the approach described by McLaughlin and Jih (1988), using the elastic reciprocity theorem (e.g. Aki and Richards, 1980) to find the far-field displacements due to a point explosive source by calculating the dilatation at the desired source point caused by an incident plane wave (see Figure 2).…”

An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

“…The staggered-grid technique (Virieux, 1986) and the parameteraveraging scheme increase the implementation complexity of the free-surface boundary condition in the presence of surface topography. Jih et al (1988) introduce a technique to decompose an irregular free surface into line segments to handle surface topography. Tessmer et al (1992) propose a coordinate mapping method including surface topography.…”

Three-Dimensional Wave-Field Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface

“…For example, in the context of the deep seismic soundings to explore the crustal structure, seismic experiments are usually carried out across: (a) orogenic belts for understanding the mechanisms; (b) basins to understand the formation mechanisms; (c) transition zones for the study of its interaction (Al-Shukri et al, 1995;Ashford et al, 1997;Boore, 1972;Jih et al, 1988;Levander, 1990;Robertsson, 1996;Zhang et al, 2010). In oil/gas seismic exploration, seismologists also have a similar problem with the undulating topography along the survey line.…”

“…These schemes include finite element method (Rial et al, 1992;Toshinawa and Ohmachi, 1992), spectral element method Tromp, 1999, 2002), pseudo-spectral method (Nielsen et al, 1994;Tessmer et al, 1992;Tessmer and Kosloff, 1994), boundary element method (Bouchon et al, 1989;Campillo and Bouchon, 1985;Sánchez-Sesma and Campillo, 1993;Sánchez-Sesma et al, 2006), finite difference method (Frankel and Vidale, 1992;Gao and Zhang, 2006;Ruud, 1994, 1998;Jih et al, 1988;Lombard et al, 2008;Robertsson, 1996;Zhang and Chen, 2006), and also a hybrid approach which combines the staggered-grid finite difference scheme with the finite element method (Galis et al, 2008;Moczo et al, 1997). Both the spectral element and the finite element methods satisfy boundary conditions on the free surface naturally.…”