Elastic finite‐difference method for irregular grids

Abstract

Finite-difference (FD) modeling of complicated structures requires simple algorithms. This paper presents a new elastic FD method for spatially irregular grids that is simple and, at the same time, saves considerable memory and computing time. Features like faults, low-velocity layers, cavities, and/or nonplanar surfaces are treated on a fine grid, while the remaining parts of the model are, with equal accuracy, represented on a coarse grid. No interpolation is needed between the fine and coarse parts due to the rectangular grid cells. Relatively abrupt transitions between the small and large grid steps produce no numerical artifacts in the present method. Planar or nonplanar free surfaces, including underground cavities, are treated in a way similar to internal grid points but with consideration of the zero-valued elastic parameters and density outside the free surface (vacuum formalism). A theoretical proof that vacuum formalism fullfills the free-surface conditions is given. Numerical validation is performed through comparison with independent methods, comparing FD with explicitly prescribed boundary conditions and finite elements. Memory and computing time needed in the studied models was only about 10 to 40% of that employing regular square grids of equal accuracy. A practical example of a synthetic seismic section, showing clear signatures of a coal seam and cavity, is presented. The method can be extended to three dimensions.

Equivalent Medium Parameters for Numerical Modeling in Media with Near-Surface Low Velocities

“…able mesh-size algorithms may substantially increase the computational complexity (Moczo et al, 1997;Opršal and Zahradník, 1999) or mesh-generating algorithms (Komatitsch and Tromp, 1999). In this study, we propose a method in which the low-velocity regions are modified in a manner that is consistent with the surface-wave dispersion but allows coarser sampling of the medium.…”

“…One solution to this problem is to use a variable mesh size to reduce the number of calculations while preserving the accuracy (Moczo et al, 1997;Komatitsch and Tromp, 1999;Opršal and Zahradník, 1999). However, accurate vari-*Present address: Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, CB12PR, U.K.…”

Frequency‐Domain Finite‐Difference Acoustic Modeling with Free Surface Topography using Embedded Boundary Method

“…Most of these investigations assume a flat surface, as waveform modeling in the presence of free surface topography is usually difficult for the finite difference method. Staircase discretization is often adopted as an approximation to the tilted free surface (Oprsal and Zahradnik, 1999;Ohminato and Chouet, 1997;Robertsson, 1996). However, this approximation generates artificial scattering unless the discretization length is small, e.g., 1/60 of the dominant wavelength (Bleibinhaus and Rondenay, 2009).…”

Scalar Wave Equation Modeling with Time-Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes

“…To improve the accuracy of FDMs for seismic wave propagation numerical modeling, many variants of the methods have been advanced-these include difference schemes of staggered grid (Virieux, 1984(Virieux, , 1986Kindelan et al, 1990), variable grid (Wang and Schuster, 1996;Hayashi and Burns, 1999), irregular grid (Opršal and Zahradník, 1999), variable time step (Tessmer, 2000), high-order accuracy (Dablain, 1986;Fornberg, 1987;Crase, 1990;Liu and Wei, 2008;Liu and Sen, 2009d), and implicit formulas (e.g., Emerman et al, 1982;Kosloff et al, 2008;Sen, 2009a, 2009b).…”