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Some unintended features of elastic finite‐difference models
Author(s) -
Wild A.J.,
Singh S.C.
Publication year - 1998
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1046/j.1365-2478.1998.810316.x
Subject(s) - eigenfunction , finite difference , finite difference method , isotropy , mathematics , mathematical analysis , operator (biology) , instability , matrix (chemical analysis) , eigenvalues and eigenvectors , physics , quantum mechanics , biochemistry , chemistry , materials science , repressor , transcription factor , composite material , gene
While finite‐difference methods have been used extensively for many years to model wave propagation in elastic media, some of the more subtle effects observable in such models are very inadequately documented in the geophysical literature, especially in regard to their practical numerical consequences. In addition to the intended travelling waves, and the undesirable exponential instability revealed by the von Neumann test, typical second‐order‐time finite‐difference equations also support drifting linear solutions, as can be verified, both theoretically and by numerical experiment. The necessity of these solutions, and their relationship to the incompleteness of the set of travelling‐wave eigenfunctions of the finite‐difference operator, can be exposed by a matrix‐based analysis, and exact expressions for them can be obtained by using standard algebraic techniques. A further peculiarity of the finite‐difference formulation is numerical anisotropy, which emerges in a grid of more than one spatial dimension, even when the modelled medium is intended to be isotropic. This anisotropy can be explained and quantified in terms of the exact eigenfunction solutions to the finite‐difference equation, which, it is found, can be obtained in a simple, closed form, for a typical modern 3D staggered scheme.