The pseudospectral method: Accurate representation of interfaces in elastic wave calculations

When finite‐difference methods are used to solve the elastic wave equation in a discontinuous medium, the error has two dominant components. Dispersive errors lead to artificial wave trains. Errors from interfaces lead to circular wavefronts emanating from each location where the interface appears “jagged” to the rectangular grid. The pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy. With this method, dispersive errors are essentially eliminated. The mappings introduced in this paper also eliminate the other dominant error source. Test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost. Although the mapping method is described here in connection with the pseudospectral method, it can also be used with high‐order finite‐difference approximations.

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Elastic wave propagation using fully vectorized high order finite‐difference algorithms

Geophysics ◽

10.1190/1.1443235 ◽

1992 ◽

Vol 57 (2) ◽

pp. 218-232 ◽

Cited By ~ 20

Author(s):

A. Vafidis ◽

F. Abramovici ◽

E. R. Kanasewich

Keyword(s):

Finite Difference ◽

Elastic Wave ◽

Second Order ◽

Finite Difference Schemes ◽

Difference Operators ◽

Two Dimensional ◽

Elastic Wave Equation ◽

The Matrix ◽

Comparison Of The Results ◽

High Order Finite Difference

Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

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The pseudospectral method: Comparisons with finite differences for the elastic wave equation

Geophysics ◽

10.1190/1.1442319 ◽

1987 ◽

Vol 52 (4) ◽

pp. 483-501 ◽

Cited By ~ 272

Author(s):

Bengt Fornberg

Keyword(s):

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

Difference Scheme ◽

Finite Differences ◽

Finite Difference Scheme ◽

Pseudospectral Method ◽

Two Dimensions ◽

Seismic Modeling ◽

Elastic Wave Equation

The pseudospectral (or Fourier) method has been used recently by several investigators for forward seismic modeling. The method is introduced here in two different ways: as a limit of finite differences of increasing orders, and by trigonometric interpolation. An argument based on spectral analysis of a model equation shows that the pseudospectral method (for the accuracies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points compared to a fourth‐order finite‐difference scheme and one‐sixteenth the number of points as a second‐order finite‐difference scheme. For the total number of points in two dimensions, these factors become 1/16 and 1/256, respectively; in three dimensions, they become 1/64 and 1/4 096, respectively. In a series of test calculations on the two‐dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinuous interfaces.

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