ACCURACY OF FINITE‐DIFFERENCE MODELING OF THE ACOUSTIC WAVE EQUATION

Recent interest in finite‐difference modeling of the wave equation has raised questions regarding the degree of match between finite‐difference solutions and solutions obtained by the more classical analytical approaches. This problem is studied by means of a comparison of seismograms computed for receivers located in the vicinity of a 90-degree wedge embedded in an infinite two‐dimensional acoustic medium. The calculations were carried out both by the finite‐difference method and by a more conventional eigenfunction expansion technique. The results indicate the solutions are in good agreement provided that the grid interval for the finite‐difference method is sufficiently small. If the grid is too coarse, the signals computed by the finite‐difference method become strongly dispersed, and agreement between the two methods rapidly deteriorates. This effect, known as “grid dispersion,” must be taken into account in order to avoid erroneous interpretation of seismograms obtained by finite‐difference techniques. Both second‐order accuracy and fourth‐order accuracy finite‐difference algorithms are considered. For the second‐order scheme, a good rule of thumb is that the ratio of the upper half‐power wavelength of the source to the grid interval should be of the order of ten or more. For the fourth‐order scheme, it is found that the grid can be twice as coarse (five or more grid points per upper half‐power wavelength) and good results are still obtained. Analytical predictions of the effect of grid dispersion are presented; these seem to be in agreement with the experimental results.