Application of normalized pseudo-Laplacian to elastic wave modeling on staggered grids

摘要

We extended the pseudo-Laplacian to staggered gridsbased on the concept of normalized pseudo-Laplacian andapplied it to constructing the pseudoanalytical formulationsfor the variable-density acoustic wave equation and the elasticwave equation. Acoustic wavefields only contain P-wavesand therefore only P-wave pseudo-Laplacians are requiredfor acoustic wave propagation. In comparison, two sets ofstaggered grid pseudo-Laplacians are needed in the elasticcase in order to properly compensate for time stepping errorsfor both P-waves and S-waves.We gave a thorough derivationof the pseudoanalytical method for the elastic wave equation,based on normalized pseudo-Laplacians implemented onstaggered grids, and presented the resulting complete discretizedformulas. We proved that the staggered grid pseudo-Laplacian reduces to the pseudo-Laplacian for the scalarwave equation on standard grids. When using zero compensationvelocities for normalized pseudo-Laplacians, the pseudoanalyticalformulas simply reduce to the pseudospectralequations. We demonstrated with numerical examples thatstaggered grid pseudo-Laplacians effectively compensate forsecond-order time stepping errors and help generate highlyaccurate acoustic and elastic wave solutions in variabledensitymedia.