The pseudo-analytical method: application of pseudo-Laplacians to acoustic and acoustic anisotropic wave propagation

We generalize the pseudo-spectral method for the acoustic wave equation to create analytical solutions to the constant velocity acoustic wave equation in an arbitrary number of space dimensions. We accomplish this by modifying the Fourier Transform of the Laplacian operator so that it compensates exactly for the error due to the second-order finite-difference time marching scheme used in the conventional pseudo-spectral method. Of more practical interest, we show that this modified or pseudo-Laplacian is a smoothly varying function of the parameters of the acoustic wave equation (velocity most importantly) and thus can be further generalized to create near-analytically accurate solutions for the variable velocity case. We call this new method the pseudo-analytical method. We further show that by applying this approach to the concept of acoustic anisotropic wave propagation, we can create scalar-mode VTI and TTI wave equations that overcome the disadvantages of previously published methods for acoustic anisotropic wave propagation. These methods should be ideal for forward modeling and reverse time migration applications.