Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation

Lowrank Fourier Finite Differences

Song and Fomel (2011) proposed FFD approach to solve the two-way wave equation. The FFD operator is a chain operator that combines FFT and FD, analogous to the concept introduced previously for one-way wave extrapolation by Ristow and Ruhl (1994). The FFD method adopts the pseudo-analytical solution of the acoustic wave equation, shown in equation 5. It first extrapolates the current wavefield with some constant reference velocity and then applies FD to correct the wavefield according to local model parameter variations. In the TTI case, the FD scheme in FFD is typically a 4th-order operator, derived from Taylor's expansion around $k=0$. However, it may exhibit some dispersion caused by the inaccuracy of the FD part. We propose to replace the original FD operator with lowrank FD in order to increase the accuracy of FFD in isotropic and anisotropic media. We call the new operator lowrank Fourier Finite Differences (LFFD).

Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation