A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

P-wave extrapolation in anisovopic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme. and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.