Reducing two-way splitting error of FFD method in dual domains

Abstract

The Fourier finite-difference (FFD) method is very popular in seismic depth migration. But its straightforward 3D extension creates two-way splitting error due to ignoring the cross terms of spatial partial derivatives. Traditional correction schemes, either in the spatial domain by the implicit finite-difference method or in the wavenumber domain by phase compensation, lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts. We propose compensating the two-way splitting error in dual domains, alternately in the spatial and wavenumber domains via Fourier transform. First, we organize the expanded square-root operator in terms of two-way splitting FFD plus the usually ignored cross terms. Second, we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions. Finally, we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator. In implementation, the compensation terms are similar to the high-order corrections of the generalized-screen method, but their functions are to compensate the two-way splitting error rather than the expansion error. Numerical experiments show that optimized one-term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1% error for all azimuths is improved up to 60° from 35°. Compared with traditional single-domain methods, our scheme can handle lateral velocity variations (even for strong velocity contrasts) much more easily with only one additional Fourier transform based on the two-way splitting FFD method, which helps retain the computational efficiency.