Efficient 3-D wavefield extrapolation with Fourier finite-differences and helical boundary conditions

Fourier finite-difference (FFD) migration combines the complementary advantages of the phase-shift and finite-difference migration methods. However, as with other implicit finite-difference algorithms, direct application to 3-D problems is prohibitively expensive. Rather than making the simple x y splitting approximation that leads to extensive azimuthal operator anisotropy, I demonstrate an alternative approximation that retains azimuthal isotropy without the need for additional correction terms. Helical boundary conditions allow the critical 2-D inverse-filtering step to be recast as 1-D inverse-filtering. A spectral factorization algorithm can then factor this 1-D filter into a (minimum-phase) causal component and a (maximum-phase) anti-causal component. This factorization provides an LU decomposition of the matrix, which can then be inverted directly by backsubstitution. The cost of this approximate inversion remains O(N ) where N is the size of the matrix.