Implicit finite difference in time-space domain with the helix transform

Previous implementations of the helix

Fomel and Claerbout (1997) show how spectral factorization of an implicit finite-difference scheme and application of the resulting filter coefficients with the helix operator can recursively solve the heat conduction equation. They also present an implicit helix-based 3D velocity continuation method for post-stack data.

In Rickett et al. (1998) the one-way wave equation is iteratively solved in the frequency-space domain to downward-continue a wavefield. The extrapolation is robust and efficient. The shortcoming is that since factorization is done for coefficients in the frequency-space domain, the velocity is factorized into the resulting filter coefficients. This forces a workaround for dealing with lateral velocity variations. The suggested method is to create several filters, each with a reference velocity, to be applied to different parts of the wavefield.