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

Conclusion and future work

It seems, at least in principle, that the general methodology described above for 2-way wave extrapolation with implicit finite difference by deconvolution using spectral factorization and the helix should be possible. The decrease in runtime in comparison to standard implicit finite difference application must be properly measured to assess the desirability of the method, although in my preliminary tests it is very evident that the method presented here is much faster than when using a standard solver.

The problem of dispersion of the wavefield when using the weights in Equation 10 must be overcome. It is not yet obvious to me what part of the observed dispersion can be attributed to the filter coefficients and what part to the finite-difference scheme. If the factorization is the main contributor, then one possible solution is to slightly increase the value of the diagonal element of matrix $U$ in Equation 10, in order to improve the accuracy of the spectrally factorized filter coefficients. The connection between that and the number of factorized coefficients used in the filter is also as of yet unclear.

Another method of reducing the dispersion is to reformulate the implicit scheme with different weights. I have made one preliminary test in one dimension which suggests that doubling the value of the weights of matrix $V$ in Equation 10 removes some of the dispersion. In any case, A more rigorous analysis of the phase velocity as a function of frequency of this or any other implicit scheme must follow.
Polynomial division is a necessarily sequential operation, as each output is dependent on the previous outputs (Equation 16). This seems to preclude any possibilty of parallelization of the process within each time step. I do not see a way around this at the moment, but where there's a will there's a way.

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