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

Standard implicit propagation Vs. helical implicit propagation

Implicit finite-difference propagation of a wavefield is the solution of the system shown in Equation 17. I tested wavefield propagation with constant velocity using a standard linear equation solver, and compared it to the wavefield propagated by repeated deconvolutions with spectrally factorized coefficients of the same finite-difference weights (Equation 14). Figure 7 shows this comparison - standard linear system solver is on the left, deconvolutions with spectrally factorized coefficients are on the right. The bottom figures were created with a larger time steps than the top ones. For the same time step size, the methods produce similar images. Both of the bottom images show greater dispersion than the top ones. This suggests that the dispersion (visible also in Figure 6) is an intrinsic property of the finite-difference weights derived by the formulation in Equation 9, and is not the result of the spectral factorization method.

Propagation with a linear equation solver is time consuming, which is why the wavefields in Figure 7 are rather small (only 100x100 elements). To test whether this method is stable over longer periods, I compared explicit wavefield propagation to the helical implicit propagation. These are shown in Figure 8. The figure is after 4 seconds of propagation, with a time step almost equal to the stability limit of the 2nd order explicit finite-difference scheme. The explicit propagation exhibits less dispersion. However, it diverges for greater time step sizes, whereas the implicit solution does not. The dispersion problem however does get worse for the implicit solver as the time step increases.

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