Implementing implicit finite-difference in the time-space domain using spectral factorization and helical deconvolution

Summary of current implementation

An increase in the time step size causes the wavefield to diverge after a certain number of propagation steps. This divergence can be avoided - by artificially increasing the value of the central finite-difference weight. This correspondingly increases the zero-lag coefficient of the factorized filter, making it more dominant in comparison to the other filter coefficients. The result is stable propagation, albeit with much dispersion owing to the finite-difference approximation itself. The exact minimum value required for $\epsilon$ which ensures stable propagation is difficult to ascertain. Too large a value and an odd dispersion pattern unlike that of standard numerical dispersion begins to appear. The addition of $\epsilon$ to the central finite-difference weight also has the rather unfortunate effect of ruining the propagation kinematics.