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

Introduction

Implicit finite-difference methods are inherently more stable than explicit ones. This attribute enables us to increase the time step size (and consequently decrease computation time) while retaining stability of the wavefield. In the previous SEP report (Barak, 2010) I showed that by using spectral factorization and the helix transform, the propagation of a wavefield using an implicit finite-difference approximation of the two-way acoustic wave equation can be achieved by a set of deconvolution operations of filter coefficients applied to the wavefield. Through testing, I have found that despite the theoretical stability advantage of the implicit finite-difference scheme which I used for propagation, the resulting wavefield becomes more dispersive as the time step increases (to the point that the wavefield is no longer useful), and also that beyond a certain time step size - the wavefield diverges.

The increased dispersion of the implicit finite-difference scheme in comparison to an explicit scheme is an attribute of the scheme itself. This is not a fundamental problem, since some of this dispersion can be alleviated simply by using a higher order approximation. However, the causes of the instability of the wavefield beyond a certain time step size remained unclear. In order to understand the reasons behind the instability, I tested several hypotheses for its causes. These were:

1. The implicit finite-difference approximation itself.
2. The precision of the floating point representation of the filter coefficients.
3. The number of filter coefficients.
4. The spectral factorization method.

First I will review the method by which wave propagation can be done by deconvolutions with spectrally factorized filters of a finite-difference approximation, and then I will go over the various tests I carried out to try and determine the causes for the instability problem.

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