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

Wilson-Burg vs. Kolmogoroff spectral factorization

Last on the checklist was the spectral factorization algorithm itself. In Rickett (2001) the Kolmogoroff spectral factorization method is shown to be successful for modeling seismic activity on the surface of the Sun. I used the SEPlib ccrosskolmog module, and compared wavefield propagation when the spectral factorization was done by the Kolmogoroff method vs. the Wilson-Burg method. The comparison is shown in Figure 10. On the left are wavefields propagated with Wilson factorization, and on the right - Kolmogoroff. The time step is $\Delta t = .5ms$ in the top Figures. When the time step is increased to $1 ms$ , the propagation with Kolmogoroff coefficients diverges. However, if $\epsilon = 10^{-4}$ is added to the central finite-difference coefficient prior to factorization (bottom right), propagation is successful and appears similar to propagation by Wilson factorized coefficients.

This result indicates that the Kolmogoroff factorization method is even less suitable than the Wilson method for this finite-difference scheme, since the addition of a small value to the central FD coefficient when using Wilson is necessary only at greater time step sizes.

The ten Wilson filter coefficients used to create the center panels in Figure 10 were:

$$\begin{array}{cccc} 1.0 & -2.4875777E-03 & 0.0000000E+00 & -6... ...9443045E-31 \\ 0.0000000E+00 & -1.5046328E-36 & & .\end{array}$$

The Kolmogoroff coefficients were:

$$\begin{array}{cccc} 1.002494 & -2.4938183E-03 & 4.7695384E-08... ...8970848E-09 \\ -9.8089106E-12 & 4.9380566E-09 & & .\end{array}$$

Other than the zero-lag coefficient not being equal to 1, a striking difference is that the Kolmogoroff coefficients do not drop off quickly as do the Wilson coefficients. This has a degrading effect on the filter correlation. The Wilson filter's correlation is:

$$\begin{array}{cccc} 1.005 & -2.5000004E-03 & 1.6940662E-23 & ... ...40020E-31 \\ 0.0000000E+00 & -1.5121468E-36 & & \\ \end{array}$$

The Kolmogoroff's filter correlation is:

$$\begin{array}{cccc} 1.005 & -2.4999434E-03 & 8.8091141E-08 & ... ...9342E-09 \\ -6.6459863E-12 & -2.8234270E-09 & & \\ \end{array}$$

The finite-difference coefficients for the parameter set of the wavefields in Figure 10 are $U_0 = 1.005, U_1 = -2.5E-03$ . The Wilson filter's correlation recreates these weights precisely, while the Kolmogoroff filter's correlation does not. Also, the drop-off in the magnitude of the filter correlation at lags which do not represent finite-difference weights (i.e. not lag 0 or lag 1) is much better for the Wilson filter.

wil-vs-kol
Figure 10. 1D helical implicit finite-difference with Wilson-Burg spectral factorization (left), and Kolmogoroff spectral factorization (right). $\Delta t = .5ms$ (top), $1 ms$ (center and bottom). $\epsilon = 1e^{-4}$ only on the bottom right Figure, otherwise $\epsilon = 0$ . Velocity = $1000 m/s$ , $\Delta x = 10 m$ .[ER]

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