Implementing implicit finite-difference in the time-space domain using spectral factorization and helical deconvolution |
Figures 1(a)-1(f) show how wave propagation in one dimension using the implicit scheme from Eq. 2 and spectral factorization fails when the time step size is increased beyond a certain limit. The horizontal axis is time, and the vertical is distance. On the left the propagation is done using a linear equation system solver, and on the right is the result of deconvolving the wavefield with the filter coefficients obtained from spectral factorization. At smaller time steps, the two solutions are similar. The increasing dispersion with increasing time step size is apparent in both solutions. However, once the time step exceeds , the wavefield propagated by deconvolution diverges, whereas the wavefield propagated by the ''standard`` linear system solver exhibits additional dispersion, yet remains stable.
Figures 2(a)-2(f) show the same kind of comparison as Figures 1(a)-1(f), except that here a small
value was added to the central finite-difference weight (
) which was sent as an input to the spectral factorizer:
This results in a filter with slightly different coefficients, and with this filter the propagation is stable (with added dispersion). The value of required to maintain stability increases as the time step size increases. So far I've been unable to determine the relation between the value of the time step and the value of , but I know it is not arbitrary. If is too large, the result is a low-frequency dispersion which seems to initially precede the wavefield, as shown in Figure 3. Afterwards, the wavefield loses amplitude until eventually it disappears altogether.
While the addition of some value does stabilize the wavefield, the flip side is that it causes wrong propagation kinematics. This is a direct result of the artificial increase of the central finite weight. The incorrect kinematics can be seen in Figure 2(d) when looking at the wavelet as it reaches the edge at the 4 second mark. The arrival time of the wavelet is retarded as increases.
A similar phenomena occurs in 2D. In Figure 4 the effect of increasing the time step size from to is shown. The increase causes the wavefield to diverge. Adding to the central finite-difference weight, as in Figure 5, alters the filter coefficients obtained by spectral factorization, and enables stable propagation, with a slight time retardation of the wavefront. If is too large, then an unusual dispersion pattern appears. As the time step is increased further (Figure 6), the value of required for stable propagation increases as well, as does the time retardation of the wavefront. With too large an value the unusual dispersion pattern appears.
impA-a,impA-b,impA-c,impA-d,impA-e,impA-f
Figure 1. 1D Implicit (left) vs. Helical Implicit (right) finite-difference with constant velocity = . Horizontal axis is time, and the vertical axis is distance. Source is a Ricker wavelet with central frequency = . The time step size is for the top Figures, for the center Figures, and for the bottom Figures. .[ER] |
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impB-a,impB-b,impB-c,impB-d,impB-e,impB-f
Figure 2. 1D Implicit (left) vs. Helical Implicit (right) finite-difference with constant velocity = . Horizontal axis is time, and the vertical axis is distance. Source is a Ricker wavelet with central frequency = . The time step size is for the top Figures, for the center Figures, and for the bottom Figures. Top right ; center right ; bottom right . .[ER] |
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imp-vs-helimp-1d-bigeps
Figure 3. 1D Helical implicit finite-difference propagation with constant velocity = . The time step size is . for the top Figure, for the bottom Figure.[ER] | |
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helimp-2d
Figure 4. 2D helical implicit finite-difference with constant velocity = . Wavefields are after 2 seconds of propagation. Source is a Ricker wavelet with central frequency = . The time step size is for the left Figure, and for the right Figure. .[ER] |
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helimp-2d-eps
Figure 5. 2D helical implicit finite-difference with constant velocity = . Wavefields are after 2 seconds of propagation. Source is a Ricker wavelet with central frequency = . The time step size is . for the left Figure, and for the right Figure. .[ER] |
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helimp-2d-eps2
Figure 6. 2D helical implicit finite-difference with constant velocity = . Wavefields are after 2 seconds of propagation. Source is a Ricker wavelet with central frequency = . The time step size is . for the left Figure, and for the right Figure. .[ER] |
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Implementing implicit finite-difference in the time-space domain using spectral factorization and helical deconvolution |