next up previous [pdf]

Next: Wilson-Burg vs. Kolmogoroff spectral Up: Barak: Implicit helical finite-difference Previous: Effect of floating point

Effect of number of spectrally factorized coefficients

I had initially assumed that the number of filter coefficients would be the most dominant factor in determining the accuracy of the propagation. I had supposed that the more filter coefficients used in the spectral factorization, the closer would be the value of their correlation to the finite-difference weights. Indeed, the instinctive response I had to the divergence problem was to increase the number of coefficients in the spectral factorization parameters. This, unfortunately, had no effect. Furthermore, the correlation of the filter coefficients created by the spectral factorizer (the Wilson-Burg algorithm) produced accurate finite-difference weights even when very few filter coefficients were present.

An example of the lack of the effect of number of coefficients on the propagation is shown in Figure 9. This 1D example shows how propagation using 2 spectrally factorized filter coefficients is basically identical to propagation when using 50 filter coefficients. Another indication comes from observing the filter coefficients themselves. This example was produced by a 2nd order scheme, which means that there are only 2 finite-difference weights. When factorizing using only 2 filter coefficients, the Wilson-Burg algorithm (for the propagation parameters used in Figure 9) yielded:


\begin{displaymath}\begin{array}{cc}
1.000000 & -5.5728100E-02.\end{array}\end{displaymath}      

The coefficients are displayed in order of lags, so the $ 1.0$ is the zero-lag filter coefficient. Correlating these coefficients, we get:

\begin{displaymath}\begin{array}{cc}
1.125 & -6.2500007E-02\end{array}\end{displaymath}      

at lag 0 and lag 1, which are equal to the floating point representations of the finite-difference weights for Figure 9.

Factorizing using 50 filter coefficients produced (only the first four coefficients are shown, the rest are in A-5 and A-6):


\begin{displaymath}\begin{array}{cccc}
1.000000 & -5.5728100E-02 & -2.7755576E-17 & 1.7347235E-18.\end{array}\end{displaymath}      

After lag 20, the coefficients are all zeros. Note that the first two coefficients are identical to the ones produced by the factorizer when requesting only two coefficients.The correlation of this filter is:


\begin{displaymath}\begin{array}{cccc}
1.125000 & -6.2500007E-02 & -3.1236769E-17 & 1.9455218E-18\end{array}\end{displaymath}      

This correlation again shows the accurate representation of the finite-difference coefficients at lag 0 and lag 1. In addition, the correlation produces a set of other values at later lags, which are much smaller than the weights themselves.

The fact that two filter coefficients were sufficient to produce the finite-difference weights by correlation was interesting, but what is more important is to test what effect the change in the number of coefficients might have on the deconvolution process. Correlating the coefficients is like convolving them over a spike, once in the forward direction and once in reverse order. In order to test the exact effect that a change in the number of coefficients had on the deconvolution, I tested the result of deconvolving the coefficients over a spike. Here as well, the result was identical. I shall spare displaying the numbers themselves for this case.

In summary, I could not a find a combination of parameters (of propagation or of factorization) for which wavefield propagation was more stable if more filter coefficients were used.

helimp-ncoeffs
Figure 9.
1D helical implicit finite-difference with 2 spectrally factorized filter coefficients (top), and 50 coefficients (bottom). Velocity = $ 1000 m/s$ , $ \Delta t = 5 ms$ , $ \epsilon = 0$ , $ \Delta x = 10 m$ .[ER]
helimp-ncoeffs
[pdf] [png]


next up previous [pdf]

Next: Wilson-Burg vs. Kolmogoroff spectral Up: Barak: Implicit helical finite-difference Previous: Effect of floating point

2010-11-26