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# Example

In a first example, we applied the method to deconvolution on the helix Claerbout (1997) using the factors obtained with the Wilson-Burg spectral factorization. We take the auto-correlation to be the negative of the Laplacian operator, and convolve it with a spike placed in the middle of each panel in Figure 3. We use the Wilson-Burg method to find the wavelet with this auto-correlation and then deconvolve (do polynomial division) on the helix to find back the input spike.

autolapfac
Figure 3
Wilson factorization of the Laplacian. From left to right: the input filter; its auto-correlation; the factors obtained with the Wilson-Burg method; the result of the deconvolution using the Wilson-Burg factors.

In another example, we analyzed the rate of convergence of the Wilson-Burg method. We selected a simple polynomial which is the cross-correlation of two triangle functions,
 (14)
Table 1 shows the quadratic rate of convergence, defined using a relation similar to equation (8) for the coefficients of the two factors, A and B.

 iter A B 1 0.0364715122 0.0442032255 2 0.0029259326 0.0002011458 3 0.0000014305 0.0000000199 4 0.0000000894 0.0000000199 5 0.0000000596 0.0000000000 6 0.0000000000 0.0000000000

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Stanford Exploration Project
7/5/1998