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# Nonlinear PEF Estimation

Once an initial estimate of the PEF is made, this estimate can be used to interpolate the data, from which a new PEF can be estimated. This nonlinear approach can be repeated until it converges to a final solution, shown in Figure 5. Unfortunately, like most nonlinear methods, the choice of a starting guess is crucial to the success and efficiency of the method. When the starting interpolation is far from the ideal solution, convergence to the best solution is not likely. Figure 6 shows the nonlinear method used on the data, with various different original guesses.

nlflow
Figure 5
Flow chart for nonlinear PEF estimation.

nlfigure
Figure 6
Nonlinear PEF estimation with various starting guesses. From left to right: inverse impulse response of PEF; filled data; envelope of Fourier Transform of filled data, all after 10 nonlinear iterations. From top to bottom: filled with PEF obtained from original data; filled with Laplacian; filled with single scale (32x32) PEF, filled with second single scale (20x20) PEF, filled with multiscale PEF.

The nonlinear method appears to create wildly different solutions depending on the starting guess. Both the Laplacian interpolation as well as one of the single scale PEF interpolations produced poor results. Both the multiscale PEF and the other single scale PEF produced pleasing results, which are both very close to the ideal solution. The ideal solution is where the PEF is calculated from the fully sampled data, and is then used as a starting guess in the nonlinear estimation. The multiscale result was obtained without knowing the solution, while the second single scale result was obtained by comparing various results with the original filled data, and selecting the most similar.

Next: Conclusions Up: Curry and Brown: Multiscale Previous: Multiscale PEF Estimation
Stanford Exploration Project
9/18/2001