Prediction error filters to enhance differences

Precision

Fu et al. (2007) implements some of the core algorithms in reverse time migration and downward continuation based migration on a FPGA. In many situations compute speed can be traded for precision on a FPGA. Clapp (2007) showed that migration is well suited for reduced precision given the summation implied by the process.

Fu et al. (2007) uses a non-linear, computer driven scheme to test whether a given number representation effects the computed result. In order to use this scheme, some test of whether a result changed in a meaningful way had to be designed. The methodology discussed above offers that potential. By summing up the errors resulting from applying equation 5 a single number that represents how well the covariance of an image has been preserved can obtained.

To test this methodology I applied the same precision limiting scheme described in Clapp (2007).The rows of Figure 7 shows the result of limiting the precision of the FK, FFT, and FX portions of the downward continuation process to 3,5, and 9 bits. The left panel is the zero subsurface offset image in each case. The center panel the error calculation using the PEF method. The right panel shows the result using the histogram matching scheme. In each case the migration result is compared against the full precision image. Note how the errors drastically decreases between 3 and 5 bits. Figure 8 show the total error for 3-9 bits. In this case, both methods seem to be an effective mechanism for testing accuracy. Note how the curve dramatically decreases between 3 and 6 bits then remains relatively constant for the PEF method scheme. The elbow in each curve curve represents the best tradeoff between bit precision and accuracy.

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Figure 7. The shows the result of limiting the precision of the FK and FX portions of the downward continuation process to 2,4, and 9 bits. The left panel is the zero subsurface offset image in each case. The center (PEF) and right (histogram based) panel are the error calculation. In each case the migration result is compared; against the full precision image.

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Figure 8. Error vs. bit representation for PSPI migration. Note the elbow in the curve represents the best tradeoff between bit precision and accuracy.

Prediction error filters to enhance differences