WHITENING HYPERBOLAS WITH 2-D PREDICTION-ERROR FILTERS

For comparison, we tried whitening three types of hyperbolas: a purely kinematic hyperbola, a hyperbola with some amplitude corrections, and a hyperbola with the amplitude corrections and a half-derivative filter. A hyperbola with only the kinematics was built by placing points on a grid along a hyperbolic path with a little interpolation. The next best approximation to a correct hyperbola was to add cosine and spherical divergence terms. Finally, a half-derivative filter was added to the amplitude corrected hyperbola. Hyperbola whitening was applied to these three cases with a 2-D prediction-error filter. If this version of hyperbola whitening works, we expect similar results from all three cases, and we expect the operator applied to the hyperbola with amplitude and half-derivative corrections to do almost nothing.

Each hyperbola was passed through the 2-D prediction-error filter, and the input and the output spectrums were displayed. The prediction-error filters were 3 elements in the trace direction and 25 elements in the time direction. The results are shown in Figures   to  .

 

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Figure 1 A hyperbola without amplitude corrections.

Figure   shows some whitening of the spectrum of the hyperbola without amplitude corrections, but the evanescent zone shows some noise buildup. The spectrum shows a spike at the origin, indicating that some undesired D.C. is being passed, but the results are encouraging. The ridges in the spectrum may be caused by a filter that was too short in the time direction.

 

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Figure 2 A hyperbola with amplitude corrections.

Figure  , with the amplitude corrected hyperbola, shows that the amplitudes of the limbs of the hyperbola are significantly lowered after the prediction-error filter. This is likely to be the result of the buildup of the operator in the evanescent zone. While the spectrums in Figure   and Figure   appear similar, and the amplitudes of the filtered hyperbolas are close, there are some differences, especially on the limbs.

 

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Figure 3 A hyperbola with amplitude and half-derivative corrections.

The spectrum of the hyperbola with amplitude and half-derivative corrections has noise inside the evanescent zone after the application of the prediction-error filter, as seen in Figure  . It is disappointing to see the spectrum is less flat after filtering than before. The prediction-error filter has done significant damage to this hyperbola, where little change was expected. Once again, the damage may be attributed to the buildup of the zone. It is interesting to note that the filtered hyperbolas and their spectrums look similar in each of the figures.