Contrary to my initial theoretical expectation, prediction error is not a reliable discontinuity attribute. In particular, I expected the combination of the three two-dimensional filters to remove any plane wave and to yield a sharp prediction error where they encountered anything but a plane wave. I was naively wrong on two accounts.
First, I failed to realize that prediction error indicates patches that contain a discontinuity but does not pinpoint the discontinuity within the patch. I disregarded the fundamental difference between the prediction of two superposed plane waves and two adjacent plane waves. Patching breaks the initial nonstationary image into small patches. Patches that contain discontinuities are, however, still nonstationary since they contain two adjacent plane-wave volumes. In general, a single prediction-error filter cannot perfectly predict and remove the statistically distinct image regions separated by the discontinuity. Instead, the filter generates considerable error amplitudes anywhere in the patch. The resolution of a prediction-error map is consequently its patch size. For example, the blurred fault of the synthetic image cases of Figures 28 and 41 illustrate the resolution loss. In this figure the patch size is about an eighth of the image size. Similarly, the plane-wave misfit of the previous chapter detects nonstationary patches that are not approximated by a single plane-wave.
Second, I incorrectly assumed that faults were sharp pixel-to-pixel boundaries, as I stylized them in the synthetic example. Instead, image processing is often unable to deliver sharp images of discontinuities. Such smooth transition zones do not cause large prediction errors. In particular, if the size of a fault zone is similar to the size of patch, the filter will be able to predict and remove components of the fault.
The patch size has to be large enough for the reliable estimation of of the prediction-error filter coefficients. On the other hand, the patch has to be small enough to contain stationary data in most cases and to yield a high-resolution output image. For the various examples, a typical patch size is (nx,ny,nz) = (3,3,10).
In the various test cases, the norm of the three 2-D prediction-error filters broadly delineates seismic discontinuities as fuzzy lines (Figure 42). In contrast, the combination of the forward and adjoint of the prediction-error filter pinpoints the discontinuities better. It even successfully highlights features within the central salt region (Figure 44).
In general, I found prediction error an unreliable and noisy discontinuity attribute. Prediction-error fails to detect smooth fault zones. However, prediction error delineates sharp faults. The discontinuity maps are intricate and reveal details in regions with little amplitude contrast. However, prediction-error maps are obscured by noise and are neither quickly and nor easily interpreted.