The use of prediction-error filters in the problem of detecting local discontinuities was suggested by Claerbout (1992b, 1993, 1999) and further refined by Schwab et al. (1996a,b) and Schwab (1998). Bednar (1997) used simple plane-destructor filters in a similar setting to compute coherency attributes.
To test the performance of the improved plane-wave destructors, I chose several examples from Claerbout (1992b). Figure 2 introduces the first example. The left plot of the figure shows a synthetic model, which resembles sedimentary layers with a plane unconformity and a curvilinear fault. The right plot shows the corresponding ``texture'' Brown (1999); Claerbout and Brown (1999), obtained by convolving a field of random numbers with the inverse plane-wave destructor filters. The inverse filters were constructed with the B-spline regularization technique Fomel (2000b), while the dip field was estimated by the linearization method of the previous section. The dip field itself and the prediction residual [the left-hand side of equation (13)] are shown in the left and right plots of Figure 3 respectively. We observe that the texture plot does reflect the dip structure of the input data, which indicates that the dip field was estimated correctly. The fault and unconformity are clearly visible both in the dip estimate and in the residual plots. Anywhere outside the slope discontinuities and the boundaries, the residual is close to zero. Therefore, it can be used directly as a fault detection measure. Comparing the residual plot in Figure 3 with the analogous plot of Claerbout (1992b) establishes a superior performance of the improved finite-difference destructors in comparison with that of the local T-X prediction-error filters.
Figure 4 shows a simpler synthetic test. The model is composed of linear events with two conflicting slopes. A regularized dip field estimation attempts to smooth the estimated dip in the places where it is not constrained by the data (the left plot of Figure 5.) The corresponding residual (the right plot of Figure 5) shows suppressed linear events and highlights the places of their intersection.
The left plot in Figure 6 shows a real shot gather (a portion of Yilmaz and Cumro data set 27). The initial dip in the dip estimation program was set to zero. Therefore, the texture image (the right plot in Figure 6) contains zero-dipping plane waves in the places of no data. Everywhere else the dip is accurately estimated from the data. The data contain a missing trace at about 0.7 km offset and a slightly shifted (possibly mispositioned) trace at about 1.1 km offset. The mispositioned trace is clearly visible in the dip estimate (the left plot in Figure 7), and the missing trace is emphasized in the residual image (the right plot in Figure 7). Additionally, the residual image reveals the forward and back-scattered surface waves, hidden under more energetic reflections in the input data.
Figure 8 shows a stacked time section from the Gulf of Mexico and its corresponding texture. The texture plot demonstrates that the estimated dip (the left plot of Figure 9) reflects the dominant local dip in the data. After the plane waves with that dip are removed, many hidden diffractions appear in the residual image (the right plot in Figure 9.) The enhanced diffraction events can be used, for example, for estimating the medium velocity Harlan et al. (1984).
Overall, the examples of this subsection show that the finite-difference plane-wave destructors are a reliable tool for enhancement of discontinuities and conflicting slopes in seismic images. The estimation step of the fault detection procedure produces an image of the local dip field, which may have its own interpretational value. An extension to 3-D is possible, as outlined by Claerbout (1993), Schwab (1998), Fomel (1999), and Clapp (2000a).