Steep-dip explicit methods
Although implicit schemes are always stable and naturally suitable to accommodate lateral velocity variations, they have been known to have some unfavorable aspects. Specifically, dispersive noise caused by the differencing approximations to differential operators does indeed deteriorate the quality of migration of real data (Figures 4.3-8, 4.3-9, 4.4-10, and 4.4-11). An equally important concern about implicit schemes is the fact that they never meet the expected theoretical dip accuracy. A 45-degree algorithm can actually provide only a 5-degree acccuracy for certain frequencies and velocities. Because of a complex interaction of temporal and spatial sampling rates, dip, velocity, and frequency, an optimum depth step size that yields minimum phase and amplitude errors for an entire data set is never easy to specify (Section D.6).
While implicit schemes are guaranteed to be stable whatever the depth step size, stability of explicit schemes requires a depth step that is sufficiently small (Section D.6). Putting aside the stability issue, explicit schemes implemented in the frequency-space domain (Section D.5) can alleviate some of these deficiencies of implicit schemes.
Refer to the impulse responses shown in Figure 4.4-18 and note that explicit schemes do not produce any visible dispersion along steep dips. Explicit schemes involve convolution of a symmetric complex filter in the frequency-space domain with the wavefield to perform extrapolation in depth. In contrast, implicit schemes can be computationally intensive for large volumes of data since they require solving complex tridiagonal equations associated with the differencing in the x direction.
Steep-dip accuracy in explicit schemes is attained by increasing the extrapolation filter length. The impulse responses shown in Figure 4.4-18 are associated with three different extrapolators — a 7-point filter with a 30-degree accuracy, an 11-point filter with a 50-degree accuracy, and a 25-point filter with a 70-degree accuracy. Note that the explicit extrapolation filter cuts off beyond the specified dip limit. In contrast, the implicit extrapolation treats the evanescent energy as though it is part of the propagating energy (Figure 4.4-1).