Prestack layer replacement
As with depth migration, layer replacement after stack can remove the effect of a complex overburden, provided the input section accurately represents a zero-offset section. However, the complex overburden causes raypath distortions that generate anomalous, nonhy-perbolic moveout patterns in prestack data. Poststack depth migration does not produce a completely accurate image of the subsurface, even when the velocity-depth model is known accurately. Similarly, poststack layer replacement does not remove the effect of complex overburden entirely, even if its geometry is known accurately, because the input stacked section differs from the zero-offset section. Nevertheless, based on simple ray tracing we can determine whether poststack layer replacement can delineate the underlying structure. If the effort yields unsatisfactory results, layer replacement should be performed before stack.
Complex moveout is evident in the modeled common-shot and CMP gathers in Figures 8.1-4a and 8.1-5a, respectively. These gathers were modeled from the velocity-depth model in Figure 8.1-1a by using ray tracing that neither includes diffractions nor properly modeled amplitudes. The offset range is 50 to 2387.5 m with 12.5-m receiver spacing. A total of 437 shot gathers was generated, each with 192 traces. The coverage is uniform along the line and is 96-fold.
Starting with the common-shot gathers, prestack layer replacement involves the following steps:
- Downward continue all receivers to the output datum using the overburden velocity.
- Sort the data to common receiver gathers.
- Downward continue all shots to the same output datum using the overburden velocity.
- Upward continue all shots back to the surface using the velocity of the substratum.
- Sort the data back to common-shot gathers.
- Upward continue all receivers back to the surface using the substratum velocity.
This series of operations eliminates the traveltime distortions associated with the water bottom, as seen in the common-shot and CMP gathers (Figures 8.1-4b and 8.1-5c, respectively). Despite the undesirable effects caused by the limitations of modeling with ray tracing, the complexity of the reflections associated with the two events (horizons 3 and 4 in Figure 8.1-1a) was reduced by layer replacement. Once the complex overburden effect is removed, these events have the hyperbolic moveout as seen in Figure 8.1-4b. Events A, B, and C are associated with horizons 2, 3, and 4, respectively, as shown in Figure 8.1-1a. Note that horizon 3 is flat. Therefore, the apex of the hyberbola is at the near-offset trace (event B). On the other hand, horizon 4 dips down to the right (Figure 8.1-1a). Therefore, the apex of the hyberbola shifts up-dip (as indicated by the arrow in Figure 8.1-4b).
Compare the velocity spectra in Figures 8.1-5b and d, and note the improvement in the velocity estimates after layer replacement for reflections beneath the complex overburden. The velocity analysis before layer replacement yields a good pick for the water-bottom reflection A. However, picks B and C, which are associated with the deeper layers (horizons 3 and 4 in Figure 8.1-1a), are not distinct. Similarly, events B and C also are indistinct on the maximum correlation curve plotted on the right side of the velocity spectrum (Figure 8.1-5b). After layer replacement (Figure 8.1-5d), note that the picks (denoted by ×) associated with the three events are distinct in the velocity spectrum and the correlation curve. The dipping water-bottom reflection A now has considerably higher moveout velocity, 2300 m/s (Figure 8.1-5d), as compared to the original velocity 1600 m/s (Figure 8.1-5b). The velocity pick for the flat event B from Figure 8.1-5d is 2000 m/s, which is the velocity of the medium above this reflector after layer replacement.
Figure 8.1-4 (a) A modeled common-shot gather from the complicated part of the velocity-depth model in Figure 8.1-1a before (a) and after (b) layer replacement. Limitations in modeling with ray tracing cause abrupt terminations along the moveout curves and amplitude glitches in (a). In turn, these caused the spurious diffractions in (b) during the upward continuation steps.
The mathematical details of wavefield extrapolation based on the Kirchhoff integral are given in Section H.1. For poststack layer replacement, we assume that the stack is a zero-offset section. A zero-offset section is equivalent to the exploding reflectors model, provided the medium velocity is halved (introduction to migration). We assume that sources already are situated along the reflectors in the subsurface. Thus, we only need to move the receivers from one datum to another during downward and upward continuation steps. For prestack layer replacement, each common-shot and common-receiver gather is extrapolated, independently. In particular, the wavefield at a point on the output datum is computed using all the traces in the input gather. The output gather should be computed beyond the lateral extent of the input gather to prevent possible loss of steeply dipping events. For prestack layer replacement, the velocity used in the extrapolation is that of the medium between the input datum and the output datum.