Field data example
Now consider the field data example in Figure 8.0-3a. Layer replacement will be done on these data before and after stack to remove the effect of the irregular water bottom. First, we must define the geometry of the overburden, in this case, the water-bottom topography. Since the overburden velocity is constant (1475 m/s), we migrate the CMP-stacked section (Figure 8.0-3a) using the constant-velocity Stolt method, digitize the water bottom, and convert it to depth as shown in Figure 8.1-6a.
First, consider poststack layer replacement. We downward continue the CMP stack (Figure 8.0-3a) by using the water velocity from the surface to the water bottom defined in Figure 8.1-6a to get the horizon-flattened section in Figure 8.1-6b. As usual, we assume that the CMP stack is a zero-offset wavefield. The water-bottom reflection is approximately flat and is situated at t = 0, which indicates that the velocity-depth model for the water layer in Figure 8.1-6a is fairly accurate. Although incorrect in shape, substratum reflections are quite continuous on this section, indicating that we achieve a focusing effect by downward extrapolation. The bottom of the section reflects the mirror image of the water-bottom topography.
The next step is to take this wavefield back to the surface, this time using the velocity of the substratum (2500 m/s). Luckily, the velocity derived from the seismic data is fairly constant across the section for the substratum. The section after upward continuation is the result of layer replacement and is shown in Figure 8.1-6c. After eliminating the effect of the complex overburden, only time migration is needed to image this section (Figure 8.1-6d).
The results of prestack layer replacement now are examined. Starting with the common-shot gathers and using the velocity-depth model in Figure 8.1-6a, we perform downward and upward continuations of common-shot and common-receiver gathers. The intermediate steps for selected common-shot gathers are shown sequentially in Figure 8.1-7. Note the arrival time of the water-bottom reflection at t = 0 on the gathers in step 2 after downward continuing both the shots and receivers to the water bottom.
Figure 8.1-6 (a) Velocity-depth model derived from the constant-velocity (1475 m/s) Stolt migration of the CMP stack in Figure 8.0-3a. (b) The stacked section in Figure 8.0-3a after extrapolating from the surface to the water bottom [Horizon 1 in (a)] using the water velocity. The water-bottom reflection is at t = 0. This is the first step in poststack layer replacement. (c) Upward continuation of the wavefield in (b) from the water bottom back to the surface using the substratum velocity just below the water bottom. This is the second step in poststack layer replacement. (d) Time migration of the section in (c). The replaced water layer has a velocity of 2500 m/s.
Figure 8.1-7 Steps involved in prestack layer replacement. These shot gathers are associated with the stacked data in Figure 8.0-3a. The near-surface model is shown in Figure 8.1-6a. (0) Common-shot gathers at the surface, (1) downward continuation of receivers to the water bottom with v = 1475 m/s, (2) downward continuation of shots to the water bottom with v = 1475 m/s, (3) upward continuation of shots to the surface with v = 2500 m/s, (4) upward continuation of receivers to the surface with v = 2500 m/s.
Figure 8.1-9 Velocity analysis in the vicinity of midpoint C in Figure 8.0-3a (a) before and (b) after layer replacement.
Figure 8.1-10 (a) CMP stack derived from the gathers (Figure 8.1-8b) after layer replacement. Compare this with the conventional stack and poststack layer replacement results in Figures 8.0-3a and 8.1-6c, respectively. (b) Time migration of the stacked section in (a). Compare this with time migration of the poststack layer replacement result in Figure 8.1-6d.
Figure 8.0-3 (a) Conventional CMP stack from an area with irregular water-bottom topography, (b) time migration. (Data courtesy Hispanoil.)
A selected set of CMP gathers before and after layer replacement, sorted from the shot gathers in steps 0 and 4, respectively, is shown in Figure 8.1-8. Because the data contain strong diffracted multiples, it is difficult to evaluate the layer replacement results. As in any other one-way extrapolation technique, wave-equation datuming treats multiples as primaries. From the velocity analysis in Figure 8.1-9, we are encouraged by the picking we can do on the velocity spectrum after layer replacement. However, the stack has the final say and is shown in Figure 8.1-10a. Again, once time-migrated (Figure 8.1-10b), the result of layer replacement should provide an accurate image of the substratum, free from the effects of complex overburden (compare Figure 8.1-10b with Figures 8.0-3d and 8.1-6d).
The interpretation of the unconformity T from the results of prestack layer replacement (Figure 8.1-10b) closely agrees with the proposed velocity-depth model in Figure 8.0-3c. The unconformity is continuous beneath midpoint C, where there is a structural high. Below and to the left of midpoint A, the unconformity extends down to the right with some tensional faults into the ancient continental shelf below midpoint B, then finally extends to the continental slope of that age to the right of the structural high below midpoint C. The present-day situation is just the opposite, with the continental shelf on the right. Note that shallow reflector R now is conformable with the rest of the shallow section above the unconformity.
For the irregular water-bottom case, it was shown that prestack layer replacement followed by NMO correction, stack, and time migration after stack with depth conversion basically is equivalent to depth migration before stack. The situation may be more complicated when dealing with a complex overburden problem that involves more than one interface. In principle, datuming can be applied layer by layer and the effect of the overburden can be removed. Nevertheless, there is no practical alternative to depth migration under those circumstances.