Following the study by Larner et al. 1989, we selected a dataset that includes steep dips in order to test the accuracy of our algorithms. The data is courtesy of Elf Aquitaine, was recorded in the North Sea, and shows a salt dome (Figure 3). Figure 1 shows (a) the data after NMO-stack and (b) after poststack Stolt migration, using a constant velocity of 2000 m/s. We notice that Stolt's method obviously yields undermigrated events on both sides of the salt body. Using a higher velocity to focus them better would have created overmigration artifacts at shallow reflectors. Stolt-stretch migrated section (c) using W=0.5 should be compared with figure 2a.
Using the Stolt-stretch method with the optimal choice for W derived from equation (9) yields a better focusing of events at all depths (Figure 2a), compared to other values of W (Figures 1b and 1c, respectively for W equals 1.0 and 0.5). The v(z) model used for migration is shown in Figure 4a and was obtained by averaging laterally the reference velocity model.
The reference method of migration for our study is the phase-shift approach proposed by Gazgag 1978. It is known to be perfectly accurate for all dips up to in a v(z) velocity field. A comparison between the phase-shift migration result (Figure 2b) and the section migrated with the Stolt-stretch approach shows almost no difference for flat events. However, a more detailed analysis reveals significant errors for steep events inside and around the salt body. The approximation made by stretching the time axis breaks for recovering steep events.
A way to overcome the difficulties encountered by Stolt's migration is to divide the whole process into a cascade, as suggested by Beasley et al. 1988. The theory of cascaded migration proves that f-k migration algorithms with a v(t) velocity model like Stolt-stretch can be performed sequentially as a cascade of n migrations with smaller interval velocities ,such as . At a given vertical traveltime t, all the successive velocity models have to be constant, except the last one Larner and Beasley (1987). Typically, the first stage is done with a constant velocity model and can be computed using Stolt's algorithm, which is then accurate for all dips. Figure 4 illustrates such a cascade of velocity models in our particular case, with 3 and 6 stages.
As a consequence of this decomposition, each intermediate velocity model shows not only a smaller velocity but also less vertical heterogeneity. In other words, the Stolt-stretch parameter W estimated for each stage tends to be closer to 1.0, thus reducing the migration errors due to the approximation. Figure 2c shows the migration result using a 6-stage cascaded scheme. All the successive values of W were greater than 0.8. There are almost no differences with the phase-shift result (Figure 2b).