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.

Figure 1

Figure 2

Figure 3

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.

Figure 4

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).

10/25/1999