Even if the parameter *W* estimated from equation (9) is
optimal in the sense that it minimizes migration errors for the
Stolt-stretch method, no single choice of *W* yields acceptable
results for all times and all dips Beasley et al. (1988): some
events are undermigrated, others overmigrated. Instead, the use of
cascaded Stolt-stretch migration allows a reduction of the apparent
dip perceived in each stage, since the migration velocity used is
reduced to a fraction of the original model. A 20-stage cascade of
migration with an algorithm accurate for dips up to can
yield accurate results for events dipping up to
Larner and Beasley (1987).

Figure 5 shows a close-up of the salt body region for all migration algorithms. The methods have a different accuracy with respect to steep dips. We notice a gradual improvement of the result from Stolt-stretch to phase-shift as we increase the number of velocities in the cascaded Stolt-stretch scheme. In theory, the migration errors in the cascaded approach can be made as small as desired by increasing the number of stages. At the limit, it corresponds to the velocity continuation concept Fomel (1996).

In our case, six stages were enough
to obtain a result comparable to phase-shift. In their comparative study
on time migration algorithms, Larner et
al. 1989 have shown that four-stage cascaded
*f-k* migration is accurate for dips up to , which is
almost comparable to phase-shift, accurate for all dips.
It is worth noting the
computational cost difference between the two: on our example,
phase-shift migration
is about 80 times more expensive than Stolt-stretch!

Another way to look at the problem is to compare the impulse responses
of the different algorithms (Figure 6), generated
using the same velocity model as before
(Figure 4a). There is a kinematic
difference in the impulse response of Stolt-stretch compared to
phase-shift. While Gazdag's phase-shift honor ray bending in any *v*(*z*)
model, Stolt-stretch is not that accurate. Both methods address
non-hyperbolic moveout, but Stolt's
stretching function is only designed to make the fitting curve look like an
hyperbola close to the apex Levin (1983), and therefore
induces residual migration errors.
As seen in Figure 2a, Stolt-stretch
result displays residual hyperbolic migration artifacts that are
due to this fundamental kinematic difference. Cascading Stolt-stretch
makes the impulse response of the migration converge towards the one
of phase-shift.

Figure 5

Now that we are familiar with the role of *W* in the algorithm, a word
should be said about *v _{0}*, which is the second arbitrary parameter of
the method. As introduced in equation (4),

Another technical aspect, the division by the Jacobian in equation (2), usually induces high amplitude artifacts for waves close to being evanescent, unless a threshold is introduced. Similarly, evanescent waves need to be scaled down to prevent migration artifacts. We used a simple linear weighting.

Figure 6

10/25/1999