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![]() | Evaluating the Stolt-stretch parameter | ![]() |
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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 dataset is courtesy of Elf Aquitaine. It was recorded
in the North Sea over a salt-dome structure.
Figure 2 shows the data after NMO-stack and
after post-stack Stolt migration, using a constant velocity of 2000
m/s. The Stolt method creates visible 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 using
is shown in in
Figure 2c. It should be compared with an
improved result shown in Figure 3a.
Using the Stolt-stretch method with the optimal choice for
estimated from equation (17) yields a better focusing of
events at all depths (Figure 3a),
compared to other values of
(Figures 2b
and 2c, respectively for
equals 1.0 and
0.5). The
model used for migration is shown in
Figure 4a and was obtained by averaging laterally
the reference velocity model.
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data-stolt-ststr
Figure 2. (a) Section of the North Sea data, after NMO-stack. (b) Section migrated using Stolt's method with ![]() ![]() |
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data-ststr-pshift-casc
Figure 3. (a) Section migrated with the Stolt-stretch method using the optimal value ( ![]() ![]() |
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The reference method of migration for our study is the phase-shift
method (Gazdag, 1978). It is known to be perfectly accurate
for all dips up to
in a
velocity field. A comparison
between the phase-shift migration result
(Figure 3b) 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
velocity model like
Stolt-stretch can be performed sequentially as a cascade of
migrations with smaller interval velocities
, such then
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velocities
Figure 4. (a) Interval velocity model ![]() ![]() ![]() |
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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
estimated for each stage tends to be closer to 1.0, thus reducing the
migration errors due to the
approximation. Figure 3c shows the
migration result using a 5-stage cascaded scheme. All the successive
values of
were greater than 0.8. There are almost no
differences with the phase-shift result
(Figure 3b).
An accuracy of the cascaded stolt-stretch migration is additionally
verified by comparing its impulse response with that of the
phase-shift migration (Figure 5). The impulses are
generated using the same velocity model as shown in
Figure 4a. Figure 6 provides a
more detailed comparison. We can see a kinematic difference in the
impulse response of Stolt-stretch compared to phase-shift. While
Gazdag's phase-shift honor ray bending in any
model,
Stolt-stretch 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 3a, 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 7 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, 1994,1997).
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.
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imp-mig3
Figure 5. 3-D impulses responses of the cascaded Stolt-stretch (a) and phase-shift (b) operators. |
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imp-mig
Figure 6. Impulses responses of the different operators. (a) Stolt-stretch. (b) Phase-shift. (c) and (d) Cascaded Stolt-stretch, with 3 and 5 velocities, respectively. |
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dip-zoom
Figure 7. Zoom in the salt body area where steep dips are located. (a) Migration with the Stolt-stretch method using optimal ![]() ![]() |
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![]() | Evaluating the Stolt-stretch parameter | ![]() |
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