Application

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.

 

data-stolt-ststr
Figure 1 (a) Section of the North Sea data, after NMO-stack. (b) Section migrated using Stolt's method with v₀=2000 m/s. (c) Section migrated using Stolt-stretch with an arbitrary value W=0.5 for the paramater of heterogeneity.

 

data-ststr-pshift-casc
Figure 2 (a) Section migrated with the Stolt-stretch method using the optimal value ($\approx 0.67$) for the parameter W. (b) Section migrated with the phase-shift method. (c) Section migrated using the cascaded Stolt-stretch approach (6 velocities).

 

vel-model
Figure 3 2-D smooth 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 $90^\circ$ 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 $v_i(t) \; , \; i=1,\ldots,n$,such as $v^2(t) = \sum_{i=1,n} v_i^2(t)$. 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.

 

velocities
Figure 4 (a) Interval velocity model v(t) estimated from the 2-D reference model. (b) Decomposition in a cascade of 3 models, such as v² = v₁² + v₂² + v₃². (c) Decomposition in a cascade of 6 models, such as v² = v₁² + v₂² + v₃² + v₄² + v₅² + v₆²

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).