Evaluating the Stolt-stretch parameter

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 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 $W=0.5$ 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 $W$ estimated from equation (17) yields a better focusing of events at all depths (Figure 3a), compared to other values of $W$ (Figures 2b and 2c, 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.

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Figure 2. (a) Section of the North Sea data, after NMO-stack. (b) Section migrated using Stolt's method with $v_0$ =2000 m/s. (c) Section migrated using Stolt-stretch with an arbitrary value $W=0.5$ for the parameter of heterogeneity.

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Figure 3. (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 (5 velocities).

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 $90^\circ$ in a $v(z)$ 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 $v(t)$ 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 then

$$v^2(t) = \sum_{i=1}^{n} v_i^2(t)\;.$$ (23)

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 method, which is then accurate for all dips. Figure 4 illustrates such a cascade of velocity models in our particular case, with 3 and 5 stages.

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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 $v^2 = v_1^2 + v_2^2 + v_3^2$ . (c) Decomposition in a cascade of 5 models, such as $v^2 = v_1^2 + v_2^2 + v_3^2 + v_4^2 + v_5^2$

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 3c shows the migration result using a 5-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 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 $v(z)$ 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 $85^\circ$ , 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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Figure 5. 3-D impulses responses of the cascaded Stolt-stretch (a) and phase-shift (b) operators.

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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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Figure 7. Zoom in the salt body area where steep dips are located. (a) Migration with the Stolt-stretch method using optimal $W$ . (b) Migration with the phase-shift method. (c) Migration with the Stolt-stretch method using $W=0.5$ . (d) Migration with the cascaded Stolt-stretch approach, using 5 velocities.

Evaluating the Stolt-stretch parameter