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Figure 7 shows the migration result in the subsurface-offset domain. The front face corresponds to the image at zero subsurface offset, and the side face corresponds to the subsurface-offset gather at a position inside the shadow zone ( ft). Since the primaries are illuminated at few reflection angles and from a dominant slanted wave-propagation direction, their signature in the subsurface-offset domain is a slanted line. Because of the multiple bounces on dipping interfaces (15
watter-bottom and 30
salt flank), multiple energy is spread horizontally in the ODCIG, resembling a zero-reflection-angle event.
Figure 8 shows the migration result transformed to the reflection-angle domain. The front face corresponds to the image at the
reflection angle, and the side face corresponds to the reflection-angle gather at a position inside the shadow zone (
ft). Notice how multiples are concentrated around the
reflection angle, and primaries are mapped at higher reflection angles. This behavior of the multiples and primaries can also be found in field datasets (Valenciano, 2008).
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migang1
Figure 8. Sigsbee2b shot-profile migration (reflection angle) using the cross-correlation imaging condition. The front face corresponds to the image at the ![]() ![]() |
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We used a three-stage pre-processing strategy to attenuate the multiple energy before inversion. It includes discriminating the multiple energy in the plane (to generate a model of the multiples); performing amplitude and phase correction of the discriminated multiples in a least-square sense; and subtracting them from the original data.
The discrimination of the multiple energy relies on the differences in dip patterns as described previously: multiples show up in the plane at low
and high
, and primaries at high
and low
. Therefore, we build a model of the primaries by submitting the migrated data to a
filter at every depth step. Figure 9 shows a depth slice at
ft of the migration in the
plane, and Figure 10 shows the result of filtering. The model for the multiples (Figure 11) is obtained after subtraction of the filtered result (Figure 10) from the migration (Figure 9).
not-filtered
Figure 9. Depth slice at ![]() ![]() |
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filtered
Figure 10. Primaries after bandpassing the migration (Figure 9) in the ![]() |
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diff-filtered
Figure 11. The model for the multiples in the ![]() |
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Since the separation of primaries and multiples is not perfect, some of the multiples are present in the model of primaries, and some primary energy leaks to the model of multiples. Because of this cross-talk between the estimates of primaries and multiples, it is desirable to control the amount of attenuation. Therefore, the impact on the primaries of the multiple-attenuation process can be ameliorated if the amplitude and phase of the multiples model are adjusted, minimizing cross-talk and the differences from the multiples present in the original data in a least-squares sense. To perform this adjustment, we use simultaneous adaptive matching of primaries and multiples, formulated by Alvarez and Guitton (2007).
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migoff1-filt
Figure 12. Sigsbee2b shot-profile migration (subsurface-offset) using the cross-correlation imaging condition, after multiple attenuation. The front face corresponds to the image at zero subsurface-offset, and the side face corresponds to the subsurface-offset gather at ![]() |
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After pre-processing, the multiples have been largely attenuated, making the migration image more suitable for inversion. Figure 12 shows the filtered migration in the subsurface-offset domain. As in Figure 7, the front face corresponds to the image at zero subsurface offset, and the side face corresponds to the subsurface-offset gather at a position inside the shadow zone ( ft). The angle domain corroborates this hypothesis (Figure 13). The front face corresponds to the image at the
reflection angle, and the side face corresponds to the reflection-angle gather at a position inside the shadow zone (
ft).
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migang1-filt
Figure 13. Sigsbee2b shot-profile migration (reflection angle) using the cross-correlation imaging condition, after multiple attenuation. The front face corresponds to the image at the ![]() ![]() |
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Inversion was computed using equation 3 for two different right-hand-side vectors. One is the migrated image at zero subsurface offset without filtering (front panel, Figure 7), and the other is the migrated image at zero subsurface offset after filtering (front panel, Figure 12).
The results of the inversion are shown in Figures 14 and 15 (unfiltered and filtered input), and should be compared with the migrations shown in Figures 7 and 12, respectively. The inversion with the unfiltered migration is more unstable. This should require the use of a high value of the regularization parameter, which reduces the effectiveness of the inversion outside the shadow zones.
invz
Figure 14. Inversion in the poststack image domain with the unfiltered migration as the input (equation 3) after seven iterations of a conjugate-gradient iterative solver and no regularization. [ER] |
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invz-filt
Figure 15. Inversion in the poststack image domain with a filtered migration as the input to equation 3. [ER] |
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The inversion of the filtered migration shows a better convergence behavior. Again, no regularization was applied. Notice the well-collapsed diffractor at (35000,17000), in contrast to the unfiltered version (Figure 14). For the latter, conjugate-gradient iterations focused on reducing the multiple energy in the space of the residuals, decreasing the overall efficacy of inversion.
Figures 16 and 17 show the residuals of the inversion with the unfiltered migration and with the filtered migration, respectively, after seven iterations of the conjugate-gradient solver. Except for the residuals in the salt body, the highest residual amplitudes of the unfiltered inversion correspond to the multiples in the shadow zone. Notice that some amount of the diffractor energy (35000,17000) is still present in the residuals. On the other hand, the residuals of the inversion with the filtered data do not show the diffractor energy. Additionally, the residuals of the fault close to the diffractor are smaller than that of the unfiltered version. Unfortunately the multiples were not completely removed by filtering, so again the inversion procedure increases their amplitudes.
Overall, the inversion results after pre-processing have more balanced amplitudes, allowing the continuation of the reflector inside the shadow zones with much-improved kinematics. The reflectors also gain more vertical and horizontal resolution, particularly seen at the two faults present in the reflectivity model (Figure 2) and at the diffractor mentioned above.
resinvz
Figure 16. Residuals of the inversion in the poststack image domain with the unfiltered migration as the input after seven iterations. [ER] |
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resinvz-filt
Figure 17. Residuals of the inversion in the poststack image domain with a filtered migration as the input after seven iterations. [ER] |
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