Synthetic data examples

To test the analytical results presented in the previous section, I modeled and migrated several data sets that combined SODCIGs extracted from the prestack images of the synthetic data set described above. The SODCIGs were uniformly spaced with four choices for their distance $\Delta x_\xi^{i}$:640 meters, 320 meters, 160 meters, and 80 meters. The original midpoint spacing of the image, $\Delta x_\xi$, was 10 meters. Therefore, the first data set required 64 independent modeling experiments, the second 32, the third 16 and the fourth 8.

Figure  compares the SODCIG extracted from the starting prestack image (Figure a) with the corresponding SODCIGs extracted from the images obtained by migrating the four combined data sets with the correct velocity. All the SODCIGs have been extracted at the same horizontal location. As predicted by equation 23, the images obtained by the combined data sets are affected by cross talk along the offset domain. The images obtained from the smaller data set that had only 8 independent experiments (Figure e) is completely degraded by the cross-talk. Whereas the larger data sets ($\Delta x_\xi^{i}$ equal to 320 and 640 meters) preserve the velocity information present in the original SODCIG and allow the computation of ADCIGs uncontaminated by artifacts, after the cross-talks are removed by limiting the offset aperture.

Figure  shows the same SODCIGs shown in Figure  after the larger subsurface offsets are zeroed. Because the distance between cross-talks decreases with decreasing $\Delta x_\xi^{i}$, the windows around zero offset also decreases in width. For Figure b the window was 410 meters wide, for Figure c it was 170 meters wide, for Figure d it was 110 meters wide, for Figure e it was 70 meters wide.

 

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Figure 8 Panel a): SODCIG extracted from source-receiver migration of synthetic data set migrated with correct velocity. Panels b) to e): SODCIGs obtained from migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m.

 

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Figure 9 Same panels shown in Figure  after zeroing the larger subsurface offsets to maximally eliminate the cross-talk before transformation to angle domain (Figure ).

Figure  shows the ADCIGs obtained by transforming into the angle domain the SODCIGs shown Figure . The ADCIGs computed by imaging the larger data sets ($\Delta x_\xi^{i}$ equal 320 and 640 meters) preserve the velocity information contained in the original ADCIG (Figure a), whereas the ADCIG computed from the data set with only 8 independent experiments (Figure e), is completely overwhelmed by artifacts.

 

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Figure 10 ADCIGs obtained by transformation of the windowed SODCIGs shown in Figure . Panel a): ADCIG computed from source-receiver migration of synthetic data set migrated with correct velocity. Panels b) to e): ADCIGs obtained from the migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m.

The amount of interference caused by the cross-talk also depends on how well the SODCIGs are focused around zero subsurface offset, in addition to the spacing between SODCIGs. When the initial migration is not perfectly focused because of velocity inaccuracies, more experiments are needed to preserve the velocity information than when the starting image is well focused. Figure , illustrating this concept, shows the SODCIGs obtained starting from the prestack image computed by source-receiver migration using a migration velocity too low by 10%. Figure  shows the original SODCIG, whereas the other panels show the SODCIG obtained with increasingly smaller data sets, as in Figure . Because of the velocity error the SODCIGs are not well focused at zero offset. In this case, only the data set with 64 independent experiments produces a SODCIG with the cross-talk sufficiently separated from zero offset not to interfere with the desired image.

This result is confirmed by the transformation to angle domain. Figure  shows the ADCIGs obtained after windowing the SODCIGs shown in Figure . The ADCIG obtained by migrating all the 64 independent experiments (Figure b) contains the same velocity information as the original ADCIG (Figure a), whereas the others are affected by artifact caused by the cross talks, increasingly so going from left to right in the figure.

 

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Figure 11 Panel a): SODCIG extracted from source-receiver migration of synthetic data set migrated with velocity too slow by 10%. Panels b) to e): SODCIGs obtained from migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m. The modeling and the migration velocities were the same and both too slow by 10%.

 

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Figure 12 ADCIGs obtained by transformation of the SODCIGs shown in Figure  after windowing. Panel a): ADCIG computed from source-receiver migration of synthetic data set migrated with velocity too slow by 10%. Panels b) to e): ADCIGs obtained from migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m.

The two previous examples display the imaging results when the modeling and migration velocity were the same. However, because the proposed modeling method would be used for MVA, which requires iterative migrations with different velocities, it is useful to evaluate the results when the modeling and migration velocities differ. Therefore, I modeled four data sets, again with decreasing $\Delta x_\xi^{i}$;I started as before with $\Delta x_\xi^{i}$= 640 meters, and went down to 320 meters, 160 meters and 80 meters. The starting image was obtained by source-receiver migration with velocity too slow by 10%. The data were modeled assuming the same low velocity, but they were migrated using the correct velocity, and thus the SODCIGs after migration are now well focused.

Figure  shows the resulting SODCIGs and compares them with the well-focused SODCIGs obtained by source-receiver migration of the original data set with the correct velocity (Figure a). As in Figure , the cross-talk artifacts in the SODCIGs obtained by migrating the data sets formed by 32 and 64 independent experiments are sufficiently far from zero offset to be easily zeroed before transformation to angle domain. Figure  shows the corresponding ADCIGs, which show flat moveout for the deep flat reflector. A small residual moveout can be observed for the shallow dipping reflector that is probably related to staircase artifacts in the initial modeling. In other words, because of the coarseness of the modeling grid, the dipping reflector behaves as a sequence of short segments of flat reflectors, instead as a continuous planar reflector dipping at 10 degrees. All ADCIGs, except the ones shown in Figure d and e are free from artifacts and provide useful velocity information.

The last example illustrates the idea that the interference between SODCIGs depends on the amount of focusing of the SODCIG after migration, not in the starting image. In other words, the ``residual propagation'' operator $\Delta G$ present in equation 22 may decrease, or increase, the amount of cross-talk artifacts, depending whether it improves, or degrades, the focusing of the image.

 

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Figure 13 Panel a): SODCIG extracted from source-receiver migration of synthetic data set migrated with velocity too slow by 10%. Panels b) to e): SODCIGs obtained from migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m. The modeling velocity was too slow by 10%, but the migration velocity equaled the correct velocity.

 

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Figure 14 ADCIGs obtained by transformation of the SODCIGs shown in Figure  after windowing. Panel a): ADCIG computed from source-receiver migration of synthetic data set migrated with velocity too slow by 10%. Panels b) to e): ADCIGs obtained from migration of data sets modeled using the proposed method, with respectively $\Delta x_\xi^{i}=$ 640 m, 320 m, 160 m and 80 m. The modeling velocity was too slow by 10%, but the migration velocity equaled the correct velocity.