I run both narrow-azimuth migration methods on the synthetic data set described above, varying the number of crossline-offset samples N_{yh} from 2 to 16. To create data with more than one crossline offset I padded the common-azimuth data with zeros. I also run a full phase-shift 3-D prestack migration on the same data. Full phase-shift 3-D prestack migration is equivalent to the first method of narrow-azimuth migration, but with the k_{yh} range centered at k_{yh}=0, instead of at the given by equation (4).
For both the full phase-shift migration and the first narrow-azimuth methods, I kept the maximum value of the crossline offset constant at 800 m for all values of N_{yh}. Therefore, according to equations (6) and (5) the sampling rate of k_{yh} was independent of N_{yh}, but the range of k_{yh} decreased as N_{yh} decreased. Both methods lose accuracy as the range of k_{yh} decreases, but the accuracy of full phase-shift migration degrades quicker than the accuracy of narrow-azimuth migration.
For all of these tests I used all the appropriate weighting factors, as discussed by Sava and Biondi 2001, except the phase shift and weights related to the stationary phase approximation. I excluded the stationary phase correction because as the range of k_{yh} increases it is not anymore necessary. Indeed, the phase of the migrated images changes as the N_{yh} increases. However, the amplitude of the deeper reflectors decreases as N_{yh} increases because of the increasing amount of zero padding.
Figure 10 shows the the results of full phase-shift migration of the synthetic data set with N_{yh}=16. It shows the same subset of the migrated cube as in Figure 4. The front face of the cube is an inline section through the stack. The other two faces are sections through the prestack image. The kinematics of the migration are correct. The events are flat in the ADCIG shown in Figure 11. I use this results as benchmark for the narrow-azimuth and the full phase-shift migration as N_{yh} decreases.
Figures 12 shows two ADCIGs, taken at the same location as the ADCIG in Figure 11, but obtained with N_{yh}=8. The ADCIG on the left (a) was obtained by full phase-shift migration, and the ADCIG on the left (a) was obtained by the first method for narrow-azimuth migration. For narrow azimuth-migration I used km/s and km/s. The kinematics of the narrow-azimuth migration are correct. On the contrary, the results of full phase shift migration begin to degrade at larger p_{xh}.
Figures 13 and 14 shows the same ADCIGs as in Figure 12, but with respectively N_{yh}=4 and N_{yh}=2. The kinematics of the narrow-azimuth migration are correct for N_{yh}=4 and show only a slight degradation at large p_{xh} for N_{yh}=2. On the contrary, the results of full phase shift migration are poor even at small p_{xh}.
Finally, I compare the results of using the two methods for narrow-azimuth migrations that I presented in the previous section. Figure 15 shows the ADCIGs taken at the same location as in Figure 12, also obtained with N_{yh}=8. The ADCIG on the left is obtained using the second narrow-azimuth method; that is, when using equation (7) to determine the sampling rate for k_{yh}. The ADCIG on the right is obtained using the first narrow-azimuth method; that is, when using equation (6) to determine the sampling rate for k_{yh}. The kinematics of the two images are equivalent. However, the image obtained using the second method has less artifacts caused by the boundaries along the crossline-offset axis.
PS-16-nhy-WKBJ-vp
Figure 10 Subset of the results of full phase-shift migration of the synthetic data set with N_{yh}=16. The front face of the cube is an inline section through the stack. The other two faces are sections through the prestack image. |
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Figure 11 An ADCIG extracted from the same migrated image shown in Figure 10. The three events in the figure correspond to the planes dipping at 30, 45 and 60 degrees. Notice that the events are perfectly flat. |