We applied our preconditioning method to a 2-D line from the 3-D Elf North Sea dataset. The results are encouraging. Figure 11 is the migrated stack of the 2-D line. Figure 13 is the result of 1 iteration of the preconditioned inversion. Note the the inversion result is lower frequency than the migration. Also, by comparing the inversion result with the picked reflectors (Figure 12) we can see that the smoothing is too strong and may smooth poorly where there are no reflectors picked. Overall, the result looks too artificial.
Performing more iterations helps to increase the frequency content of the inversion result. Figure 14 shows the result of five iterations. Although it is higher frequency than Figure 13, it still looks artificial.
We can also consider the results of the smoothing along the angle axis. Figure 15 shows the angle gather that results from migration. Figure 16 is the angle gather after 3 iterations of preconditioned inversion. The preconditioned result is lower frequency and smoothes energy all across the gather, but it is possible to see where the true events end on the preconditioned results. This shows where the linear operator from equations (1) and (2) stops operating and where the preconditioner takes over. The events change in character as they are smoothed across angles where our survey geometry does not provide large angle information.
Figure 15 Migrated angle gather
Figure 16 Angle gather after 3 iterations of preconditioned inversion