Tests on synthetic data set

To test the theoretical result reached in the previous section I migrated one shot record from a synthetic data modeled over a medium with strong lateral velocity variations that cause the image produced by migrating a single shot to have coherent artifacts. The test is useful both to demonstrate that in the source-receiver migration the downward continuation along the shot axis and along the receiver axis do indeed commute in presence of strong lateral variations. It is also interesting to confirm that the migration artifacts produced by source-receiver migration and shot profile migration are the same. The data set was kindly provided by Bill Symes of Rice University, and it has been used to study the artifacts produced by different kind of migrations Stolk and Symes (2002). The reflector geometry is a simple flat reflector, but a strong velocity anomaly above it creates severe multipathing that challenges different migration schemes.

The theoretical result is based on the assumption that the numerical algorithm used to propagate the wavefield is exactly the same. Unfortunately, SEP still lacks a downward-continuation shot-profile migration capable of handling severe velocity variations, though Brad Artman is close to succeeding in getting one up and running 2002. Therefore, for the moment I had to run an ``imperfect'' test and I compare the results of migrating synthetic data by source-receiver downward continuation and reverse-time shot-profile migration Biondi and Shan (2002).

Figure 1 shows the shot with source location at .5 kilometers used for the test. Figure 2 shows the zero offset (stack) image produced by both migrations methods. The panel on the left shows the image produced by shot-profile migration, and the panel on the right shows the image produced by source-receiver migration. The two images are similar, except for a small difference in frequency content caused by the fact that I did not enter a perfect impulsive source in the shot-profile migration to avoid dispersion. Not only the flat reflector is imaged similarly in the two images, but also the strong ``ghost'' reflectors caused by the triplication of the wavepath Valenciano and Biondi (2002), visible between the surface locations of 0 and 1 km, are almost identical.

Figure 3 shows the subsurface offset-domain common image gathers at the surface location of 300 meters: panel a) shot-profile migration, panel b) source-receiver migration). Again the images are similar for both the ``true'' reflector and the ``ghost'' reflectors. Figure 4 shows the angle-domain common image gathers obtained from the offset-domain gathers shown in Figure 3 after a slant stack transformation Sava et al. (2001). Notice that the ``true'' reflector gets imaged at both positive and negative aperture angle because of the wavepath triplications. The ``ghost'' reflectors get imaged in the aperture-angle gap between the two branches of the true reflector.

Figure 5 demonstrates that the artifacts disappear if the whole data set (400 shots) is imaged. It shows the zero offset image [panel a)] and the angle-domain common image gather [panel b)] obtained by source-receiver migration when all the shot records are included in the data. I did not migrate all the shots by reverse time migration, because it would have taken considerable computer resources.

 

Shot-trip Figure 1 Shot profile used for the tests.

 

Mig-trip-both
Figure 2 Zero-offset sections of the migrated cubes obtained using: a) shot-profile reverse-time migration, b) source-receiver downward-continuation migration.

 

Cig-trip-100-both Figure 3 Offset-domain common image gathers obtained by slicing the migrated cubes at the surface location of 100 meters: a) shot-profile reverse-time image, b) source-receiver downward-continuation image.

 

Ang-Cig-trip-100-both Figure 4 Angle-domain common image gathers obtained by slicing the migrated cubes at the surface location of 100 meters: a) shot-profile reverse-time image, b) source-receiver downward-continuation image.

 

Mig-Ang-Cig-trip-300-sr-full Figure 5 Stacked image (panel a)) and angle-domain common image gather at surface location of 100 meters (panel b)) obtained by source-receiver downward-continuation migration of the whole data set (400 shots).

I have proven theoretically that source-receiver migration is exactly equivalent to downward-continuation shot-profile migration.

The results of the migration tests that I show strongly support this theoretical result, though they are not the ultimate proof, since I was limited to run a reverse-time shot-profile migration instead of a downward-continuation shot-profile migration.

I would like to thank Bill Symes of Rice University for kindly providing the synthetic data set that was used for the tests shown in the paper.