(f, x) Migration — Some Side Issues

Conventional methods of treating boundaries in finite difference (f, x) migration, namely padding in both space and time, have the disadvantage of increasing the data volume. This increase, whilst acceptable for 2-D processing, becomes uneconomical in the 3-D arena. Dipping events move up-dip during the migration process. Dips close to the sides of the data that move outside the data range can be reflected back if the boundary conditions are poorly set. This results in misleading events appearing in the migrated data. The addition of extra zero trace padding allows these events to move out of the data and so avoids these artefacts.

The alternative to padding with zero traces is to use absorbing side boundary conditions. Absorbing sides have been proposed by several authors. A method developed by Clayton and Engquist is reviewed in detail, and implementation issues are discussed. Although this method is exact for only one propagation angle, it is found to attenuate significantly the reflections of other dipping events.

Time padding is used to avoid the wraparound that causes shallow data to reappear as noise at later times in the migrated result. Wraparound is inherent in (f, x) finite difference migration, because of the cyclic nature of the fast Fourier transform. The time padding increases the run time of both the fast Fourier transform and the migration.

Clearly, it is desirable to find a technique to remove the wraparound and so avoid the need for costly time padding. Kjartansson showed a method for wraparound removal for the case when the sampling interval is equal to the migration time step or depth interval. In practice, however, a coarser time step is usually employed. A new generalised method has been developed for wraparound removal under these conditions of coarse time step. The technique is applied at each time step extrapolation. Although this obviously involves more computation, the method is more efficient than using extra time padding.

Both these techniques, the absorbing side boundaries and the wraparound removal, can be used in 2-D and one-pass 3-D implementations. Without the need for extra padding, migration run times and hence costs are reduced. The methods are demonstrated on synthetic data. The results compare well with those obtained using conventional padding, with a considerable reduction in run time.