Downward continuation methods are usually more computationally efficient than Kirchhoff methods for full-volume imaging, and they are straightforward to implement. In 3-D, the cost of Kirchhoff migration grows with the cube of the image depth, because the migration aperture typically grows linearly with depth. On the contrary, because of the recursive nature of downward continuation methods, their migration aperture increases with depth automatically, resulting in a much more attractive linear increase of cost with migrated depth. Unfortunately, 3-D surveys are typically recorded with a sparse coverage of the surface in both the source and the receiver domain, while wave extrapolation must be performed on a densely sampled grid to avoid aliasing of the wavefield. Because the wavefield sampling criteria are independent from the actual surface sampling of the recorded traces, a straightforward application of 3-D downward continuation to 3-D prestack data would thus result in a method that wastes a large fraction of the computations propagating dead traces and wavefield components that never contribute to the final image. Due to these large computational inefficiencies, downward continuation methods, to my knowledge, have never been used to perform 3-D prestack migration of realistically-sized field data. One notable exception being the migration of data sets recorded with unusual geometries, such as vertical marine cables Roberts et al. (1996).
The migration results that I present in this paper have been obtained by prestack migrating a 3-D marine data set using a new downward continuation method (); Biondi and Palacharla (1995) that overcomes the computational problems of 3-D prestack downward continuation and actually produces full-volume images more efficiently than Kirchhoff migration. This method takes advantage of the narrow range of the source-receiver azimuth typical of marine data by selectively propagating only a portion of the whole prestack wavefield. It can be shown that from an imaging perspective the propagated portion of the wavefield is the main component of the wavefield. Although modern marine data sets are not exactly common-azimuth due to the use of multiple streamers and the occurrences of cable feathering, they can be transformed to common-azimuth by application of partial-prestack migration operators, such as AMO Biondi et al. (1996a) or DMO followed by inverse DMO Canning and Gardner (1996).
Common-azimuth downward continuation is not theoretically exact because it selectively propagates part of the wavefield; the error that it introduces depends on the amount of ray-bending caused by velocity variations. However, previous tests on synthetic data, which were generated assuming both strong vertical and lateral velocity gradients, showed that the method is robust with respect to velocity variations (); Biondi and Palacharla (1995). These encouraging results motivated the study presented in this paper. The data set used for the tests was collected in the southern sector of the North Sea and presents challenging depth migration problems caused by shallow velocity variations created by uneven thinning of a high-velocity chalk layer, and deeper velocity variations created by a high-velocity salt layer Hanson and Witney (1995). A typical in-line geological section of the area is shown in Figure .
I compare the results of common-azimuth 3-D prestack migration with the results of multi-line 2-D depth prestack migration, where, for the purpose of this paper, I define multi-line 2-D migration as the combination of independent 2-D migrations of common-azimuth data as a set of parallel 2-D lines. The comparison with multi-line 2-D migration show that common-azimuth migration correctly focuses the energy in the cross-line direction, as well in the in-line direction. Reflectors in the target zone below the salt layer are focused and they seem to be correctly positioned. However, I think that the migration results of deeper reflectors could be improved by further refinements in the velocity model. I am planning to generalize the 2-D methodology of depth-focusing analysis MacKay and Abma (1992) to 3-D common-azimuth migration to estimate the necessary refinements in the velocity function. Further analysis on the data should also indicate whether multi-pathing affected the propagation of the reflected energy, and in the affirmative case, whether common-azimuth migration has the capability of improving the imaging of these reflections over Kirchhoff migration.