Introduction

In a previous report Biondi (1999a) I showed that common-azimuth prestack migration produces better sub-salt images than a single arrival Kirchhoff migration. In this report Vaillant and Sava 1999 show excellent preliminary results of common-azimuth migration of a salt body in the North Sea, and again favorable comparison with the results obtained using a most-energetic arrival Kirchhoff-like migration. The evidence is mounting that wave-equation migration, even when applied using approximate methods as common-azimuth, has the potential to produce better images than Kirchhoff methods when strong lateral velocity variations cause multi-pathing of the reflected energy. Full-volume imaging of deep targets when steep reflectors require huge migration aperture is another area of applications for efficient wave-equation migration methods that is gaining interest. In these cases the cost of Kirchhoff migration is large, and downward continuation methods become attractive from the computational point of view.

Offset plane wave migration Mosher et al. (1997); Ottolini and Claerbout (1984) is another approximate wave-equation method that has been recently applied to the migration of 3-D prestack marine data. Offset plane wave migration is related to common-azimuth migration, and has similar computational complexity. Therefore, a comparison between the two is of both practical and theoretical interest. Both methods have been applied to marine data transformed to common-azimuth data, and achieve computational efficiency by restricting the computational domain to a 4-D space from the 5-D space that is required by full downward continuation. Offset plane wave migration has the additional computational advantage that it can be performed as several independent migrations of 3-D cubes, while common-azimuth migration requires, at least in principle, to be performed on the whole 4-D data set simultaneously. This difference means that offset-plane wave migration requires less computations (about 10%) and has lower minimum-memory requirements to run efficiently. Though, for both methods the memory requirements are manageable on modern computers because the computational domain is further decomposed in temporal-frequency components. On the other hand, downward continuing the offset plane waves separately introduces errors when migration velocity has strong lateral variations, as in the case of sub-salt imaging. In this paper I show example of migration errors related to this approximation.

Another approximation introduced by offset plane wave migration is neglecting the cross-line component of the offset plane wave ray parameter vector and setting its value equal to zero. In this paper, I study the effects of this approximation with a theoretical analysis and with migration results. The approximation affects mostly the migration accuracy of reflections recorded at large offset from shallow reflectors with dips oriented at an angle with respect to the acquisition axes. The demonstration of this phenomenon is quite simple. It is based on the analytical proof that in constant velocity neglecting the cross-line component of the offset plane wave ray parameter vector is equivalent to reversing the correct order of two-pass migration. The correct order for two-pass prestack migration is: in-line prestack migration followed by cross-line zero-offset migration Biondi (1999b); Rosa et al. (1999). On the contrary, offset plane wave migration is equivalent to performing a cross-line zero offset migration followed by an in-line prestack migration.

The reversing of the correct order of two-pass migration produces the largest errors for shallow reflectors and large offsets. The errors become negligible for deep reflectors. Offset plane wave migration is thus a valuable tool for producing full-volume images of deep targets below relatively mild velocity functions, such as the imaging of salt flanks in deep waters. On the contrary, when shallow reflectors are important, or when strong lateral velocity variations are present, common-azimuth migration produces better images.