Analysis of common-azimuth migration for vertically layered media shows that downward continuing the data in a narrow strip around the zero crossline offset should yield a kinematically correct migration scheme. I introduce two migration methods that exploit the common-azimuth equations to define an optimal range of crossline-offset wavenumbers and thus to minimize the number of crossline offsets that are necessary to sample adequately the crossline-offset dips. Tests on synthetic data generated assuming a vertically layered medium confirm the theoretical analysis and suggest further testing on data sets with complex velocity functions.
Common-azimuth is an attractive alternative to shot-profile migration for wave-equation 3-D prestack migration. It is computationally more tractable Biondi and Palacharla (1996) and it can easily generate Angle-Domain Common Image Gathers (ADCIG) Prucha et al. (1999). In the past few years, we have shown that it produces good images both with challenging synthetic data (SEG/EAGE salt data set) Biondi (2000) and real data Vaillant et al. (2000). However, in variable velocity, common-azimuth migration is not exact. In this paper, I generalize common-azimuth migration to overcome this limitation, following some of the ideas that we previously explored Vaillant and Biondi (1999, 2000). My aim is to define a method that is accurate in presence of arbitrary velocity variations.
I attack the problem by studying the simple case of vertically layered media, because the general case is difficult to analyze and numerical tests are expensive. On the contrary, in layered media, I can easily analyze the accuracy limitations of common-azimuth migration with the help of raytracing modeling and a synthetic data set modeled over five dipping planes.
I compare ``raytracing migration'' of an event with both the correct dispersion relation and its common-azimuth approximation. This analysis of the kinematics is consistent with the migration results of the synthetic data. It confirms previous results that even in the worst-case scenario for common-azimuth migration, (reflector's dip oriented at 45 degrees with respect to the acquisition geometry, and one reflections path close to overturn) the kinematics of common-azimuth migration are a good approximation of the kinematics of the exact migration.
The raytracing modeling also confirms that the departure of the exact raypaths from the common-azimuth assumption is small. For the particular example analyzed, the maximum crossline offset at depth is less than 200 m, when the full inline offset is 2.9 km. These results suggest that a narrow-azimuth extension to common-azimuth should be capable of handling correctly all the events in the data.
I thus define and test two narrow-azimuth schemes. Both schemes downward continue the data along a narrow crossline-offset strip, and take advantage of the common-azimuth equations to define the proper range of crossline-offset dips. The adaptation of the first schemes to lateral velocity variations is straightforward, and thus it is a good candidate for future tests on data with complex velocity. The second scheme is less affected by artifacts caused by the boundaries along the crossline-offset axis. It may have potential for fast and accurate migration when the velocity is only slowly varying as a function of the horizontal coordinates.
Tests on the synthetic data show that a narrow crossline-offset range (e.g. four or even two crossline offsets) is sufficient to obtain accurate migration results of the most steeply dipping (60 degrees) reflector in the synthetic data. These encouraging results suggest further test on more challenging data.