In principle, downward continuation of 3-D prestack data should be carried out in the 5-D space of full 3-D prestack geometry (recording time, source surface location, and receiver surface location), even when the data sets to be migrated have fewer dimensions, as in the case of common-azimuth data sets that are only four dimensional. This increase in dimensionality of the computational space causes a severe increase in the amount of computations required for migrating the data. Unless this computational efficiency issue is solved, 3-D prestack migration methods based on downward continuation cannot compete with Kirchhoff methods. We address this problem by presenting a method for downward continuing common-azimuth data in the original 4-D space of the common-azimuth data geometry. The method is based on a new common-azimuth downward-continuation operator derived by a stationary-phase approximation of the full 3-D prestack downward-continuation operator expressed in the frequency-wavenumber domain. Although the new common-azimuth operator is exact only for constant velocity, a ray-theoretical interpretation of the stationary-phase approximation enables us to derive an accurate generalization of the method to media with both vertical and lateral velocity variations. The proposed migration method successfully imaged a synthetic data set that was generated assuming strong lateral and vertical velocity gradients. The common-azimuth downward-continuation theory also can be applied to the derivation of a computationally efficient constant-velocity Stolt migration of common-azimuth data. The Stolt migration formulation leads to the important theoretical result that constant-velocity common-azimuth migration can be split into two exact sequential migration processes: 2-D prestack migration along the inline direction, followed by 2-D zero-offset migration along the cross-line direction.

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