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Kinematic analysis of ADCIGs by integral migration

The analysis shown in the previous section provides the fundamental equations to relate the offset and midpoint dips measured from prestack images to the phase angles at the reflection point. However, the previous analysis is not directly applicable to the analysis of the kinematic properties of events in the ADCIGs because it is based on plane waves. An important application of ADCIGs is the measurements of residual moveout along the aperture angle (i.e. departure from flatness) caused by velocity errors. To achieve this goal we must relate traveltime errors accumulated during the propagation in the overburden to movements of the migrated events in the ADCIG. This task is easier in the ray domain than in the plane-wave domain because traveltime errors are naturally evaluated along rays, which are related to group velocity and angles. To overcome this obstacle toward the use of ADCIGs for velocity estimation, in this section I introduce an integral formulation of the methodology to compute angle gathers that enables a simple link between ADCIGs and kinematics. The following analysis has also the theoretical value of being independent from the migration method applied to compute the prestack images (integral method or wavefield-continuation method) and thus of providing a conceptual link between the angle gathers obtained using different migration methods.

My analysis is based on the conceptual generalization of integral (Kirchhoff) migration to the computation of sub-surface offset gathers. Integral migration is defined by the summation surfaces over which the data are integrated to compute the image at every point in the image space. The shapes of these summation surfaces are usually computed as the sum of the time delays from the image point $(z_\xi,m_\xi)$in the subsurface to the source and receiver locations at the surface. The basic idea underlying the generalization I introduce in this paper, is that we can compute the summation surfaces by evaluating the time delays starting not from the same point in the subsurface for both the source and receiver rays, but starting from two points horizontally shifted by $\pm h_\xi$ with respect to the image point. The summation of data along these surfaces produces a prestack image as a function of the subsurface offset that is kinematically equivalent to the image created by wavefield-continuation migrations such as source-receiver downward continuation, or shot-profile migration in conjunction to the generalized imaging condition discussed by Rickett and Sava (2002). Therefore, the kinematic analysis that follows, and its conclusions, are independent from the migration method applied to compute the prestack images. An interesting observation is that the ADCIGs computed using this generalization of integral migration should be immune from the artifacts that affect angle gathers computed by conventional integral migration and discussed by Stolk and Symes (2003).