3D angle-domain common-image gathers for migration velocity analysis

Angle-domain common-image gathers (ADCIGs) are an essential tool for migration velocity analysis (MVA). We present a method for computing ADCIGs in 3D from the results of wavefield-continuation migration. The proposed methodology can be applied before or after the imaging step in a migration procedure. When computed before imaging, 3D ADCIGs are functions of the offset ray parameters (p(xh), p(yh)); we derive the geometric relationship that links the offset ray parameters to the aperture angle and the reflection azimuth. When computed after imaging, 3D ADCIGs are directly produced as functions of gamma and phi. The mapping of the offset ray parameters (p(xh), p(yh)) into the angles (gamma,phi) depends on both the local dips and the local interval velocity; therefore, the transformation of ADCIGs computed before imaging into ADCIGs that are functions of the actual angles is difficult in complex structure. By contrast, the computation of ADCIGs after imaging is efficient and accurate even in the presence of complex structure and a heterogeneous velocity function. On the other hand, the estimation of the offset ray parameters (p(xh), p(yh)) is less sensitive to velocity errors than the estimation of the angles (gamma,phi). When ADCIGs that are functions of the offset ray parameters (p(xh), p(yh)) are adequate for the application of interest (e.g. ray-based tomography), the computation of ADCIGs before imaging might be preferable. Errors in the migration velocity cause the image point in the angle domain to shift along the normal to the apparent geological dip. By assuming stationary rays (i.e. small velocity errors), we derive a quantitative relationship between this normal shift and the traveltime perturbation caused by velocity errors. This relationship can be directly used in an MVA procedure to invert depth errors measured from ADCIGs into migration velocity updates. In this paper, we use it to derive an approximate 3D residual moveout (RMO) function for measuring inconsistencies between the migrated images at different gamma and phi. We tested the accuracy of our kinematic analysis on a 3D synthetic data set with steeply dipping reflectors and a vertically varying propagation velocity. The tests confirm the accuracy of our analysis and illustrate the limitations of the straight-ray approximation underlying our derivation of the 3D RMO function.