f Accurate 3D time migration focusing in regions of lateral velocity gradients

Many 3D zero offset time migration algorithms allow lateral velocity variations. However, these methods are generally derived by employing simplifying assumptions about the medium velocity. It should not be expected that, if the derivation is based on a constant velocity or v(z) medium, the resulting algorithm will be accurate in regions with lateral gradients. It is well known that time migration mispositions events relative to their true subsurface position whenever lateral gradients are present. Here, I address only the issue of focusing. Insight into this aspect of the problem may be gained by analyzing the scalar wave equation in the natural coordinates of the time migration output space: image ray surface location and two-way traveltime.