Migration and the zero-distance phase front

Migration with an erroneous velocity gives a curve along which the energy is smeared. In a single-shot experiment, this curve, or the apparent “reflector”, can be constructed as the envelope of ellipsoidal curves of equal traveltime, with the source and receiver at the foci of the ellipses. Associated with this reflector is another curve, the caustic, enveloped by the reflected rays. When data from a single-point diffractor in a constant velocity medium are considered, construction of the zero-distance phase front of reflection reveals that the position of the caustic is independent of the locations of the source and receivers. The caustic is cusped, and the position of the cusp suggests how to modify the imaging principle, or the velocity, so as to improve the quality of migration output.