Time migration algorithms do not yield the correct image when velocity varies laterally. The correct image is best obtained by prestack depth migration. By studying the errors in time migration, it may be possible to link time migration with depth migration. This is desirable since time migration algorithms are generally robust, well understood and much less expensive than prestack depth migration.
Hubral 1977 argued that the image ray connects time migration with depth migration. He started with the incorrect assumption that time migration moves an event to the apex of the true-diffraction curve, the zero-offset response to a point diffractor. The apex of the true-diffraction curve is located at the end of the image ray, the ray that strikes the earth's surface at normal incidence. Since the image ray connects the top of the true-diffraction curve with the subsurface point, Hubral claimed that the image ray is the connection between time and depth migration. A correction method based on this view of the image ray can be applied to time migrated data Hatton et al. (1981); Larner et al. (1981).
This argument and correction is valid for events with small dips but fails for steeply dipping events. This failure is due to the fact that time migration moves an event to the apex of a time migration curve which is not the same as the true-diffraction curve for a medium with lateral velocity variation Black and Brzostowski (1993).
In this paper we study the effect of lateral velocity gradients on two dimensional zero-offset time migration. We use a simple dipping-layer earth model with an implicit velocity gradient. The main point of this paper is to derive the response of time migration to a point diffractor in a v(x) medium. We show that the response of time migration in a v(x) medium is a cusp-like shape that we call a ``plume.'' Although we devote a great deal of attention to the exact nature of the plume, its existence is not a new discovery since examples of plumes can be found in Yilmaz's chapter on depth migration Yilmaz (1987). The shape and location of the plume is not as simple as predicted by Black and Brzostowski's first-order theory, but it can be explained using ray-tracing and geometrical constructions.