It is well known that in an anisotropic medium, an isotropic migration
with the right velocity locates dipping reflections in an erroneous
vertical and lateral position Alkhalifah (1997b); Uzcategui (1995). Moreover, the
final migrated image looks undermigrated. Alkhalifah (1997b)
presented a 2-D prestack time migration based on the stationary phase
approximation of the in-line offset wavenumber. In addition, he
presented a 3-D poststack time DSR operator for P-waves as a function
of and the vertical NMO velocity *V*_{NMO}. This last
result is very important because a vertical transversely depends on
only two parameters.

Alkhalifah et al. (1997) introduce the basis of time processing for heterogeneous anisotropic media based on the NMO-velocity and the parameter. They show that the vertical velocity is necessary for the time-to-depth conversion but it is not necessary for imaging (or mapping). In order to avoid working in depth, Alkhalifah et al. (1997) redefine the P-wave equation in vertical time or pseudo-depth.

This paper extends the DSR operator to handle lateral velocity variations for anisotropic media for P-waves presented by Alkhalifah (1997). Using the extended split-step method, the DSR operator is approximated to include lateral velocity variations, through the definition of reference velocities for every downward continuation step Malcotti and Biondi (1998). The results in this paper are obtained by keeping constant the parameter during the downward continuation and by defining a number of reference in the same fashion as the velocity. A more general approach will be addressed in future papers.

The anisotropic migration algorithm presented in this paper is a modified version of the extended isotropic split-step depth migration that I also present in this report Malcotti and Biondi (1998). The main modifications are the inclusion of the parameter, and the downward continuation is in pseudo-depth, in the same fashion as isotropic migrations work Claerbout (1985).

7/5/1998