The 3-D prestack DSR operator in frequency and wave number domain is Biondi and Palacharla (1996); Claerbout (1985)
Alkhalifah (1997) derives the dispersion equation for transversely anisotropic media. He assumes that the vertical S-wave velocity is equal to zero (VS0=0) and that the dispersion relation for the 3-D prestack DSR can be rewritten in a more general equation as follows:
Equation (2) is expressed in pseudo-depth by the following equation:
Rewriting the DSR equation equation (2) in pseudo-depth () for the 2-D case, we have :
Applying the split-step approximation to equation (2), gives the DSR used in a 3-D prestack depth migration Malcotti and Biondi (1998) as follows:
where vSref(z) and vGref(z) are the reference velocities defined by the geometry of the survey and depth. In the case of 3-D prestack migration algorithm, this DSR is rewritten using the common-azimuth approximations Biondi and Palacharla (1996).
Equation (5) shows that in anisotropic media, the split-step approximation is obtained by assuming that is constant for every depth step. In the seismic synthetic examples that I show in the results section, I use two different approaches to migrate the anisotropic Marmousi data set. In the first approach, I use a constant . In the second approach I define a number of reference 's, in order to obtain a better migrated image of the dipping reflectors. This is an approach that gives impressive results but it should be generalized to handle lateral variations, where variations are the velocity variation.
Other possible solution to include lateral variation is using different 's defined in the same fashion that reference velocities are defined but migrate with just one reference velocity (Alkhalifah 1998, personal communication).
Alkhalifah and Tsvankin (1994), assuming that the vertical S-wave velocity is equal to zero vS0=0, introduce an anisotropic parameter called . This anisotropic parameter can be written as a function of Thomsen's parameters ( and ) as follows:
In this paper I set to avoid working with the ratio of the vertical velocity and migration velocity in equation (5).
The variable is expressed as a function of the NMO velocity (VNMO) as follows: