Though the vertical axis of the earth subsurface is measured in units of distance, seismic images of the earth subsurface are usually presented in units of time. The process, referred to as time imaging, is practiced far more often than depth imaging. The reason for the preference of time imaging over the depth one is that we are simply unable to accuratly position reflecors in depth. Some researchers attribute this shortcoming to the presence of strong lateral inhomogeneity, but in areas of smooth velocity variations, the only proper explanation for the depth inaccuracies is the presence of anisotropy Banik (1984).
Time imaging algorithms, however, are based on the lateral homogeneity assumption, a condition that will restrict their use in areas of complex velocity structures. As a result, depth migration is used in these regions despite the inherent ambiguity of estimating depth from surface seismic data. In transversely isotropic media with vertical symmetry axis (VTI media), this implies that we need to estimate the vertical velocity in order to do depth migration. However, the vertical velocity cannot be estimated from surface seismic data. More information is needed to estimate the vertical velocity (i.e., well log or check shot data).
Alkhalifah et al. (1997) derived an eikonal and ray tracing equations for VTI media in the (x-)-domain. These equations are slightly more complicated than their depth counterparts, yet they can be solved as efficiently using standard numerical methods. Alkhalifah et al. (1997) also show that these traveltime equations are some-what independent of the vertical velocity in VTI media. Instead, these equations depend on two parameters: the NMO velocity for a horizontal reflector and the anisotropy parameter, referred to as . Conveniently, these two parameters can be estimated from surface seismic P-wave data Alkhalifah and Tsvankin (1995).
In this paper, I use Alkhalifah et al. (1997) traveltime equations, specifically the ray tracing ones, to build traveltime tables in the (x-)-domain. These traveltime tables are subsequently used to implement prestack Kirchhoff time migration. The accuracy of the migration is demonstrated on the isotropic, as well as the anisotropic, Marmousi dataset.