The depth axis has always been a source of uncertainty
in seismic processing. Geophysicist have shied away
from predicting depths from surface seismic *P*-wave data.
Typically, well-log data are used for such a task.
However, since well-log
data are rare and sparse, seismically based
interpolation of well-log information is commonly used.
Although the conventional isotropic theory suggests that
depth can be resolved using the velocity field that focuses
the seismic image, field data have rarely agreed with this isotropic principal.
Anisotropy, on the other hand, suggests that depth cannot
be resolved using surface seismic data. The velocity
needed to resolve depth is the vertical velocity, which is
different from the imaging velocity (the velocity
that yields the best image). This difference accords with the typical field data experience.
In fact, in VTI media, processing is controlled by three velocities: one
responsible for depth mapping, another for stacking, and the third for migration.
Although this is a simplistic representation and theory suggests that there is
more interaction between these velocities
and their influences, such a representation is close to what actually happens in practice.
Two of these velocities are resolvable
from surface seismic data, or, in a general inhomogeneous case, two combinations of these velocities are
resolvable, which implies the existence of a null space in the three-parameter representation of VTI media.

Considering that depth in VTI media is determined by multiplying half of the vertical traveltime with the vertical velocity, it seems that representing data with the vertical time, instead of depth, could absorb the vertical velocity influence. This has been shown to be the case for vertically inhomogeneous media Alkhalifah and Tsvankin (1995) but has yet to be shown for more general inhomogeneity. In the next section, we replace the depth axis with vertical time to represent more general, arbitrarily inhomogeneous media.

10/9/1997