The main feature of the anisotropic parameter representation suggested by
Alkhalifah and Tsvankin (1995) is that time processing--normal moveout
correction (NMO), dip moveout (DMO), and time migration-become
independent of vertical *P*--wave velocity, a parameter
necessary to resolve reflector depth. As a result, estimating the vertical velocity
is unnecessary for time processing, which depends on only two parameters:
the normal-moveout (NMO) velocity for a horizontal reflector and an anisotropy
parameter denoted by .
However, this rather
fortunate behavior of seismic waves in transversely isotropic media with a
vertical symmetry axis (VTI media) seems to hold only for vertically inhomogeneous
media. When lateral inhomogeneity exists, three parameters are needed
to characterize the medium and implement processing.

Our goal is to implement time processing that truly
honors the lateral inhomogeneity of the medium, and yet is independent of the vertical *P*-wave
velocity.
Separating the *P*-wave vertical velocity, *v*_{v},
from the image processing stage helps in avoiding the intrinsic ambiguity that this velocity
introduces into the problem of estimating parameters in VTI models. This separation allows us to
correct for the depth whenever such information becomes available, for example, well-log data.

This report shows that certain lateral inhomogeneities fall
into this fortunate category of independence from vertical *P*-wave
velocity when we replace the depth axis
with the vertical time.
We refer to such an inhomogeneity as being *factorized laterally*.
The term *factorized* was introduced by Shearer and Chapman (1988) to
describe a medium in which the ratio between the different elastic
coefficients remains constant throughout the medium. In the case of
our new coordinate system, this
constraint is needed only between the NMO velocity and vertical velocity
and it is needed only laterally. In other words, , defined as
the ratio between the vertical and NMO *P*-wave velocity, can
change only vertically. This condition still allows for data processing in media of any
lateral inhomogeneity, but does not allow for applying any depth conversion.
In fact, this condition is extremely convenient considering that reflector depth is typically resolved
at only one location along a given seismic line (at the well), and that we can therefore
use this , extracted from the well, to estimate depths. When varies
laterally, the accuracy of the processing depends on the size of the variation. Our analysis shows
that such dependency is small for typical variations and, as a result, can be ignored.

The term *time processing* implies that an image of the subsurface is obtained with
its vertical axis given in time rather than in depth. Traditionally,
only vertical inhomogeneity was treated in the processing of this image. Such processing might include
approximations to treat mild inhomogeneities, but nothing that could come close to properly
imaging complex data such as the Marmousi model.
*Time processing* takes on a quite different meaning in this paper. It
includes exact treatment for media with any lateral
inhomogeneity. Specifically, we develop ray-theoretical solutions of wave propagation in the time
domain, including the eikonal and raytracing equations that can handle any lateral inhomogeneity.
An acoustic wave equation
constrains all other aspects (such as amplitudes) of wave propagation in the -domain.

We also show numerical results of raytracing and examine its dependence on only two parameters in VTI media.

10/9/1997