Introduction

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 $\eta$. 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, $\alpha$, 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 $\alpha(z)$, extracted from the well, to estimate depths. When $\alpha$ 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 $(x-\tau)$-domain.

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