Introduction

A well-known paradox in seismic imaging is that the detailed information about the subsurface velocity is required before a reliable image can be obtained. In practice, this paradox leads to an iterative approach to building the image. It looks attractive to relate small changes in velocity parameters to inexpensive operators perturbing the image. This approach has been long known as residual migration . A classic result is the theory of residual post-stack migration Rothman et al. (1985), extended to the prestack case by Etgen (1990). In a recent paper, Fomel (1996) introduced the concept of velocity continuation as the continuous model of the residual migration process. All these results were based on the assumption of the isotropic velocity model.

Recently, emphasis has been put on the importance of considering anisotropy and its influence on data. Alkhalifah and Tsvankin (1995) demonstrated that, for TI media with vertical symmetry axis (VTI media), just two parameters are sufficient for performing all time-related processing, such as normal moveout (NMO) correction (including non-hyperbolic moveout correction, if necessary), dip-moveout (DMO) correction, and prestack and poststack time migration. One of these two parameters, the short-spread NMO velocity for a horizontal reflector, is given by  

$$v_{{\rm nmo}}(0)=v_{v} \sqrt{1+2 \delta} \, ,$$ (1)

where v_v is the vertical P-wave velocity, and $\delta$ is one of Thomsen's anisotropy parameters Thomsen (1986). Taking v_h to be the P-wave velocity in the horizontal direction, the other parameter, $\eta$, is given by  

$$\eta \equiv 0.5(\frac{v_h^2}{v_{{\rm nmo}}^2(0)}-1)=\frac{\epsilon-\delta}{1+2 \delta} \, ,$$ (2)

where $\epsilon$ is another of Thomsen's parameters. In addition, Alkhalifah (1997) has showed that the dependency on just two parameters becomes exact when the vertical shear wave velocity (V_S0) is set to zero. Setting V_S0=0 leads to remarkably accurate kinematic representations. It also results in much simpler equations that describe P-wave propagation in VTI media. Throughout this paper, we use these simplified, yet accurate, equations, based on setting V_S0=0, to derive the continuation equations. Because we are only considering time sections, and for the sake of simplicity, we denote $v_{\rm nmo}$ by v. Thus, time processing in VTI media, depends on two parameters (v and $\eta$), whereas in isotropic media only v counts.

In this paper, we generalize the velocity continuation concept to handle VTI media. We define anisotropy continuation as the process of seismic image perturbation when either v or $\eta$ change as migration parameters. This approach is especially attractive, when the initial image is obtained with isotropic migration (that is with $\eta=0$). In this case, anisotropy continuation is equivalent to introducing anisotropy in the model without the need of repeating the migration step.

For the sake of simplicity, we start from the post-stack case and purely kinematic description. We define however the guidelines for moving to the more complicated and interesting cases of prestack migration and dynamic equations. The results are preliminary, but they open promising opportunities for seismic data processing in presence of anisotropy.