# The basic components of residual migration in VTI media using anisotropy continuation

## Abstract

We introduce anisotropy continuation as a process which relates changes in seismic images to perturbations in the anisotropic medium parameters. This process is constrained by two kinematic equations, one for perturbations in the normal-moveout (NMO) velocity and the other for perturbations in the dimensionless anisotropy parameter η. We consider separately the case of post-stack migration and show that the kinematic equations in this case can be solved explicitly by converting them to ordinary differential equations using the method of characteristics. When comparing the results of kinematic analytical computations with synthetic numerical experiments confirms the theoretical accuracy of the method.

### Keywords

Velocity continuation Residual migration Anisotropy## 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 relatively 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.

*v*

_{v}is the vertical

*P*-wave velocity, and δ is one of Thomsen’s anisotropy parameters (Thomsen 1986). Taking

*v*

_{h}to be the

*P*-wave velocity in the horizontal direction, the other anisotropy parameter,

*η*, is given by

*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 with respect to conventional data processing objectives, 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*

_{nmo}by

*v*. Thus, time processing in VTI media, depends on two parameters (

*v*and η), whereas in isotropic media only

*v*counts. To emphasize the importance of anisotropy to the dip moveout process, Alkhalifah (2005) introduced residual dip moveout for VTI media.

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 *η* change as migration parameters. This approach is especially attractive, when the initial image is obtained with isotropic migration (that is with *η* = 0). In this case, anisotropy continuation is equivalent to introducing anisotropy in the model without the need for 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 open promising opportunities for seismic data processing in the presence of anisotropy.

## The general theory

*l*, from the source to the reflection point is related to the two-way zero-offset time,

*t*, by the simple equation

*v*

_{g}is the group velocity, best expressed in terms of its components, as follows:

*v*

_{gx}denotes the horizontal component of group velocity,

*v*

_{v}is the vertical

*P*-wave velocity, and

*v*

_{gτ}is the

*v*

_{v}-normalized vertical component of the group velocity. Under the assumption of zero shear-wave velocity in VTI media, these components have the following analytic expressions:

*p*

_{x}is the horizontal component of slowness, and

*p*

_{τ}is the normalized (again by the vertical

*P*-wave velocity

*v*

_{v}) vertical component of slowness. The two components of the slowness vector are related by the following eikonal-type equation (Alkhalifah 1998):

*v*and

*η*as imaging parameters (migration velocity and migration anisotropy coefficient), the ray length

*l*can be fixed through the imaging process. This implies that the partial derivatives of with respect to the imaging parameters are zero. Therefore,

*t*, which leads to the equations

*τ*and the reflection slope given by \(\frac{\partial \tau}{\partial x},\) instead of

*t*and

*p*

_{x}, respectively, we need to eliminate

*p*

_{x}from Eqs. (12) and (13). This task can be achieved with the help of the following explicit relation, derived in Appendix 1,

*τ*

_{x}= \(\frac{\partial \tau}{\partial x}\), and

*v*and

*η*. In summary, these equations have the form

Equations of the form (15) and (16) contain all the necessary information about the kinematic laws of anisotropy continuation in the domain of zero-offset migration.

### Linearization

*η*equal to zero in the right hand side of the equations. Under this approximation, Eq. (15) leads to the kinematic velocity-continuation equation for elliptically anisotropic media, which has the following relatively simple form:

*v*=

*v*

_{v}, yields Fomel’s expression for isotropic media (Fomel 1996) given by

*z*is depth, we immediately obtain the equation

*η*= 0 and

*v*=

*v*

_{v}in Eq. (16) leads to the following kinematic equation for

*η*-continuation:

We include more discussion about different aspects of linearization in Appendix 2. The next section presents the analytic solution of Eq. (17). Later in this paper, we compare the analytic solution with a numerical synthetic example.

## Ordinary differential equation representation: anisotropic rays

*m*stands for either

*v*or

*η*,

*τ*

_{x}= \(\frac{\partial\tau}{\partial x}\), \(f_m =\frac{\partial \tau}{\partial m}\). To trace the

*v*and

*η*rays, we must first identify the initial values

*x*

_{0},

*τ*

_{0},

*τ*

_{x0}, and

*τ*

_{m0}from the boundary conditions. The variables

*x*

_{0}and

*τ*

_{0}describe the initial position of a reflector in a time-migrated section,

*τ*

_{x0}describes its migrated slope, and

*τ*

_{m0}is simply obtained from Eqs. (15) or (16).

Using the exact kinematic expressions for *f*, the results in rather complicated representations of the ordinary differential equations. The linearized expressions, on the other hand, are simple and allow for a straightforward analytical formulation of the ray tracing system.

### From kinematics to dynamics

*η*-continuation equation (17) corresponds to the following linear fourth-order dynamic equation

*t*coordinate refers to the vertical traveltime

*τ*, and

*P*(

*t*,

*x*,

*η*) is the migrated image, parameterized in the anisotropy parameter

*η*. To find the correspondence between Eqs. (17) and (21), it is sufficient to apply a ray-theoretical model of the image

*t*=

*τ*(

*x*,

*η*) is the anisotropy continuation “wavefront”—the image of a reflector for the corresponding value of

*η*, and the function

*A*is the amplitude. Substituting the trial solution into the partial differential equation (21) and considering only the terms with the highest asymptotic order (those containing the fourth-order derivative of the wavelet

*f*), we arrive at the kinematic equation (17). The next asymptotic order (the third-order derivatives of

*f*) gives us the linear partial differential equation of the amplitude transport, as follows:

*τ*

_{x}= 0 and

*τ*

_{xx}= 0), equation (23) reduces to the equality

*η*. This is of course a reasonable behavior in the case of a flat reflector. It does not guarantee although that the amplitudes, defined by Eq. (23), behave equally well for dipping and curved reflectors. The amplitude behavior may be altered by adding low order terms to Eq. (21). According to the ray theory, such terms can influence the amplitude behavior, but do not change the kinematics of the wave propagation.

An appropriate initial value condition for Eq. (21) is the result of isotropic migration that corresponds to the *η* = 0 section in the (*t*, *x*, *η*) domain. In practice, the initial value problem can be solved by a finite-difference technique.

## Synthetic test

Residual post-stack migration operators can be obtained by generating synthetic data for a model consisting of diffractors for given medium parameters and then migrating the same data with different medium parameters. For example, we can generate diffractions for isotropic media and migrate those diffractions using an anisotropic migration. The resultant operator describes the correction needed to transform an isotropically migrated section to an anisotropic one, that is the *anisotropic residual migration* operator.

*η*). Despite the inherent accuracy of the synthetic operators, they suffer from the lack of aperture in modeling the diffractions, and therefore, beyond a certain angle the operators vanish and start to deviate. The agreement between the synthetic and calculated operators for small angles, especially for the

*η*= 0.1 case, promises reasonable results in future dynamic implementations.

## Conclusions

*η*for VTI media. Despite the fact that we have considered the simple case of post-stack migration separately, the exact kinematic equations describing the continuation process are anything, but simple. However, useful insights into this problem are deduced from linearized approximations of the continuation equations. These insights include the following observations:

The leading order behavior of the velocity continuation is proportional to

*τ*_{x}^{2}, which corresponds to small or moderate dips.The leading order behavior of the η continuation is proportional to

*τ*_{x}^{4}, which corresponds to moderate or steep dips.Both leading terms are independent of the strength of anisotropy (

*η*).

In practical applications, the initial migrated section is obtained by isotropic migration, and, therefore, the residual process is used to correct for anisotropy. Setting *η* = 0 in the continuation equations for this type of an application is a reasonable approximation, given that *η* = 0 is the starting point and we consider only weak to moderate degrees of anisotropy (*η* ≈ 0.1). Numerical experiments with synthetically generated operators confirm this conclusion.

## Notes

### Acknowledgments

Tariq Alkhalifah would like to thank KAUST and KACST for their financial support, and Sergey Fomel likes to thank the University of Texas, Austin for its support.

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