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 normalmoveout (NMO) velocity and the other for perturbations in the dimensionless anisotropy parameter η. We consider separately the case of poststack 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 AnisotropyIntroduction
A wellknown 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 poststack 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.
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 poststack 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
Equations of the form (15) and (16) contain all the necessary information about the kinematic laws of anisotropy continuation in the domain of zerooffset migration.
Linearization
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
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
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 finitedifference technique.
Synthetic test
Residual poststack 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.
Conclusions

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.
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.
Appendix 1
Relating the zerooffset and migration slopes
Appendix 2
Linearized approximations
Although the exact expressions might be sufficiently constructive for actual residual migration applications, linearized forms are still useful, because they give us valuable insights into the problem. The degree of parameter dependency for different reflector dips is one of the most obvious insights in the anisotropy continuation problem. Perturbation of a small parameter provides a general mechanism to simplify functions by recasting them into power series expansion over a parameter that has small values. Two variables can satisfy the small perturbation criterion in this problem: The anisotropy parameterη (η ≪ 1) and the reflection dip τ_{ x } (τ_{ x }v≪ 1 or p_{ x }v ≪ 1).
Copyright information
This article is published under license to BioMed Central Ltd. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.