Kinematic and dynamic raytracing in inhomogeneous, anisotropic media has been traditionally formulated in terms of elastic parameters. Such a formulation is inefficient for computation as it requires evaluating complicated right-hand-side functions and solving an eigenvalue problem at each ray step. It also requires that a medium be specified with elastic parameters. This is inconsistent with the common practice in seismic data processing where anisotropy is usually described with Thomsen (1986) parameters. This inconsistency may result in ambiguity in specifying the elastic parameters. To overcome these difficulties, we have reformulated the kinematic and dynamic raytracing systems in terms of phase velocity. The new formulation is much simpler and computationally more efficient than the previous elasticparameter based formulations. Solution of the eigenvalue problem at each ray step is no longer required. As the medium for raytracing is now specified with phase velocity, the possible ambiguity in specifying elastic parameters is also eliminated. The kinematic and dynamic raytracing systems developed in this study have been used to implement Gaussian beam depth migration in anisotropic media. Numerical results show that our formulation is efficient and accurate and has greatly speeded up the depth migration in anisotropic media.
Kinematic and dynamic raytracing in inhomogeneous, anisotropic media is an essential building block for seismic modeling and imaging with ray methods. Kinematic raytracing in anisotropic media has traditionally been formulated in terms of elastic parameters (Cerveny, 1972, 2001). Such a formulation is, however, computationally cumbersome (Cerveny, 1989). It also requires a medium to be specified with elastic parameters. The common practice in seismic data processing, on the other hand, is to describe anisotropy with the Thomsen (1986) parameters. This inconsistency may cause problems in medium specification. The elastic-parameter based formulation for anisotropic dynamic raytracing is even more complicated as it now involves differentiation of kinematic raytracing system with respect to ray parameters (Hanyga, 1986; Cerveny, 2001). The purpose of this study is to formulate the kinematic and dynamic raytracing systems in anisotropic media in terms of phase velocity. This formulation overcomes some of difficulties of the elasticparameter based formulation, and is especially useful for transversely isotropic (TI) and orthorhombic media where simple analytic expressions for phase velocity have been derived in terms of the Thomsen parameters (Thomsen, 1986; Tsvankin, 2001).
Dynamic raytracing equations in anisotropic media are commonly expressed in Cartesian coordinates (e.g., Cerveny 1972, 2001). This leads to a system of six linear first-order ordinary differential equations. For many applications such as Gaussian beam calculation, it is convenient to use raycentered coordinates. The dynamic raytracing system also takes the simplest form in such coordinates, reducing the number of differential equations in the system from six to four. Here we will first formulate dynamic raytracing equations in terms of phase velocity in Cartesian coordinates, and then transform them to a ray-centered coordinate system.
Gaussian beam migration (GBM) was originally developed by Hill (1990, 2001) for both poststack and prestack migration in isotropic media, and was extended by Alkhalifah (1995) for anisotropic poststack migration.