Traveltime computation with the linearized eikonal equation for Anisotropic media

A linearized eikonal equation is developed for transversely isotropic (TI) media with vertical symmetry axis (VTI). It is linear with respect to perturbations in the horizontal velocity or the anisotropy parameter n. An iterative linearization of the eikonal equation is used as the bases for an algorithm of finite-difference traveltime computations. A practical implementation of this iterative technique suggests starting from a background model of an elliptically anisotropic, inhomogeneous, nature. Especially since, for such media, traveltimes are calculated efficiently using eikonal solvers valid for the isotropic case. This constrains the perturbation to changes in the anisotropy parameter n (the parameter most responsible for imaging improvements in anisotropic media). The iterative implementation includes repetitive calculation of n from traveltimes, which is then used to evaluate the perturbation needed for the next round of traveltime perturbation calculation using the linearized eikonal equation. Unlike for isotropic media, interpolation is needed to estimate n in areas where the traveltime field is independent of n, like in areas where the wave probagates vertically. Typically, two to three iterations are enough to approach sufficient accuracy in traveltimes for imaging applications. The cost of each iteration is slightly less than the cost of a typical eikonal solver. However, this method will ultimately provide us with traveltime solutions for VTI media. The main limitation of the method is that some smoothness of the medium is required for the iterative implementation to work. Especially since we evaluate derivatives of the traveltime field as part of the iterative approach. If only a single perturbation is sufficient for the traveltime calculation, which may be the case for week anisotropy, no smoothness of the medium is necessary. Traveltime calculation tests demonstrate the robustness and effeciency of this approach.