Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.