Curved Rays Tomography updates background anisotropy velocity parameters in the time-migrated domain. The tomography uses image gathers generated by Anisotropy Curved Rays Kirchhoff Time Migration. A locally varying 1D Vertical Transverse Isotropy (VTI) model is assumed. The background anisotropy parameters are the instantaneous (interval) vertical compression velocity V and the two Thomsen anisotropy parameters δ and ε Interval velocity (or alternatively δ) is updated from short offsets reflection events, while ε is updated from the available long offset data. Two complementary approaches are presented in this study: local and global. In the local approach, the medium parameters are updated from top down, layer by layer, one parameter at a time. The residual anisotropy parameters, that best fit the residual moveout curves, are picked. The residual moveout includes overburden and current layer components. In the global approach, all parameters are inverted simultaneously. Due to a large number of offsets, the problem becomes over-defined, and we solve it by a constrained least-squares minimization. The cost function accounts for data and model variances, which reflect the reliability of the data and control parameter variations, respectively. The updated parameters are constrained to a feasible range.