Diffracted waves carry valuable information that can help improve our velocity modeling capability for better imaging of the subsurface. They are especially useful for anisotropic media as they inherently possess a wide range of dips necessary to resolve the angular dependence of velocity. We develop a scheme for diffraction traveltime computations based on perturbation theory for transverse isotropic media with tilted axis of symmetry (TTI). The formulation has advantages on two fronts: firstly it alleviates the computational complexity associated with solving the TTI eikonal equation and secondly it provides a mechanism to scan for the best fit anellipticity parameter without the need for repetitive modeling of traveltimes. The accuracy of such an expansion is further enhanced by the use of Shanks transform. We establish the effectiveness of the proposed formulation with tests on a homogeneous TTI model and the BP TTI model.