The horizontal transversely isotropic model, with arbitrary symmetry axis orientation, is the simplest effective representative that explains the azimuthal behaviour of seismic data. Estimating the anisotropy parameters of this model is important in reservoir characterisation, specifically in terms of fracture delineation. We propose a travel-time-based approach to estimate the anellipticity parameter η and the symmetry axis azimuth φ of a horizontal transversely isotropic medium, given an inhomogeneous elliptic background model (which might be obtained from velocity analysis and well velocities). This is accomplished through a Taylor's series expansion of the travel-time solution (of the eikonal equation) as a function of parameter η and azimuth angle φ. The accuracy of the travel time expansion is enhanced by the use of Shanks transform. This results in an accurate approximation of the solution of the non-linear eikonal equation and provides a mechanism to scan simultaneously for the best fitting effective parameters η and φ, without the need for repetitive modelling of travel times. The analysis of the travel time sensitivity to parameters η and φ reveals that travel times are more sensitive to η than to the symmetry axis azimuth φ. Thus, η is better constrained from travel times than the azimuth. Moreover, the two-parameter scan in the homogeneous case shows that errors in the background model affect the estimation of η and φ differently. While a gradual increase in errors in the background model leads to increasing errors in η, inaccuracies in φ, on the other hand, depend on the background model errors. We also propose a layer-stripping method valid for a stack of arbitrary oriented symmetry axis horizontal transversely isotropic layers to convert the effective parameters to the interval layer values.