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Expanding the traveltime solutions of the TI eikonal equation in a power series in terms of independent parameters, like the tilt angle $\theta $ in 2D, provides an efficient tool to estimate $\theta $ in a generally inhomogeneous background medium. Additional expansions from a background elliptical anisotropic medium with a vertical symmetry axis allows us to search for the anisotropy parameter $\eta $ and the tilt angles simultaneously, even in 3D media. For a homogeneous background, I obtain analytic nonhyperbolic moveout equations for anisotropic media that are generally simple, and yet accurate. Nevertheless, the formulations provide $\eta $ and tilt-angles estimation capabilities for a general inhomogeneous background medium.