  Numerical solutions of the eikonal (Hamilton-Jacobi) equation for
transversely isotropic (TI) media are essential for imaging and traveltime
tomography applications. Such solutions, however, suffer from the inherent
higher-order nonlinearity of the TI eikonal equation, which requires solving a
quartic polynomial for every grid point. Analytical solutions of the quartic
polynomial yield numerically unstable formulations. Thus, we need to utilize a
numerical root finding algorithm, adding significantly to the computational
load. Using perturbation theory we approximate, in a first order discretized
form, the TI eikonal equation with a series of simpler equations for the
coefficients of a polynomial expansion of the eikonal solution, in terms of the
anellipticity anisotropic parameter. Such perturbation, applied to the
discretized form of the eikonal equation, does not impose any restrictions on
the complexity of the perturbed parameter field. Therefore, it provides
accurate traveltime solutions even for models with complex distribution of
velocity and anisotropic anellipticity parameter, such as that for the
complicated Marmousi model. The formulation allows for large cost reduction
compared to using the exact TI eikonal solver. Furthermore, comparative tests
with previously developed approximations illustrate remarkable gain in accuracy
in the proposed algorithm, without any addition to the computational cost.
