  In this paper we show that in anisotropic elasticity, in the particular case
of transversely isotropic media, under appropriate convexity conditions,
knowledge of the qSH wave travel times determines the tilt of the axis of
isotropy as well as some of the elastic material parameters, and the knowledge
of qP and qSV travel times conditionally determines a subset of the remaining
parameters, in the sense if some of the remaining parameters are known, the
rest are determined, or if the remaining parameters satisfy a suitable
relation, they are all determined, under certain non-degeneracy conditions.
Furthermore, we give a precise description of the additional issues, which are
a subject of ongoing work, that need to be resolved for a full treatment.
