Abstract: I will discuss the travel time tomography problem for the

elastic wave equation, where the aim is to recover elastic

coefficients in the interior of an elastic medium given the travel

times of the corresponding elastic waves. I will consider in

particular the transversely isotropic case, which provides a

reasonable seismological model for the interior of the Earth or other

planets. By applying techniques from boundary rigidity problems, our problem is reduced to the

microlocal analysis of certain operators obtained from a

pseudo-linearization argument. These operators are not quite elliptic,

but they strongly resemble parabolic operators, for which a symbol

calculus first constructed by Boutet de Monvel can be applied. I will

describe how to use this calculus to solve the problem given certain

global assumptions, and if time permits I will discuss current work to

modify this calculus in order to solve the problem more locally.