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.