  We propose a double obstacle phase field methodology for binary recovery of
the slowness function of an Eikonal equation found in first traveltime
tomography. We treat the inverse problem as an optimization problem with
quadratic misfit functional added to a phase field relaxation of the perimeter
penalization functional. Our approach yields solutions as we account for well
posedness of the forward problem by choosing regular priors. We obtain a
convergent finite difference and mixed finite element based discretization and
a well defined descent scheme by accounting for the non-differentiability of
the forward problem. We validate the phase field technique with a $\Gamma$ -
convergence result and numerically by conducting parameter studies for the
scheme, and by applying it to a variety of test problems with different
geometries, boundary conditions, and source - receiver locations.
