  In this paper a compact Riemannian manifold with strictly convex boundary is
reconstructed from its partial travel time data. This data assumes that an open
measurement region on the boundary is given, and that for every point in the
manifold, the respective distance function to the points on the measurement
region is known. This geometric inverse problem has many connections to
seismology, in particular to micro seismicity. The reconstruction is based on
embedding the manifold in a function space. This requires the differentiation
of the distance functions. Therefore this paper also studies some global
regularity properties of the distance function on a compact Riemannian manifold
with strictly convex boundary.
