  Consider the geometric inverse problem: There is a set of delta-sources in
spacetime that emit waves travelling at unit speed. If we know all the arrival
times at the boundary cylinder of the spacetime, can we reconstruct the space,
a Riemannian manifold with boundary? With a finite set of sources we can only
hope to get an approximate reconstruction, and we indeed provide a discrete
metric approximation to the manifold with explicit data-driven error bounds
when the manifold is simple. This is the geometrization of a seismological
inverse problem where we measure the arrival times on the surface of waves from
an unknown number of unknown interior microseismic events at unknown times. The
closeness of two metric spaces with a marked boundary is measured by a labeled
Gromov--Hausdorff distance. If measurements are done for infinite time and
spatially dense sources, our construction produces the true Riemannian manifold
and the finite-time approximations converge to it in the metric sense.
