  Full waveform inversion is a successful procedure for determining properties
of the earth from surface measurements in seismology. This inverse problem is
solved by a PDE constrained optimization where unknown coefficients in a
computed wavefield are adjusted to minimize the mismatch with the measured
data. We propose using the Wasserstein metric, which is related to optimal
transport, for measuring this mismatch. Several advantageous properties are
proved with regards to convexity of the objective function and robustness with
respect to noise. The Wasserstein metric is computed by solving a Monge-Ampere
equation. We describe an algorithm for computing its Frechet gradient for use
in the optimization. Numerical examples are given.
