Communications in Mathematical Sciences

Volume 14 (2016)

Number 8

Optimal transport for seismic full waveform inversion

Pages: 2309 – 2330



Björn Engquist (Department of Mathematics and ICES, University of Texas, Austin, Tx., U.S.A.)

Brittany D. Froese (Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, N.J., U.S.A.)

Yunan Yang (Department of Mathematics and ICES, University of Texas, Austin, Tx., U.S.A.)


Full waveform inversion is a successful procedure for determining properties of the Earth from surface measurements in seismology. This inverse problem is solved by 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–Ampère equation. We describe an algorithm for computing its Fréchet gradient for use in the optimization. Numerical examples are given.


optimal transport, Wasserstein metric, computational seismology, full waveform inversion

2010 Mathematics Subject Classification

35J96, 49N45, 65K10, 65N06, 86A15

Published 26 October 2016