Communications in Information and Systems

Volume 19 (2019)

Number 2

Seismic imaging and optimal transport

Pages: 95 – 145



Bjorn Engquist (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Yunan Yang (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)


Seismology has changed character since 50 years ago when the full wavefield could be determined. Partial differential equations (PDE) started to be used in the inverse process of finding properties of the interior of the earth. In this paper, we will review earlier techniques focusing on Full Waveform Inversion (FWI), which is a large-scale non-convex PDE constrained optimization problem. The minimization of the objective function is usually coupled with the adjoint state method, which also includes the solution to an adjoint wave equation. The least-squares ($L^2$) norm is the conventional objective function measuring the difference between simulated and measured data, but it often results in the minimization trapped in local minima. One way to mitigate this is by selecting another misfit function with better convexity properties. Here we propose using the quadratic Wasserstein metric ($W_2$) as a new misfit function in FWI. The optimal map defining $W_2$ can be computed by solving a Monge–Ampère equation. Theorems pointing to the advantages of using optimal transport over $L^2$ norm will be discussed, and several large-scale computational examples will be presented.


seismic imaging, full-waveform inversion, optimal transport, Monge-Ampère equation

2010 Mathematics Subject Classification

65K10, 86A15, 86A22

Received 12 June 2018

Published 19 September 2019