Methods and Applications of Analysis

Volume 26 (2019)

Number 2

Special Issue in Honor of Roland Glowinski (Part 1 of 2)

Guest Editors: Xiaoping Wang (Hong Kong University of Science and Technology) and Xiaoming Yuan (The University of Hong Kong)

Seismic inversion and the data normalization for optimal transport

Pages: 133 – 148



Björn Engquist (Department of Mathematics and ICES, 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.)


Full waveform inversion (FWI) has recently become a favorite technique for the inverse problem of finding properties in the earth from measurements of vibrations of seismic waves on the surface. Mathematically, FWI is PDE constrained optimization where model parameters in a wave equation are adjusted such that the misfit between the computed and the measured dataset is minimized. In a sequence of papers, we have shown that the quadratic Wasserstein distance from optimal transport is to prefer as misfit functional over the standard $L^2$ norm. Datasets need however first to be normalized since seismic signals do not satisfy the requirements of optimal transport. There has been a puzzling contradiction in the results. Normalization methods that satisfy theorems pointing to ideal properties for FWI have not performed well in practical computations, and other scaling methods that do not satisfy these theorems have performed much better in practice. In this paper, we will shed light on this issue and resolve this contradiction.


optimal transport, seismic inversion, Monge–Ampère equation, optimization, data normalization

2010 Mathematics Subject Classification

49N45, 65K10, 86A22

Dedicated to Professor Roland Glowinski on the occasion of his eightieth birthday

The first author was partially supported by NSF DMS-1620396.

Received 7 January 2018

Accepted 30 July 2019

Published 2 April 2020