The convexity of optimal transport-based waveform inversion for certain structured velocity models

TL;DR

This paper investigates the convexity properties of the squared Wasserstein distance as an objective function in seismic waveform inversion, demonstrating convexity for certain structured velocity models in 1D and 2D, which can improve inversion stability.

Contribution

It proves the convexity of the squared Wasserstein distance with respect to velocity parameters in specific structured models, offering insights into its advantages over traditional norms in seismic inversion.

Findings

Convexity holds for constant, piecewise increasing, and linearly increasing velocity models in 1D.

Convexity is established for linearly increasing velocity models in 2D within large regions.

The squared Wasserstein distance can be more convex than the squared L2 norm for waveform inversion.

Abstract

Full--waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared $W_2$ distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared $W_2$ distance with respect to the velocity parameter on the interval from zero to the true value of the velocity parameter when the source function is a probability measure. Furthermore, we consider a two--dimensional model where velocity is linearly increasing as a function of depth and prove the convexity of the squared $W_2$ distance in the velocity parameter on large regions containing the true value. We discuss the convexity of the squared $W_2$ distance compared with the convexity of the squared $L^2$ norm, and we discuss the relationship between frequency and convexity of these respective distances. We also discuss multiple approaches to optimal transport for non--probability measures by first converting the wave data into probability measures.