Geophysical Inversion and Optimal Transport

TL;DR

This paper introduces a novel optimal transport-based method for comparing oscillatory time series, improving convergence in seismic inverse problems by leveraging Wasserstein distance and a new transformation.

Contribution

The paper presents a differentiable optimal transport measure for time series comparison, enhancing inverse problem solutions with better convergence properties.

Findings

Superior convergence in seismic source inversion.

Effective application of Wasserstein distance to oscillatory signals.

Potential links between optimal transport and Bayesian inference.

Abstract

We propose a new approach to measuring the agreement between two oscillatory time series, such as seismic waveforms, and demonstrate that it can be employed effectively in inverse problems. Our approach is based on Optimal Transport theory and the Wasserstein distance, with a novel transformation of the time series to ensure that necessary normalisation and positivity conditions are met. Our measure is differentiable, and can readily be employed within an optimization framework. We demonstrate performance with a variety of synthetic examples, including seismic source inversion, and observe substantially better convergence properties than achieved with conventional $L_2$ misfits. We also briefly discuss the relationship between Optimal Transport and Bayesian inference.