Seismic Inversion and the Data Normalization for Optimal Transport

TL;DR

This paper investigates the use of optimal transport-based misfit functions in seismic full waveform inversion, focusing on how data normalization impacts the effectiveness of these methods and resolving a contradiction between theory and practice.

Contribution

It analyzes the effects of different data normalization methods on optimal transport-based FWI, clarifying why some practical approaches outperform theoretically ideal ones.

Findings

Normalization significantly influences FWI performance.

Certain practical normalization methods outperform theoretically ideal ones.

The paper provides insights into aligning theory with practical results.

Abstract

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.