Seismic Inversion and the Data Normalization for Optimal Transport
For geophysicists using full waveform inversion, this clarifies a practical issue with optimal transport misfit functionals, improving the reliability of seismic imaging.
The paper resolves a contradiction in seismic inversion where theoretically ideal normalization methods for optimal transport underperform in practice, while empirically better methods lack theoretical support. It explains why this occurs and provides a resolution.
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.