Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion

Abstract

Conventional full-waveform inversion (FWI) using the least-squares norm (L 2 ) as a misfit function is known to suffer from cycle skipping. This increases the risk of computing a local rather than the global minimum of the misfit. In our previous work, we proposed the quadratic Wasserstein metric (W 2 ) as a new misfit function for FWI. The W 2 metric has been proved to have many ideal properties with regards to convexity and insensitivity to noise. When the observed and predicted seismic data are regarded as two density functions, the quadratic Wasserstein metric corresponds to the optimal cost of rearranging one density into the other, where the transportation cost is quadratic in distance. The difficulty of transforming seismic signals into nonnegative density functions is discussed. Unlike the L 2 norm, W 2 measures not only amplitude differences, but also global phase shifts, which helps to avoid cycle skipping issues. In this work, we build on our earlier method to cover more realistic high-resolution applications by embedding the W 2 technique into the framework of the adjoint-state method and applying it to seismic relevant 2D examples: the Camembert, the Marmousi, and the 2004 BP models. We propose a new way of using the W 2 metric trace-by-trace in FWI and compare it to global W 2 via the solution of the Monge-Ampère equation. With corresponding adjoint source, the velocity model can be updated using the l-BFGS method. Numerical results show the effectiveness of W 2 for alleviating cycle skipping issues and sensitivity to noise. Both mathematical theory and numerical examples demonstrate that the quadratic Wasserstein metric is a good candidate for a misfit function in seismic inversion.

The quadratic Wasserstein metric for inverse data matching

“…In Figure 8, we show a true wave speed model that we used to generate synthetic data, an initial guess as well as the inversion results obtained with the L 2 metric and the W 2 metric. The computational details of this numerical experiment can be found in [35]. Here we are mainly interested in demonstrating the smoothing effect of the quadratic Wasserstein metric at the initial guess that is far from the true wave speed field.…”

Passive Adjoint Tomography of the Crustal and Upper Mantle Beneath Eastern Tibet With a W2‐Norm Misfit Function

“…For actual seismic data processing, if the length of a seismogram is T 0 , the exact expression of the quadratic Wasserstein distance in one dimension is as follows (Yang et al, ): …”

“…If the misfit between the synthetic and observed data is examined from another point of view, a mapping can be considered to exist between them (Ma & Hale, ; Engquist & Froese, ; Métivier et al, ; Yang et al, ). Recently, a novel measure of misfit related to optimal transport was proposed by Engquist and Froese ().…”

Diagnosing Forward Operator Error Using Optimal Transport

“…Our work is motivated in part by [7,8,34], where the authors use the quadratic Wasserstein metric to solve Full-Waveform Inversion (FWI) problems. In particular, it is demonstrated that the quadratic Wasserstein metric, as opposed to the L 2 norm, provides an effective measure of the misfit between given data and computed solution.…”

“…Our work extends that of [7,8,34] by considering different inverse problems, a more general noise model, and we use a different Wasserstein metric. See section 4.4 for more detail.…”