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.

It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the second boundary value problem for the Monge-Ampère equation. Viscosity solutions are a powerful tool for analysing and approximating fully nonlinear elliptic equations. However, we demonstrate that this nonlinear elliptic equation does not satisfy a comparison principle and thus existing convergence frameworks for viscosity solutions are not valid. We introduce an alternative PDE that couples the usual Monge-Ampère equation to a Hamilton-Jacobi equation that restricts the transportation of mass. We propose a new interpretation of the optimal transport problem in terms of viscosity subsolutions of this PDE. Using this reformulation, we develop a framework for proving convergence of a large class of approximation schemes for the optimal transport problem. Examples of existing schemes that fit within this framework are discussed.

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