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.

Abstract. Full waveform inversion is a successful procedure for determining properties of the earth from surface measurements in seismology. This inverse problem is solved by a PDE constrained optimization where unknown coefficients in a computed wavefield are adjusted to minimize the mismatch with the measured data. We propose using the Wasserstein metric, which is related to optimal transport, for measuring this mismatch. Several advantageous properties are proved with regards to convexity of the objective function and robustness with respect to noise. The Wasserstein metric is computed by solving a Monge-Ampère equation. We describe an algorithm for computing its Frechet gradient for use in the optimization. Numerical examples are given.

Full waveform inversion (FWI) with 2 norm objective function often suffers from cycle skipping that causes the solution to be trapped in a local minimum, usually far from the true model. We introduce a new norm based on the optimal transport theory for measuring the data mismatch to overcome this problem. The new solution uses an exponential encoding scheme and enhances the phase information when compared with the conventional 2 norm. The adjoint source is calculated trace-wise based on the 1D Wasserstein distance. It uses an explicit solution of the optimal transport over the real line. It results in an efficient implementation with a computational complexity of the adjoint source proportional to the number of shots, receivers and the length of recording time. We demonstrate the effectiveness of our solution by using the Marmousi model. A second example, using the BP 2004 velocity benchmark model, illustrates the benefit of the combination of the new norm and Total Variation (TV) regularization.

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Full-waveform inversion has evolved into a powerful computational tool in seismic imaging. New misfit functions for matching simulated and measured data have recently been introduced to avoid the traditional lack of convergence due to cycle skipping. We have introduced the Wasserstein distance from optimal transport for computing the misfit, and several groups are currently further developing this technique. We evaluate three essential observations of this new metric with implication for future development. One is the discovery that trace-by-trace comparison with the quadratic Wasserstein metric works remarkably well together with the adjoint-state method. Another is the close connection between optimal transport-based misfits and integrated techniques with normalization as, for example, the normalized integration method. Finally, we study the convexity with respect to selected model parameters for different normalizations and remark on the effect of normalization on the convergence of the adjoint-state method.