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Imaging the interior of the Earth by recording seismic signals at its surface is a challenging problem. Extracting any piece of information from the entire time traces is mandatory and is performed through a linearized inversion scheme, known as full waveform inversion (FWI). The misfit between observed and calculated data is minimized following quasi-Newton methods
adapted to the large-scale aspect of the problem as we consider millions of unknowns. The calculated data is obtained through seismic wave propagation modeling. Conventional FWI strategies rely on a definition of the misfit function based on the L2 norm. This results in a multi-modal misfit function
as we must fit oscillatory signals (the cycle ambiguity problem).

Numerous modification of this misfit function have been promoted to face this long-standing problem: cross-correlation, deconvolution, normalized integration are examples of such techniques. Our proposition is to measure the discrepancy between observed and predicted data through the solution of an optimal transport problem based on the Kantorovich-Rubinstein (KR) distance. Therefore, instead of comparing each sample in space and in time by each corresponding prediction, as underlaid by the usage of the L2 distance, the optimal transport approach offers a global comparison of data and synthetics, taking into account simultaneously the spatial and time coherency of data panels. We shall illustrate the formulation of the comparison over time and receivers for each source although we are not limited to this selection. The corresponding optimal transport problem is reformulated as a non-smooth convex optimization problem, which can be solved efficiently through proximal splitting techniques. As for other modifications of the misfit function, the optimal transport distance is integrated naturally in the FWI workflow through a modification of the adjoint source term we shall describe. This modified adjoint source is simply the solution of the optimal transport problem.

Synthetic illustrations on realistic reference examples such as the Marmousi2 model or the benchmark Chevron model will be presented. We shall highlight how we mitigate the sensitivity to the definition of the initial model from which we start the linearized Newton-based optimization.

Co-authors of the study: Ludovic Métivier, Romain Brossier, Quentin Mérigot
and Edouard Oudet