A graph space optimal transport distance as a generalization of Lp distances: application to a seismic imaging inverse problem

Optimal transport distance is an appealing tool to measure the discrepancy between datasets in the frame of inverse problems, for its ability to perform global comparisons and its convexity with respect to shifted patterns in the compared quantities. However, solving inverse problems might require to compare signed quantities, while the optimal transport theory has been developed for the comparison of probability measures. In this study we propose to circumvent this difficulty by applying optimal transport to the comparison of the graph of the data rather than the data itself. We investigate this approach in the frame of seismic imaging, where each channel of the oscillatory data is interpreted as a discrete point cloud. We demonstrate that the corresponding misfit function can be computed through the solution of series of linear sum assignment problem (LSAP), while, based on the adjoint state technique, its gradient can be computed from the assignment solution of these LSAP. We illustrate how this approach yields a convex misfit function in the frame of seismic imaging using the full waveform. We show how an efficient strategy, based on a specific LSAP solver, the auction algorithm, can be designed. We illustrate the interest of the approach on a realistic 2D visco-acoustic seismic imaging problem. The proposed strategy relaxes the constraint on the accuracy of the initial model, outperforming the conventional least-squares approach and a previously proposed optimal transport based approach. The computational time increases by only a few percents compared with the least-squares approach, opening the way to applications to 3D field data in the near future.