On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging

International audienceFull waveform inversion is a PDE-constrained nonlinear least-squares problem dedicated to the estimation of the mechanical subsurface properties with high resolution. Since its introduction in the early 80s, a limitation of this method is related to the non-convexity of the misfit function which is minimized when the method is applied to the estimation of the subsurface wave velocities. Recently, the definition of an alternative misfit function based on an optimal transport distance has been proposed to mitigate this difficulty. In this study, we review the difficulties for exploiting standard optimal transport techniques for the comparison of seismic data. The main difficulty is related to the oscillatory nature of the seismic data, which requires to extend optimal transport to the transport of signed measures. We review three standard possible extensions relying on a decomposition of the data into its positive and negative part. We show how the two first might not be adapted for full waveform inversion and focus on the third one. We present a numerical strategy based on the dual formulation of a particular optimal transport distance yielding an efficient implementation. The interest of this approach is illustrated on the 2D benchmark Marmousi model