An optimal transport approach for seismic tomography: application to 3D full waveform inversion

The use of optimal transport distance has recently yielded significant progress in image processing for pattern recognition, shape identification, and histograms matching. In this study, the use of this distance is investigated for a seismic tomography problem exploiting the complete waveform; the full waveform inversion. In its conventional formulation, this high resolution seismic imaging method is based on the minimization of the L distance between predicted and observed data. Application of this method is generally hampered by the local minima of the associated L misfit function, which correspond to velocity models matching the data up to one or several phase shifts. Conversely, the optimal transport distance appears as a more suitable tool to compare the misfit between oscillatory signals, for its ability to detect shifted patterns. However, its application to the full waveform inversion is not straightforward, as the mass conservation between the compared data cannot be guaranteed, a crucial assumption for optimal transport. In this study, the use of a distance based on the Kantorovich–Rubinstein norm is introduced to overcome this difficulty. Its mathematical link with the optimal transport distance is made clear. An efficient numerical strategy for its computation, based on a proximal splitting technique, is introduced. We demonstrate that each iteration of the corresponding algorithm requires solving the Poisson equation, for which fast solvers can be used, relying either on the fast Fourier transform or on multigrid techniques. The development of this numerical method make possible applications to industrial scale data, involving tenths of millions of discrete unknowns. The results we obtain on such large scale synthetic data illustrate the potentialities of the optimal transport for seismic imaging. Starting from crude initial velocity models, optimal transport based Inverse Problems Inverse Problems 32 (2016) 115008 (36pp) doi:10.1088/0266-5611/32/11/115008 0266-5611/16/115008+36$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1 inversion yields significantly better velocity reconstructions than those based on the L distance, in 2D and 3D contexts.