Constructing misfit function for full waveform inversion based on sliced Wasserstein distance

It has been proven that full waveform inversion (FWI) based on optimal transport (OT) is robust against noise and contains less local minima. The accurate solution to OT problem on the real line (1D) is analytical with the selected distance function which can be effectively and non-iteratively solved by numerical interpolation. Currently finding an accurate optimal transport plan in higher dimension is known to be expensive even if it exists. The seismic data are naturally sampled in higher dimension and typically considered to be large scale. Given shot gathers from 2D survey, when calculating the misfit along one direction only, the information in other directions would be under-determined. In this paper, we introduce sliced Wasserstein distance to calculate the transport plan function in the lower dimensions of transformed domain such that other directions are considered implicitly. The extra cost caused by radon transform as compared to its counterpart based on $L_2$ norm is negligible. Numerical examples indicate that misfit based on sliced Wasserstein distance can improve the efficiency of FWI and shows the capacity for salt velocity building.