Misfit function for full waveform inversion based on the Wasserstein metric with dynamic formulation

Abstract

Conventional full waveform inversion (FWI) using least square distance (L2 norm) between the observed and predicted seismograms suffers from local minima. Recently, the Wasserstein metric (W1 metric) has been introduced to FWI to compute the misfit between two seismograms. Instead of comparisons bin by bin, the W1 metric allows to compare signal intensities across different coordinates. This measure has great potential to account for time and space shifts of events within seismograms. However, there are two main challenges in application of the W1 metric to FWI. The first one is that the compared signals need to satisfy nonnegativity and mass conservation assumptions. The second one is that the computation of W1 metric between two seismograms is a computationally expensive problem. In this paper, a strategy is used to satisfy the two assumptions via decomposition and recombination of original seismic data. In addition, the computation of the W1 metric based on dynamic formulation is formulated as a convex optimization problem. A primal-dual hybrid gradient method with linesearch has been developed to solve this large-scale optimization problem on GPU device. The advantages of the new method are that it is easy to implement and has high computational efficiency. Compared to the L2 norm based FWI, the computation time of the proposed method will approximately increase by 11% in our case studies. A 1D time-shift signals case study has indicated that the W1 metric is more effective in capturing time shift and makes the misfit function more convex. Two applications to synthetic data using transmissive and reflective recording geometries have demonstrated the effectiveness of the W1 metric in mitigating cycle-skipping issues. We have also applied the proposed method to SEG 2014 benchmark data, which has further demonstrated that the W1 metric can mitigate local minima and provide reliable velocity estimations without using low frequency information in the recorded data.