A Graph-Space Optimal Transport Approach for Full Waveform Inversion

The use of optimal transport distances has been recently promoted to mitigate cycle skipping issues in full waveform inversion. This distance is convex with respect to shifted patterns between compared data. This is the reason why it has attracted interest for full waveform inversion as the convexity with respect to time shifts can be seen as a proxy for the convexity with respect to wave velocities. The main difficulty for the application of optimal transport to seismic data is related to their non positivity: the optimal transport theory is developed for the comparison of positive quantities. We present a review of the strategies proposed to overcome this issue, and explain their limitations. They are either not adapted for the comparison of realistic data, or lose the convexity property. On this basis, we propose a novel approach based on a graph space interpretation of the seismic traces. Synthetic and observed traces are seen as point clouds in a 2D space (the graph space), which are compared using a 2D optimal transport technique. This strategy overcomes the positivity issue while preserving the convexity property. An application of this strategy on the Marmousi 2 model illustrates its interest for mitigating cycle skipping.