Stereo optimal transport of the matching filter

A recently introduced optimal transport of the matching filter (OTMF) provided us with a robust misfit function for reducing cycle skipping in Full-Waveform Inversion (FWI). Unlike the conventional L2-norm approach, OTMF computes a matching filter first by deconvolution of the predicted data with the measured ones and constructs the misfit function by measuring the Wasserstein distance W2 between the resulting preconditioned matching filter and the target, i.e., the Dirac delta function. Compared to the conventional application of the optimal transport (OT) misfit in the data domain directly, OTMF applies the OT to the resulting matching filter avoiding the modification of the amplitude or the phase of the original seismic data. Measuring the distance between the resulting matching filter and a Dirac delta function using Wasserstein metric W2 suggests a convex misfit function with respect to the time shifted signal. We propose a misfit function by stereo optimal transport of the matching filter (SOTMF), which takes the space coherency of the resulting matching filter into consideration. Compared to OTMF, SOTMF has an extra regularization term, which controls the variations of the resulting matching filter along the space (offset) axis, and the regularization term is formulated by Wasserstein distance between the matching filters of the nearby traces. Thus, in the framework of OT, SOTMF tries to focus the resulting matching filter to be a Dirac delta function in time and enhance its space coherency as well. We use the Marmousi example to show that SOTMF can reduce the cycle-skipping, and at the same time its result shows less artifacts than OTMF. A result using an anisotropic version of SOTMF waveform inversion on a real dataset also demonstrates the good performance of the approach.