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A graph-space optimal transport objective function based on q-statistics to mitigate cycle-skipping issues in FWI


 Full-waveform inversion (FWI) is a powerful seismic imaging methodology to estimate geophysical parameters that honor the recorded waveforms (observed data), and it is conventionally formulated as a least-squares optimization problem. Despite many successful applications, least-squares FWI suffers from cycle skipping issues. Optimal transport (OT) based FWI has been demonstrated to be a useful strategy for mitigating cycle skipping. In this work, we introduce a new Wasserstein metric based on q-statistics in the context of the OT distance. In this sense, instead of the data themselves, we consider the graph of the seismic data, which are positive and normalized quantities similar to probability functions. By assuming that the difference between the graphs of the modeled and observed data obeys the q-statistics, we introduce a robust q-generalized graph-space OT objective function in the FWI context namely q-GSOT-FWI, in which the standard GSOT-FWI based on l2-norm is a particular case. To demonstrate how the q-GSOT-FWI deals with cycle skipping, we present two numerical examples involving 2D acoustic wave-equation modeling. First, we investigate the convexity of q-GSOT objective function regarding different time shifts, and, secondly, we present a Brazilian pre-salt synthetic case study, from a crude initial model which generates significant cycle-skipping seismic data. The results reveal that the q-GSOT-FWI is a powerful strategy to circumvent cycle skipping issues in FWI, in which our objective function proposal presents a smoother topography with a wider attraction valley to the optimal minimum. They also show that q-statistics leads to a significant improvement of FWI objective function convergence, generating higher resolution acoustic models than classical approaches. In addition, our proposal reduces the computational cost of calculating the transport plan as the q-value increases.