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In our previous work [Chen el al., J. Comput. Phys., 373(2018)], thequadratic Wasserstein metric is successfully applied to the earthquake locationproblem. The actual earthquake hypocenter can be accurately recovered startingfrom initial values very far from the true ones. However, the seismic wavesignals need to be normalized since the quadratic Wasserstein metric requiresmass conservation. This brings a critical difficulty. Since the amplitude of aseismogram at a receiver is a good representation of the distance between thesource and the receiver, simply normalizing the signals will cause theobjective function in optimization process to be insensitive to the distancebetween the source and the receiver. When the data is contaminated with strongnoise, the minimum point of the objective function will deviate and lead to alow accurate location result.  To overcome the difficulty mentioned above, we apply theWasserstein-Fisher-Rao (WFR) metric [Chizat et al., Found. Comput. Math.,18(2018)] to the earthquake location problem. The WFR metric is one of thenewly developed metric in the unbalanced Optimal Transport theory. It does notrequire the normalization of the seismic signals. Thus, the amplitude ofseismograms can be considered as a new constraint, which can substantiallyimprove the sensitivity of the objective function to the distance between thesource and the receiver. As a result, we can expect more accurate locationresults from the WFR metric based method compare to those based on quadraticWasserstein metric under high-intensity noise. The numerical examples alsodemonstrate this.

The Wasserstein-Fisher-Rao metric for waveform based earthquake location