  In our previous work [Chen el al., J. Comput. Phys., 373(2018)], the
quadratic Wasserstein metric is successfully applied to the earthquake location
problem. The actual earthquake hypocenter can be accurately recovered starting
from initial values very far from the true ones. However, the seismic wave
signals need to be normalized since the quadratic Wasserstein metric requires
mass conservation. This brings a critical difficulty. Since the amplitude of a
seismogram at a receiver is a good representation of the distance between the
source and the receiver, simply normalizing the signals will cause the
objective function in optimization process to be insensitive to the distance
between the source and the receiver. When the data is contaminated with strong
noise, the minimum point of the objective function will deviate and lead to a
low accurate location result.
  To overcome the difficulty mentioned above, we apply the
Wasserstein-Fisher-Rao (WFR) metric [Chizat et al., Found. Comput. Math.,
18(2018)] to the earthquake location problem. The WFR metric is one of the
newly developed metric in the unbalanced Optimal Transport theory. It does not
require the normalization of the seismic signals. Thus, the amplitude of
seismograms can be considered as a new constraint, which can substantially
improve the sensitivity of the objective function to the distance between the
source and the receiver. As a result, we can expect more accurate location
results from the WFR metric based method compare to those based on quadratic
Wasserstein metric under high-intensity noise. The numerical examples also
demonstrate this.
