The auxiliary function method for waveform based earthquake location

This paper introduces the auxiliary function method (AFM), a novel, fast and simple approach for waveform based earthquake location. From any initial hypocenter and origin time, we can construct the auxiliary function, whose zero set contains the real earthquake hypocenter and the origin time. In most of situations, there are very few elements in this set. The overall computational cost of the AFM is significantly less than that of the iterative methods. According to our numerical tests, even for large noise, the method can still achieve good location results. These allow us to determine the earthquake hypocenter and the origin time extremely fast and accurate.Comment: 20 pages, 14 figure

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

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.Comment: 20 pages, 42 figure