The quadratic Wasserstein metric for earthquake location

TL;DR

This paper applies the quadratic Wasserstein metric to earthquake location, demonstrating its effectiveness in accurately and efficiently determining earthquake hypocenters even with noisy data, by leveraging a trace-by-trace comparison approach.

Contribution

It introduces a concise analytic expression for the Fréchet gradient of the Wasserstein metric tailored for earthquake signals, enabling efficient optimization in earthquake location problems.

Findings

Few iterations needed for convergence.

Robust to noise in seismic data.

Convexity observed in the misfit function.

Abstract

In [Engquist et al., Commun. Math. Sci., 14(2016)], the Wasserstein metric was successfully introduced to the full waveform inversion. We apply this method to the earthquake location problem. For this problem, the seismic stations are far from each other. Thus, the trace by trace comparison [Yang et al., arXiv(2016)] is a natural way to compare the earthquake signals. Under this framework, we have derived a concise analytic expression of the Fr\`echet gradient of the Wasserstein metric, which leads to a simple and efficient implementation for the adjoint method. We square and normalize the earthquake signals for comparison so that the convexity of the misfit function with respect to earthquake hypocenter and origin time can be observed numerically. To reduce the impact of noise, which can not offset each other after squaring the signals, a new control parameter is introduced. Finally, the LMF (Levenberg-Marquardt-Fletcher) method is applied to solve the resulted optimization problem. According to the numerical experiments, only a few iterations are required to converge to the real earthquake hypocenter and origin time. Even for data with noise, we can obtain reasonable and convergent numerical results.