The quadratic Wasserstein metric for earthquake location

Under this framework, we have derived a concise analytic expression of the Frèchet gradient of the Wasserstein metric, which leads to a simple and efficient implementation of 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 realized and observed numerically. To reduce the impact of noise, which does not offset each other after the signals are squared, 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.