Comparison between the Hamiltonian Monte Carlo method and the Metropolis–Hastings method for coseismic fault model estimation

Abstract

A rapid source fault estimation and quantitative assessment of the uncertainty of the estimated model can elucidate the occurrence mechanism of earthquakes and inform disaster damage mitigation. The Bayesian statistical method that addresses the posterior distribution of unknowns using the Markov chain Monte Carlo (MCMC) method is significant for uncertainty assessment. The Metropolis–Hastings method, especially the Random walk Metropolis–Hastings (RWMH), has many applications, including coseismic fault estimation. However, RWMH exhibits a trade-off between the transition distance and the acceptance ratio of parameter transition candidates and requires a long mixing time, particularly in solving high-dimensional problems. This necessitates a more efficient Bayesian method. In this study, we developed a fault estimation algorithm using the Hamiltonian Monte Carlo (HMC) method, which is considered more efficient than the other MCMC method, but its applicability has not been sufficiently validated to estimate the coseismic fault for the first time. HMC can conduct sampling more intelligently with the gradient information of the posterior distribution. We applied our algorithm to the 2016 Kumamoto earthquake (MJMA 7.3), and its sampling converged in 2 × 104 samples, including 1 × 103 burn-in samples. The estimated models satisfactorily accounted for the input data; the variance reduction was approximately 88%, and the estimated fault parameters and event magnitude were consistent with those reported in previous studies. HMC could acquire similar results using only 2% of the RWMH chains. Moreover, the power spectral density (PSD) of each model parameter's Markov chain showed this method exhibited a low correlation with the subsequent sample and a long transition distance between samples. These results indicate HMC has advantages in terms of chain length than RWMH, expecting a more efficient estimation for a high-dimensional problem that requires a long mixing time or a problem using nonlinear Green’s function, which has a large computational cost.Graphical