Estimation of source parameters in the Bayesian framework by Markov Chain Monte Carlo method

Accuracy of stress drop estimate has been considered as a big problem. To estimate stress drop, two seismic parameters (corner frequency and seismic moment) are obtained through the comparison of theoretical source spectrum representation with the spectrum of the observed waveforms. Estimation error of these parameters decrease the reliability of the stress drop estimates. In this study, we use Markov Chain Monte Carlo method in the Bayesian framework for stress drop analysis instead of grid search to evaluate the estimation accuracy and trade-off among parameters. Spectral ratio of two co-located earthquakes is utilized for the calculation that is represented by chi-square distribution. Therefore, we use F-distribution as a probability density function instead of normal distribution.

We focus on a cluster consisted of 36 earthquakes that occurred from 2015 May to 2016 November in Oklahoma. The spectral ratios between the large event (M_L = 4.1) and eGf events (2.2 < M_L < 3.7; < 2 km from the large event) are formed to remove path effects. We analyze 5.12 seconds after twice the S-arrival time with the band-pass filter of 0.5 to 30 Hz. We use as many stations as possible and all spectral ratios are simultaneously used to calculate likelihood. We update the value of moment ratio and two corner frequencies with 200,000 iterations.

Sampling distribution shows strong trade-off among all three parameters. Moment ratio has a negative correlation with f_c1 (corner frequency of large event) and f_c2 (corner frequency of eGf) while f_c1 has a positive correlation with f_c2. The histogram of the sampling showed a single peak in most cases, but some events showed multimodal histogram. We compared the results obtained from the calculation with F-distribution and normal distribution. Both calculations showed almost similar estimates, but the sampling distribution was showed different pattern. In this analysis, F-distribution is suitable for a probability density function than normal distribution because the sampling distribution showed reasonable pattern.

NY, TM and TS are supported by JST CREST Grant Number JPMJCR1763. NY is supported by JSPS KAKENHI, Grant Number 19K14812.