Our previous study proposed a Bayesian framework to enhance the approach by the renewal process to forecast earthquakes' timing and tested it in simulated seismicity-like time series. As a first step toward applying the Bayesian approach to actual seismic activity, it is crucial to use seismic catalogs for examination of the probability density functions in Bayes' theorem, which is in its simplest form in the Bayesian approach: the inter-event time distribution and the conditional and inverse probability between inter-event times at two cutoff magnitudes. In this study, I examined the properties of these probability density functions using time series with weak inter-event correlations extracted from three seismic catalogs: stationary time series with a nearly constant occurrence rate and aftershock sequences transformed by the Omori Utsu law. I found a new scaling property related to the temporal hierarchy of seismic activity. Using this property, I derived the above three probability density functions. Regarding the inter-event time distribution, I discuss its approximate scaling universality from the viewpoint of temporal fluctuations of transformed seismic time series by instantaneous occurrence rate. The derived inverse probability enables probabilistic evaluation of the large earthquakes' timing in the simplest Bayesian approach. Finally, I discuss extending it to the general Bayesian approach toward its practical use.

We present a Bayesian updating method on the inter-event times at different magnitude thresholds in a marked point process, toward the probabilistic forecasting of an upcoming large event using temporal information on smaller events. Bayes' theorem in a marked point process that yields the one-to-one relationship between intervals at lower and upper magnitude thresholds is presented. This theorem is extended to Bayesian updating for an uncorrelated marked point process that yields the relationship between multiple consecutive lower intervals and one upper interval. The inverse probability density function and its approximation function are derived. For the former, the condition for having a peak is shown. The latter is easier to apply to the time series of the ETAS model, and it consists of the kernel part, which includes the product of the conditional probabilities, and the correction term. The maximum point of the kernel part is shown to be not significantly affected by the correction term when applying the Bayesian updating to the ETAS model time series numerically. The occurrence time of the upcoming large event is estimated using this maximum point, and its accuracy is evaluated considering the relative error with the actual occurrence time. Moreover, forecasting is evaluated to be effective by the continuity of the updates with the accuracy within an acceptable range prior to the upcoming large event. Under these conditions, the statistical analysis indicates that forecasting is relatively effective immediately or long after the last major event in which stationarity is dominant in the time series.

In this presentation, we introduce a Bayesian updating method for inter-event times of different magnitude thresholds in marked point process, apply it to time-series of the ETAS model [1], and discuss the effectiveness in probabilistic forecasting of forthcoming large event considering the information on smaller events. To investigate magnitude threshold dependence of the inter-event time distribution of earthquakes, the conditional probability between inter-event times of different magnitude thresholds is proposed [2]. This gives the one-to-one statistical relationship between inter-event times of different magnitude thresholds. Firstly, we show the Bayes’ theorem on this conditional probability and derive the representation of the inverse probability density function. Secondly, we extend it to the Bayesian updating that gives the relationship between multiple intervals for lower threshold and an interval for upper threshold. We show the derivation of the inverse probability density function and its approximation function for uncorrelated marked point process (background seismicity in the ETAS model). The condition for the inverse probability density function to have a peak is also shown. The approximation function consists of two parts, a kernel-part that determines its outline and a correction term. The former has an easy form to handle numerically and is applicable to the time-series with correlations among events. Thirdly, based on the results for uncorrelated time-series, we apply the Bayesian updating method to time-series of the ETAS model. The mode of the approximation function is numerically shown to be nearly the same as that of the kernel-part. Therefore, the mode of the kernel part is used as the estimate of the occurrence time of forthcoming large event. By using the relative error between the estimate and the actual occurrence time of large event, effectiveness of the estimation with the approximation function is statistically evaluated. As a result, it is shown that if the time-series is dominated by stationary part, immediately or long after the large event, the forecasting is effectively conducted. [1] Y. Ogata, Ann. Inst. Statist. Math. 50(2), 379 (1998). [2] H. Tanaka and Y. Aizawa, J. Phys. Soc. Jpn. 86, 024004 (2017).