A document by Yuta Tsusaka. Click on the document to view its contents.

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).

Capability of TEC ’s CoRrelation Analysis (CRA) (Iwata and Umeno, 2016) for detecting preseismic anomaly is explained from the view point of the increase in signal-to-noise ratio to {\it amplify} preseismic TEC’s small anomaly signals with multiple sensor data synchronization and correlation to respond to all the criticisms proposed recently by Ikuta et al. 2021. Furthermore, deceleration at propagation velocities of MSTID before the 2016 Kumamoto earthquake firstly observed by CRA as velocity reduction of MSTID propagation in the F Layer of the ionosphere is then elucidated as a candidate of preseismic anomalies. This paper presents three models to explain its physical relationship with preseismic anomalies before large earthquakes. In particular, Model 1 predicts that the 35 m/s change in MSTID propagation velocities estimated by TEC’s CRA requires 0.58*10^{-3} V/m electric field change in the F Layer ionosphere, which is almost consistent with the estimation (Kelley et. al. 2017) in that the E*B/B^2 rift of 12 m/s for dislocations of electrons requires 0.5*10^{-3} V/m electric field in the E Layer to explain Heki’s finding of TEC anomaly behavior before the Tohoku-Oki earthquake. The \(10000\) times amplified effect of weak signals such as 0.58 mV/m in electrical field to affect MSTID propagation velocity change as is firstly observed by Iwata and Umeno, 2017 by CRA which has significant amplified capability. Contrary to the claim by Ikuta et al. 2021, TEC’s correlation anomalies detected (Iwata and Umeno 2016 and 2017) already provided supporting evidences that physical preseismic anomalies really exist.

Capability of TEC ’s CoRrelation Analysis (CRA) (Iwata and Umeno, JGR-Space Physics, 2016) for detecting preseismic anomalies is explained from the view point of the increase in signal-to-noise ratio to amplify preseismic TEC’s small anomaly signals with multiple sensor data synchronization and correlation. Furthermore, deceleration at propagation velocities of MSTID before the 2016 Kumamoto earthquake firstly observed by CRA (Iwata and Umeno, JGR-Space Physics, 2017) as velocity reduction of MSTID propagation in the F Layer of the ionosphere is then elucidated as a candidate of preseismic anomalies. A new physical model (Umeno, Nakabayashi, Iwata, Kao, 2021,DOI: 10.4236/ojer.2021.104008 ) which is recently constructed from the first principle to explain such ionospheric anomaly (ΔV=αΔE, Linear response theory of deceleration ΔV in propagation velocities of MSTID before large earthquakes with electric field change ΔE) is also presented to characterize preseismic ionospheric TEC anomalies by associating deceleration, acceleration, moving to the inverse direction of macroscopic ionic velocities before various large earthquakes such as our existing findings on ionospheric TEC anomaly before 2016 Tainan earthquake (Goto, Uchida, Igarashi, Chen, Kao, Umeno, JGR-Space Physics, 2019). In particular, this physical model predicts that the 35 m/s change in MSTID propagation velocities estimated by TEC’s CRA for 2016 Kumamoto earthquake requires 0.58 mV/m electric field change in the F Layer ionosphere, which is almost consistent with the estimation (Kelley et. al. JGR-Space Physics, 2017) in that the E✖︎B/B^2 drift of 12 m/s for dislocations of electrons requires 0.5 mV/m electric field in the E Layer to explain Heki’s finding of TEC anomaly behavior before the Tohoku-Oki earthquake. The 10000 times amplification effect of weak signals such as 0.5-0.58 mV/m in electrical field to affect MSTID propagation velocity change as is firstly observed by Iwata and Umeno, 2017 by CRA which means a significant signal amplification capability in this multi-sensor TEC correlation analysis (CRA). To summarize, various interrelation between physical models to exhibit TEC anomalies and observed TEC anomalies as above will be presented to understand preseismic ionospheric anomaly before large earthquakes.