Long term probability of a Magnitude 8 Kanto earthquake along the Sagami Trough, central Japan
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 30 March 2012
Accepted: 17 June 2012
Published: 16 August 2012
We attempt to estimate the long-term probability of a Magnitude (M) 8 earthquake along the Sagami Trough in the Kanto subduction zone, central Japan. A Brownian passage time model is applied to sets of historical earthquakes identified in previous studies. An optimal model is obtained by the maximum likelihood method for each data set. The optimal parameters are not well constrained since each data set includes a small number of earthquakes. To obtain reliable probabilities, two weighting methods are introduced. First, we apply the weighted log-likelihood method, where the model parameters are estimated from the log-likelihood function, summed up with each log-likelihood weighted in proportion to the reliability of the data set. Second, probabilities are estimated as the weighted average of every alternative model. The weight of each model represents the normalized relative likelihood of the model. The probabilities of the weighted log-likelihood function are within the ranges of those obtained for each set. In averaging the proposed sequence and over probable parameter values, the probability of an M 8 earthquake occurring in the next 30 years is estimated to be 2.0 to 4.6%, depending on the cutoff value of the weight.
The 1923 and 1703 earthquakes in the Kanto region, central Japan, are Magnitude (M) 8 class earthquakes that were caused by the relative motion between the Philippine Sea Plate (PH) and the North American Plate (NA) along the Sagami Trough, where the PH is subducting beneath NA.
According to the Earthquake Research Committee (ERC), a committee of the government of Japan, the long-term probability of an M 8 earthquake during the next 30 years in Kanto, central Japan, is no higher than two percent. This probability is estimated using a Brownian passage time (BPT) model with two model parameters, which are assigned based on geological and geomorphological evidence, and model parameters for other areas such as off Miyagi Prefecture, the Nankai and the Tonankai areas, and major inland Quaternary active faults. Efforts have been made to produce more reliable models by using different kinds ofdatasimultaneously (Fitzenz et al., 2010; Rhoades and Van Dissen, 2003). In this paper, we discuss the validity of the current BPT model for Kanto by using historical earthquake sequences proposed by Ishibashi (1994), Shishikura (2003), and Shimazaki et al. (2011a, b). In each of these sequences, only a small number of earthquakes, specifically five (Ishibashi, 1994), four (Shishikura, 2003), and three (Shimazaki et al., 2011a, b), are listed. Therefore, we could not constrain the BPT model parameters well, but probabilities for the next 30 years could be estimated by taking the average of the probable hazard function and comparing this with the currently reported value.
2. BPT Model and Parameter Uncertainties
In an exact sense, we should take into the above log-likelihood function a factor corresponding to non-occurrence after the 1923 earthquake, which is represented by the reliability function (survivor function, complementary cumulative distribution function) at the time point of interest (Ogata, 1999). Only about 90 years have passed since the last event and the reliability, even in the smallest case, becomes 0.996, which reduces only 0.004 units of log-likelihood. For simplicity, this factor is neglected in our study.
3. Historical Earthquakes
Historical earthquake sequences used in the study. The optimal BPT parameters are listed in the second set. The AlC’s values for the optimal case and the Poisson model are in the third set. The difference in AIC, specifically the AIC of the Poisson minus that of the optimal BPT, is also given. Values of log-likelihood for the Poisson and the optimal BPT are listed in the bottom set. At the bottom, Δl refers to a value of the log-likelihood of the Poisson minus that of the optimal BPT.
Shimazaki et al. (2011b)
BPT optimal values
BPT optimal case
BPT optimal case
4. Log-likelihood and Probabilities
Probability of an M 8 earthquake in the next 30 years for various cases. Probabilities listed in the first row indicate those for the optimal BPT cases. Probabilities averaged with weighting down to Δl = −2 at every 0.5 step are given in the following lines.
Shimazaki et al.
Optimal case loge (weight)
0 ∼ −0.5
0 ∼ −1
0 ∼ −1.5
0 ∼ −2
5. Weighted Log-likelihood
The probabilities in Table 2 vary for different authors. In averaging the probabilities among the three, we have attempted to apply the weighted log-likelihood method (Wang and Zidek, 2005). In this method, the model parameters are estimated from the log-likelihood function, summed up with each log-likelihood weighted in proportion to the reliability of the data.
The historical sequences adopted here are reported by the authors from different viewpoints with independent evidence: historical documents, or geological or paleoseismo-logical evidence. However, times of earthquakes are precisely dated in historical documents, but the assignment of the events as a Kanto recurrent earthquake is the point of argument. For this reason, we do not apply the method handling uncertainties proposed in early studies (Rhoades et al., 1994; Ogata, 1999) but the method of weighted log-likelihood.
The estimated model parameters and probabilities are listed in the last column of Tables 1 and 2. Comparing these values with those obtained for each sequence, the values estimated from the weighted log-likelihood function are within the ranges of the corresponding estimates. Therefore, it is reasonable to consider that these values represent the results obtained when the BPT model is adapted to an M 8 sequence in Kanto with largely uncertain historical data.
At the bottom of Table 2, the probabilities estimated with the Poisson model are listed. Comparing each of these values with the respective ones of the BPT model, the estimates of the Poisson models are about two times larger than those of the BPT model.
6. Discussion and Summary
Considering that the current probability issued to the public was estimated by including parameter values of Δl ranging down to −2.0 (Figs. 2(a–d)), the probability of 0.046 for the weighted log-likelihood case would become an alternative to the current probability. The difference between the publicized value, and that of the present study, stems from differences in the model parameters, which are constrained to a value in the range 200 to 400 for µ and 0.17 to 0.24 for α in the former case. The optimal values of parameter α (Table 1) are obviously larger than that used in the ERC estimation. It partially contributes to larger probabilities than that of the ERC estimation (Ishizeki and Kumamoto, 2007).
The recent great earthquake in Tohoku, northeast Japan (M w = 9.0, March 11, 2011), will definitely disturb the regularity of the recurrent sequences within its source and nearby areas. For example, the source area of the off-Ibaraki earthquake (M 7.0) on May 8, 2008, is likely to be ruptured again only about three years after the last event, where values of 22 years and 0.20 have been adopted as the BPT parameters. This great earthquake suggests that the recurrent model is applicable only under limited conditions, and that having the α parameter range from 0.17 to 0.24 will be useful in these limited cases. Without this prior information, the average probability over probable parameters would become an alternative estimate to the current one.
In summary, we conclude that the BPT model is slightly superior to the stationary Poisson model based on historical earthquakes, and geologic and paleoseismological evidence, in Kanto. In averaging the proposed sequence and the average probability over probable parameters instead of constraining the α parameter, a probability of 2.0 to 4.6% could be obtained, depending on the cutoff value of the probable parameters.
The authors thank Takeo Ishibe and an anonymous reviewer for their discussions and comments on this manuscript.
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