Temporal distribution model and occurrence probability of M ≥ 6.5 earthquakes in North China Seismic Zone

The temporal distribution of earthquakes provides important basis for earthquake prediction and seismic hazard analysis. The relatively limited records of strong earthquakes have often made it difficult to study the temporal distribution models of regional strong earthquakes. However, there are hundreds of years of complete strong earthquake records in the North China Seismic Zone, providing abundant basic data for studying temporal distribution models. Using the data of M ≥ 6.5 earthquakes in North China as inputs, this paper estimates the model parameters using the maximum likelihood method with Poisson, Gamma, Weibull, Lognormal and Brownian passage time (BPT) distributions as target models. The optimal model for describing the temporal distribution of earthquakes is determined according to Akaike information criterion (AIC),and Kolmogorov–Smirnov test (K–S test). The results show that Lognormal and BPT models perform better in describing the temporal distribution of strong earthquakes in North China. The mean recurrence periods of strong earthquakes (M ≥ 6.5) calculated based on these two models are 8.1 years and 13.2 years, respectively. In addition, we used the likelihood profile method to estimate the uncertainty of model parameters. For the BPT model, the mean and 95% confidence interval of recurrence interval μ is 13.2 (8.9–19.1) years, and the mean and 95% confidence interval of α is 1.29 (1.0–1.78). For the Lognormal model, the mean value and 95% confidence interval of v is 2.09 (1.68–2.49), the mean value exp (v) corresponding to earthquake recurrence interval is 8.1 (5.4–12.1) years. In this study, we also calculated the occurrence probability of M ≥ 6.5 earthquakes in the North China Seismic Zone in the future, and found that the probability and 95% confidence interval in the next 10 years based on the BPT model is 35.3% (26.8%-44.9%); the mean value and 95% confidence interval of earthquake occurrence probability based on the Lognormal distribution is 35.4% (22.9%-49.7%); the mean probability and 95% confidence interval based on the Poisson model is 53.1% (41.1%-64%). The results of this study may provide important reference for temporal distribution model selection and earthquake recurrence period calculation in future seismic hazard analysis in North China.


Introduction
The temporal distribution models of earthquakes represent an important part of seismological research. For a long time, it has been widely assumed in probabilistic seismic hazard analysis and earthquake prediction that earthquakes follow the Poisson distribution in time (Cornell 1968;Gardner and Knopoff 1974;Schwartz and Coppersmith 1984;Frankel 1995;Console et al. 2003;Parsons and Geist 2012;Ben-Naim et al. 2013;Petersen et al. 2014). Also known as the time-independent model, the Poisson model holds that the occurrence of earthquakes does not change with time, and represents the theoretical basis for probabilistic seismic hazard analysis in some countries at present (Cornell 1968;Hu 1990;Petersen et al. 2014;Gao 1996;Pan et al. 2013).
Aside from the time-independent model, there are also time-dependent models for describing the occurrence of earthquakes. Many studies have proved that time-dependent models perform better in describing the temporal distribution characteristics of earthquakes in certain areas (Utsu 1984;Nishenko and Buland 1987;Ogata and Abe 1991;Mega et al. 2003;Tripathi 2006;Sharma and Kumar 2010). Seismologists have put forward such statistical models as Gamma, Lognormal, Weibull and Brownian passage time (BPT) functions (Utsu 1984;Matthews et al. 2002;Tripathi 2006;Pasari andDikshit 2015, 2018;Bajaj and Sharma 2019;Pasari 2019). Contrary to the Poisson model, the time-dependent models hold that the occurrence of earthquakes varies with time, exerting an important influence on the results of seismic hazard.
In addition, the Epidemic Type Aftershock Sequence (ETAS) model has also been used to study the spatio-temporal distribution of earthquakes in recent years (Ogata 1988(Ogata , 1998Zhuang et al. 2005;Ogata and Zhuang 2006), and has been proposed or used by scientists as a means for probabilistic seismic hazard analysis (e.g., Xu and Wu, 2017;Pei et al., 2022;Šipčić et al. 2022). However, for seismic hazard analysis, more attention is paid to the temporal distribution of main shocks (Michael 2011;Daub et al. 2012;Shearer and Stark 2012;Beroza 2012). In fact, in the ETAS model, background earthquakes are sometimes still regarded as following the Poisson distribution (Ogata 1988;Zhuang et al. 2005;Lombardi and Marzocchi 2007). In applying the ETAS model to probabilistic seismic hazard analysis, Šipčić et al. (2022) also identified the characteristics of the temporal changes of background seismicity as a key area for future research, which suggests that studying the temporal distribution of main shocks will also contribute to the effective use of the ETAS model.
Research shows that strong earthquakes (M ≥ 6.5) will generate fault rupture of a certain scale, which has a considerable impact on seismic hazard analysis (Schwartz and Coppersmith 1984;Frankel et al. 2002). For a long time, the study of the temporal distribution characteristics of strong earthquakes has been constrained by limited strong earthquake records. Fortunately, there are hundreds of years of complete strong earthquake records in North China, providing excellent basic data for studying temporal distribution models. The present study will focus on the temporal distribution models for strong earthquakes (M ≥ 6.5) in the North China Seismic Zone. The North China Seismic Zone is a geological zone with its boundaries determined by Chinese seismologists and geologists. In China, a seismic zone refers to an area with similar geological structures and seismicity characteristics, serving as an analysis unit for seismic hazard analysis during a certain period. Chinese seismologists have partitioned the Chinese mainland into different seismic zones, including Northeast China, North China, South China, Xinjiang, and the Qinghai-Tibet Plateau. Among these zones, the North China Seismic Zone stands out for its abundant historical earthquake data, with a complete historical earthquake record spanning over 500 years. (Gao et al., 2003;Xu and Gao 2015). In this study, based on the catalogs of M ≥ 6.5 events in the North China Seismic Zone, and using Poisson (exponential distribution), Gamma, Lognormal, Weibull and BPT as target models, we regressed the parameters of each model by using the maximum likelihood method and selected the optimal models for describing the temporal distribution of seismicity using the Akaike information criterion (AIC) and the Kolmogorov-Smirnov test (K-S test). Based on the optimal models, we calculated the occurrence probability of M ≥ 6.5 events in future in the study area. The results may provide theoretical basis for the selection of temporal distribution models and the calculation of seismicity parameters in seismic hazard analysis in the North China Seismic Zone.

Seismic catalogs
We obtained the data of historical strong earthquakes from The Catalogue of Chinese Historical Strong Earthquakes (twenty-third century B.C. ~ 1911 A.D.) (Department of Earthquake Disaster Prevention, National Earthquake Administration 1995), the data between 1912 and 1990 from The Catalogue of Chinese Earthquakes (1912Earthquakes ( ~ 1990) (Department of Earthquake Disaster Prevention, National Earthquake Administration 1999), and the data after 1990 from The Catalogue of Earthquakes in China and Adjacent Areas after 1990. (Lv et al. 2010;Xu and Gao 2014). The magnitude adopted in the catalogues is surface wave magnitude. Figure 1 shows the distribution of epicenters of M ≥ 6.5 earthquakes in the North China Seismic Zone.
As this study focuses on the temporal statistical characteristics of main shocks, it is necessary to remove aftershocks from the catalogs. At present, many methods are available for seismic declustering, such as the traditional space-time window (Gardner and Knopoff 1974), the stochastic declustering approach based on the ETAS model (Zhuang et al. 2002), and the nearest neighbor distance method (Baiesi and Paczuski 2004). Despite its good declustering effects, the stochastic declustering method based on the ETAS model requires the prior estimation of model parameters, which can be difficult in regions with limited earthquake catalogues or poor catalogue completeness. Moreover, this method requires considerable amounts of computation. Among them, the space-time window of Gardner and Knopoff (1974) is widely used in seismicity analysis and seismic hazard analysis (Shearer and Stark 2012;Daub et al. 2012;Petersen et al. 2014;Xu and Gao 2015). In this paper, the space-time window method was used to remove aftershocks from the catalogs in the Chinese mainland, in which the space window parameters calculated by Cheng et al. (2020) were adopted.
The completeness of catalogs represents an important factor affecting the analysis results. Huang et al. (1994) analyzed the completeness and reliability of the catalogs of historical earthquakes in North China (M S ≥ 4.8). Through a comparison with the characteristics of earthquake damage recorded by modern instruments, they showed that the catalogs of historical earthquakes in North China are complete and reliable for the purpose of seismological study. The complete records of earthquakes started from 1480. Xu and Gao (2015) used more statistical methods to investigate the completeness of the historical catalogs in North China, and concluded that the records of M S ≥ 4.8 earthquakes in this region are complete since 1500. The catalogs of historical earthquakes in China provide crucial data for disaster prevention, and are widely used in earthquake prediction, probabilistic seismic hazard analysis and compilation of seismic hazard maps (Huang et al. 1994;Lv et al. 2010;Pan et al. 2013;Gao 1996Gao , 2003Gao , 2015. Therefore, for this study, complete records are available for M S ≥ 6.5 earthquakes in North China since 1500. We obtained 37 M ≥ 6.5 events, which exceeds the requirement of having at least 25 events in order to distinguish temporal distribution models, as proposed by Matthews et al. (2002). Figure 2 shows the magnitude-time distribution and the histogram of the inter-event times. The largest earthquake in the seismic catalogue occurred in Huaxian, Shaanxi, with a magnitude of 8.3. The most recent event was the Tangshan earthquake of 1976, with a magnitude of

Statistical models of earthquake recurrence intervals
In this study, we analyze the statistical characteristics of earthquake recurrence intervals, defined as the time intervals between successive events. In engineering seismology, earthquake recurrence interval is also called return period. Its reciprocal, which is the frequency of earthquake occurrence per unit of time, is a crucial parameter for probabilistic seismic hazard calculation and earthquake prediction.
Commonly used models in statistical seismology at present include exponential (Poisson), Gamma, Lognormal, Weibull and BPT (Utsu 2002;Matthews et al. 2002;Bajaj and Sharma 2019). In this study, we use the above five models to analyze the statistical characteristics of recurrence intervals in North China (Table 1).

Model parameter estimation and goodness-of-fit (GOF)
The maximum likelihood method was used to estimate the parameters in the above statistical models. The maximum likelihood method was developed to estimate the range of potential parameters by using the data per se (Fisher 1922), and raised for the first time Table 1 Statistical distribution models

Model name
Probability density function and cumulative distribution function of statistical models and their parameters In the equation, is the scale parameter and is the shape parameter Gamma model is Gamma function, k and are shape and scale parameters respectively Lognormal model where Φ is the cumulative probability distribution function of normal distribution, and are the mean and standard deviation of logarithmic values of x respectively Brownian passage time model f BPT where μ is the mean of recurrence interval, α = σ/μ, and σ is the standard deviation of recurrence interval dx , Φ is the cumulative probability function of standard normal distribution such a question: given a random sample and its distribution model, which parameters are most likely to produce the sample? In other words, the parameters estimated by the maximum likelihood method can maximize the occurrence probability of the current random sample. As the intervals between earthquakes in this study are definite, the maximum likelihood method may be used to estimate the parameters and their uncertainties. The aforementioned five models all have probability density functions, and the likelihood function is the joint probability density function of random variable x, which can be written as: where represents one or more model parameters, N is the sample length of the random variable, and f is the probability density function of the statistical model. The natural logarithm of the likelihood function is obtained and the coefficient is derived. The parameter values in the model can be obtained by solving the likelihood equation (group).
In this study, the distribution of earthquake recurrence interval is unknown. To know whether they conform to a statistical model, we used the K-S test to test the difference between the observed values of recurrence intervals and their theoretical distributions. The statistic of the K-S test represents the biggest difference in cumulative distribution probability between observed and theoretical values: where O i is the cumulative probability of observed values, and E i is the cumulative probability of the theoretical model. Smaller D N results in better goodness-of-fit.
To select the optimal model, we used AIC to determine the goodness-of-fit. AIC is defined as: where k is the number of model parameters; L is the likelihood function. In this study, due to the limited sample size of the catalogue, we adopted the Akaike Information Criterion (AIC) corrected for small sample sizes, as proposed by Hurvich and Tsai (1989), so as to accurately select the optimal model. The corrected Akaike Information Criterion (AICc) is defined as: N is the sample size. Generally, the model with the smallest AICc value is chosen as the optimal model from available alternatives.
In addition, a significance test was performed for the K-S test statistic, which will help determine whether the empirical distributions are different from the theoretical distributions. We take the observation data consistent with the theoretical model introduced in the paper as the null hypothesis, and calculate the p-value with a confidence level of 0.05. If the p-value is greater than 0.05, the null hypothesis is accepted, and the larger the p-value is, the better the observation data conforms to the theoretical model. See Gibbons and Chakraborti (2003) for a detailed description of relevant test methods.  Table 2 shows the parameter values of each model regressed using the seismic catalogs of magnitude 6.5 or above in the North China Seismic Zone and GOF parameter values. Table 2 shows that the Lognormal distribution has the lowest AICc and D N values, indicating the best goodness-of-fit among the five models. We calculated the Δ i for each model, i.e., Δ i = AICc i -AICc min . Generally speaking, models with ∆ i ≤ 2 have substantial evidence that they fit the data better, models with 4 ≤ ∆ i ≤ 7 have considerably less evidence, and models with ∆ i > 10 have essentially no evidence supporting them (Burnham and Anderson 2004). According to the aforementioned criteria, both the Weibull and Gamma distributions have ∆ i values greater than or equal to 4, indicating a lack of compelling evidence to support the notion that the temporal distribution of strong earthquakes in the North China region aligns well with these two models. The Poisson model has a ∆ i value of 2.2, surpassing the threshold of 2, further suggesting insufficient evidence to demonstrate the suitability of the Poisson model for the temporal distribution of earthquakes in the North China region. On the other hand, the BPT model has a ∆ i value of 0.4, indicating its ability to effectively describe temporal distribution of earthquakes in the North China region, as similar to the Lognormal model. In addition, the results of the K-S significance test show that the Lognormal distribution and BPT models exhibit relatively large p-values, suggesting a favorable agreement between the theoretical models and the observed data. Conversely, the p-values associated with the remaining three models are relatively small, signifying notable disparities between the theoretical models and empirical data. We also use the bootstrap method (Efron and Tibshirani 1993) to calculate the uncertainty of the AICc and D N , and find that even considering 1 times the standard deviation of AICc and D N , it does not change the above conclusion. So the results in this paper are reasonable and reliable. Furthermore, the cumulative distribution curves in Fig. 3 shows that the BPT and Lognormal models are in better agreement with empirical data. From the above analysis, it can be concluded that Lognormal and BPT distributions can better describe the temporal statistical distribution characteristics of earthquakes in the North China Seismic Zone, and are suitable for describing the temporal distribution of M ≥ 6.5 earthquakes in the North China Seismic Zone. From the above analysis, it is evident that the temporal distribution of strong earthquakes in the North China Seismic Zone align better with the Lognormal distribution and BPT model. In the statistical sense, these two time-dependent models allow us to make more accurate probability prediction in the North China Seismic Zone. In physics, the BPT model holds strong physical significance as it is able to describe the processes of stress accumulation and release within the study area. By employing the BPT model, more precise and physically meaningful probability predictions can be achieved for the North China Seismic Zone.

Temporal distribution models and parameters
The maximum likelihood method was used to estimate the values of the model parameters. The limited length of seismic event samples leads to the uncertainty of the statistical parameters. In actual seismic hazard analysis, seismologists pay much attention to the range of such uncertainties, which are often estimated based on normal distribution. In real cases, however, the model parameters may not follow normal distribution. In this paper, we calculate the uncertainty range of parameters of BPT and Lognormal models according to the likelihood surface with the model parameters and the method of likelihood profiles proposed by Biasi et al. (2015).
We calculated the likelihood ratio, which is the ratio of the maximum likelihood value on the likelihood surface to the logarithm of the likelihood values at other parameter pairs L ML ∕L( ) . The logarithm of the likelihood ratio log L ML ∕L( ) can be approximated to follow a chi-square distribution (Biasi et al. 2015). Therefore, for a twotailed test with 2 degrees of freedom, the logarithm of the likelihood ratio corresponding to the parameter θ within the 95% confidence interval is 2 log L ML ∕L( ) < 5.99. The confidence interval of a single parameter in the BPT and Lognormal models can be obtained by plotting a profile on the likelihood surface (Biasi et al. 2015). On the likelihood surface, a profile running parallel to the vertical and horizontal axes and passing through the point of the maximum likelihood value is plotted, so as to produce the curve  Biasi et al. (2015), the range of each parameter in a certain confidence interval can be obtained. Figure 4 presents the likelihood profile plots of parameters μ and α in the BPT model. Above the gray dotted line is the 95% confidence interval of parameters. The mean and 95% confidence interval of recurrence interval μ is 13.2 (8.9-19.1) years, and the mean and 95% confidence interval of α is 1.29 (1.0-1.78). Figure 5 shows the likelihood profile plots of parameters v and σ of the Lognormal model, where the mean value and 95% confidence interval of v is 2.09 (1.68-2.49), the mean value exp (v) corresponding to earthquake recurrence interval

Probability of occurrence of strong earthquakes
The probability of strong earthquakes, which often cause serious casualties and damage to buildings, has been an important source of concern among scientists. In the above, we found that both the BPT and Lognormal distributions can describe the temporal distribution characteristics of M ≥ 6.5 earthquakes in North China relatively well. In this study, we calculated the future occurrence probability of M ≥ 6.5 events in the study area based on the above two distribution models. For comparison, we also calculated the occurrence probability based on the Poisson model. The probability of earthquake occurrence in the future can be calculated when the elapsed time and recurrence intervals of large earthquakes and their uncertainties are known. Let T e be the time before the last event occurred, that is, the elapsed time, then the conditional probability of at least one earthquake occurring in the future ΔT time is (Matthews et al. 2002): where F T e = ∫ T e 0 f (t)dt is the cumulative distribution function of recurrence interval. f (t) is the probability density function of the seismicity temporal distribution model introduced above.
The most recent earthquake with a magnitude of 6.5 or higher in the North China Seismic Zone was the Tangshan earthquake of 1976, approximately 46 years ago from 2022, Fig. 6 Variation of occurrence probability of M ≥ 6.5 earthquakes in North China with time with an elapsed time of 46 years. Based on the Poisson model, Lognormal model and BPT model, we calculate the probability of earthquake elapsed time greater than 46 years in the North China Seismic Zone, which is 3%, 6% and 7% respectively. This means that the quiescent period of seismic activity in the North China Seismic Zone has been long enough. Based on the elapsed time and the coefficients in Table 2, we used Eq.
(1) to calculate the earthquake occurrence probability in the North China Seismic Zone for a future time frame. Figure 6 shows the variation of earthquake occurrence probability in the North China Seismic Zone over time calculated based on Poisson, BPT, and Lognormal models. The probability of earthquake occurrence calculated based on different models is different (Table 3), with the probability calculated based on the Poisson model being the largest and the probability calculated based on BPT model being the smallest. Among them, the probability of M ≥ 6.5 events in North China in the next 10 years calculated based on the BPT model is 35.3%, and the 95% confidence interval is 26.8%-44.9%. The average probability and 95% confidence interval calculated based on the Lognormal distribution is 35.4% (22.9-49.7%), and the average probability and 95% confidence interval calculated based on the Poisson model is 53.1% (41.1-64%). It is clear the occurrence probability values based on the BPT and Lognormal models are similar, and significantly smaller than that based on the Poisson model. The calculations above represent the probability of earthquakes occurring within different future time frames, given a specific elapsed time. Additionally, we calculated the probability of earthquakes at various elapsed times for specific future time frames (1 year and 10 years). Figure 7 depicts the variation of earthquake occurrence probability as the elapsed time intervals increase ( Fig. 7a for a 1-year time frame and Fig. 7b for a 10-year time frame). For the 1-year conditional probability, the calculated values based on the BPT and Lognormal models show the following characteristics: When the elapsed time is short, the calculated occurrence probability is smaller than that of the Poisson model; with the increase of elapsed time, the calculated occurrence probability begins to exceed that of the Poisson model; when the elapsed time is relatively long (relative to the recurrence interval), the calculated occurrence probability is again smaller than that of the Poisson model. For the 10-year conditional probability, when the elapsed time of the earthquake is short, the occurrence probability calculated based on the BPT and Lognormal models is greater than that of the Poisson model, and when the elapsed time of the earthquake is greater than the recurrence interval, the earthquake occurrence probability calculated based on the two models is less than that of Poisson model. The black vertical dashed line in Fig. 7 is the current time. It can be seen that the occurrence probability calculated based on the BPT and Lognormal models is about half of the value calculated based on the Poisson model. Figure 7 shows that the earthquake occurrence probability calculated based on the BPT and Lognormal models initially increase with increasing elapsed time, followed by a slow decline as the elapsed time further extends. In the case of the BPT model, this behavior can be attributed to the continued accumulation of tectonic stress as elapsed time increases, leading to a progressive increase in the probability of seismic events. However, due to the relatively high coefficient of variation of earthquake recurrence intervals in the North China region, the earthquake occurrence probability slowly decreases with further increases in the elapsed time. This indicates that the earthquake sequence in this region is primarily influenced by noise (Matthews et al., 2002). It may be attributed to the influence of complex stress from multiple directions and sources, where stress interactions and dissipation occur, and the seismogenic structures in the region are not subjected to sustained stress loading, eventually reaching a constant stress state, where the earthquake probability approaches a constant value as time increases. For the Lognormal distribution, the longer the occurrence interval, the longer the waiting time (Matthews et al., 2002), as indicated by decreasing earthquake occurrence probability as the interval increases. From a physical perspective, as the elapsed time lengthens, the tectonic stress in the study area dissipates through various mechanisms, and the seismogenic faults may "fail" and lose their ability to generate earthquakes after multiple seismic events. It is evident that as the occurrence interval increases, the earthquake occurrence probability calculated using the Lognormal model gradually tends towards zero.

Conclusions
The abundant records of historical strong earthquakes in North China make it possible to investigate the corresponding temporal distribution models. This study provides a reliable and effective method for determining the optimal models for the recurrence intervals of M ≥ 6.5 earthquakes in North China. Using Poisson, Gamma, Weibull, Lognormal and BPT models as target models, we regressed the model parameters by the maximum likelihood method, and evaluated the goodness-of-fit using the K-S test and AIC values. The results show that the BPT and Lognormal models outperform other models in describing the temporal distribution of M ≥ 6.5 earthquakes in North China. Regarding the uncertainty of earthquake recurrence intervals, we used the likelihood profile method introduced by Biasi et al. (2015) to calculate parameter uncertainties of the BPT and Lognormal models. For the BPT model, the calculated recurrence interval and 95% confidence interval is 13.2 (8.9-19.1) years. For the Lognormal model, the mean value and 95% confidence interval of v is 2.09 (1.68-2.49), and the mean value exp (v) corresponding to the recurrence interval is 8.1 (5.4-12.1) years. In China's fifth-generation seismic source model proposed by Gao (2015), the recurrence interval of earthquakes with a magnitude of 6.5 or higher in the North China Seismic Zone is approximately 10 years. This interval is shorter than the 13.2 years calculated using the BPT model in our study, but longer than the 8.1 years calculated using the Lognormal model. Thus our calculations may provide reference for future seismic hazard analysis in the North China Seismic Zone. Based on the BPT, Lognormal and Poisson models, we also calculated the probability of future M ≥ 6.5 events in North China: the probability and 95% confidence interval in the next 10 years based on the BPT model is 35.3% (26.8%-44.9%); the mean value and 95% confidence interval of earthquake occurrence probability based on the Lognormal distribution is 35.4% (22.9-49.7%); the mean probability and 95% confidence interval based on the Poisson model is 53.1% (41.1-64%). The BPT and Lognormal models yield similar probability values, while that based on the Poisson model is significantly greater.
This study shows that M ≥ 6.5 events in North China conform better to the time-dependent BPT and Lognormal statistical models. The Lognormal model shows that the probability of earthquake occurrence decreases and ultimately approaches zero as the time since the last earthquake increases. If the Lognormal model is physically correct, it means that as time elapses, the tectonic stress in the study area dissipates through alternative mechanisms, resulting in a lack of stress accumulation on active faults. This seems physically implausible. Another possibility is those fault segments, after experiencing multiple earthquakes, become "seismically quiescent" and lose their ability to generate further seismic events (Matthews et al., 2002). Normally, for the BPT model, the calculated occurrence probability should be greater than that of Poisson model in case of a long elapsed time (relative to the return period). However, the occurrence rate of earthquakes calculated based on the time-dependent models in this study is lower than that calculated by the Poisson model, which is associated with the large aperiodicity. The coefficient of variation of earthquake recurrence intervals calculated based the on BPT model is 1.29 (1.0-1.78). For the BPT model, Matthews et al. (2002) showed that the earthquake sequence is dominated by noise when α is greater than 2 −1∕2 . This could be attributed to the complex tectonic stress in the region originating from multiple directions and sources. The interaction and dissipation of the stresses may prevent the seismogenic structures in the region from experiencing continuous stress accumulation. Consequently, they remain in a relatively constant stress state, leading to a seismic probability that tends to stabilize over time (Fig. 7). When the elapsed time is relatively long (relative to the earthquake recurrence interval), the waiting time for the next event based on the BPT model will exceed that of the Poisson model, i.e., the occurrence probability of the former will be smaller than that of the latter.
The results of this study may serve as important reference for seismic hazard analysis and earthquake prediction in North China.