Earthquake Reaearch in China  2018, Vol. 32 Issue (4): 533-548
Seismic Inventory Simulation and Conformance Test Based on the Monte Carlo Method-A Case Study of the Fenhe-Weihe Belt
Shao Xiaoyi1,2, Wang Xiaoqing1, Dou Aixia1, Yuan Xiaoxiang1
1. Institute of Earthquake Forecasting, China Earthquake Administration, Beijing 100036, China;
2. Key Laboratory of Active Tectonics and Volcano, Institute of Geology, China Earthquake Administration, Beijing 100029, China
Abstract: Artificial earthquake catalogue simulation is one of the ways to effectively improve the incompleteness of the existing earthquake catalogue, the scarcity of large earthquake records and the improvement of seismological research.Based on the Poisson distribution model of seismic activity and the Gutenberg-Richter magnitude-frequency relationship, the Monte Carlo method which can describe the characteristics of the stochastic nature and the physical experiment process is used.This paper simulates the future seismic catalogues of the Fenhe-Weihe seismic belt of different durations and conducts statistical tests on them. The analysis shows that the simulation catalogue meets the set seismic activity parameters and meets the Poisson distribution hypothesis, which can obtain a better simulated earthquake catalogues that meets the seismic activity characteristics.According to the simulated earthquake catalogues, future earthquake trends in this region are analyzed to provide reference for seismic hazard analysis.
Key words: Monte Carlo method     Simulated earthquake catalogues     Seismic activity     Fenhe-Weihe seismic belt

INTRODUCTION

The earthquake catalogue is the most important collection of data for seismic activity research and seismic hazard analysis. The catalogue of earthquakes recorded by modern instruments is only a few decades old. Although historical records and ancient earthquake studies can extend the time of the earthquake catalogue, it is not enough to study large earthquake events with long periods and low probability. Whether it is based on statistical methods to recreate earthquakes with a period of 100 years or use earthquake precursors which is based on certain physical model assumptions to test and correct, it is still not enough to realize earthquake prediction and seismic zoning studies (Zhou Shiyong, 2008). Therefore, it is of great significance to carry out seismic catalogue simulation research.

Artificial earthquake catalogue simulations have been in development for decades. It can be roughly divided into mechanical experiments, numerical simulations, statistics and the combination of mechanical and statistical models. Mogi K., (1962) recorded the acoustic emission generated by micro-fracture events through rock fracture mechanics experiments and simulated the earthquake catalogue based on the sequence of micro-rupture events. According to the sequence of rupture events (which may include multiple "pre-seismic" or "aftershocks"), the rupture processes of the main shock, pre-seismic-main shock-aftershock and group-seismic types were studied (Tang Chun'an, 1997; Tang Chun'an et al., 1997). These simulation results have important implications for the mechanism of earthquake preparation and occurrence, but it is impossible to simulate a long-term earthquake catalogue composed of a series of independent seismic events.

Seismic catalogue simulation based on a combination of mechanical and statistical models is a hypothetical basis which is based on a theoretical model of the mechanism of earthquake occurrence (Sólnes J. et al., 1997; Shi Yaolin et al., 1998; Console R. et al., 2015; Zhou Shiyong et al., 2006). Such as the basic spring-slider model (Burridge R. et al., 1967) and the modified spring-slider-damper model (Zhu Yuanqing et al., 1991) and the fault (interaction) system based on the modified spring-slider-damper model (Jin Xin et al., 2017; Zhou Shiyong, 2008), and based on the stress-strain process reflected by these mechanical models, combined with cellular automaton model (Liu Guiping et al., 1995, 2000), coupled stress release statistical model (AIC)(Li Hong et al., 2015), Brown time process model (BPT model), time predictable model or empirical model (Working Group on California Earthquake Probabilities, 2003;Hainzl S. et al., 2007), EEPAS model (Rhoades D.A. et al., 2011;Bi Jinmeng et al., 2017) to simulate the earthquake catalogue. The spring-slider model or the fault system model has a relatively clear physical meaning and a relatively strict theoretical basis, but there are many factors affecting the occurrence of earthquakes. The setting of the model parameters is different from the actual fracture structure. The more complex the model, the more hypothetical conditions, resulting in greater uncertainty in the simulated earthquake catalogue, and the simulation results have certain difficulties (Li Hong et al., 2015).

The statistical-based earthquake catalogue simulation method mainly uses the random test method to simulate the earthquake catalogue based on the probability distribution characteristics of seismic activity, such as the Poisson model (Zhao Hong et al., 2015) and the lognormal model (Nishenko S.P. et al., 1987), Weibull distribution model (Hagiwara Y., 1974), Bernoulli model (Fu Zhengxiang et al., 1995), ETAS model (Jiang Changsheng et al., 2013; Xu Weijin et al., 2017). These methods are not directly combined with the cause of the earthquake, but by combining the characteristics of time activities such as seismic activity or group activities, spatial clustering, and the characteristics of the active structural block and its marginal activities. To a certain extent close to the actual seismic space-time activity law.

The Monte Carlo method uses random sampling techniques to solve physical problems (Huh U. et al., 2016), which more realistically describes the characteristics of the things with a random nature and the physical experimental process. In this paper, we will adopt the Poisson model, based on the Monte Carlo random independent trial, combined with the earthquake magnitude distribution and time distribution law, simulate the regional earthquake catalogue, and conduct the conformity test on the generated catalogue, and based on this catalogue, analyze the future seismic hazard of the study area.

1 SEISMIC CATALOGUE SIMULATION METHOD BASED ON THE MONTE CARLO RANDOM INDEPENDENT REPEATED TEST

The Monte Carlo method, also known as the statistical simulation method or random sampling technique, is a numerical calculation method based on probability and statistical theory. The method has better processing for small probability events, limited by geometric conditions, and the convergence speed is independent of the dimension of the problem. It can be applied to earthquakes, earthquake disaster analysis and prediction, seismic hazard and seismic risk assessments (Liu Shanqi et al., 2013; Liao Jinggao et al., 2014; Iervolino I. et al., 2016). It can also be used in seismic activity (Ebel J.E. et al., 1999; Musson R.M.W., 2000; Shaw D. et al., 2007; Yazdani A. et al., 2012; Assatourians K. et al., 2013; Mohammed T. et al., 2014; Guo Xing, 2015; Guo Xing et al., 2016), seismic hazard analysis (Wang J.P. et al., 2014; Pavel F. et al., 2017; Setiawan B., 2017) and seismic risk analysis (Weatherill G.A., 2009; Yazdani A. et al., 2012; Bourne S.J. et al., 2015; Liu Jiamei et al., 2016).

1.1 The Monte Carlo Random Independent Test Method

For a given physical (or mathematical) system, if the relationship between a core variable (s) and its probability distribution function F(s) is known, i.e.

 $F(s)=r$ (1)

The function F(s) is a monotonically increasing function of the range [0, 1]. Then, by using random independent sampling to determine the uniform random number r of the interval [0, 1], the inverse transformation of the equation (1) is obtained.

 $s=F^{-1}(r)$ (2)

The value of the core variable s can be obtained. Through multiple independent repeated sampling, a series of core variable values can be determined, and a random simulation test of the system parameters is completed. The above method, called inverse sampling, is one of the basic methods of the Monte Carlo randomized trials (Huh U. et al., 2016).

1.2 Determination of the Magnitude Distribution within Specified Years

Assuming that the magnitude distribution of the earthquake satisfies the truncated G-R relationship (Cornell C.A., 1968), the earthquake magnitude distribution function F(m) is (Youngs R.R. et al., 1985):

 $F\left(m \right) = \left\{ \begin{array}{l} \frac{{1 - {e^{ - \beta (m - {m_0})}}}}{{1 - {e^{ - \beta ({m_{uz - }}{m_0})}}}}, \;\;\;{m_0} \le m \le {m_{uz}}\\ 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{other}}\;{\rm{conditions}} \end{array} \right.$ (3)

Where β=2.3b, b is the coefficient of the G-R relationship in the study area, m0 is the lower limit of the magnitude, and muz is the upper limit of the magnitude. F(m) is a monotonically increasing function of the interval [0, 1], let F(m)=r, transform the equation (3),

 $m = {m_0} - \ln \left\{ {1 - r*\left[ {1 - {e^{ - \beta \left({{m_{uz}} - {m_0}} \right)}}} \right]} \right\}/\beta$ (4)

The value of F(m) is determined by generating a random number r of the interval [0, 1], and the magnitude of the earthquake m is obtained according to the equation (4).

1.3 The Number of Earthquakes within the Specified Years and the Interval between Two Adjacent Earthquakes Are Determined

Assuming that the seismic activity follows the Poisson distribution, for a region with an annual average rate of v0, the probability of occurrence of k times earthquakes greater than m0 during the [0, t] time period is (Pan Hua et al., 2013):

 $P\left({n = k} \right) = \frac{{{{({v_0} \cdot t)}^k}}}{{k!}} \cdot {e^{ - {v_0}t}}, \;\;\;\;\;\;\left({k = 1, 2, 3 \cdots \cdots } \right)$ (5)

Adding the formula (5), the probability distribution function of the number of earthquake occurrences is:

 $P\left({n \le k} \right) = \mathop \sum \limits_{n = 0}^k P\left({n = k} \right) = \mathop \sum \limits_{n = 0}^k \frac{{{{({v_0} \cdot t)}^k}}}{{k!}} \cdot {e^{ - {v_0}t}}, \;\;\;\;\;\;\;\;\left({k = 1, 2, 3 \cdots \cdots } \right)$ (6)

A random number between [0, 1] is determined by random sampling as the value of the probability distribution function P(nk). According to the annual occurrence rate v0 and time t, according to equation (6), the inverse sampling method can be used, we can determine the number of occurrences of earthquakes (k) during this time period.

Since the earthquake obeys the Poisson distribution, the time interval of any two earthquakes τj=tj-tj-1 follows the exponential distribution(Hu Yuxian, 2006), and its distribution function is:

 $F\left({{t_j}} \right) = \left\{ \begin{array}{l} 0, \;\;\;\;\;\;\;\;\;\;\;\;{t_j} < 0\\ 1 - {e^{ - {v_0}{t_j}}}, \;\;\;{t_j} \ge 0 \end{array} \right.$ (7)

Following the inverse sampling method, a time interval sequence of seismic events can be obtained. Starting from the first earthquake, the time rj of the earthquake of the jth earthquake event can be obtained according to the time interval tj

Therefore, through Monte Carlo random independent trials, the regional earthquake magnitude distribution and the number of earthquake events k are determined based on the annual rate v0 and b-value. On this basis, for each earthquake event, the magnitude m and the earthquake time t are determined, and a simulated earthquake catalogue that satisfies the Poisson distribution assumption is obtained (Fig. 1).

 Fig. 1 Technical flow chart for earthquake catalog generation
2 EARTHQUAKE CATALOGUE SIMULATIONS IN THE FENHE-WEIHE SEISMIC BELT

The Fenhe-Weihe seismic belt is located in the east and south margins of the Ordos block, from the Weihe Basin in the south and to the Huailai-Yanqing Basin in the north (Wang Ji et al., 2002). As of 2014, there were 130 earthquakes with magnitudes 5.0 or above in the region, of which 23 occurred in the 6.0-6.9 earthquake, 7 in the 7.0-7.9 earthquake, and 2 in the magnitude 8.0 earthquake (the earthquake of M8¼ in Huaxian County on February 2, 1556 and the magnitude 8.0 earthquake in Hongdong on September 25, 1303). Strong earthquakes mainly occurred in the Luding Basin, the Linfen Basin and the eastern part of the Weihe Fault Depression Belt in the Shanxi Fault Depression (Fig. 2). The fifth generation ground motion parameter zoning map is based on the seismic data collected from the Fenhe-Weihe seismic belt and the analysis of seismic activity characteristics. The seismic data incompleteness and cognitive uncertainty are fully considered, and the historical seismic activity level and characteristics are determined (Department of Earthquake Disaster Prevention of China Earthquake Administration, 1999; Institute of Geophysics, China Earthquake Administration, 2015; Earthquake Administration of Shaanxi Province, 2005). The seismic activity parameters of the Fenhe-Weihe seismic belt are b=0.78, v0=2.5, m0=4, muz=8.5 (Table 1, Fig 3).

 Fig. 2 The epicentral distribution of historical destructive earthquakes in the Fenhe-Weihe seismic belt (From 780 to 2014, M≥4.6, F1 is the Changzhi fault, F2 is the Taigu fault, F3 is the Jiaocheng fault, F4 is the Xidanshan fault, F5 is the Sunzhuangzi-Wulonggou fault, F6 is the Hengshan fault, F7 is the Kouquan fault, F8 is the Qinling fault, F9 is the Lishi fault, F10 is the Hancheng fault)

Table 1 The average annual activity rate of earthquakes occurrence in different periods of the earthquake statistical area

 Fig. 3 The b-value and annual incidence rate v0 of the Fenhe-Weihe seismic belt

According to the aforementioned earthquake catalogue simulation method, the earthquake catalogue (30 years, 50 years, 70 years, 100 years, etc.) of different time scales in the future of the Fenhe-Weihe earthquake belt is simulated. Each time scale is simulated 20, 000 sets. Fig. 4 shows the m-t relationship of randomly selected simulated earthquake catalogs with six time scales of 50 years.

 Fig. 4 The 6 simulate catalogues of M-t

In the 20, 000-group, the 50-year simulated earthquake catalogue, the number of earthquakes with magnitude 4.0 or higher has an average of about 125(v0×T), which is very close to the theoretical value. The frequency distribution is counted at intervals of 10, and it can be seen from Fig. 5 that the number of earthquakes corresponding to the peak ranges from 120 to 130, and the maximum frequency is 0.17.

 Fig. 5 Frequency distribution of seismic number of 20, 000 sets of simulated earthquake catalogs in the 50-year in the Fenhe-Weihe seismic belt
3 SIMULATION OF PARAMETERS AND ERROR ESTIMATION

Mathematical expectation, variance, and standard deviation are other important characteristics of random variables for sequence xi

Mathematical expectation:

 $X = \frac{1}{N}\mathop \sum \limits_{I = 1}^N {x_i}, \;\;\;\;\;\;i = 1, 2, 3 \cdots N$ (8)

Variance:

 ${S^2} = \frac{1}{N}\mathop \sum \limits_{I = 1}^N {({x_i} - X)^2}, \;\;\;\;\;\;i = 1, 2, 3 \cdots N$ (9)

The standard deviation, the sum of the squared residuals and the deterministic coefficients (parameters that determine the goodness of fitting) generally determine the fitting effect.

Standard error:

 $MS = \sqrt {{S^2} + {{\left({X - \hat X} \right)}^2}}$ (10)

Residual sum of squares:

 $RRS = \mathop \sum \limits_i^k {[{y_k} - \hat y]^2},$ (11)

where yk is the measured value and $\hat y$ is the predicted value.

Coefficient of determination:

 $R = 1 - \frac{{RRS}}{{SST}}$ (12)

where SST is the Total sum of square, $SST = \mathop \sum \limits_{i = 1}^k {({y_k} - \bar y)^2}$, y is the average of the values to be fitted.

3.1 Test of G-R Relationship

In order to verify whether the generated earthquake catalogue meets the set seismic activity parameters and whether the seismic activity poisson distribution assumption is met, the number of earthquakes of magnitude 4.0 or higher per year in each group of catalogues is counted, and 50 samples are obtained for each group (T=50 years). When the earthquake occurs i times (i=0, 1, 2, …), the probability density of the number of occurrences of the earthquake ρ(i) is

 $\rho \left(i \right) = \frac{i}{{50}}$ (13)

We randomly select six groups of sub-catalogues in the Fenhe-Weihe seismic belt as an example to perform parameter fitting. Fig. 6 shows the Poisson distribution fitting diagram of the number of annual earthquakes obtained by the non-linear fitting method. Table 2 shows the v0 fitted value to each group and its error parameters. The goodness of fit between the simulated earthquake occurrence probability density and the fitted Poisson probability density value is above 0.8, and the sum of squared residuals is in the order of 10-2. It can be seen that the catalogue generated by the random test can better match the Poisson distribution.

 Fig. 6 Poisson distribution fitting diagram

Table 2 Simulation catalogues of v0 value test parameter

According to the above method, thev0 value of each group of catalogues in the 20, 000-group earthquake catalogue is calculated, and the mean value is 2.46, and the standard deviation is 0.22. The distribution is counted at intervals of 0.2. It can be seen from Fig. 7(a) that the v0 corresponding to the peak value of the curve is 2.4 to 2.6, which is substantially close to the set value (2.5). Fig. 7(b) shows the number distribution of earthquake catalogs corresponding to different v0 values and different earthquake numbers (20, 000 earthquake catalogues). When the number of earthquakes is 120-130 and the v0 value is 2.4-2.6, the number of earthquake catalogue groups are the most, with the total of 2, 120 groups.

 Fig. 7 The relationship between the number of earthquakes in the annual sample and the distribution of v0 values (T=50 years)

In order to analyze the influence of different catalogue durations on the results, v0 is calculated for earthquake catalogues of 20, 000 sets of different durations from 30 to 150 years, and the mean, median, 4-interval, extremum and standard error of different durations are calculated. It can be seen from Fig. 8 and Table 3 that the longer the catalogue time, the more convergent the result, the closer the average value of the data is to the theoretical value, and the smaller the standard error.

 Fig. 8 Simulation catalogue different time b-value statistics

Table 3 Simulation catalogue different duration v0 value test parameter table
3.2 The b-value Test

Table 4 shows the random selection of 4 sets of earthquake catalogues using the least squares fitting method to calculate the b-value and parameters, the goodness of fit is above 0.9, and the sum of squared residuals is less than 0.4.

Table 4 The b-value fitting test parameters of earthquake catalog

According to the above method, the b-value of each group of catalogues in the 20, 000-group earthquake catalogue is calculated, and the mean value is 0.76, and the standard deviation is 0.13.

The distribution is counted at intervals of 0.2. It can be seen from Fig. 9(a) that the b-value corresponding to the peak value of the curve is 0.76, which is substantially close to the set value (0.78). Fig 9(b) shows the number distribution of earthquake catalogues corresponding to different b values and different earthquake numbers (20, 000 earthquake catalogues). When the number of earthquakes is 120-130 and the b-value is 0.8, the number of earthquake catalogue groups is at most, a total of 540 groups.

 Fig. 9 The relationship between the number of earthquakes in the annual sample and the distribution of b values (T=50 years)

In order to analyze the influence of different catalogue durations on the results, b-value is calculated for earthquake catalogues of 20, 000 sets of different durations from 30 years to 150 years, and the mean, median, 4-interval, extremum and standard error of different durations are calculated.

It can be seen from Fig. 10 and Table 5 that the longer the catalog time, the more convergent the result, the closer the average value of the data is to the theoretical value, and the smaller the standard error.

 Fig. 10 Simulation catalogue different time b-value statistics

Table 5 Simulation catalogue different duration b-value test parameter table

The simulation results show that the longer the simulated earthquake catalogue time, the smaller the standard error of the parameters, and the closer the mean value is to the true value, the more representative of the seismic activity law described by the Poisson model.

Medium- and long-term earthquake prediction is an estimation and prediction of the strong earthquake situation and risk in a certain area in the next few years to decades. The results can be directly served to the society and the basis for short-term forecasting. Therefore, using the above simulation to generate a 20, 000-group, 100-year earthquake catalogue, the number of catalogues with the specified magnitude is counted as a percentage of the total catalogue. The probability of earthquakes occurrence of magnitude 7.0 or higher and magnitude 8.0 or higher in the next 100 years is 0.66 and 0.19, respectively. The results calculated by the Poisson model theory is basically the same as 0.67 and 0.18.

There was no earthquake of magnitude 6.5 or higher in 202 years since the earthquake of the Pinglu MS6 ¾ earthquake in 1815. In 1998, there was no earthquake of magnitude 6.0 or above in the 19 years since the Zhangbei MS6.2 earthquake. Using the Monte Carlo method to generate 20, 000 sets of the 212-year earthquake catalogue, the magnitude 6.5 earthquake was found in the first 202 years in the simulation catalogue, and take the last ten years of the simulated sequence in the catalogue, and count the earthquakes with magnitude 6.5 or higher in these sequences. According to the proportion of the catalogue, the conditional probability of the earthquake with a magnitude of 6.5 or higher in the next 10 years is 0.22 in the absence of an earthquake of magnitude 6.5 or higher in 202 years. And we also can obtain the conditional probability of the earthquake with a magnitude of 6.2 or higher in the next 10 years is 0. 0.31 in the absence of an earthquake of magnitude 6.2 or higher in 19 years.

4 CONCLUSION AND DISCUSSION

The Monte Carlo simulation makes random and independent sampling, so that the distribution of a certain parameter satisfies the known or set probability distribution, and finally realizes the simulation of the research problem. Through a large number of repeated experiments, it can truly approach the actual physical process, subject to geometric conditions. The limit is small, and it is better for small probability events (Huh U. et al., 2016). In this paper, by changing the magnitude relationship of earthquake events, the time distribution model, the Monte Carlo simulation can be used to obtain simulated earthquake catalogues under various complex model conditions. The simulation catalogue fills the vacancies of the earthquake catalogue caused by insufficient monitoring methods or incomplete historical records (no time) and vacancies in the records, which makes the earthquake sequences in those long-term inactive areas more complete.

Based on the Poisson distribution model of seismic activity and the Gutenberg-Ricart magnitude-frequency (G-R) relationship hypothesis, the anti-sampling method and the direct sampling method are used to generate a random number to simulate the earthquake catalogue, based on the seismic activity parameters of the Fenhe-Weihe seismic belt. The earthquake catalogue simulation of different time lengths of the Fenhe-Weihe seismic belt is simulated. The test results show that the earthquake catalogue obtained by this method reflects the statistical characteristics of seismic activity. On this basis, the probability of earthquakes with magnitude 7.0 or higher and magnitude 8.0 or higher in the next 100 years is 0.66 and 0.19, respectively, using the simulated earthquake catalogue calculation. On the condition of earthquakes of magnitude 6.5 or higher in the 202 years since the earthquake of the Pinglu MS6 ¾ earthquake in 1815. In 1998, there was no earthquake of magnitude 6 or above in the 19 years since the Zhangbei MS6.2 earthquake, the conditional probability of occurrence of earthquakes of magnitude 6.5 or higher and magnitude 6.2 or higher in the next 10 years is 0.22 and 0.31, respectively.

Because the Poisson model used in the paper and real seismic activity often does not coincide with the cluster model, considering the mutual excitation problem between earthquakes (Poisson + cluster), the sampling function can be changed in the future to introduce the ETAS model (Marzocchi W. et al., 2011), which provides more accurate results in areas with less ancient seismic records and complex geological structures (Tom P., 2008).

ACKNOWLEDGEMENT

Thanks to the fifth-generation zoning map provided by the Institute of Geophysics, China Earthquake Administration. We're grateful to the Shaanxi Earthquake Agency for their support and assistance in collecting information, and thank the reviewers for their comments and suggestions.

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1. 中国地震局地震预测研究所，北京市复兴路63号 100036