# G-R Curves Based on Weibull Distribution

Authors: Yubo Wang; Jie Yu; Gang Shen; Tiebin Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

China Earthquake time intervals have been calculated and passed the hypothesis test based on Weibull distribution. Comparative analyses have been carried out with the China Earthquake Catalogue according to different time intervals and different zones. From the results, we can see that the shape parameters are enlarged with the year’s extrapolation.b value curves and Amira images of the global plate subduction zones have been drawn, and the parameters of Weibull Distribution have been calculated. The relationship diagrams of b values and the parameters of Weibull Distribution of the whole subduction zones have been drawn.


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## Body

## 1. Introduction

Weibull distribution possesses rich predictions contents which can be applied in the analysis of the reliability and risks of the systems [1–4]. Weibull distribution contains three parameters such as shape parameter, scale parameter, and position parameter. Exponential distribution and normal distribution can be looked as the special cases of Weibull distribution which is applicable of all types of tests and is used widely in all areas [5–9].We analyzed the interval between earthquakes in Taiwan (From Jan. 1st 1973 to Dec. 31st 2010) and drew the scatter diagram of density function and distribution function. We think that the distribution type is Weibull distribution. The parameters of Weibull distribution are estimated, and the linear correlation test is carried out to obtain the regression equation [10, 11]. Finally, the hypothesis test is carried out. The distribution of earthquake interval time conforms to the hypothetical Weibull distribution, while the exponential distribution does not pass the hypothesis test. The Weibull distribution parameters in Taiwan are obtained as follows: shape parameter: k^=1.011 and scale parameter: b^=115.55.


## 2. Earthquake Data Based on Weibull Distribution

We analyze the earthquake data of China from USGS, and we draw the conclusion that the earthquake time intervals obey Weibull distribution [12, 13]. The time of the catalog is from Jan. 1, 1973 to Dec. 31, 2015, and the earthquake magnitude is above 4.0 and the depth of focus is from 0 to 70 km.We group the earthquake catalog of China according to the areas and study the rules followed of the different parts. These are Taiwan areas, Kunlun-Qilian Mountains-Qinling areas, Tibet and Xinjiang areas, Tianshan areas, Sichuan and Yunnan areas, Erdos-Qilian Mountains-Qinling areas, North of China, and Northeast of China. Then, we analyze the laws followed by the whole earthquake catalog.We group the earthquakes that occurred in the areas according to intervals of each year and get forty-three groups.The earthquake intervals of these areas all accord to Weibull distribution and pass through hypothesis tests. We calculate the parameters of Weibull distribution of other areas which are given as below:From Table1, we can see that the estimated parameters b^ and k^ of different areas are different and the correlation parameters are also different. There are always some states showing a single peak in the curve of density function, and the shape parameter k^ changes around 1.0.Table 1
The parameters of Weibull distribution of these areas.
The areasρ^ραbThe estimated parameters (1973-2015) (t=516)Taiwan areas0.99150.27050.97k^=1.021b^=126.43Kunlun-Qlianshan-Qinling areas0.98960.27050.90k^=1.169b^=128.26Tibet and Xinjiang areas0.98720.27050.98k^=1.152b^=135.71Tianshan areas0.97220.27050.97k^=1.099b^=132.87Chuandian areas0.98110.27050.92k^=1.206b^=129.21Erdos-Qilianshan-Qinling areas0.99210.27051.07k^=1.079b^=117.33North of China0.96010.27050.65k^=1.301b^=169.65The total of the eight areas0.98220.27050.94k^=1.152b^=137.68Mainland and Taiwan areas0.98010.27050.98k^=1.011b^=105.97From the above analysis, we can think that the earthquake intervals obey Weibull distribution [14, 15]. We can discuss the changes of b and its laws when k^ is about 1.0.We analyze the earthquake data in China earthquake catalog from Jan. 1, 1973 to Dec. 31, 2010 with the magnitude lower limit of 4.0. The parameters andb values are given in Table 2. We contrast the results with the data from Jan. 1, 1973 to Dec. 31, 2015.Table 2
The contrast analysis of China earthquake catalog.
The areasThe estimated parameters (1973-2015) (t=516 m)b (the magnitude lower limit of 4.0)b (the magnitude lower limit 5.0)The estimated parameters (1973-2010) (t=456 m)b (the magnitude lower limit 4.0)b (the magnitude lower limit 5.0)Taiwan areask^=1.021b^=126.43b=0.97b=0.95k^=1.011b^=115.55b=0.99b=1.02Kunlun-Qlianshan-Qinling areask^=1.169b^=128.26b=0.90b=0.88k^=1.233b^=136.65b=0.87b=0.80Tibet and Xinjiang areask^=1.152b^=135.71b=0.98b=0.95k^=1.172b^=140.63b=0.99b=1.00Tianshan areask^=1.099b^=132.87b=0.97b=0.96k^=1.077b^=135.52b=0.98b=0.94Chuandian areask^=1.206b^=129.21b=0.92b=0.89k^=1.125b^=130.19b=0.90b=0.89Erdos-Qilianshan-Qinling areask^=1.079b^=117.33b=1.07b=0.99k^=1.109b^=126.65b=1.18b=1.86North of Chinak^=1.301b^=169.65b=0.65b=0.53k^=1.299b^=160.08b=0.68b=0.57The total of the eight areask^=1.152b^=137.68b=0.95b=0.95k^=1.149b^=135.55b=0.95b=0.92Mainland and Taiwan areask^=1.011b^=105.97b=0.99b=0.96k^=1.009b^=107.26b=0.99b=1.03From Table2, we can see that b values from 1973 to 2010 are lower than those from 1973 to 2015 and the shape parameters k^ correspondingly deduce.Shape parameters can be used to describe the possibility of earthquake occurrence over time. When the shape parameter is less than 1, it indicates that the possibility of earthquake decreases with time. When the shape parameter is equal to or approximate to 1, it means that the possibility of earthquake will not change with time, and the occurrence of earthquake is accidental.


## 3.B Value Analysis Based on Weibull Distribution of the Global Plate Subduction Zone

According to the seismic catalog which comes from USGS during 1st Jan. 1973 and 31st December 2015. We get earthquakes above level 4.0, and we draw the temporal clustering of global seismic image. x-axis shows the subduction zones, and y-axis shows the time. Different earthquake levels show with different color balls. Blue balls show the levels from 4.0 to 4.9. Green balls show the levels from 5.0 to 5.9. Pink balls show the levels from 6.0 to 6.9. Yellow balls show the levels from 6.0 to 6.9. Red balls show the levels above 8.0.From Figure1, we can see that the global subduction zones consisting of several main seismic clustering and the central pacific plates cluster intensive earthquakes and the central pacific zones often have earthquake and erupt frequently [16, 17]. There are a series of trenches, islands, and volcanoes on the central pacific plates with violent plate movements. The Alpine-Himalayan belt is clustered earthquakes where are mainly east-west huge mountains across the central and southern Eurasia and northern Africa.Figure 1
Temporal clustering of global seismic image.The earthquakes are also active in mid-Atlantic ridge extending south-north of the Pacific ocean which lies in the middle of the ocean basin and extends to the series of mainland shore between flat abyssal plains. Figure2 is the 3D image of the global catalog with the depth. From the depth, we can see that the earthquakes cluster intensively around the east ridge and the west ridge of the Pacific plates and distribute widely in the depth.Figure 2
3D image of the global catalog with the depth.The seismic catalog of the whole Pacific seismic belt, Eurasian seismic zone, and Ridge earthquake belt shows the obvious earthquake cluster properties in terms of time and space. Therefore, we can get the main activity properties of the main plate belts of the whole global from the probability and statistics. With the supplement and improvement of the seismic catalog, we can adopt the visualization of the earthquake clusters and Weibull distribution to estimate the distribution of precursor of the main seismic initially which can be used as the seismic prediction.From Figures3 and 4 we can see that the fitting curves have no data dots below the fitting lines. We can draw the results that the seismic catalog records above level 4 and level 5 are complete. These data provide a reliable basis for the following statistics analysis.Figure 3
b value curve of the global catalog (limit level: 4.0).Figure 4
b value curve of the global catalog (limit level: 5.0).Weibull parameters of the global catalog are given as below:k^=1.110,b^=128.59,t=463m.The shape parameter is greater than 1, and we can think that the occurrence of earthquake is accidental.We divide the global plate subduction zone into 338 parts (as shown in Figure5). For the parts with no records, we do not analyze. For some seismic zones, we cannot draw b value curves because the earthquake levels below 5.0. The data are also from USGS during 1st Jan. 1973 and 31st December 2015.Figure 5
The global seismic map (1/1/1973-1/12/2015).We select representative subduction zones and drawb value curves and 3D images and calculate Weibull parameters [18–20]. We analyze the physical meanings of the fault zones.Subduction 10 (longitude: 142-180, latitude: 60-51):b value curve and 3D image are given as Figures 6 and 7.Figure 6
b value of subduction 10.Figure 7
3D image of subduction 10.We get Weibull parameters of subduction 10:k^=1.05,b^=114.31.Subduction 177 (longitude: 118.93-135.18, latitude: 12.52-3.05):b value curve and 3D image are given as Figures 8 and 9.Figure 8
b value of subduction 177.Figure 9
3D image of subduction 177.We get Weibull parameters of subduction 177:k^=1.03,b^=105.09.From Figure6–9, the shape parameters are all close to 1.0, and we can think that the occurrence of earthquake is accidental.We draw the curves of shape parametersk^ and b values of each subduction zones with the horizontal direction of b values and vertical direction of shape parameters k^.From Figure10, we can see that k^ is always greater than 1.0 which is consistent to the unimodal state of the density function curve. When b values change near 1.0, shape parameters also change near 1.0. For the subduction zones with large changes of b values, shape parameters change obviously.Figure 10
b values and shape parameters k^.We draw the curves of scale parametersb^ and b values of each subduction zones with horizontal direction of b values and vertical direction of shape parameters b^.From Figure11, we can see that scale parameters b^ are in fluctuating state with the changes of b values. There is no correlation between the b values of the G-R law and the scale parameters b^ of Weibull distribution.Figure 11
b values and shape parameters b^.


## 4. Conclusions

We contrast the data in the China earthquake catalog according to different time intervals and areas. From the results, we can see that the shape parametersk^ increase with the years increased (only with the magnitude lower limit of 4.0). The conclusions are correct or do not need to be proven with a lot of calculations and facts. We will develop this part in the near future.During the process of the hypothesis testing, we equally divide the time intervals for the convenience of calculation which are considerable discrepancy between the fault intervals of the engineering. So we think it is necessary to divide the earthquake time randomly and then carry out the hypothesis testing of Weibull distribution. Thus, the conclusions will own actual meaning and implications of science.The paper calculatesb values and draws the figures of b values of the subduction zones. The paper draws 3D images of typical subduction zones and calculates parameters of Weibull distribution of the subduction zones with enough data.The result of this paper is very close to the conclusion of seismology, which verifies that the model hypothesis is correct and feasible.For the further relationship of Weibull distribution parameters andb values, we will carry out in the following study.



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*Source: 7741865-2022-03-14.xml*