The Study of Sample Size on b-value Statistics in the Gutenberg-Richter's Law
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摘要: b值是研究地震活动的重要指标，其广泛应用于地震危险性分析和地震预测研究之中，与实际资料的完整性、样本量的大小、计算方法等因素有着重要的关系。常见的b值计算方法有最小二乘法和最大似然法，样本量的大小对这2种方法影响很大。本文利用蒙特卡罗模拟地震目录和汾渭地震带实际目录作为样本，从中抽取不同大小的样本量进行计算，研究不同样本量下这2种方法计算得到的b值与设定值或真实值之间的差别。结果表明，最小二乘法需要的最低样本量为1000，最大似然法为200；当样本量达不到要求时，计算出的b值是不可靠的；由于对样本量的要求不同，前者适用于计算区域的整体b值，而后者在研究某区域b值在时间轴上的变化方面更有优势。本研究为确定2种b值计算方法对样本量的最低要求提供了参考依据。Abstract: The b-value is an important indicator for evaluating the level of seismicity, and is widely used in seismic hazard analysis and earthquake prediction research. The b-value is basically effected by some factors, such as the actual data integrity, the quantity of earthquake samples, calculation methods and so on. The most widely used two methods for calculating b values include the least square method (LSM) and the maximum likelihood method (MLM), and the results from both methods can remarkably vary with different the quantity of earthquake samples. In this paper, based on the real catalog of the Fen-Wei seismic zone and the Monte-Carlo simulated earthquake catalog, we use different number of samples to calculate the b values and try to find the differences between the calculated results and the set values/true values. It turns out that the threshold value of samples is 1000 and 200 for LSM and MLM, respectively. When the sample size does not meet the requirements, the calculated b-value is not reliable. It suggets that LSM is suitable for calculating the b-value of the whole region, while MLM is more advantageous in studying the b-value of a region on the time axis. This study provides a reference for the threshold values of the samples for two methods, which is of great significance to the seismic hazard analysis and earthquake prediction.
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Key words:
- b-value /
- The least square method /
- The maximum likelihood method /
- Sample size
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图 1 汾渭地震带范围及震中分布
Figure 1. The distribution of earthquakes in the Fen-Wei seismic tectonic zone
图 2 不同样本量下最大似然法计算得到的平均b值拟合图
Figure 2. The fitting curve of mean b values calculated by maximum likelihood method under different sample sizes
图 3 不同样本量下最小二乘法计算得到的平均b值拟合图
Figure 3. The fitting curve of mean b values calculated by least squares method under different sample sizes
图 4 不同样本量的b值分布直方图
Figure 4. The histogram of the b value calculated under different sample sizes
表 1 最大似然法模拟结果
Table 1. Simulation results by the maximum likelihood method
N b1 Pr- Pr Pr+ σ 10 1.221 0.46 0.14 0.40 0.42 20 1.211 0.46 0.25 0.29 0.38 50 1.145 0.44 0.30 0.16 0.16 100 1.061 0.53 0.37 0 0.13 200 1.039 0.52 0.48 0 0.11 400 1.036 0.47 0.53 0 0.09 500 1.025 0.45 0.55 0 0.08 1000 1.003 0.30 0.70 0 0.08 
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表 2 最小二乘法模拟结果
Table 2. Simulation results by the least squares method
N b1 Pr- Pr Pr+ σ 10 0.870 0.79 0.08 0.13 0.52 20 0.895 0.76 0.07 0.17 0.48 50 0.918 0.68 0.27 0.05 0.42 100 0.941 0.64 0.28 0.08 0.37 200 0.942 0.46 0.48 0.06 0.32 400 0.955 0.48 0.52 0 0.28 500 0.982 0.45 0.50 0.05 0.16 1000 0.996 0.20 0.80 0 0.12
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表 3 汾渭地震带地震分档统计和年平均发生率
Table 3. The annual average incidence of different magnitude-class in Fen-Wei seismic zone
震级档M 地震个数 年平均发生率 2.0—2.4 5038 119.95 2.5—2.9 2006 47.76 3.0—3.4 734 17.48 3.5—3.9 230 5.61 4.0—4.4 110 2.62 4.5—4.9 91 2.17 5.0—5.4 15 0.36 5.5—5.9 10 0.23 6.0—8.5 3 0.07 注：年平均发生率指震级≥M的年均地震数，代表地震活动水平。
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表 4 汾渭地震带b值拟合情况(1500—2010年)
Table 4. The b value fitting of earthquakes from 1500—2010 in the Fen-Wei seismic zone
样本量 均匀抽样计算b值 最小二乘法 最大似然法 10 0.60 0.897 50 0.62 0.923 100 0.63 0.816 200 0.63 0.728 300 0.65 0.767 500 0.69 0.752 700 0.70 0.756 1000 0.72 0.746 2000 0.74 0.751 5000 0.75 0.753
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