Application of generalized extreme value distribution based on profile likelihood estimation in long term earthquake prediction
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摘要: 为描述强震预测的不确定性，在地震预报极值分析模型的参数估计中，引入轮廓似然估计法。对广义极值分布中形状参数和地震重现水平的轮廓似然估计原理及数值算法进行了详细地阐述，并利用构建的广义极值分布模型对东昆仑地震带进行了地震危险性分析。关于形状参数和重现水平的点估计，以及10年以内的重现水平置信区间的估计，轮廓似然估计法与极大似然估计法效果基本相同，但在中长期地震重现水平置信区间的预测中，轮廓似然估计法得到的关于置信水平不对称的置信区间，在强震水平下对预测震级的不确定性表达更准确，预测结果更加有效。Abstract: To describe the uncertainty of strong earthquake prediction, we introduced the profile likelihood estimation into parameter estimation of extreme value model for earthquake prediction. It is elaborated that the profile likelihood estimation principle and numerical algorithm of shape parameters and earthquake return level in generalized extreme value distribution. Meanwhile, a model of generalized extreme value distribution was created and was used to analyze the seismic risk of the East Kunlun seismic belt. The results showed that profile likelihood estimation and maximum likelihood estimation generated basically the same effect in point estimation of shape parameters and return level as well as the estimation of confidence interval of earthquake return level within 10 years. However, in the confidence interval estimation of moderate and long interval earthquake return level, the asymmetric confidence interval of return level obtained through the profile likelihood estimation can more accurately express the uncertainty of predicted magnitude of a strong earthquake and more effectively predict the outcome.
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图 1 东昆仑地震带的地震分布规律
（a） 地震空间分布；（b） M-t图
Figure 1. Distribution law of earthquakes in East Kunlun seismic zone
（a） Spatial distribution of earthquakes；（b） M-t diagram
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图 2 形状参数与轮廓对数似然函数之间的关系
Figure 2. Relationship between shape parameters and profile log likelihood function
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图 3 GEV分布模型适应性检验图
（a） 密度曲线与直方图；（b） P-P 检验
Figure 3. Adaptability test of GEV distribution model
（a） Density curves and histograms；（b） P-P test
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图 4 重现水平及置信区间的轮廓似然估计
（a） 20年重现期；（b） 50年重现期；（c） 100年重现期；（d） 500年重现期
Figure 4. The reappearance level and confidence interval of the profile likelihood estimation
（a） 20-year return period；（b） 50-year return period；（c）100-year return period；（d） 500-year return period
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图 5 重现水平的轮廓似然估计与极大似然估计对比
Figure 5. Comparation of the reproduction level between profile likelihood estimation and maximum likelihood estimation
表 1 轮廓似然估计与极大似然估计GEV的分布结果对比
Table 1 Comparation of the profile likelihood estimation and maximum likelihood estimationof GEV distribution
模型分析项目 轮廓似然估计 极大似然估计 参数估计 （−0.204 0，0.847 5，4.834 5） （−0.204 4，0.847 8，4.834 8） 形状参数置信区间 ［ −0.259 0，−0.134 0 ］ ［ −0.268 8，−0.140 1 ］ 震级理论上限 MS8.989 1 MS8.981 9 地震带最大震级均值 MS5.178 9 MS5.179 0 
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表 2 极大似然估计与轮廓似然估计的重现水平对比
Table 2 Comparation of the recurrence level between profile likelihood estimation and maximum likelihood estimation
重现期
/年极大似然估计
重现水平极大似然估计95%
置信区间轮廓似然估计
重现水平轮廓似然估计95%
置信区间轮廓估计重现水平
两侧区间长度比1 MS5.13 ［ 4.97，5.29 ］ MS5.13 ［ 4.98，5.29 ］ 1.07 5 MS6.36 ［ 6.15，6.57 ］ MS6.36 ［ 6.17，6.59 ］ 1.21 10 MS6.72 ［ 6.48，6.96 ］ MS6.72 ［ 6.51，7.00 ］ 1.33 20 MS7.03 ［ 6.75，7.30 ］ MS7.03 ［ 6.79，7.37 ］ 1.42 50 MS7.36 ［ 7.03，7.69 ］ MS7.36 ［ 7.10，7.80 ］ 1.69 100 MS7.58 ［ 7.20，7.95 ］ MS7.58 ［ 7.29，8.10 ］ 1.79 500 MS7.97 ［ 7.48，8.46 ］ MS7.97 ［ 7.63，8.68 ］ 2.09
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