利用震级均值特性建立有上限震级概率模型对2017年九寨沟7.0级地震的余震概率预测
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摘要
震级均值ma是震级m以上事件的震级平均值.研究了九寨沟7. 0级地震余震序列的震级均值ma之后,发现两个与ma有关的线性关系:(1) lg N'=a'-b'ma,(2) ma=c+dm.第一个关系式不仅与G-R关系的形式相同,并且也可以用来估计完备震级Mc.令ΔN=lg N-lg N',lg N从目录统计得到,lg N'是(1)式给出的预测值.在ΔN与ma的散点图上,ΔN第一次突破0轴对应的ma值可以很好地估计完备性震级Mc.第二个关系式称为震均关系(A-M关系),只能从实际目录统计,不能从已有的统计关系推导出来.若震级是连续随机变量且G-R关系在完备震级Mc以上的任意的震级范围[m,∞]成立,将A-M关系代入极大似然法求b值的Aki-Utsu公式,可得到b值函数B(m)=-r/[c-(1-d) m].对B(m)积分可得(3) lg N=p+klg(c-qm),其中q=1-d.从(3)式可导出(4) F(M≥m)=[(cqm)/(c-qm0]k.(3)和(4)分别是有上限的震级-频度关系及其条件概率分布函数.按以下步骤确定参数c、q、p、k:第一步,拟合线性关系(1) lg N'=a'-b'ma,用ΔN=lg N-lg N'确定完备震级Mc,并统计Mc以上余震序列的震级均值mca.第二步,利用余震序列的最大震级不可能超过主震震级Mm的特点,假定余震序列震级上限mu=Mm+0. 5,通过点(mu,mu)和点(Mc,mca)求解A-M关系(2) ma=c+dm,获得参数c、q.第三步,拟合lg(c-qm)和相应的lg N,确定参数p、k.根据以上步骤,用震后2 h的余震目录拟合各项参数,获得概率分布函数F(M≥m)=[(0. 62-0. 0827m)/(0. 62-0. 0827m0]8. 8602.据此估计在后续累计发生0级以上余震10000次的条件下,至少发生一次的概率为63%、10%和1%对应的的震级分别为4. 8、5. 4、5. 9级.据蒋海昆等(2006)的多震型序列定义,后续须至少发生一次6. 4级以上地震,才能认为九寨沟7. 0级余震序列是多震型序列,而6. 4级以上地震至少发生一次的概率仅为4. 02E-4.根据这些概率计算,在震后2 h可以作出的预测意见为:九寨沟7. 0级地震的余震序列类型为主震-余震型,最大余震震级可能在5级左右或5～6级.迄今为止的事实表明,虽然随着余震的不断发生,以后的参数c、q、p、k会发生一定的波动,但不至于改变震后2 h作出余震预测意见.这表明将A-M关系和G-R关系结合起来,可以在震后最短时间内为预测余震序列类型和最大可能震级提供可靠判据.
The average magnitude m_a is defined as the mean magnitude of earthquakes with magnitude m and larger. After investigating the aftershock sequence of Jiuzhaigou earthquake, M_S 7, we have found out two linear relationships of m_a:(1) lgN′=a′-b′m_a,(2) m_a=c+dm. The first relation has the same form as the G-R relation, and can be used to estimate the completeness magnitude M_c. Let ΔN=lgN-lgN′, where lgN can be directly obtained from the statistics of sequence catalog and lgN′ can be calculated from equation(1). Then the completeness magnitude M_c can be estimated by the value of m_(a )where the ΔN just crosses the 0-axis of the graph of ΔN versus m_a.The second relation is called A-M relationship. It can only be obtained from observing the statistical graphs of m_a vs. m, and cannot be derived from any existing statistical relations. Suppose that G-R relation is applicable for any continuous range [m,∞] where m is above the completeness magnitude M_c, a function of b-value will be written as B(m)=-r/[c-(1-d)m] by replacing the mean magnitude with A-M relation in the Aki-Utsu's formula for b-value. Then equation(3) lgN=p+klg(c-qm) can be obtained from the integral of B(m), and equation(4) F(M≥m)=[(c-qm)/(c-qm_0)]k will be derived from equation(3). In equations(3) and(4), the parameter q=1-d. Equation(3) is a new magnitude-frequency relationship with upper-bound magnitude and equation(4) is the conditional probability distribution function for limited magnitude range. The parameters c, q, p and k can be determined by following procedures. The first step, fit the linear relationship(1) lgN′=a′-b′m_a and determine the complete magnitude M_c by observing the graph of ΔN versus m_a. Then estimate m_(ca), the mean magnitude of the aftershocks with magnitude M_c and larger. Next, assume m_u, the upper-limit magnitude of the aftershock sequence as M_m+0.5, in the light of the characteristic that the maximum magnitude of the aftershock sequence do not exceed the magnitude of the mainshock, M_m. Solve the A-M relation m_a=c+dm by fitting points(m_u,m_u) and(M_c,mca) for estimating these parameters c, q. Finally, the parameters p, k can be assessed by fitting lg(c-qm) and its corresponding lgN. By above steps, parameters c, q, p, k, will be estimated as 0.62, 0.0827, 4.8628, 8.8602, respectively, with the aftershock catalog of 2 hours just after Jiuzhaigou mainshock. Then the probability distribution function will be determined as F(M≥m)=[(0.62-0. 0827 m)/( 0. 62-0. 0827 m0) ]8. 8602. Given that 10000 successive aftershocks cumulatively occur with magnitude above 0,the at-least-once probabilities for magnitudes 4. 8,5. 4,5. 9 will be63%,10% and 1%,respectively. According to the definition of the type of multiplet mainshocks proposed by Jiang Haikun et al.( 2006),the sequence of Jiuzhaigou earthquake MS7. 0 will never be identified as the type of multiplet mainshocks unless the shock with magnitude above 6. 4 occurs at least once in successive sequence. But the at-least-once probability of magnitude 6. 4 is only4. 02 E-4 calculated by the probability distribution function F( M≥m) = [( 0. 62-0. 0827 m)/( 0. 62-0. 0827 m0) ]8. 8602. These probabilities imply that the sequence of Jiuzhaigou earthquake MS7 will be the type of mainshock-aftershock and the largest aftershock will be magnitude about 5 or between 5 and 6. The successive events by now proves that this prediction is stable and reliable,although parameters c,k,p,q fluctuate somewhat with successive occurrence of aftershocks. From this study, we conclude that combining A-M relation with G-R relation will give a new magnitude-frequency distribution with upper-bound magnitude. And this distribution will provide a method for predicting the type and the maximum possible magnitude of an aftershock sequence as soon as possible after the mainshock.
The average magnitude m_a is defined as the mean magnitude of earthquakes with magnitude m and larger. After investigating the aftershock sequence of Jiuzhaigou earthquake, M_S 7, we have found out two linear relationships of m_a:(1) lgN′=a′-b′m_a,(2) m_a=c+dm. The first relation has the same form as the G-R relation, and can be used to estimate the completeness magnitude M_c. Let ΔN=lgN-lgN′, where lgN can be directly obtained from the statistics of sequence catalog and lgN′ can be calculated from equation(1). Then the completeness magnitude M_c can be estimated by the value of m_(a )where the ΔN just crosses the 0-axis of the graph of ΔN versus m_a.The second relation is called A-M relationship. It can only be obtained from observing the statistical graphs of m_a vs. m, and cannot be derived from any existing statistical relations. Suppose that G-R relation is applicable for any continuous range [m,∞] where m is above the completeness magnitude M_c, a function of b-value will be written as B(m)=-r/[c-(1-d)m] by replacing the mean magnitude with A-M relation in the Aki-Utsu's formula for b-value. Then equation(3) lgN=p+klg(c-qm) can be obtained from the integral of B(m), and equation(4) F(M≥m)=[(c-qm)/(c-qm_0)]k will be derived from equation(3). In equations(3) and(4), the parameter q=1-d. Equation(3) is a new magnitude-frequency relationship with upper-bound magnitude and equation(4) is the conditional probability distribution function for limited magnitude range. The parameters c, q, p and k can be determined by following procedures. The first step, fit the linear relationship(1) lgN′=a′-b′m_a and determine the complete magnitude M_c by observing the graph of ΔN versus m_a. Then estimate m_(ca), the mean magnitude of the aftershocks with magnitude M_c and larger. Next, assume m_u, the upper-limit magnitude of the aftershock sequence as M_m+0.5, in the light of the characteristic that the maximum magnitude of the aftershock sequence do not exceed the magnitude of the mainshock, M_m. Solve the A-M relation m_a=c+dm by fitting points(m_u,m_u) and(M_c,mca) for estimating these parameters c, q. Finally, the parameters p, k can be assessed by fitting lg(c-qm) and its corresponding lgN. By above steps, parameters c, q, p, k, will be estimated as 0.62, 0.0827, 4.8628, 8.8602, respectively, with the aftershock catalog of 2 hours just after Jiuzhaigou mainshock. Then the probability distribution function will be determined as F(M≥m)=[(0.62-0. 0827 m)/( 0. 62-0. 0827 m0) ]8. 8602. Given that 10000 successive aftershocks cumulatively occur with magnitude above 0,the at-least-once probabilities for magnitudes 4. 8,5. 4,5. 9 will be63%,10% and 1%,respectively. According to the definition of the type of multiplet mainshocks proposed by Jiang Haikun et al.( 2006),the sequence of Jiuzhaigou earthquake MS7. 0 will never be identified as the type of multiplet mainshocks unless the shock with magnitude above 6. 4 occurs at least once in successive sequence. But the at-least-once probability of magnitude 6. 4 is only4. 02 E-4 calculated by the probability distribution function F( M≥m) = [( 0. 62-0. 0827 m)/( 0. 62-0. 0827 m0) ]8. 8602. These probabilities imply that the sequence of Jiuzhaigou earthquake MS7 will be the type of mainshock-aftershock and the largest aftershock will be magnitude about 5 or between 5 and 6. The successive events by now proves that this prediction is stable and reliable,although parameters c,k,p,q fluctuate somewhat with successive occurrence of aftershocks. From this study, we conclude that combining A-M relation with G-R relation will give a new magnitude-frequency distribution with upper-bound magnitude. And this distribution will provide a method for predicting the type and the maximum possible magnitude of an aftershock sequence as soon as possible after the mainshock.
引文
Aki K. 1965. Maximum likelihood estimate of b in the formula log(N)=a-bM and its confidence limits [J]. Bulletin of Earthquake Research Institute, 43(1965): 237-239.
Berrill J B, Davis R O. 1980. Maximum entropy and the magnitude distribution [J]. Bulletin of the Seismological Society of America, 70(5): 1823-1831.
Chen Shijun, Wang Zhicai, Tao Jiuqin. 1998. Non-linear Magnitude-Frenqucy relationship and Two Mechanisms for Earthquake Occurrence [J]. Earthquake Science (in Chinese), 20(2): 174-184.
Cornell C A, Vanmarcke E. 1969. The major influences on seismicrisk. Proc 4th World Conference on Earthquake Engineering. Santiago de Chile.15-19, January.
Healy J H, Keilis-Borok V I, Lee W H K. 1997. IASPEI software library [M]. Published by IASPEI in collaboration with Seismological Society of America, (6): 80-94.
JIANG Chang-Sheng, ZHUANG Jian-Cang, WU Zhong-Liang, et al. 2017. Application and comparison of two short-term probabilistic forecasting models for the 2017 Jiuzhaigou, Sichuan, MS 7.0 earthquake [J]. Chinese Journal of Geophysics (in Chinese), 60(10): 4132- 4144, doi: 10.6038/cjg20171038.
Jiang H K, Qu Y J, Li Y L, et al. 2006. Some statistic features of aftershock sequences in Chinese mainland [J]. Chinese Journal of Geophysics (in Chinese), 49(4): 1110-1117.
Kagan Y. 1997. Seismic moment-frequency relation for shallow earthquakes: Regional comparison [J]. Journal of Geophysical Research, 102(B2): 2835-2852.
Main I. 1996. Statitical physics.seismogenesis, and seismic hazard [J]. Reviews of Geophysics, 34(4): 433- 462.
Main I, Burton P W. 1984. Information theory and the earthquake frequency-magnitude distribution [J]. Bulletin of the Seismological Society of America, 74(4): 1409-1426.
Main I, Burton P W. 1986. Long-term earthquake recurrence constrained by tectonic seismic moment release rates [J]. Bulletin of the Seismological Society of America, 76(1): 297-304.
Mignan A, Woessner J. 2012. Estimating the magnitude of completeness in earthquake catalogs, Community Online Resource for Statistical Seismicity Analysis. doi: 10.5078/corssa- 00180805. Available at http: //www. corssa.org.
Mogi K. 1967. Regional variations in magnitude-frequency relation of earthquakes [J]. Bulletin of Earthquake Research Institute, 45(1967): 313-325.
Ogata Y. 1988. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes [J]. Journal of the American Statistical Association, 83(401): 9-27.
Ogata Y. 1989. Statistical model for standard seismicity and detection of anomalies by residual analysis [J]. Tectonophysics, 169(1-3): 159-174.
Purcaru, G. 1975. A new magnitude-frequency relation for earthquakes and a classification of relation types [J]. Geophys. J. R. Astr. Soc., 42(1): 61-79.
Reasenberg P A, Jones L M. 1989. Earthquake hazard after a mainshock in California [J]. Science, 243(4895): 1173-1176.
Ren Xuemei, Gao Mengtan, Yu Yanxiang. 2012. Modification of magnitude-frequency relation and magnitude limit determination based on MGR model for moderate-strong earthquakes in Chinese mainland [J]. Acta seismologica Sinica (in Chinese), 34(3): 331-338, doi: 10.3969/j.issn.0253-3782.2012.03.005.
Tan Y P, Chen J F, Cao J Q et al. 2015. Catalogue completeness analysis of aftershock sequence of the 2013 Minxian-Zhangxian, Gansu, MS 6.6 earthquake based on location and magnitude estimation of single-station earthquake events [J]. Acta Seismologica Sinica (in Chinese), 37(5): 806-819, doi: 10.11939/jass.2015.05.009.
Utsu, T. 1965. A method for determining the value of b in a formula logn=a-bM showing the magnitude frequency for earthquakes, Geophysical Bulletin of Hokkaido University, (1965): 99-103.
Utsu T. 1971. Aftershocks and Earthquake Statistics (3): Analyses of the Distribution of Earthquakes in Magnitude, Time and Space with Special Consideration to Clustering Characteristics of Earthquake Occurrence(1). Journal of the Faculty of Science, Hokkaido University. Series 7, Geophysics, 3(5): 379- 441.
Ustu T. 1978. Estimation of Parameters in Formulas for Frequency-Magnitude Relation of Earthquake Occurrence: In Cases Involving a Parameter c for the Maximum Magnitude [J]. Journal of the Seismological Society of Japan, 31(2): 367-382.
Utsu T. 1999. Representation and analysis of the earthquake size distribution: a historical review and some new approaches [J]. Pure appl. Geophys. 155(2- 4): 509-535.
Wang Wei, Dai Weile, Huang Bing shu. 1994. Statistical Distribution of Earthquake Magnitude and the Anomalous Change of Earthquake Magnitude Factor Mf Value before Mid-strong Earthquakes Occured in North China. Earthquake Research in China (in Chinese), 10(Suppl.): 95-110.
Wiemer S, Wyss M. 2000. Minimum magnitude of completeness in earthquake catalogs: Examples from Alaska, the western United States, and Japan [J]. Bulletin of the Seismological Society of America, 90(4): 859-869.
Woessner, J., Wiemer S. 2005. Assessing the quality of earthquake catalogues: Estimating the magnitude of completeness and its uncertainty [J]. Bulletin of the Seismological Society of America, 95(2): 684- 698, doi: 110.1785/0120040007.
Wu Zhongliang, Zhu Chuanzhen, Jian Changsheng, et al. 2008. Beyond Controversies and Techiques: Fundamental Problems of Statistical Seismology. Earthquake Research in China (in Chinese), 24(3): 197-206.
Ye Youqing, Shi Bingxin, Zhu Yongli. 2016. The characteristics and application of mean of magnitude random variables. 2016 Chinese Earth Science Association Symposium (seventeen) special 36: earthquake mechanism, seismogenic environment and seismicity analysis. Special 37: strong earthquake disaster characteristics and its social influence (in Chinese).
陈时军, 王志才, 陶九庆. 1998. 非线性震级频度关系与两类地震活动系统[J]. 地震学报, 20(2): 174-184.
蒋长胜, 庄建仓, 吴忠良, 等. 2017. 两种短期概率预测模型在2017年九寨沟7.0级地震中的应用和比较研究[J]. 地球物理学报, 60(10): 4132- 4144, doi: 10.6038/cjg20171038.
蒋海昆, 曲延军, 李永莉, 等. 2006. 中国大陆中强地震余震序列的部分统计特征[J]. 地球物理学报, 49(4): 1110-1117.
任雪梅, 高孟潭, 俞言祥. 2012. 基于MGR模型修正我国大陆中强以上地震的震级-频度关系和确定震级极限值[J], 地震学报, 34(3): 331-338, doi: 10.3969/j.issn.0253-3782.2012.03.005.
谭毅培, 陈继锋, 曹井泉, 等. 2015. 2013年甘肃岷县—漳县MS 6.6地震余震序列目录完备性研究—基于对单台记录地震事件震中与震级的估计[J]. 地震学报, 37(5): 806-819, doi: 10.11939/jass.2015.05.009.
王炜, 戴维乐, 黄冰树. 1994. 地震震级的统计分布及其地震强度因子Mf在华北中强以上地震前的异常变化[J]. 中国地震, 10(增刊): 95-110.
吴忠良, 朱传镇, 蒋长胜, 等. 2008. 统计地震学的基本问题[J]. 中国地震, 2008, 24(3): 197-206.
叶友清, 史丙新, 朱永莉. 2016. 震级随机变量的均值特性及其应用[J]. 2016中国地球科学联合学术年会论文集(十七)专题36: 强震机理、孕育环境与地震活动性分析.专题37: 强震震害特点及其社会影响.
Berrill J B, Davis R O. 1980. Maximum entropy and the magnitude distribution [J]. Bulletin of the Seismological Society of America, 70(5): 1823-1831.
Chen Shijun, Wang Zhicai, Tao Jiuqin. 1998. Non-linear Magnitude-Frenqucy relationship and Two Mechanisms for Earthquake Occurrence [J]. Earthquake Science (in Chinese), 20(2): 174-184.
Cornell C A, Vanmarcke E. 1969. The major influences on seismicrisk. Proc 4th World Conference on Earthquake Engineering. Santiago de Chile.15-19, January.
Healy J H, Keilis-Borok V I, Lee W H K. 1997. IASPEI software library [M]. Published by IASPEI in collaboration with Seismological Society of America, (6): 80-94.
JIANG Chang-Sheng, ZHUANG Jian-Cang, WU Zhong-Liang, et al. 2017. Application and comparison of two short-term probabilistic forecasting models for the 2017 Jiuzhaigou, Sichuan, MS 7.0 earthquake [J]. Chinese Journal of Geophysics (in Chinese), 60(10): 4132- 4144, doi: 10.6038/cjg20171038.
Jiang H K, Qu Y J, Li Y L, et al. 2006. Some statistic features of aftershock sequences in Chinese mainland [J]. Chinese Journal of Geophysics (in Chinese), 49(4): 1110-1117.
Kagan Y. 1997. Seismic moment-frequency relation for shallow earthquakes: Regional comparison [J]. Journal of Geophysical Research, 102(B2): 2835-2852.
Main I. 1996. Statitical physics.seismogenesis, and seismic hazard [J]. Reviews of Geophysics, 34(4): 433- 462.
Main I, Burton P W. 1984. Information theory and the earthquake frequency-magnitude distribution [J]. Bulletin of the Seismological Society of America, 74(4): 1409-1426.
Main I, Burton P W. 1986. Long-term earthquake recurrence constrained by tectonic seismic moment release rates [J]. Bulletin of the Seismological Society of America, 76(1): 297-304.
Mignan A, Woessner J. 2012. Estimating the magnitude of completeness in earthquake catalogs, Community Online Resource for Statistical Seismicity Analysis. doi: 10.5078/corssa- 00180805. Available at http: //www. corssa.org.
Mogi K. 1967. Regional variations in magnitude-frequency relation of earthquakes [J]. Bulletin of Earthquake Research Institute, 45(1967): 313-325.
Ogata Y. 1988. Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes [J]. Journal of the American Statistical Association, 83(401): 9-27.
Ogata Y. 1989. Statistical model for standard seismicity and detection of anomalies by residual analysis [J]. Tectonophysics, 169(1-3): 159-174.
Purcaru, G. 1975. A new magnitude-frequency relation for earthquakes and a classification of relation types [J]. Geophys. J. R. Astr. Soc., 42(1): 61-79.
Reasenberg P A, Jones L M. 1989. Earthquake hazard after a mainshock in California [J]. Science, 243(4895): 1173-1176.
Ren Xuemei, Gao Mengtan, Yu Yanxiang. 2012. Modification of magnitude-frequency relation and magnitude limit determination based on MGR model for moderate-strong earthquakes in Chinese mainland [J]. Acta seismologica Sinica (in Chinese), 34(3): 331-338, doi: 10.3969/j.issn.0253-3782.2012.03.005.
Tan Y P, Chen J F, Cao J Q et al. 2015. Catalogue completeness analysis of aftershock sequence of the 2013 Minxian-Zhangxian, Gansu, MS 6.6 earthquake based on location and magnitude estimation of single-station earthquake events [J]. Acta Seismologica Sinica (in Chinese), 37(5): 806-819, doi: 10.11939/jass.2015.05.009.
Utsu, T. 1965. A method for determining the value of b in a formula logn=a-bM showing the magnitude frequency for earthquakes, Geophysical Bulletin of Hokkaido University, (1965): 99-103.
Utsu T. 1971. Aftershocks and Earthquake Statistics (3): Analyses of the Distribution of Earthquakes in Magnitude, Time and Space with Special Consideration to Clustering Characteristics of Earthquake Occurrence(1). Journal of the Faculty of Science, Hokkaido University. Series 7, Geophysics, 3(5): 379- 441.
Ustu T. 1978. Estimation of Parameters in Formulas for Frequency-Magnitude Relation of Earthquake Occurrence: In Cases Involving a Parameter c for the Maximum Magnitude [J]. Journal of the Seismological Society of Japan, 31(2): 367-382.
Utsu T. 1999. Representation and analysis of the earthquake size distribution: a historical review and some new approaches [J]. Pure appl. Geophys. 155(2- 4): 509-535.
Wang Wei, Dai Weile, Huang Bing shu. 1994. Statistical Distribution of Earthquake Magnitude and the Anomalous Change of Earthquake Magnitude Factor Mf Value before Mid-strong Earthquakes Occured in North China. Earthquake Research in China (in Chinese), 10(Suppl.): 95-110.
Wiemer S, Wyss M. 2000. Minimum magnitude of completeness in earthquake catalogs: Examples from Alaska, the western United States, and Japan [J]. Bulletin of the Seismological Society of America, 90(4): 859-869.
Woessner, J., Wiemer S. 2005. Assessing the quality of earthquake catalogues: Estimating the magnitude of completeness and its uncertainty [J]. Bulletin of the Seismological Society of America, 95(2): 684- 698, doi: 110.1785/0120040007.
Wu Zhongliang, Zhu Chuanzhen, Jian Changsheng, et al. 2008. Beyond Controversies and Techiques: Fundamental Problems of Statistical Seismology. Earthquake Research in China (in Chinese), 24(3): 197-206.
Ye Youqing, Shi Bingxin, Zhu Yongli. 2016. The characteristics and application of mean of magnitude random variables. 2016 Chinese Earth Science Association Symposium (seventeen) special 36: earthquake mechanism, seismogenic environment and seismicity analysis. Special 37: strong earthquake disaster characteristics and its social influence (in Chinese).
陈时军, 王志才, 陶九庆. 1998. 非线性震级频度关系与两类地震活动系统[J]. 地震学报, 20(2): 174-184.
蒋长胜, 庄建仓, 吴忠良, 等. 2017. 两种短期概率预测模型在2017年九寨沟7.0级地震中的应用和比较研究[J]. 地球物理学报, 60(10): 4132- 4144, doi: 10.6038/cjg20171038.
蒋海昆, 曲延军, 李永莉, 等. 2006. 中国大陆中强地震余震序列的部分统计特征[J]. 地球物理学报, 49(4): 1110-1117.
任雪梅, 高孟潭, 俞言祥. 2012. 基于MGR模型修正我国大陆中强以上地震的震级-频度关系和确定震级极限值[J], 地震学报, 34(3): 331-338, doi: 10.3969/j.issn.0253-3782.2012.03.005.
谭毅培, 陈继锋, 曹井泉, 等. 2015. 2013年甘肃岷县—漳县MS 6.6地震余震序列目录完备性研究—基于对单台记录地震事件震中与震级的估计[J]. 地震学报, 37(5): 806-819, doi: 10.11939/jass.2015.05.009.
王炜, 戴维乐, 黄冰树. 1994. 地震震级的统计分布及其地震强度因子Mf在华北中强以上地震前的异常变化[J]. 中国地震, 10(增刊): 95-110.
吴忠良, 朱传镇, 蒋长胜, 等. 2008. 统计地震学的基本问题[J]. 中国地震, 2008, 24(3): 197-206.
叶友清, 史丙新, 朱永莉. 2016. 震级随机变量的均值特性及其应用[J]. 2016中国地球科学联合学术年会论文集(十七)专题36: 强震机理、孕育环境与地震活动性分析.专题37: 强震震害特点及其社会影响.