震级-频度关系中b值的极大似然法估计及其影响因素分析
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摘要
b值在地震活动性研究和地震危险性分析中起着十分重要的作用,通常拟合b值的方法有最小二乘法(Least Squares Method)和极大似然法(Maximum Likelihood Estimation)。最小二乘法简单易行,得到了广泛的应用。然而很多研究表明该方法存在一定的局限性,极大似然法在特定条件下可以作为最小二乘法的一种可行的替代或补充方法。前人对极大似然法的研究非常繁杂,提出了各种各样的方程式,每个方程式的隐含假设和求解方式各不相同。文中对主要方程式进行了简要的回顾,并按照是否考虑震级的归档效应、是否设定有限最大震级、是否对不同震级档数据取不同的观察时段和是否具有解析解这4个方面,对这些方程式进行了分类和总结。进而对震级的归档效应、震级的测量误差、样本量、震级跨度、最小完整震级和前余震共6个可能影响极大似然法估计b值的因素进行了分析和总结。最后对正确使用这些方程式提出了合理的建议。文中的分析和总结有助于更准确地理解和使用不同的极大似然法估计b值的方程式,以供相关研究者参考。
b-value in the magnitude-frequency( G-R) relationship plays a vital role in seismicity research and seismic hazard analysis,and the most commonly used techniques to simulate it are least square approach and maximum likelihood method. Least square method is simple and easy to apply,therefore widely used in China. However,many researches show that there exist some limits in least square estimation of b-value. Earthquakes with different magnitudes are not equally weighted in this method,and larger events have higher weights,so b-value is vulnerable to the fluctuation of several big earthquakes; meanwhile,least square method needs to divide magnitude intervals artificially. With a small sample size,data points could be not enough if the magnitude interval is too wide,and events in a magnitude interval may be lacking if it is divided to be too narrow. Especially for incremental G-R relationship,it is possible that N( Mi) equals 0 in an interval with large magnitude,so log( N( Mi))loses meaning and has to be ignored,resulting in a low b-value. Therefore,under certain conditions,maximum likelihood method is recommended as an effective substitution or supplementary for least square estimation of b-value. Among numerous previous researches on maximum likelihood estimation of b-value,lots of equations have been provided,based on varied implicit assumptions and different ways of solution. A brief overview is first presented for these equations,and classification and summary are provided based on whether taking account of the effect of binned magnitude,with finite maximum magnitude,using unequal observation periods for different magnitude intervals,and with analytic solution or not. Following this,a total of 6 influential factors are analyzed,such as binning magnitude,measurement errors of magnitude,sample size,magnitude span,minimum completeness magnitude and fore-and aftershocks. At last,reasonable suggestions are provided for using those equations properly. The equations of Aki( 1965),Utsu( 1965),Page( 1968) and Kijko and Smit( 2012) are based on assumption that magnitudes are continuous random variables,and have no corrections for this,so these equations are not recommended here. For simplicity,the equations of Utsu( 1966) or Tinti and Mulargia( 1987) can be used,but magnitude span should be greater than 2. 5 due to without finite maximum magnitude in the formulas. For researchers having capability to write code and calculate numerically,Weichert( 1980) or Bender( 1983) 's algorithm could be utilized.Especially when it is required to apply data with different observation periods for varied magnitudes,the formula of Weichert( 1980) is recommended. This study contributes to more accurately understand and use different formulas of estimating b-value by maximum likelihood technique,which can be used as reference for peers.
b-value in the magnitude-frequency( G-R) relationship plays a vital role in seismicity research and seismic hazard analysis,and the most commonly used techniques to simulate it are least square approach and maximum likelihood method. Least square method is simple and easy to apply,therefore widely used in China. However,many researches show that there exist some limits in least square estimation of b-value. Earthquakes with different magnitudes are not equally weighted in this method,and larger events have higher weights,so b-value is vulnerable to the fluctuation of several big earthquakes; meanwhile,least square method needs to divide magnitude intervals artificially. With a small sample size,data points could be not enough if the magnitude interval is too wide,and events in a magnitude interval may be lacking if it is divided to be too narrow. Especially for incremental G-R relationship,it is possible that N( Mi) equals 0 in an interval with large magnitude,so log( N( Mi))loses meaning and has to be ignored,resulting in a low b-value. Therefore,under certain conditions,maximum likelihood method is recommended as an effective substitution or supplementary for least square estimation of b-value. Among numerous previous researches on maximum likelihood estimation of b-value,lots of equations have been provided,based on varied implicit assumptions and different ways of solution. A brief overview is first presented for these equations,and classification and summary are provided based on whether taking account of the effect of binned magnitude,with finite maximum magnitude,using unequal observation periods for different magnitude intervals,and with analytic solution or not. Following this,a total of 6 influential factors are analyzed,such as binning magnitude,measurement errors of magnitude,sample size,magnitude span,minimum completeness magnitude and fore-and aftershocks. At last,reasonable suggestions are provided for using those equations properly. The equations of Aki( 1965),Utsu( 1965),Page( 1968) and Kijko and Smit( 2012) are based on assumption that magnitudes are continuous random variables,and have no corrections for this,so these equations are not recommended here. For simplicity,the equations of Utsu( 1966) or Tinti and Mulargia( 1987) can be used,but magnitude span should be greater than 2. 5 due to without finite maximum magnitude in the formulas. For researchers having capability to write code and calculate numerically,Weichert( 1980) or Bender( 1983) 's algorithm could be utilized.Especially when it is required to apply data with different observation periods for varied magnitudes,the formula of Weichert( 1980) is recommended. This study contributes to more accurately understand and use different formulas of estimating b-value by maximum likelihood technique,which can be used as reference for peers.
引文
陈培善,白彤霞,李保昆.2003.b值和地震复发周期[J].地球物理学报,46(4):510-519.CHEN Pei-shan,BAI Tong-xia,LI Bao-kun.2003.b-value and earthquake occurrence period[J].Chinese Journal of Geophysics,46(4):510-519(in Chinese).
黄玮琼,时振梁,曹学锋.1989.b值统计中的影响因素及危险性分析中b值的选取[J].地震学报,11(4):351-361.HUANG Wei-qiong,SHI Zhen-liang,CAO Xue-feng.1989.Factors influencing the estimation of b-value and the selection of b-value in hazard analysis[J].Acta Seismologica Sinica,11(4):351-361(in Chinese).
黄亦磊,周仕勇,庄建仓.2016.基于地震目录估计完备震级方法的数值实验[J].地球物理学报,59(4):1350-1358.HUANG Yi-lei,ZHOU Shi-yong,ZHUANG Jian-cang.2016.Numerical tests on catalog-based method to estimate magnitude completeness[J].Chinese Journal of Geophysics,59(4):1350-1358(in Chinese).
雷建成,高孟潭,吕红山,等.2010.地震带划分方案对地震活动性参数的影响[J].地震学报,32(4):457-465.LEI Jian-cheng,GAO Meng-tan,LHong-shan,et al.2010.Effects of seismic belt division scheme on seismicity parameters[J].Acta Seismologica Sinica,32(4):457-465(in Chinese).
李全林,陈锦标,于渌,等.1978.b值时空扫描:监视破坏性地震孕育过程的一种手段[J].地球物理学报,21(2):101-125.LI Quan-lin,CHEN Jin-biao,YU Lu,et al.1978.Time and space scanning of b-value:A method for monitoring the development of catastrophic earthquakes[J].Chinese Journal of Geophysics,21(2):101-125(in Chinese).
李正芳,周本刚.2014.利用断裂带上的低b值识别凹凸体方法的探讨:以龙门山断裂带和鲜水河断裂带为例[J].震灾防御技术,9(2):213-225.LI Zheng-fang,ZHOU Ben-gang.2014.Asperity identification based on low b-value:Application to the Longmenshan and Xianshuihe fault zone[J].Technology for Earthquake Disaster Prevention,9(2):213-225(in Chinese).
刘静伟,吕悦军.2016.川滇地区b值空间分布特征及其与震源类型关系的初步探讨[J].震灾防御技术,11(3):561-572.LIU Jing-wei,LYue-jun.2016.Spatial distribution of b values and its relationship with the type of focal mechanism in the Sichuan-Yunnan area[J].Technology for Earthquake Disaster Prevention,11(3):561-572(in Chinese).
龙锋,闻学泽,倪四道.2009.区域最小完整性震级时空分布的确定:以龙门山断裂带为例[J].地震,29(3):27-36.LONG Feng,WEN Xue-ze,NI Si-dao.2009.Determination of temporal-spatial distribution of the regional minimum magnitudes of completeness:Application to the Longmenshan fault zone[J].Earthquake,29(3):27-36(in Chinese).
潘华,高孟潭,谢富仁.2013.新版地震区划图地震活动性模型与参数确定[J].震灾防御技术,8(1):11-23.PAN Hua,GAO Meng-tan,XIE Fu-ren.2013.The earthquake activity model and seismicity parameters in the new seismic hazard map of China[J].Technology for Earthquake Disaster Prevention,8(1):11-23(in Chinese).
潘华,李金臣.2006.地震统计区地震活动性参数b值及v4不确定性研究[J].震灾防御技术,1(3):218-224.PAN Hua,LI Jin-chen.2006.Study on uncertainties of seismicity parameters b andυ4in seismic statistical zones[J].Technology for Earthquake Disaster Prevention,1(3):218-224(in Chinese).
秦长源.2000.地震震级误差对b值的影响[J].地震学报,22(4):337-344.QIN Chang-yuan.2000.The effect of the magnitude uncertainty on the b-value[J].Acta Seismologica Sinica,22(4):337-344(in Chinese).
任雪梅,高孟潭,冯静.2011.地震目录的完整性对b值计算的影响[J].震灾防御技术,6(3):257-268.REN Xue-mei,GAO Meng-tan,FENG Jing.2011.Effect of completeness of earthquake catalogue on calculating bvalue[J].Technology for Earthquake Disaster Prevention,6(3):257-268(in Chinese).
邵延秀,袁道阳,王爱国,等.2011.西秦岭北缘断裂破裂分段与地震危险性评估[J].地震地质,33(1):79-90.doi:10.3969/j.issn.0253-4967.2011.01.008.SHAO Yan-xiu,YUAN Dao-yang,WANG Ai-guo,et al.2011.The segmentation of rupture and estimate of earthquake risk along the north margin of western Qinling fault zone[J].Seismology and Geology,33(1):79-90(in Chinese).
吴果.2014.基于多模型的概率地震危险性分析方法实例研究[D].北京:中国地震局地质研究所.WU Guo.2014.Case study of probabilistic seismic hazard analysis method based on multi-model[D].Institute of Geology,CEA,Beijing(in Chinese).
吴果,周庆,冉洪流.2014.中亚地震目录震级转换及其完整性分析[J].震灾防御技术,9(3):368-383.WU Guo,ZHOU Qing,RAN Hong-liu.2014.Magnitude conversion of earthquake catalog in central Asia and the completeness analysis[J].Technology for Earthquake Disaster Prevention,9(3):368-383(in Chinese).
易桂喜,闻学泽,苏有锦.2008.川滇活动地块东边界强震危险性研究[J].地球物理学报,51(6):1719-1725.YI Gui-xi,WEN Xue-ze,SU You-jin.2008.Study on the potential strong-earthquake risk for the eastern boundary of the Sichuan-Yunnan active faulted-block,China[J].Chinese Journal of Geophysics,51(6):1719-1725(in Chinese).
易桂喜,闻学泽,辛华,等.2013.龙门山断裂带南段应力状态与强震危险性研究[J].地球物理学报,56(4):1112-1120.YI Gui-xi,WEN Xue-ze,XIN Hua,et al.2013.Stress state and major-earthquake risk on the southern segment of the Longmen Shan fault zone[J].Chinese Journal of Geophysics,56(4):1112-1120(in Chinese).
张建中,宋良玉.1981.地震b值的估计方法及其标准误差:应用蒙特卡罗方法估计b值精度[J].地震学报,3(3):78-87.ZHANG Jian-zhong,SONG Liang-yu.1981.On the method of estimating b-value and its standard error:The Monte Carlo method of estimating the accuracy of b-value[J].Acta Seismologica Sinica,3(3):78-87(in Chinese).
张盛峰,吴忠良,房立华.2014.双差(DD)定位地震目录能用于地震序列的统计地震学参数计算吗?---云南鲁甸MS6.5地震序列b值的空间分布[J].地震地质,36(4):1244-1259.doi:10.3969/j.issn.0253-4967.2014.04.024.
ZHNAG Sheng-feng,WU Zhong-liang,FANG Li-hua.2014.Can the DD-relocation earthquake catalogue be used for the statistical parameters of an earthquake sequence?A case study of the spatial distribution of b-value for the aftershocks of the 2014 Ludian MS6.5 earthquake[J].Seismology and geology,36(4):1244-1259(in Chinese).
Aki K.1965.Maximum likelihood estimate of b in the formula log N=a-b M and its confidence limits[J].Bulletin of the Earthquake Research Institute,University of Tokyo,43(2):237-239.
Beauval C,Scotti O.2004.Quantifying sensitivities of PSHA for France to earthquake catalog uncertainties,truncation of ground-motion variability,and magnitude limits[J].Bulletin of the Seismological Society of America,94(5):1579-1594.
Bender B.1983.Maximum likelihood estimation of b-values for magnitude grouped data[J].Bulletin of the Seismological Society of America,73(3):831-851.
Cao A,Gao S S.2002.Temporal variation of seismic b-values beneath northeastern Japan island arc[J].Geophysical Research Letters,29(9):48-1-48-3.
Cornell C A.1968.Engineering seismic risk analysis[J].Bulletin of the Seismological Society of America,58(5):1583-1606.
De Santis A,Cianchini G,Favali P,et al.2011.The Gutenberg-Richter law and entropy of earthquakes:Two case studies in Central Italy[J].Bulletin of the Seismological Society of America,101(3):1386-1395.
Dong W M,Bao A B,Shah H C.1984.Use of maximum entropy principle in earthquake recurrence relationships[J].Bulletin of the Seismological Society of America,74(2):725-737.
Felzer K R.2009.Simulated aftershock sequences for a M7.8 earthquake on the southern San Andreas Fault[J].Seismological Research Letters,80(1):21-25.
Gardner J K,Knopoff L.1974.Is the sequence of earthquakes in Southern California,with aftershocks removed,Poissonian?[J].Bulletin of the Seismological Society of America,64(5):1363-1367.
Gerstenberger M,Wiemer S,Giardini D.2001.A systematic test of the hypothesis that the b-value varies with depth in California[J].Geophysical Research Letters,28(1):57-60.
Gibowicz S J.1973.Variation of the frequency-magnitude relation during earthquake sequences in New Zealand[J].Bulletin of the Seismological Society of America,63(2):517-528.
Gutenberg B,Richter C F.1944.Frequency of earthquakes in California[J].Bulletin of the Seismological Society of America,34(4):185-188.
Gutenberg B,Richter C F.1949.Seismicity of the Earth and Associated Phenomena[M].Princeton University Press:273.
Helmstetter A,Kagan Y Y,Jackson D D.2006.Comparison of short-term and time-independent earthquake forecast models for southern California[J].Bulletin of the Seismological Society of America,96(1):90-106.
Helmstetter A,Kagan Y Y,Jackson D D.2007.High-resolution time-independent grid-based forecast for M≥5earthquakes in California[J].Seismological Research Letters,78(1):78-86.
Helmstetter A,Werner M J.2014.Adaptive smoothing of seismicity in time,space,and magnitude for time-dependent earthquake forecasts for California[J].Bulletin of the Seismological Society of America,104(2):809-822.
Kagan Y Y,Knopoff L.1987.Statistical short-term earthquake prediction[J].Science,236(4808):1563-1567.
Karnik V,Klima K.1993.Magnitude-frequency distribution in the European-Mediterranean earthquake regions[J].Tectonophysics,220(1-4):309-323.
Kijko A,Smit A.2012.Extension of the Aki-Utsu b-value estimator for incomplete catalogs[J].Bulletin of the Seismological Society of America,102(3):1283-1287.
Knopoff L.2000.The magnitude distribution of declustered earthquakes in Southern California[J].Proceedings of the National Academy of Sciences,97(22):11880-11884.
Kutliroff J R.2017.Estimating the proportions of large to small earthquakes in seismic regions with a short span of earthquake magnitudes[D].The University of Memphis,America.
Main I.2000.Apparent breaks in scaling in the earthquake cumulative frequency-magnitude distribution:Fact or artifact?[J].Bulletin of the Seismological Society of America,90(1):86-97.
Marzocchi W,Sandri L.2003.A review and new insights on the estimation of the b-value and its uncertainty[J].Annals of Geophysics,46(6):1271-1282.
Miao Q,Langston C A.2007.Empirical distance attenuation and the local-magnitude scale for the central United States[J].Bulletin of the Seismological Society of America,97(6):2137-2151.
Mignan A,Woessner J.2012.Estimating the magnitude of completeness for earthquake catalogs[EB/OL],Community Online Resource for Statistical Seismicity Analysis.doi:10.507 8/corssa-00180805.
Mogi K.1962a.Study of elastic shocks caused by the fracture of heterogeneous materials and its relations to earthquake phenomena[J].Bulletin of the Earthquake Research Institute,University of Tokyo,40(1):125-173.
Mogi K.1962b.Magnitude-frequency relation for elastic shocks accompanying fractures of various materials and some related problems in earthquakes(2nd paper)[J].Bulletin of the Earthquake Research Institute,University of Tokyo,40(4):831-853.
Mogi K.1963.The fracture of a semi-infinite body caused by an inner stress origin and its relation to the earthquake phenomena(1st paper)[J].Bulletin of the Earthquake Research Institute,University of Tokyo,41:595-614.
Ogata Y,Zhuang J.2006.Space-time ETAS models and an improved extension[J].Tectonophysics,413(1-2):13-23.
Okal E A,Kirby S H.1995.Frequency-moment distribution of deep earthquakes;implications for the seismogenic zone at the bottom of slabs[J].Physics of the Earth and Planetary Interiors,92(3-4):169-187.
Pacheco J F,Scholz C H,Sykes L R.1992a.Changes in frequency-size relationship from small to large earthquakes[J].Nature,355(6355):71-73.
Pacheco J F,Sykes L R.1992b.Seismic moment catalog of large shallow earthquakes,1900 to 1989[J].Bulletin of the Seismological Society of America,82(3):1306-1349.
Page R.1968.Aftershocks and microaftershocks of the great Alaska earthquake of 1964[J].Bulletin of the Seismological Society of America,58(3):1131-1168.
Papazachos B,Delibasis N,Liapis N,et al.1967.Aftershock sequences of some large earthquakes in the region of Greece[J].Annals of Geophysics,20(1):143.
Petersen M D,Frankel A D,Harmsen S C,et al.2008.Documentation for the 2008 update of the United States national seismic hazard maps[R].United States Geological Survey Open File Report,2008-1128.
Petersen M D,Moschetti M P,Powers P M,et al.2015.The 2014 United States national seismic hazard model[J].Earthquake Spectra,31(S1):1-30.
Reasenberg P.1985.Second-order moment of central California seismicity,1969-1982[J].Journal of Geophysical Research:Solid Earth,90(B7):5479-5495.
Rhoades D A.1996.Estimation of the Gutenberg-Richter relation allowing for individual earthquake magnitude uncertainties[J].Tectonophysics,258(1-4):71-83.
Robinson R.1979.Variation of energy release,rate of occurrence and b-value of earthquakes in the main seismic region,New Zealand[J].Physics of the Earth&Planetary Interiors,18(3):209-220.
Sandri L,Marzocchi W.2007.A technical note on the bias in the estimation of the b-value and its uncertainty through the least squares technique[J].Annals of Geophysics,50(3):329-339.
Scholz C H.1968.The frequency-magnitude relation in microfracturing in rock and its relation to earthquakes[J].Bulletin of the Seismological Society of America,58:399-415.
Scholz C H.1997.Size distributions for large and small earthquakes[J].Bulletin of the Seismological Society of America,87(4):1074-1077.
Schorlemmer D,Wiemer S,Wyss W.2005.Variations in earthquake-size distribution across different stress regimes[J].Nature,437(7058):539-542.
Schwartz D P,Coppersmith K J.1984.Fault behavior and characteristic earthquakes:Examples from the Wasatch and San Andreas fault zones[J].Journal of Geophysical Research:Solid Earth,89(B7):5681-5698.
Shi Y,Bolt B A.1982.The standard error of the magnitude-frequency b-value[J].Bulletin of the Seismological Society of America,72(5):1677-1687.
Smith W D.1981.The b-value as an earthquake precursor[J].Nature,289(5794):136-139.
Suyehiro S.1966.Difference between aftershocks and foreshocks in the relationship of magnitude to frequency of occurrence for the great Chilean earthquake of 1960[J].Bulletin of the Seismological Society of America,(1):185-200.
Suyehiro S,Asada T,Ohtake M.1964.Foreshocks and aftershocks accompanying a perceptible earthquake in central Japan:On the peculiar nature of foreshocks[J].Papers in Meteorology&Geophysics,15(1):71-88.
Suzuki Z.1958.A statistical study on the occurrence of small earthquakesⅢ[J].Science Reports of the Tohoku University Ser 5,Geophysics,10(1):15-27.
Talebi M,Zare M,Peresan A,et al.2017.Long-term probabilistic forecast for M≥5.0 earthquakes in Iran[J].Pure and Applied Geophysics,174(4):1561-1580.
Tinti S,Mulargia F.1985.Effects of magnitude uncertainties on estimating the parameters in the Gutenberg-Richter frequency-magnitude law[J].Bulletin of the Seismological Society of America,75(6):1681-1697.
Tinti S,Mulargia F.1987.Confidence intervals of b-values for grouped magnitudes[J].Bulletin of the Seismological Society of America,77(6):2125-2134.
Triep E G,Sykes L R.1997.Frequency of occurrence of moderate to great earthquakes in intracontinental regions:Implications for changes in stress,earthquake prediction,and hazards assessments[J].Journal of Geophysical Research:Solid Earth,102(B5):9923-9948.
Utsu T.1965.A method for determining the value of b in formula log N=a-b M showing the magnitude-frequency relation for earthquakes[J].Geophysical bulletin of Hokkaido University,(13):99-103.
Utsu T.1966.A statistical significance test of the difference in b-value between two earthquake groups[J].Journal of Physics of the Earth,14(2):37-40.
Utsu T.1999.Representation and analysis of the earthquake size distribution:A historical review and some new approaches[J].Pure and Applied Geophysics,155(2-4):509-535.
Wang Q,Jackson D D,Kagan Y Y.2011.California earthquake forecasts based on smoothed seismicity:Model choices[J].Bulletin of the Seismological Society of America,101(3):1422-1430.
Weichert D H.1980.Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes[J].Bulletin of the Seismological Society of America,70(4):1337-1346.
Wiemer S.2001.A software package to analyze seismicity:ZMAP[J].Seismological Research Letters,72(3):373-382.
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.
Working Group on California Earthquake Probabilities.2003.Earthquake probabilities in the San Francisco Bay region:2002-2031[R].United States Geological Survey Open File Report,03-214.
Wyss M,Wiemer S.2000.Change in the probability for earthquakes in southern California due to the Landers magnitude 7.3 earthquake[J].Science,290(5495):1334-1338.
Youngs R R,Coppersmith K J.1985.Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates[J].Bulletin of the Seismological Society of America,75(4):939-964.
黄玮琼,时振梁,曹学锋.1989.b值统计中的影响因素及危险性分析中b值的选取[J].地震学报,11(4):351-361.HUANG Wei-qiong,SHI Zhen-liang,CAO Xue-feng.1989.Factors influencing the estimation of b-value and the selection of b-value in hazard analysis[J].Acta Seismologica Sinica,11(4):351-361(in Chinese).
黄亦磊,周仕勇,庄建仓.2016.基于地震目录估计完备震级方法的数值实验[J].地球物理学报,59(4):1350-1358.HUANG Yi-lei,ZHOU Shi-yong,ZHUANG Jian-cang.2016.Numerical tests on catalog-based method to estimate magnitude completeness[J].Chinese Journal of Geophysics,59(4):1350-1358(in Chinese).
雷建成,高孟潭,吕红山,等.2010.地震带划分方案对地震活动性参数的影响[J].地震学报,32(4):457-465.LEI Jian-cheng,GAO Meng-tan,LHong-shan,et al.2010.Effects of seismic belt division scheme on seismicity parameters[J].Acta Seismologica Sinica,32(4):457-465(in Chinese).
李全林,陈锦标,于渌,等.1978.b值时空扫描:监视破坏性地震孕育过程的一种手段[J].地球物理学报,21(2):101-125.LI Quan-lin,CHEN Jin-biao,YU Lu,et al.1978.Time and space scanning of b-value:A method for monitoring the development of catastrophic earthquakes[J].Chinese Journal of Geophysics,21(2):101-125(in Chinese).
李正芳,周本刚.2014.利用断裂带上的低b值识别凹凸体方法的探讨:以龙门山断裂带和鲜水河断裂带为例[J].震灾防御技术,9(2):213-225.LI Zheng-fang,ZHOU Ben-gang.2014.Asperity identification based on low b-value:Application to the Longmenshan and Xianshuihe fault zone[J].Technology for Earthquake Disaster Prevention,9(2):213-225(in Chinese).
刘静伟,吕悦军.2016.川滇地区b值空间分布特征及其与震源类型关系的初步探讨[J].震灾防御技术,11(3):561-572.LIU Jing-wei,LYue-jun.2016.Spatial distribution of b values and its relationship with the type of focal mechanism in the Sichuan-Yunnan area[J].Technology for Earthquake Disaster Prevention,11(3):561-572(in Chinese).
龙锋,闻学泽,倪四道.2009.区域最小完整性震级时空分布的确定:以龙门山断裂带为例[J].地震,29(3):27-36.LONG Feng,WEN Xue-ze,NI Si-dao.2009.Determination of temporal-spatial distribution of the regional minimum magnitudes of completeness:Application to the Longmenshan fault zone[J].Earthquake,29(3):27-36(in Chinese).
潘华,高孟潭,谢富仁.2013.新版地震区划图地震活动性模型与参数确定[J].震灾防御技术,8(1):11-23.PAN Hua,GAO Meng-tan,XIE Fu-ren.2013.The earthquake activity model and seismicity parameters in the new seismic hazard map of China[J].Technology for Earthquake Disaster Prevention,8(1):11-23(in Chinese).
潘华,李金臣.2006.地震统计区地震活动性参数b值及v4不确定性研究[J].震灾防御技术,1(3):218-224.PAN Hua,LI Jin-chen.2006.Study on uncertainties of seismicity parameters b andυ4in seismic statistical zones[J].Technology for Earthquake Disaster Prevention,1(3):218-224(in Chinese).
秦长源.2000.地震震级误差对b值的影响[J].地震学报,22(4):337-344.QIN Chang-yuan.2000.The effect of the magnitude uncertainty on the b-value[J].Acta Seismologica Sinica,22(4):337-344(in Chinese).
任雪梅,高孟潭,冯静.2011.地震目录的完整性对b值计算的影响[J].震灾防御技术,6(3):257-268.REN Xue-mei,GAO Meng-tan,FENG Jing.2011.Effect of completeness of earthquake catalogue on calculating bvalue[J].Technology for Earthquake Disaster Prevention,6(3):257-268(in Chinese).
邵延秀,袁道阳,王爱国,等.2011.西秦岭北缘断裂破裂分段与地震危险性评估[J].地震地质,33(1):79-90.doi:10.3969/j.issn.0253-4967.2011.01.008.SHAO Yan-xiu,YUAN Dao-yang,WANG Ai-guo,et al.2011.The segmentation of rupture and estimate of earthquake risk along the north margin of western Qinling fault zone[J].Seismology and Geology,33(1):79-90(in Chinese).
吴果.2014.基于多模型的概率地震危险性分析方法实例研究[D].北京:中国地震局地质研究所.WU Guo.2014.Case study of probabilistic seismic hazard analysis method based on multi-model[D].Institute of Geology,CEA,Beijing(in Chinese).
吴果,周庆,冉洪流.2014.中亚地震目录震级转换及其完整性分析[J].震灾防御技术,9(3):368-383.WU Guo,ZHOU Qing,RAN Hong-liu.2014.Magnitude conversion of earthquake catalog in central Asia and the completeness analysis[J].Technology for Earthquake Disaster Prevention,9(3):368-383(in Chinese).
易桂喜,闻学泽,苏有锦.2008.川滇活动地块东边界强震危险性研究[J].地球物理学报,51(6):1719-1725.YI Gui-xi,WEN Xue-ze,SU You-jin.2008.Study on the potential strong-earthquake risk for the eastern boundary of the Sichuan-Yunnan active faulted-block,China[J].Chinese Journal of Geophysics,51(6):1719-1725(in Chinese).
易桂喜,闻学泽,辛华,等.2013.龙门山断裂带南段应力状态与强震危险性研究[J].地球物理学报,56(4):1112-1120.YI Gui-xi,WEN Xue-ze,XIN Hua,et al.2013.Stress state and major-earthquake risk on the southern segment of the Longmen Shan fault zone[J].Chinese Journal of Geophysics,56(4):1112-1120(in Chinese).
张建中,宋良玉.1981.地震b值的估计方法及其标准误差:应用蒙特卡罗方法估计b值精度[J].地震学报,3(3):78-87.ZHANG Jian-zhong,SONG Liang-yu.1981.On the method of estimating b-value and its standard error:The Monte Carlo method of estimating the accuracy of b-value[J].Acta Seismologica Sinica,3(3):78-87(in Chinese).
张盛峰,吴忠良,房立华.2014.双差(DD)定位地震目录能用于地震序列的统计地震学参数计算吗?---云南鲁甸MS6.5地震序列b值的空间分布[J].地震地质,36(4):1244-1259.doi:10.3969/j.issn.0253-4967.2014.04.024.
ZHNAG Sheng-feng,WU Zhong-liang,FANG Li-hua.2014.Can the DD-relocation earthquake catalogue be used for the statistical parameters of an earthquake sequence?A case study of the spatial distribution of b-value for the aftershocks of the 2014 Ludian MS6.5 earthquake[J].Seismology and geology,36(4):1244-1259(in Chinese).
Aki K.1965.Maximum likelihood estimate of b in the formula log N=a-b M and its confidence limits[J].Bulletin of the Earthquake Research Institute,University of Tokyo,43(2):237-239.
Beauval C,Scotti O.2004.Quantifying sensitivities of PSHA for France to earthquake catalog uncertainties,truncation of ground-motion variability,and magnitude limits[J].Bulletin of the Seismological Society of America,94(5):1579-1594.
Bender B.1983.Maximum likelihood estimation of b-values for magnitude grouped data[J].Bulletin of the Seismological Society of America,73(3):831-851.
Cao A,Gao S S.2002.Temporal variation of seismic b-values beneath northeastern Japan island arc[J].Geophysical Research Letters,29(9):48-1-48-3.
Cornell C A.1968.Engineering seismic risk analysis[J].Bulletin of the Seismological Society of America,58(5):1583-1606.
De Santis A,Cianchini G,Favali P,et al.2011.The Gutenberg-Richter law and entropy of earthquakes:Two case studies in Central Italy[J].Bulletin of the Seismological Society of America,101(3):1386-1395.
Dong W M,Bao A B,Shah H C.1984.Use of maximum entropy principle in earthquake recurrence relationships[J].Bulletin of the Seismological Society of America,74(2):725-737.
Felzer K R.2009.Simulated aftershock sequences for a M7.8 earthquake on the southern San Andreas Fault[J].Seismological Research Letters,80(1):21-25.
Gardner J K,Knopoff L.1974.Is the sequence of earthquakes in Southern California,with aftershocks removed,Poissonian?[J].Bulletin of the Seismological Society of America,64(5):1363-1367.
Gerstenberger M,Wiemer S,Giardini D.2001.A systematic test of the hypothesis that the b-value varies with depth in California[J].Geophysical Research Letters,28(1):57-60.
Gibowicz S J.1973.Variation of the frequency-magnitude relation during earthquake sequences in New Zealand[J].Bulletin of the Seismological Society of America,63(2):517-528.
Gutenberg B,Richter C F.1944.Frequency of earthquakes in California[J].Bulletin of the Seismological Society of America,34(4):185-188.
Gutenberg B,Richter C F.1949.Seismicity of the Earth and Associated Phenomena[M].Princeton University Press:273.
Helmstetter A,Kagan Y Y,Jackson D D.2006.Comparison of short-term and time-independent earthquake forecast models for southern California[J].Bulletin of the Seismological Society of America,96(1):90-106.
Helmstetter A,Kagan Y Y,Jackson D D.2007.High-resolution time-independent grid-based forecast for M≥5earthquakes in California[J].Seismological Research Letters,78(1):78-86.
Helmstetter A,Werner M J.2014.Adaptive smoothing of seismicity in time,space,and magnitude for time-dependent earthquake forecasts for California[J].Bulletin of the Seismological Society of America,104(2):809-822.
Kagan Y Y,Knopoff L.1987.Statistical short-term earthquake prediction[J].Science,236(4808):1563-1567.
Karnik V,Klima K.1993.Magnitude-frequency distribution in the European-Mediterranean earthquake regions[J].Tectonophysics,220(1-4):309-323.
Kijko A,Smit A.2012.Extension of the Aki-Utsu b-value estimator for incomplete catalogs[J].Bulletin of the Seismological Society of America,102(3):1283-1287.
Knopoff L.2000.The magnitude distribution of declustered earthquakes in Southern California[J].Proceedings of the National Academy of Sciences,97(22):11880-11884.
Kutliroff J R.2017.Estimating the proportions of large to small earthquakes in seismic regions with a short span of earthquake magnitudes[D].The University of Memphis,America.
Main I.2000.Apparent breaks in scaling in the earthquake cumulative frequency-magnitude distribution:Fact or artifact?[J].Bulletin of the Seismological Society of America,90(1):86-97.
Marzocchi W,Sandri L.2003.A review and new insights on the estimation of the b-value and its uncertainty[J].Annals of Geophysics,46(6):1271-1282.
Miao Q,Langston C A.2007.Empirical distance attenuation and the local-magnitude scale for the central United States[J].Bulletin of the Seismological Society of America,97(6):2137-2151.
Mignan A,Woessner J.2012.Estimating the magnitude of completeness for earthquake catalogs[EB/OL],Community Online Resource for Statistical Seismicity Analysis.doi:10.507 8/corssa-00180805.
Mogi K.1962a.Study of elastic shocks caused by the fracture of heterogeneous materials and its relations to earthquake phenomena[J].Bulletin of the Earthquake Research Institute,University of Tokyo,40(1):125-173.
Mogi K.1962b.Magnitude-frequency relation for elastic shocks accompanying fractures of various materials and some related problems in earthquakes(2nd paper)[J].Bulletin of the Earthquake Research Institute,University of Tokyo,40(4):831-853.
Mogi K.1963.The fracture of a semi-infinite body caused by an inner stress origin and its relation to the earthquake phenomena(1st paper)[J].Bulletin of the Earthquake Research Institute,University of Tokyo,41:595-614.
Ogata Y,Zhuang J.2006.Space-time ETAS models and an improved extension[J].Tectonophysics,413(1-2):13-23.
Okal E A,Kirby S H.1995.Frequency-moment distribution of deep earthquakes;implications for the seismogenic zone at the bottom of slabs[J].Physics of the Earth and Planetary Interiors,92(3-4):169-187.
Pacheco J F,Scholz C H,Sykes L R.1992a.Changes in frequency-size relationship from small to large earthquakes[J].Nature,355(6355):71-73.
Pacheco J F,Sykes L R.1992b.Seismic moment catalog of large shallow earthquakes,1900 to 1989[J].Bulletin of the Seismological Society of America,82(3):1306-1349.
Page R.1968.Aftershocks and microaftershocks of the great Alaska earthquake of 1964[J].Bulletin of the Seismological Society of America,58(3):1131-1168.
Papazachos B,Delibasis N,Liapis N,et al.1967.Aftershock sequences of some large earthquakes in the region of Greece[J].Annals of Geophysics,20(1):143.
Petersen M D,Frankel A D,Harmsen S C,et al.2008.Documentation for the 2008 update of the United States national seismic hazard maps[R].United States Geological Survey Open File Report,2008-1128.
Petersen M D,Moschetti M P,Powers P M,et al.2015.The 2014 United States national seismic hazard model[J].Earthquake Spectra,31(S1):1-30.
Reasenberg P.1985.Second-order moment of central California seismicity,1969-1982[J].Journal of Geophysical Research:Solid Earth,90(B7):5479-5495.
Rhoades D A.1996.Estimation of the Gutenberg-Richter relation allowing for individual earthquake magnitude uncertainties[J].Tectonophysics,258(1-4):71-83.
Robinson R.1979.Variation of energy release,rate of occurrence and b-value of earthquakes in the main seismic region,New Zealand[J].Physics of the Earth&Planetary Interiors,18(3):209-220.
Sandri L,Marzocchi W.2007.A technical note on the bias in the estimation of the b-value and its uncertainty through the least squares technique[J].Annals of Geophysics,50(3):329-339.
Scholz C H.1968.The frequency-magnitude relation in microfracturing in rock and its relation to earthquakes[J].Bulletin of the Seismological Society of America,58:399-415.
Scholz C H.1997.Size distributions for large and small earthquakes[J].Bulletin of the Seismological Society of America,87(4):1074-1077.
Schorlemmer D,Wiemer S,Wyss W.2005.Variations in earthquake-size distribution across different stress regimes[J].Nature,437(7058):539-542.
Schwartz D P,Coppersmith K J.1984.Fault behavior and characteristic earthquakes:Examples from the Wasatch and San Andreas fault zones[J].Journal of Geophysical Research:Solid Earth,89(B7):5681-5698.
Shi Y,Bolt B A.1982.The standard error of the magnitude-frequency b-value[J].Bulletin of the Seismological Society of America,72(5):1677-1687.
Smith W D.1981.The b-value as an earthquake precursor[J].Nature,289(5794):136-139.
Suyehiro S.1966.Difference between aftershocks and foreshocks in the relationship of magnitude to frequency of occurrence for the great Chilean earthquake of 1960[J].Bulletin of the Seismological Society of America,(1):185-200.
Suyehiro S,Asada T,Ohtake M.1964.Foreshocks and aftershocks accompanying a perceptible earthquake in central Japan:On the peculiar nature of foreshocks[J].Papers in Meteorology&Geophysics,15(1):71-88.
Suzuki Z.1958.A statistical study on the occurrence of small earthquakesⅢ[J].Science Reports of the Tohoku University Ser 5,Geophysics,10(1):15-27.
Talebi M,Zare M,Peresan A,et al.2017.Long-term probabilistic forecast for M≥5.0 earthquakes in Iran[J].Pure and Applied Geophysics,174(4):1561-1580.
Tinti S,Mulargia F.1985.Effects of magnitude uncertainties on estimating the parameters in the Gutenberg-Richter frequency-magnitude law[J].Bulletin of the Seismological Society of America,75(6):1681-1697.
Tinti S,Mulargia F.1987.Confidence intervals of b-values for grouped magnitudes[J].Bulletin of the Seismological Society of America,77(6):2125-2134.
Triep E G,Sykes L R.1997.Frequency of occurrence of moderate to great earthquakes in intracontinental regions:Implications for changes in stress,earthquake prediction,and hazards assessments[J].Journal of Geophysical Research:Solid Earth,102(B5):9923-9948.
Utsu T.1965.A method for determining the value of b in formula log N=a-b M showing the magnitude-frequency relation for earthquakes[J].Geophysical bulletin of Hokkaido University,(13):99-103.
Utsu T.1966.A statistical significance test of the difference in b-value between two earthquake groups[J].Journal of Physics of the Earth,14(2):37-40.
Utsu T.1999.Representation and analysis of the earthquake size distribution:A historical review and some new approaches[J].Pure and Applied Geophysics,155(2-4):509-535.
Wang Q,Jackson D D,Kagan Y Y.2011.California earthquake forecasts based on smoothed seismicity:Model choices[J].Bulletin of the Seismological Society of America,101(3):1422-1430.
Weichert D H.1980.Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes[J].Bulletin of the Seismological Society of America,70(4):1337-1346.
Wiemer S.2001.A software package to analyze seismicity:ZMAP[J].Seismological Research Letters,72(3):373-382.
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.
Working Group on California Earthquake Probabilities.2003.Earthquake probabilities in the San Francisco Bay region:2002-2031[R].United States Geological Survey Open File Report,03-214.
Wyss M,Wiemer S.2000.Change in the probability for earthquakes in southern California due to the Landers magnitude 7.3 earthquake[J].Science,290(5495):1334-1338.
Youngs R R,Coppersmith K J.1985.Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates[J].Bulletin of the Seismological Society of America,75(4):939-964.