Open House, ISM in Tachikawa, 2014.6.13統計数理研究所オープンハウス（立川）、H26.6.13ポスター発

Jiancang Zhuang

庄 建倉

English

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

震源過程とVere-Jonesの 分枝モデル間の類似特性庄 建倉 モデリング研究系 准教授2014年6月13日 統計数理研究所 オープンハウス【Abstract】Vere-Jones’ branching crack model was developed in 1970s. In this model, the earthquake source is regarded as the result from the total population of crack elements in a critical or near-critical branching process, where the crack propagates through a series of steps. At each step, each crack element simply terminates or generates several other crack elements nearby. Regarding the total number of steps (generations) as the duration time and the total number of crack elements as the total energy released, the following similarities are found between earthquake sources and this model: 1. The distribution of energies is asymptotically a Pareto distribution (power law) for the critical case, or a tapered Pareto distribution (tapered power law, Kagan distribution) for the subcritical case.2. VJ’s branching crack model explains why the EQ magnitude cannot be determined or predicted before the whole source process finishes.   3. The number of crack elements at each generation (time step) show similar patterns of earthquake source time functions.【On earthquake sources】Summary statistics• Magnitude• Seismic moment, Energy• Moment tensorModels• Slip distribution (spatial or spatiotemporal)• Fault geometry• Source time function• Stress drop distributionComplexity• Accompanying (sub) faults• Fault thickness ….【Vere-Jones’ branching crack model 】• Each earthquake starts from an individual crack.• Each crack triggers further cracks independently in the same manner (probability distribution).       Y: Number of cracks triggered by a crack  Pr 	cracks	are	triggered	by	a	given	crack ,   1, 2, …		 	  	 ∑ 	 !"  #$%&	  ' (  '  )'  ∞.				• The energy (moment) of the earthquake can be emulated by the total number of cracks.• The duration time can be emulated by the total number of generations.【Total Energies】 It is reasonable to assume the energy , of the earthquake can be emulated by the total number - of cracks in VJ’s branching model.  (Dwass theorem) For a general branching process with a single time 0 ancestor and offspring distribution  and total population size -:Pr -  . "/Pr " 0 ' 0⋯0 /  . ( 1where ", ', … , / are independent copies of . Using the Dwass theorem and central limit theorem, we can provePr -  2 ∼"4 '5467exp (4 ":; :" 7'467∼ 2:<7 	exp (4 ":; :" 7'467				When   # <1, 						Pr -  2 ∼ 2:<7 	exp (":; 7'672 				 (subcritical		regime)When   # =1, 						Pr -  2 ∼ 2:<7	 (critical	regime) Asymptotes of the survival functions of magnitudes (B) and  moments (,)B 'Clog"E, ( 2.9• When   # <1 (subcritical		regime), Pr magnitude G B ∼ const ⋅ 10:E.JKL:M	"ENO	PPr moment G , ∼ const ⋅ ,:E.K exp((,/,R) (tapered	Pareto	distribution)• When   # =1 (critical	regime), 		Pr magnitude G B ∼ const ⋅ 10:E.JKL i. e., S  0.75Pr moment G , ∼ const ⋅ ,:E.K (Pareto	distribution,	G-R	law),R : corner magnitude     Z, [:	constants   • The critical case gives the famous Gutenberg-Richter magnitude-frequency relation, and the subcritical gives the more proper distribution for earthquake sizes (Kagan, 1993).  Kagan (2010):  S=0.75 is universal for earthquakes!【Source-time functions】 The source time function represents the release rate of seismic moment as a function of time. In Vere-Jones’s branching crack model,  it is assumed: Each generation is a time step, and energy at each time step is the number of cracks at this generation.  Some examples of source time functions. Simulation examples of big earthquakes Regarding  “generation” as time step, VJ’s branching crack model has a similar source time function as an earthquake.  If the branching process does not stop at a certain time step, any number of cracks could be produced in its continuation. This randomness of VJ’s branching model explains why the EQ magnitude cannot be determined unless the recorded waveforms contain information of the whole source process. 【The duration-moment relation 】 The centroid time is the time at which an earthquake has released half its energy. VJ’s branching crack model has a similar relationship between source duration time and magnitude as earthquakes. 【Conclusions】Please see Abstract. 【Acknowledgements】This poster is based on a joint research with Dr. Dun Wang from Earthquake Research Institute, University of Tokyo. I thank Prof. David Vere-Jones for stimulating discussions. 【Main references】Kagan, Y. (1993). Statistics of characteristic earthquakes. Bulletin of the Seismological Society of America. 83, 7-24. Kagan, Y. (2010)  Earthquake size distribution: power law with exponent \  1/2?. Tectonophysics, 490(1–2), 103–114.Vere-Jones, D. (1977). Statistical theories of crack propagation. Mathematical Geology, 9(5), 455-481.Vere-Jones, D. (1976). A branching model for crack propagation. Pure and Applied Geophysics, 114(4), 711-725. The 09/29/2009 Mw 8.0 Samoa Earthquake (USGS, 2009)The 02/06/2013 Mw 8.0 -10.7377,165.1378 Earthquake (USGS)The 3/11/2011 Tohoku-oki Earthquake (Lee et al, 2011)Plots of some typical source time functions. They are all different! Big errors

震源過程とVere-Jonesの分枝モデル間の類似特性

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震源過程とVere-Jonesの分枝モデル間の類似特性

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震源過程とVere-Jonesの分枝モデル間の類似特性

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