fŵ𗘗pBPTߒ̎OzɂxCY^\ɂ

A Bayesian predictor based on prior distributions of BPT model with slip rates

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The Institute of Statistical Mathematics

nkψiERCjɂ{̊fwf[^PjڍׂɌJāCefŵRONm\Ȃǂ\ĂDnk\ł͎Brownian Passage Time iBPTjzɂXVߒf𓖂Ă͂߂Đ肵Ăi}PQƁjDʂƃ͕ςƂ΂\p^łD̍Ŗސl (MLE) p^ɑė\zivOC\j

@@@@@@@                               (1)

{񍐂ł BPT z̃p^ʂƃ͊fwƂɈقȂlƂȂCŜƂĂ鎖Oz甭ĂƍlD̊fw瓾ꂽp^̐挱iOzjƁCX̊fw̒nkf[^瓾p^̏iޓx֐j̗p㕪zgāCfw̒nkm͎̊֐ixCY\zjɂ萄łD

(2)

ŁCœKȎOz\邱ƁioIxCY@jdvłD̂߂ɊfŵP̒nk̗̂UݐϓIȂ̑x V ŏϊԊu T = U / V Ŋ ERC 36 f[^܂ 79 f[^𓯎ɖޓxɎgC/ T  ƃ̎Ož̒ ԒrxCYʋKABIC ŏ悤ȎOzIԁDC̏񂪖c 43 f[^͎ۂ̊Ԋuf[^𒼐ڎgCϊԊu̎Ozɂ 1/ ʂɎgDʂƂă/T  ƃ̍ŗǂ̎Oz\P̃f̒ł͋ Weibull złi\PR}QƁjCwzΑ卷͂ȂD傫΂Ă0.34𕽋ςƂ悤ɕzĂ邱ƂD= 0.34̃vOC\ł95%EiP}̑Ίpjł95%EEُlDɑ΂ăxCY\ł͍I95%EiP}̑ΊpԐj^ĂD

đSʂ̎OziR}jgČvZꂽefw̎㕪z̃̕ϒlixCYlj͋[ʂĂiS}QƁjDႦΓǩǗAXyeB̑nk̓CŊfw̍ݍĂ钆n啪̒naтł̓l傫ȂĂDfwԂ̒nk̑ݍpƃl̂΂͊֌WĂDCP}玦悤ɁCp^Brownian Passageߒ̗h炬̕W΍ɔႵĂCꂪfwԂ̃XgXω̑ݍp𔽉fĂD

XɁC㕪zɂ (2) ̗\zƎۂ̃f[^̔rT}ɎĂD

L@ō\Oz△̎Ozɂ\zƌs ERC ɂ\zɂāCV~[Vɂ\x𕽋ϑ΃Ggs[ő肵rʁiU}}jDL̃xCY^\̂ǂ̎@肵ĕϓIɗDĂ邱ƂDU}E}́CgϊԊu^̂̂0.5~2.0{̒ɔ[܂΃xCY^\̌덷͖2{ȓɎ܂邱ƂĂD

ERC \xCY^\Ŋfwтւ̓KpʂrƖwǂ̊fwтŒnkm͊TˍvCꕔ̊fwтŗLӂȍoi\PjD͕ϊԊux΂ 傫قȂf[^ƁuԊu̕ϒlvƁu̗ʁEx̊vɂ镽ϊԊu傫قȂ邽߂ɕσʂ傫قȂf[^i\PjD

i쑺r, `ǕFj

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@1) nki{nkψCfw]@http://www.jishin.go.jp/main/p_hyoka02_danso.htm@

@2) Ί֗KEG@Cfw 27 (2007)C63-73D

@3) cERVEO~EcCnk 2 60 (2007)C85-100D

@4)@Kumamoto, T. and Y, Hamada, Active Fault Research, 25 (2005), 9-22.

@5) LEOcCfw 30 (2009)C27-365)

\P@OzABIC̑Βl

Table. 1. Prior distribution models and the relative ABIC values

\Q@BPTzm, a̐ƂRONm̒nkψ\ƃxCY\̔rDijxCY㕪źuiҒlj}iW΍jv ŕ\

Table 2. Comparison of the m, a estimates and 30 year probability between the ERC and the Bayes predictor. () means average probability (}standard error)

P}@BPTzBPTߒ̊֌W

Fig. 1. Relationship between the BPT probability distribution and the BPT process

Q}@efw̒nkԊuf[^΂ꑬx҂銈Ԋũvbgyтꂼ̐덷DΊp̗ΐAAԐ͂ꂼnkρi= 0.24j̗\zAŖސʁi= 0.34j̗\zAĎ@̗\z\Ԋu95%ED

Fig. 2. Occurrence time intervals of a fault against the expected interval from the estimate of the ratio of slip size to deformation rate of a fault. Pink bars are data given in range, and the green, blue and red diagonal lines show 95% error bounds of the BPT prediction intervals from the expected time interval assuming = 0.24, 0.34 for the plug-in predictor and the proposed Bayesian predictor, respectively.

R}@̂f[^狁߂Ŗސl/ T iFXpCNjƂ̃qXgODȐ\PœꂽœKȎOmxzDc̗ΐƐ͂ꂼnkς̃̐li0.24jƑSf[^琄肵̍Ŗސʁi0.34j̈ʒuĂD

Fig. 3. The blue spikes and the histograms of the MLEs(normalized recurrence time) and obtained from respective fault segments, and the solid curves show the estimated prior densities. The green and blue vertical lines shows the value of ERC's estimate (0.24) and MLE from all data (0.34).

S}@efwBPTza  ̃xCYli㕪z̕ϒlj

Fig. 4.@Posterior mean (Bayes estimate) of the BPT a value for respective fault

T}@nkԊuBPTxCY\ziԐjBPTvOC\zia = 0.34CjERC BPT\zia = 0.24CΐjƐKnk̊Ԋuf[^̏dˍ킹̔rD(}) ꂽnkԊũXpCN\ƖxzƃqXgOC(E})̗ݐϕzD

Fig. 5.@Comparison between the Bayesian predictive distribution (red lines), plug-in predictive distributions (= 0.34, blue lines) and the ERC (= 0.24, green lines) for the empirical distributions of normalized recurrence intervals. (Left) Histogram and the corresponding probability density functions, (Right) Empirical distribution function and the corresponding cumulative distribution.

U}@i}jV~[Vɂ胿ωƂ̊e\z̗\덷iϑ΃Ggs[j̔rDERC (a = 0.24; ΐ)CMLE (a = 0.34; )C񖳂̃xCY\ijyя\i/T = 1̂ƂjgxCY\iԐjDiE}jgxCY\ / TƃωƂ̗\덷̓Dԓ_̒fʂ}̐ԋȐɑΉD

Fig. 6.@Prediction error for simulated data (Mean Relative Entropy) of the plug-in predictor with= 0.34 (blue curve), plug-in predictor with= 0.24 (green curve), Bayesian predictor using only occurrence data (purple line) and Bayesian predictor using the loading and slip data when m/T = 1 (red curve) for the sampled data with various a.