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1 c ETAS Epidemic-Type Aftershocks Sequence ETAS 1. John Milne James Ewing Thomas Gray Bolt, Alpide
2 a 4.2 bc km 2 2 b 2 c
3 Omori-Utsu19 Omori Omori, 1894 λ(t) t λ(t) K/(t + c). K Utsu Utsu, 1969 Omori-Utsu λ(t)= K (t + c). p K c p c p 1 Omori-Utsu Utsu et al Gutenberg-Richter m 0 log N( m)=a b(m m 0). N( m) m b Gutenberg- Richter b Gutenberg and Richter, 1954 Pr{M m} 10 bm = e βm. M β = b log e ETAS ETAS Daley and Vere-Jones, 2003 E{N ((t,t + δ t] (x,x + δ x] (y,y + δ y] (m,m + δ m]) H t} λ(t,x,y,m) = lim δ t 0 + δ tδ mδ xδ y δm,δx,δy 0 = µ(x,y,m)+ ξ(t,x,y,m;t i,x i,y i,m i). i:t i <t t (t i,x i,y i,m i) (t,x,y,m)
4 H t t t µ(x,y,m) m ξ(t,x,y,m;t i,x i,y i,m i) i 1 λ(t,x,y,m)=λ(t,x,y)s(m). (3.1) λ(t,x,y)=µ(x,y)+ ξ(t,x,y;t i,x i,y i,m i) i:t i <t s(m)=βe β(m mc), m m c Gutenberg-Richter p.d.f.2 ξ(t,x,y;t i,x i,y i,m i) ξ(t,x,y;t i,x i,y i,m i)=κ(m i)g(t t i)f(x x i,y y i;m i). κ(m)=ae α(m mc), m m c m g(t)= p 1 ( 1+ t ) p, t>0, c c p.d.f. ) q 1 (3.2) f(x,y;m)= (1+ x2 + y 2 q, πde γ(m mc) De γ(m mc) m c Zhuang et al. 2004, 2005 Ogata and Zhuang 2006 Ogata t (x,y) µ t ξ(t,x,y;t i,x i,y i,m i) 3.2 u(x,y) 3.2 µ(x,y)=νu(x,y). ν T log L(θ)= log λ θ (t k,x k,y k ) λ θ (t,x,y)dtdxdy {k:t k [0,T ],(x k,y k ) S} MLE ˆθ =(ˆν,Â, ˆα,ĉ, ˆp, ˆD,ˆγ) S [0,T] k Ogata S
5 Thinning procedure Lewis and Shedler, 1979; Ogata, 1981 (t j,x j,y j) j i i j ξ(t j,x j,y j;t i,x i,y i,m i), when j>i (3.3) ρ ij = λ(t j,x j,y j) 0, otherwise. j ρ j = i ρ ij, j (3.4) ϕ j =1 ρ j = µ(xj,yj) λ(t j,x j,y j) ϕ j,ρ j ρ ij j i 3.4 ˆµ(x,y)= 1 Z hj (x x j,y y j). T j T j h j h j =max{ε,inf(r : N[B(x i,y i;r)] >n p)}. ε B(x,y;r) (x,y) r n p h j n p (3.5) ˆµ(x,y)= 1 ϕ jz hj (x x j,y y j). T j j (3.6) ˆM(x,y)= 1 Z hj (x x j,y y j) T (3.7) Ĉ(x,y)= 1 (1 ϕ j)z hj (x x j,y y j) T j j
6 Zhuang et al., µ θ ϕ i 3 2 A A1. n p ε km (t j,x j,y j,m j : j =1,2,...,N) h j A2. l =0,u (l) (x,y)=1 A3. Ogata, 1998 λ(t,x,y)=νµ (l) (x,y)+ κ(m i)g(t t i)f(x x i,y y i;m i) i:t i <t κ,g f 3.1 A4. j<i(i =1,2,...,N) ρ ij,ρ i ϕ i A µ(x,y) u (l+1) (x,y) A6. ε max u (l+1) (x,y) u (l) (x,y) <ε l = l +1 A3 νu (l+1) (x,y) ρ ij,ρ i ϕ i km 2 a MLE A =0.020 event/(deg 2.day) α =1.135,c =0.040 day p =1.15 d =0.0010/deg 2 3 a d C(x,y)/M (x,y) 3 d ϕ i ρ ij
7 a 3.6 M /(deg 2 )b C /(deg 2 )3.7 c 3.5 µ /(deg 2 )dc/m 3.6 B B1. A ϕ i ρ ij i =1,2,...,N,j=1,2,...,N N B2. j (j =1,2,...,N) [0,1] U j B3. j { } k I j =max k 1:ϕ j + ρ ij U j and 0 k<j. i=1 I j =0 j j I j 4 3.5
8 ab c d e f B ϕ j ρ ij 4. ETAS 3.2 Ogata 1998 Zhuang et al. 2002, I ) q 1 (4.1) f(x,y;m)= exp ( x2 + y 2 q, πdeα(m mc) De α(m mc)
9 187 II (4.2) f(x,y;m)= q 1 ) (1+ x2 + y 2 q, πde α(m mc) De α(m mc) 2 I II Ogata 1998AIC 2 1 j i r ij = (xj x i) 2 +(y j y i) 2 De α(m i m c) I II r ij (4.3) f R(r)=2re r2, r 0, 2r(q 1) f R(r)=, r 0, (1 + r 2 ) q 4.3 Rayleigh f R(r) ( i,j ρiji r ij r < r ) 2 ˆf R(r)= r. i,j ρij r 2 ˆf R(r) f R 5 I II 5. f R (r) I II f R (r)
10 Ogata, 1998; Console et al., 2003 I II II I II ETAS 6 3 i j ( m ρiji i m < m ) 2 (4.4) ˆκ(m) =, m i j ( m ϕiρiji i m < m ) 2 (4.5) ˆκ b (m) = m, i ϕi X ETAS
11 189 (4.6) ˆκ c(m)= i j ( m (1 ϕi)ρiji i m < m 2 m i (1 ϕi) N f t N N = {t i : i =1,...,n} f(t,n) t {t k : t k <t} {t k : t k t} λ(t) N(t)=N[0,t] N(t) N(t) t t N f(t,n) f(t) 1. N λ(t) f E [ ] f(t i) = E f(t)λ(t)dt. S t i N S 2. Zhuang, 2006 N λ(t) h 1(s)h 2(t) f(s,t) h 1, h 2 D [ ] [ ] E f(t i,t j)i(i j (t i,t j) N N D) = E f(s,t)λ(s)λ(t)dsdt. D i,j 2 ETAS (5.1) λ 1(t,x,s)=λ 1(t,x,s,ω)I(ω =0)+λ 1(t,x,s,ω)I(ω =1). )
12 u(x)s 0(m), if ω =0, (5.2) λ 1(t,x,m,ω)= s 1(m) i,x i,m i,ω i), if ω =1, t i <tξ(t,x;t (5.3) { κ0(m i)g 0(t t i)f 0(x x i,m i), if ω i =0, ξ(t,x;t i,x i,m i,ω i)= κ 1(m i)g 1(t t i)f 1(x x i,m i), if ω i =1. ω =0 ω =1 (t,x,m) s 0(m) s κ 0(s) γ 1,g 0 f 0 (t,x,m) s 1(m) κ 1(m) s 1 g 1 f 1 κ 0 2 H(t,x,m;t,x,m )= s1(m )ξ(t,x ;t,x,m,ω)λ 1(t,x,m,ω)I(ω =0) λ 1(t,x,m )λ 1(t,x,m) (t,x,m;t,x,m )= s1(m )κ 0(m)g 0(t t)f 0(x x,m)u(x)s 0(m) λ 1(t,x,m )λ 1(t,x,m) λ1(t,x,m,ω)i(ω =0) h(t,x,m)= = u(x)s0(m) λ 1(t,x,m) λ. 1(t,x,m) f 1 f 2 g 1 g [ ] (5.4) E H(t i,x i,m i;t j,x j,m j)i(m i [m 0 δ,m 0 + δ]) i,j [ = E ] H(t,x,m;t,x,m )I(m [m 0 δ,m 0 + δ])dtdxdmdt dx dm = κ 0(m)s 0(m)I(m [m 0 δ,m 0 + δ])dm u(x)dtdx 2δκ 0(m 0)s 0(m 0) u(x)dtdx (5.5) [ ] E h(t i,x i,m i)i(m i [m 0 δ,m 0 + δ]) i [ = E = ] h(t,x,m)i(m [m 0 δ,m 0 + δ])dtdxdm s 0(m)I(m [m 0 δ,m 0 + δ])dm u(x)dtdx. 2δs 0(m 0) u(x)dtdx ratio-unbiased κ 0(m) i,j H(ti,xi,mi;tj,xj,mj)I(mi [m0 δ,m0 + δ]) ˆκ 0(m 0)= i h(ti,xi,mi)i(mi [m0 δ,m0 + δ]).
13 191 γ (0) 0 = γ (1) 0 =ˆγ κ (0) 0 = κ (1) 0 =ˆκ f (0) 0 = f (1) 0 = ˆf 0 ˆγ ˆκ ˆf ĝ ETAS ˆκ (1) 0 (m0)= i,j ˆϕi ˆρijI(mi [m0 δ,m0 + δ]) i ˆϕiI(mi [m0 δ,m0 + δ]). ˆϕ i ˆρ ij ϕ i,ρ ij i,j ˆκ H (t i,x i,m i;t j,x j,m j)i(m i [m 0 δ,m 0 + δ]) 1(m 0)= i [1 h(ti,xi,mi)]i(mi [m0 δ,m0 + δ]). 2 H (t,x,m;t,x,m )= s1(m )ξ(t,x ;t,x,m,ω)λ 1(t,x,m,ω)I(ω =1) λ 1(t,x,m )λ 1(t,x,m) (t,x,m;t,x,m )= s1(m )κ 1(m)g 1(t t)f 1(x x,m) λ 1(t,x,m ) 1 (m0)= i,j (1 ˆϕi)ˆρijI(mi [m0 δ,m0 + δ]) i (1 ˆϕi)I(mi [m0 δ,m0 + δ]). ˆκ (1) [ 1 u(x)s0(m) ] λ 1(t,x,m) g 1 g 2 f 1 f 2 6. ETAS Bolt, B. A Earthquakes and Geological Discovery, Scientific American Library from W. H. Freeman and Co., Salt Lake City, Utah. Console,R.,Murru,M.andLombardi,A.M Refining earthquake clustering models, Journal of Geophysical Research, 108 B , doi: /2002jb Daley, D. and Vere-Jones, D An Introduction to the Theory of Point Process, Vol. 1, Springer- Verlag, New York.
14 Gutenberg, B. and Richter, C. F Seismicity of the Earth and Associated Phenomena, 2nd ed., Princeton University Press, Princeton, New Jersey. Lewis, P. A. W. and Shedler, E Simulation of non-homogeneous Poisson processes by thinning, Naval Research Logistics Quarterly, 26, Ogata, Y On Lewis simulation method for point processes, IEEE Transactions on Information Theory, IT-27, Ogata, Y Space-time point-process models for earthquake occurrences, Annals of the Institute of Statistical Mathematics, 50, Ogata, Y. and Zhuang, J Space-time ETAS models and an improved extension, Tectonophysics, 413, Omori, F On after-shocks of earthquakes, Journal of the Faculty of Science, University of Tokyo, 7, Utsu, T Aftershock and earthquake statistics I : Some parameters which characterize an aftershock sequence and their interrelations, Journal of the Faculty of Science, Hokkaido University, Series. VII Geophysics, 3, Utsu, T., Ogata, Y. and Matsu ura, R. S The centenary of the Omori formula for a decay law of aftershock activity, Journal of Physics of the Earth, 43, Zhuang, J Second-order residual analysis of spatiotemporal point processes and applications in model evaluation, Journal of the Royal Statistical Society, Series B Statistical Methodology, 68, doi: /j x. Zhuang, J., Ogata, Y. and Vere-Jones, D Stochastic declustering of space-time earthquake occurrences, Journal of the American Statistical Association, 97, Zhuang, J., Ogata Y. and Vere-Jones D Analyzing earthquake clustering features by using stochastic reconstruction, Journal of Geophysical Research, 109, B05301, doi: /2003jb Zhuang, J., Chang, C.-P., Ogata, Y. and Chen, Y.-I A study on the background and clustering seismicity in the Taiwan region by using a point process model, Journal of Geophysical Research, 110, B05S18, doi: /2004jb
15 Proceedings of the Institute of Statistical Mathematics Vol. 57, No. 1, (2009) 193 Statistical Models for Earthquake Clustering and Declustering Jiancang Zhuang The Institute of Statistical Mathematics This review paper summarizes the statistical methods associated with modeling earthquake clusters and declustering. Earthquake clusters are well described by the epidemictype aftershocks sequence (ETAS) model. In this model, each earthquake, whether it is from the background or triggered by a previous earthquake, triggers its own child events independently according to some probability rules. The stochastic declustering method is developed by making use of the additive property of this model. With this method, each earthquake in the catalog can be identified to be a background event or be triggered from a particular previous event in probabilities. These estimated probabilities can be used to test hypotheses associated with seismicity clustering or background. Key words: Earthquake, ETAS model, cluster, declustering, second-order residual analysis.
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