Point process models for earthquakes with applications to Groningen and Kashmir data

Transcription

1 Point process models for earthquakes with applications to Groningen and Kashmir data Marie-Colette van Lieshout CWI & Twente The Netherlands Point process models for earthquakes with applications to Groningen and Kashmir data p. 1/21

2 Earthquake data Latitude (degrees north) Longitude (degrees east) Left: Groningen 2014, M 1.5. Right: Kashmir Oct 8 Nov 7, 2005, M 4.5. Point process models for earthquakes with applications to Groningen and Kashmir data p. 2/21

3 Data characteristics a region W : the gas field and a bounding box, respectively; a finite list of points in W ; additional marks (time stamps, magnitudes,...); Sometimes also information about the environment is available (covariates). Such data is called a point pattern. Point process models for earthquakes with applications to Groningen and Kashmir data p. 3/21

4 The Poisson process Write N X (A) for the number of points in A. A point process X on R d is a homogeneous Poisson process with intensity λ > 0 if N X (A) is Poisson distributed with mean λ A for every bounded Borel set A R d ; for any k disjoint bounded Borel sets A 1,...,A k, k N, the random variables N X (A 1 ),...,N X (A k ) are independent. Note: Replacing λ A by A λ(x)dx for some integrable function λ : R d R + yields an inhomogeneous Poisson process. Point process models for earthquakes with applications to Groningen and Kashmir data p. 4/21

5 First and second order moments Let X be a point process on R d. Define, for Borel sets A,B R d, α (1) (A) = EN X (A); µ (2) (A B) = E[N X (A)N X (B)]; [ ] α (2) (A B) = E 1{x A;y B}. x X y X Often, α (2) (A B) = for product density ρ (2). Normalise A B ρ (2) (x,y)dxdy to get the pair correlation function. g(x,y) = ρ(2) (x,y) ρ (1) (x)ρ (2) (y) Point process models for earthquakes with applications to Groningen and Kashmir data p. 5/21

6 Spatial regression Goal: given data pattern x = {x 1,...,x n } in W, investigate whether the intensity depends on covariate functions C j : W R, j = 1,...,p. Point process models for earthquakes with applications to Groningen and Kashmir data p. 6/21

7 Maximum likelihood for Poisson processes Model λ(u) = λ β (u) = exp [ β 0 + ] p β j C j (u). j=1 The log likelihood function L(β) reads p n L(β) = nβ 0 + β j C j (x i ) e β 0 j=1 i=1 W [ p ] exp β j C j (u) du j=1 so ˆβ solves the score equations C j (u)λ β (u)du = W n C j (x i ) i=1 for j = 0,...,p under the convention that C 0 1. Point process models for earthquakes with applications to Groningen and Kashmir data p. 7/21

8 Results Van Hove et al., 2015 Three students at Twente (Van Hove, Van Lingen and Riemens) estimated the parameters in this model and extrapolated them into the future. Remark: In 2014, the Dutch government decided to reduce gas extraction near Loppersum (NW) by some 80%. In this scenario, 18 rather than 24 quakes are predicted in 2018 Point process models for earthquakes with applications to Groningen and Kashmir data p. 8/21

9 Hypothesis tests To assess the significance of the covariates, use where f is the likelihood function. Λ(X) = sup{f(x;β) : β j = 0} f(x; ˆβ) Under β j = 0, 2logΛ(X) is approximately χ 2 1-distributed. For the composite hypothesis β j = 0 for all j = 1,...,p, use Λ(X) = sup{f(x;β) : β 1 = = β p = 0} f(x; ˆβ). Now, 2logΛ(X) is approximately χ 2 p-distributed. Van Hove et al.: the kernel smoothed fault lines, amount of gas extraction and subsidence all proved significant; the number of quakes in previous years was not. Point process models for earthquakes with applications to Groningen and Kashmir data p. 9/21

10 Kashmir data 176 spatial locations of shallow (depth less than 70 km) earthquakes of magnitude 4.5 or higher recorded October 8 November 7, 2005, in W = [72.65, 74.25] [33.70, 35.25] (Kashmir). Latitude (degrees north) Longitude (degrees east) Point process models for earthquakes with applications to Groningen and Kashmir data p. 10/21

11 Exploratory analysis Frequency Magnitude Time in hours from October 8, Time in hours from October 8, 2005 Notes: 76% of shocks happen within the first 48 hours from the main shock; two clear magnitude extremes of 7.6 (the main shock) and 6.4 ((Båth s law). Point process models for earthquakes with applications to Groningen and Kashmir data p. 11/21

12 Poisson cluster process X = x Φ(x+Z x ) where the Z x are i.i.d. clusters centred around the points x of a homogeneous Poisson process Φ. Depending on the structure of Z x, there are two main models: Trigger process: Z x consist of a Poisson number of i.i.d points; Hawkes process: Z x is a branching process. Point process models for earthquakes with applications to Groningen and Kashmir data p. 12/21

13 Trigger process Φ Poisson(κ); each parent generates Poisson(ν) offspring, i.i.d. bivariate normally distributed around the parent with covariance matrix σ 2 I. The trigger process X is stationary with intensity ρ = κν and has pair correlation function (Stoyan et al. 1995). g(r = x y ) = πκσ 2 exp[ r 2 /(4σ 2 ) ] Note: g(r) does not depend on ν, so fix ρ = κν = 176/ W. Point process models for earthquakes with applications to Groningen and Kashmir data p. 13/21

14 Hawkes process Φ Poisson(κ); its points form a subset of X, called generation zero; each point of generation j = 0,1,... acts as parent and produces Poisson(ν) offspring, i.i.d. bivariate normally distributed around the parent with covariance matrix σ 2 I with ν < 1; the combined offspring of generation j form generation j +1; the iteration terminates at the first empty generation. The Hawkes process X is stationary with intensity ρ = κ/(1 ν) and has pair correlation function g(r) = 1+ 1 ν κ (n+1)ν n 1 2πnσ exp[ r 2 /(2nσ 2 ) ] 2 n=1 (Møller and Torrisi 2007). Point process models for earthquakes with applications to Groningen and Kashmir data p. 14/21

15 Minimum contrast method Asume the pair correlation function g : R 2 R is rotation-invariant and minimise over the parameter θ. t1 t 0 ĝ(r) g(r;θ) 2 dr, Here and λ 2 g(r) = 1 2πr x X W ˆλ = N X(W) W y X W is an edge-corrected kernel estimator. 1{ ǫ r y x ǫ} 2ǫ W W y x Point process models for earthquakes with applications to Groningen and Kashmir data p. 15/21

16 Results g(r) g fit(r) g^ripley(r) g Pois(r) g(r) g fit(r) g^ripley(r) g Pois(r) Distance r (in degrees) Distance r (in degrees) Trigger process ˆκ = 0.88, ˆσ = 0.08 and ˆν = 81. Thus, 2.17 main shocks in Kashmir with 81 aftershocks each. Hawkes process ˆκ = 7.44, ˆσ = 0.03 and ˆν = 0.9. Thus, main shocks in Kashmir with 1/(1 ν) = 9.54 aftershocks each. Point process models for earthquakes with applications to Groningen and Kashmir data p. 16/21

17 Mixture model of two bivariate normal distributions with a cluster around the main shock at µ 1 = (73.59,34.54); a cluster around the location µ 2 = (73.10,34.73) of the second largest shock; all m = 69 earthquakes y i happening before the second shock belonging to the first cluster. For the other n = 105 earthquake locations x i, introduce latent variables (Z 1,...Z n ). If Z i = 1, x i is allocated to µ 1, if 2 to µ 2. Likelihood function m n [ f(y i µ 1,Σ 1 ) p1{zi =1}f(x i µ 1,Σ 1 )+(1 p)1 {zi =2}f(x i µ 2,Σ 2 ) ] i=1 i=1 where f( µ,σ) is the bivariate normal probability density function with mean vector µ and covariance matrix Σ. Point process models for earthquakes with applications to Groningen and Kashmir data p. 17/21

18 Mixture parameter estimation by EM-method Iterate p t+1 Σ 1,t+1 = = 1 n + Σ 2,t+1 = n P(Z i = 1 X i = x i ;θ t ) i=1 m i=1 (y i µ 1 )(y i µ 1 ) T m+ n i=1 P(Z i = 1 X i = x i ;θ t ) n i=1 P(Z i = 1 X i = x i ;θ t )(x i µ 1 )(x i µ 1 ) T m+ n i=1 P(Z i = 1 X i = x i ;θ t ) n i=1 P(Z i = 2 X i = x i ;θ t )(x i µ 2 )(x i µ 2 ) T n i=1 P(Z i = 2 X i = x i ;θ t ) ;, where P(Z i = 1 X i = x i ;θ t ) = p t f(x i µ 1,Σ 1,t ) p t f(x i µ 1,Σ 1,t )+(1 p t )f(x i µ 2,Σ 2,t ) and P(Z i = 2 X i = x i ;θ t ) = 1 P(Z i = 1 X i = x i ;θ t ). Point process models for earthquakes with applications to Groningen and Kashmir data p. 18/21

19 Results Latitude (degrees north) Longitude (degrees east) Allocate x i to µ 1 (triangles) if ˆpf(x i µ 1, ˆΣ 1 ) > (1 ˆp)f(x i µ 2, ˆΣ 2 ) with ˆp = 0.28, ˆΣ =, ˆΣ = and to µ 2 (crosses) otherwise. The circles indicate the y i. Point process models for earthquakes with applications to Groningen and Kashmir data p. 19/21

20 Conclusions Groningen (Van Hove et al., 2015) The number of induced earthquakes has increased over the years. An increase in magnitude is not statistically significant. Covariates on gas extraction and subsidence have a statistically significant effect on the earthquake intensity. Even if there would be no more gas extracted, there would still be earthquakes in the near future. E. van Hove, R. van Lingen en S. Riemens. Geïnduceerde aardbevingen in gasveld Groningen. Een statistische analyse. (In Dutch). B.Sc. thesis, University of Twente, Point process models for earthquakes with applications to Groningen and Kashmir data p. 20/21

21 Conclusions Kashmir A trigger process is a better fit than a Hawkes process. Gaussian mixture modelling allows for variable cluster characteristics. The cluster of aftershocks of the main shock is more widely scattered than that of the second most severe one. Longitude and latitude are negatively correlated; the variation in longitude is larger than that in latitude (reflecting the tilt of the convergence zone in northern Pakistan). K. Türkyilmaz, M.N.M. van Lieshout and A. Stein. Comparing the Hawkes and trigger process models for aftershock sequences following the 2005 Kashmir earthquake. Mathematical Geosciences, 45: , Point process models for earthquakes with applications to Groningen and Kashmir data p. 21/21