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A New Multidimensional Stress Release Statistical Model Based on Co-seismic Stress Transfer Mingming Jiang, Shiyong Zhou, Yongshun John Chen and Yinshuang Ai Institute of Theoretical and Applied Geophysics, School of Earth and Space Science, Peking University, Beijing 100871, China E-mail: zsy@pku.edu.cn Abstract Following the stress releasemodel (SRM) proposed by Vere-Jones (1978), we developed a new multidimensional SRM, which is a space–time–magnitude version based on multidimensional point processes. First, we interpreted the exponential hazard functional of the SRM as the mathematical expression of static fatigue failure caused by stress corrosion. Then, we reconstructed the SRM in multidimensions through incorporating four independent submodels: the magnitude distribution function, the space weighting function, the loading rate function and the coseismic stress transfer model. Finally, we applied the new model to analyse the historical earthquake catalogues in North China T53A-2678 Conclusions MSRM is a new space–time–magnitude version of SRM. As an essential improvement, the new model is extended to space dimensions through involving the coseismic stress transfer model and the space-weighting function. Compared with SRM and CSRM using the same catalogue data, MSRM results in the prominent AIC and AICc reductions, which reflects the advantage of MSRM in statistics. References Ogata, Y., 1998. Space-time point-process models for earthquake occurrence, Ann. Inst. Stat. Math., 50, 379–402. Vere-Jones, D., 1978. Earthquake prediction: a statistician view, J. Phys.Earth, 26, 129–146. 2012AGU Introduction The stress release model (SRM; Vere-Jones 1978) is a statistical representation of the elastic rebound hypothesis. Models based on the elastic rebound hypothesis imply that a large earthquake should be followed by a period of quiescence. Generally, the promotion and inhibition of subsequent events can exist simultaneously in space after the occurrence of a large earthquake. This kind of phenomenon is interpreted as stress transfer. These pioneering models are available for constructing new space–time physical models instead of the elastic rebound model. Figure 1. Historical destructive earthquakes in North China (−780 to 1997). The blue circles denote the historical earthquakes in the weighting catalogue. The earthquakes with beach balls belong to the expanded catalogue. The red bars denote the strikes and rupture lengths of earthquakes. The origin time is marked in the figure. Model description We define the earthquake sequences as multidimensional point processes in space, time and magnitude. The conditional probability intensity, namely the earthquake occurrence rate, is defined by (Ogata 1998), for infinitesimal dt, dx, dy and dM. We formulate MSRM as where X(t,x,y) is the scalar notional stress in space and time,Ψ[X(t,x,y)] is the hazard functional. j[M|X(t,x,y)] is the magnitude distribution. where μ and ν are the undetermined parameters, M min and M max are the lower and upper magnitude bounds of the earthquake catalogue, β is the undetermined parameter proportional to b value (β = bln10). The scalar notional stress function X(t,x,y) can be expressed as where X 0 (x,y) is the initial stress, ρ(x,y) is the loading rate function, S(t,x,y) is the accumulated earthquake-induced Coulomb stress variation. we combine μ(x,y) with X 0 (x,y) to form a space-weighting function w(x,y). where S(x,y,ξ i,ζ i ) is the Coulomb stress variations at the location (x,y) induced by the ith earthquake with the epicentre (ξ i,ζ i ) and the origin time t i. in the 3-D half-space are represented as where τ s is the shear stress variation along the slip directions of receiver faults, σ n is the normal stress variation, μ’ f is the apparent friction coefficient. Thus, the final formulation of MSRM is expressed as (1), (3), (5), (6), and we applied it in North China. Figure 2. Snapshot maps of the conditional intensity function just before and after deadly historical earthquakes in North China: Huaxian earthquake(1556.02.02, M = 8.25), Tancheng earthquake (1668.07.25, M = 8.5), Sanhe earthquake (1679.09.02, M = 8.0) and Tangshan earthquake (1976.07.28, M = 7.8). The earthquakes in the expanded catalogue are plotted. The related earthquakes are plotted with stars and the other earthquakes are plotted with circles. The filled symbols mean the earthquakes had occurred and the open symbols mean the earthquakes would occur. Application to North China We apply MSRM to analyse the historical earthquake catalogue in North China. AIC and AICc are adopted as effective tools for model comparison between MSRM and other versions of SRM and CSRM. Table 1. Model comparisons between MSRM and the previous versions

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ss= * +(a-b) ln(V/V * ) a-b > 0 stable sliding a-b < 0 slip is potentially unstable Correspond to T~300 °C For Quartzo- Feldspathic rocks Stationary.

ss= * +(a-b) ln(V/V * ) a-b > 0 stable sliding a-b < 0 slip is potentially unstable Correspond to T~300 °C For Quartzo- Feldspathic rocks Stationary.

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