## International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

### Development of stochastic heterogeneous slip distribution model for simulation of earthquake ground motion Abe, Hiroyasu; Sekimura, Naoto; Itoi, Tatsuya 2015-07

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`12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Development of Stochastic Heterogeneous Slip Distribution Model for Simulation of Earthquake Ground Motion   Hiroyasu Abe Graduate Student, Graduate School of Engineering, The University of Tokyo Naoto Sekimura Professor, Graduate School of Engineering, The University of Tokyo  Tatsuya Itoi Associate professor, Graduate School of Engineering, The University of Tokyo  ABSTRACT: Ground motion simulation is an important tool for analyzing seismic risk in engineering systems. Recently, ground motion simulations using fault model are being widely applied. Characterized fault models are conveniently used to model the heterogeneous slip distribution on fault plane, which divides the fault into two areas, i.e., asperity area and background area. The model, however, is too simplified to model the complex characteristics of actual slip. In this paper, a model to simulate the slip distribution of the fault plane is proposed.  1. INTRODUCTION Ground motion simulation is an important tool for analyzing seismic risk for engineering systems. Recently, ground motion simulations using so-called fault model are being widely used for this purpose. A simplified characterized fault model was proposed and used for that purpose. The method, however, was proposed to calculate the average characteristics of ground motion. More detailed model is needed to conduct probabilistic seismic risk assessment, which would incorporate uncertainty in ground motion prediction. In some studies, the spatial coherence is investigated by considering zero offset and nonzero offset2), or stochastic characterization of the spatial complexity of slip was proposed3). Most of the studies, however, focus on the correlation characteristics between source parameters and assume that the slip is normally distributed. In this study, a probabilistic modeling of slip distribution focusing on crustal earthquake is discussed and proposed.  2. STOCHASTIC MODEL FOR HETEROGENEOUS SLIP DISTRIBUTION ON FAULT PLANE  2.1 Probabilistic distribution of slip  In this study, the 2005 West Off Fukuoka Earthquake occurred in Japan is used as an example of crustal earthquakes. Figure 1 shows the spatial distribution of slip in the fault plane of the 2005 West Off Fukuoka Earthquake4),5). In a conventional method1), the fault plane is divided into two areas, i.e., asperity area and background area. In this study, a more detailed model for slip distribution is proposed.  The length in the slip point at Figure 1 is denoted as y={y1,･･,yn}, where n is the number of divided areas. First, probabilistic distribution for the slip displacement is investigated. Figure 2 shows the probability paper plot of the actual slip. The log normal and gamma distributions are also shown for comparison. These distribution are obtained by maximum likelihood estimation. The log-normal distribution fits the actual data than gamma distribution. Therefore, the log-normal 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 distribution is used hereafter.  Table 1 shows the average and standard deviation of log y in Figure 1.  Figure 1: Fault plane of 2005 West Off Fukuoka Earthquake4)5)    Figure 2: Cumulative distribution of slip at each element of 2005 West Off Fukuoka Earthquake   Table 1: Average and standard deviation of logarithmic slip of 2005 West Off Fukuoka Earthquake Average Standard deviation 3.90 0.87  2.2 Spatial correlation of slip  Next, the correlation structure from slip distribution during actual earthquake is analyzed. Semivariogram ‘r’ is used for that purpose. r(Yi,Yj) is defined as follows:   ])[(21),( 2jiji YYYYr  (1) where, Yi and Yj are the lengths of slip at the i-th and j-th element respectively. And, semivariogram ‘r(Yi,Yj)’ is defined  as follows:  )1(),( 2 YiYjYji YYr    (2) where, σY is the standard deviation of the fault plane and ρYiYj is the correlation coefficient between Yi and Yj . Semivariogram and correlation coefficient have a relationship that correlation coefficient is closer to 1 when r(h) is smaller, and it is closer to 0 when r(h) is larger. In this study, this correlation is assumed to depend on h, which is the distance between two slip points. The correlation coefficient 𝑟(ℎ) is defined as follows:   𝑟(ℎ) =12𝑁(ℎ)∑ (𝑌(𝕏1𝑖) − 𝑌(𝕏2𝑖))2𝑁(ℎ)𝑖=1  (3) Provided that N(h) is the number of pairs that fulfill equation (4) include (𝕏1𝑖, 𝕏2𝑖).  h −△ℎ2≦ |𝕏1𝑖 − 𝕏2𝑖| ≦ h +△ℎ2 (4) where,Δh is decided to be 1.34km by trial and error. Figure 3(a) shows r(h) obtained from Equation (3). Figure 3(b) shows N(h) for each bin. In Figure 3(a), the fitted curve (Equation (5)) that is determined from the least –squares method is also shown.   ))exp(1()( 2ahbhr   (5) where, b is equal to the dispersion σY’2 that is determined from Y. a is 0.0947 km-2, and b is 3.4819×103.    Figure 3: Semivariogram showing spatial correlation of slip obtained from slip distribution during 2005 West Off Fukuoka Earthquake  2.3 Stochastic simulation of slip distribution Simulation of slip distribution in fault plane is conducted by Monte Carlo Simulation. The slip 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 of the fault plane in i-th element is Yi. Y=[Y1,Y2,…YN]’. Y follows the log-normal distribution.  W is the normal random variable that is transformed from Y by the Rosenblatt transformation as follows:   ))((1 YFW   (6) where, Φ-1( )  is the cumulative function of a standard normal distribution, and F(Y) is the cumulative distribution function of  Y. W is determined from Equation (7) as follows:  ZW w  (7) where, Z is the vector of random variables that fulfill the normal distribution, with zero mean and unit standard deviation. ΦW is the eigenvector of covariance CWW as follows:  WWWWWC   (8) ΛW is the square matrix, where diagonal element is the eigenvalues and the others are 0. 𝐶𝑊𝑊 is expressed as follows: 𝐶𝑊𝑊 = 𝜎𝑊2 [𝜌𝑊1𝑊1 ⋯ 𝜌𝑊1𝑊𝑁⋮ ⋱ ⋮𝜌𝑊𝑁𝑊1 ⋯ 𝜌𝑊𝑁𝑊𝑁] (9) where, 𝜎𝑊 is the standard deviation of 𝑊. In this equation, 𝜌𝑊𝑖𝑊𝑗  is the correlation coefficient between Wi and Wj. In this study, it is assumed that 𝜌𝑊𝑖𝑊𝑗 is equal to 𝜌𝑌𝑖𝑌𝑗, and fulfills Equation (5). Therefore, it is assumed that 𝜌𝑊𝑖𝑊𝑗  is a function of only hij which is the distance between i and j.   𝜌𝑊𝑖𝑊𝑗 = 𝑒𝑥𝑝(−𝑎ℎ𝑖𝑗2) (10) Distribution in fault is simulated under the condition of MW 6.6 (M0=9.0×1018N・m) that is same as the 2005 West Off Fukuoka Earthquake. M0 is seismic moment, and MW is moment magnitude. Fault area S and seismic moment M0 have a relation as follows:  )105.7()10(1024.4 180217011   MMS  (11) Fault area is 420km2 and it is presumed that the depth of the area is 15km and the width is 28km. The fault area is divided into 140 elements.  The example of simulated slip distribution is shown in Figure 4. Real slip distribution of 2005 West Off Fukuoka Earthquake is shown in Figure 5.                On one hand, large slip area (red color in Figure 5) is concentrated on upper center area and below center area (Figure 1). On the other hand, large slip area is not so much concentrated in Figure 4. Though heterogeneous feature of fault slip can be modeled, the large slip area is scattered, compared with Figure 1.    Figure 4: A sample of simulated distribution     3. VALIDATION OF PROPOSED MODEL USING GROUND MOTION In this section, the proposed model is validated using ground motion simulation. Stochastic Green’s function7) method is used for ground motion simulation.  The fault geometry and receiver location are showed in Figure 5. S-wave velocity on surface is assumed to be 400m/s in this study.    Figure 5: The receiver location of calculating point and fault   12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 The seismic moment of all fault area is distributed at each element. The seismic moment at each element depends on the slip length at each element. Therefore, seismic moment M0i at each element is given as follows:    iii yyMM 00 (12) where, yi  is the slip length at the i-th element . Stress drop σi is assumed to be given as follows:   23023167iii SM  (13) where, 𝑆𝑖 is the area of each element. Simulated acceleration time history is shown in Figure 6. Ground motion is simulated for slip distribution shown in Figure 4. According to Figure 6, S-wave arrives at 3 seconds since earthquake occurrence and continues for about 17 seconds. The maximum acceleration is 17m/s2 at t = 7.0s. In Figure 7, velocity history is shown. The maximum velocity is 0.9m/s at t = 7.5(s).   Figure 6: The acceleration for the case of Figure 4.    Figure 7: The velocity for the case of Figure 4   Monte Carlo simulation is conducted to simulate different slip distribution under the same condition as 2.3. In Figure 8(a), examples of slip distribution are shown, while simulated ground motion for respective case are shown in Figure 8(b). As shown in the figure, the temporal characteristics of velocity history are different between samples. Maximum velocity increases if large slip appears between the hypocenter and the receiver as shown in the bottom figure in Figure 8.    Figure 8: Distribution slip and velocity at the ground point   To verify the proposed method, maximum acceleration and maximum velocity are compared with existing attenuation model10). Figure 10 and Figure 11 show the comparison for maximum acceleration and velocity respectively. Simulated ground motions are larger than that by the attenuation models. In these figures, logarithmic standard deviation are also shown. The logarithmic standard deviation by our proposed method is little smaller than that by the attenuation model. So, variation in other parameters such as hypocenter location is required to be considered. For example, stress drop for each element obtained by Equation (13) is required to be modified to simulate more realistic ground motion.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5  Figure 9: Comparison between maximum acceleration simulated and attenuation model (○: median, I: logarithmic standard deviation）   Figure 10: Comparison between maximum velocity simulated and attenuation model (○: median, I: logarithmic standard deviation）  4. CONCLUSIONS In this paper, a model to simulate the stochastic slip distribution of fault plane in crustal earthquake was discussed. It was demonstrated that the slip of the 2005 West Off Fukuoka Earthquake is log-normally distributed, and spatial correlation can be modeled using semivariogram. The seismic ground motion was simulated from a slip distribution of fault plane calculated by our proposed model. The simulated slip distribution was closer to real distribution than the characterized fault model. It needs, however, to be improved in the further study. One is that large slip area simulated by the proposed method scatters compared with the real slip distribution. Another issue to be solved is ground motion simulation method, e.g. how to determine stress drop for each element.  5. REFERENCES 1) Headquarter for Earthquake Research Promotion: “Recipe for Predicting Strong Ground Motion by a fault plane of earthquake”, 2009. http://www.jishin.go.jp/main/chousa/09_yosokuchizu/g_furoku3.pdf (In Japanese) (cited: 2015-03-07) 2) Song, S. G., Pitarka A., and Somerville P.: “Exploring Spatial Coherence between Earthquake Source Parameters”, Bulletin of the Seismological Society of America, Vol. 99, No. 4, 2564–2571, August 2009 3) Mai, P. M. and Beroza, G. C.: “A spatial random field model to characterize complexity in earthquake slip”, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B11, 2308 4)http://equake-rc.info/SRCMOD/searchmodels/viewmodel/s2005FUKUOK01ASAN/ (cited: 2014-07-24） 5) Asano, K. and Iwata, T.: “Source process and near-source ground motions of the 2005West Off Fukuoka Prefecture earthquake”, Earth Planets Space, 58, 93–98, 2006. 6) Irikura, K. and Miyake, H.,: “Prediction of Strong Ground Motions for Scenario Earthquakes”, Tokyo Geographical Society, 110, 849-875、2001. 7) Boore, D. M.: “Stochastic Simuration of High-Frequency Ground Motions Based on Seismological Models of the Radiated Spectra”, Bull. Seism. Soc Am., 73, No.6, 1865-1894, 1983. 8) Kamae, K., Irikura K., and Fukuchi, Y.: “Prediction of strong ground motion based on scaling law of earthquake”: By stochastic synthesis method, Journal of structural and construction engineering, 430, 1-9, 1991. 9) Eshelby, J. D.: “The determination of the elastic field of an ellipsoidal inclusion, and related problems”, Proceedings of the Royal Society, A241, 376-396, 1957. 10) Si, H. and Midorikawa, S.: “New Attenuation Relationships for Peak Ground Acceleration and Velocity Considering Effects of Fault Type and Site Condition”, Journal of structural and construction engineering, 523, 63-70. (In Japanese) `

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