Reviewing of stochastic description of the spatial variation of ground motion and coherence function model
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摘要: 本研究主要讨论地震动空间变化的随机描述. 首先给出了基于密集地震台阵记录估计相干函数的方法,并对计算中需要关注的问题给出了相应的解释;然后对现有的经验和半经验相干函数模型的建立进行了详细的梳理,并对模型在工程应用中的适用性、有效性和局限性进行了讨论;最后通过对比分析不同相干函数模型对场址地震动空间相关性的模拟结果,对相干函数模型的选择提出了建议.Abstract: This study addresses the topic of the spatial variation of seismic ground motions as evaluated from data recorded at dense instrument arrays. It concentrates on the stochastic description of the spatial variation, and focuses on spatial coherency. The estimation of coherency from recorded data and its interpretation are presented. Some empirical and semi-empirical coherency models are described, and their validity and limitations in terms of physical causes discussed. Finally, by comparing and analyzing the simulation results of different coherence function models on the spatial correlation of ground motion, some suggestions on the selection of coherence function model are put forward.
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图 2 UPSAR台站布置图(修改自Yu et al., 2011)
Figure 2. Distribution of UPSAR stations (modified from Yu et al., 2011)
图 3 迟滞相干函数随频率的变化(记录来源于自贡台阵的汶川地震记录)(修改自Yu et al., 2020)
Figure 3. Variation of hysteresis coherence function with frequency (recorded from Wenchuan earthquake records of Zigong array) (modified from Yu et al., 2020)
图 4 迟滞相干函数随台站距离的变化.(a)汶川地震记录(ZGSA)(与台站Z0相关的值用‘*’表示, 与台站Z1相关的值用‘+’表示, 其他值用‘○’表示);(b)Parkfield地震记录(UPSAR)(修改自Yu et al., 2011, 2020)
Figure 4. Variation of the hysteretic coherence function with the distance of the station. (a)Wenchuan earthquake (ZGS) (The values related to station Z0 are denoted with asterisks‘*’, the values related to station Z1 are denoted with plus symbols ‘+’, and the data from other stations are denoted with circles‘○’). (b)Parkfield earthquake (UPSAR) (modified from Yu et al., 2011, 2020)
图 5 四种相干函数模型对实际地震记录相干函数的拟合结果.(a)汶川地震记录(ZGSA)(与台站Z0相关的值用‘*’表示, 与台站Z1相关的值用‘+’表示, 其他值用‘○’表示);(b)Parkfield地震记录(UPSAR);(c)San Simeon地震记录(UPSAR)(修改自Yu et al., 2011, 2020)
Figure 5. Fitting results of four coherence function models to the coherence function of real seismic records. (a) Wenchuan earthquake (ZGSA) (The values related to station Z0 are denoted with asterisks‘*’, the values related to station Z1 are denoted with plus symbols‘+’, and the data from other stations are denoted with circles‘○’). (b) Parkfield earthquake (UPSAR). (c) San Simeon earthquake (UPSAR) (modified from Yu et al., 2011, 2020)
图 6 震中距95~140 km,Yu等(2020)模型与其它模型的对比
Figure 6. Comparison and analysis of Yu et al. (2020) model with other models in the range of 95~140 km epicenter distance
图 7 震中距50~95 km,Yu等(2020)模型与其它模型的对比
Figure 7. Comparative analysis of Yu et al. (2020) model with other models in the range of epicenter distance of 50~95 km
表 1 基于SMART-1台阵记录的迟滞相干模型
Table 1. Lagged coherence model based on SMART-1 array records
序号 模型表达式 参数说明 文献 1 $\left| {\gamma (\omega,d)} \right| = \exp (- \alpha d)$ 参数$\alpha $基于第5次地震 Loh(1985) 2 $\begin{array}{l} \left| {\gamma (\omega,d)} \right| = A\exp \left(- \frac{ {2Bd} }{ {a\nu (\omega)} }\right) + (1 - A)\exp \left(- \frac{ {2Bd} }{ {\nu (\omega)} }\right) \\ \nu (\omega) = k{\left[1 + {\left(\frac{\omega }{ {2\text{π} {f_0} } }\right)^b}\right]^{ - 1/2} };\begin{array}{*{20}{c} } {B = 1 - A + aA} \end{array} \\ \end{array}$ $A = 0.736,a = 0.147$
$k = 5\;210\;{\rm{ m}},{f_0} = 1.09\; {\rm{Hz} }$
${f_0} = 1.09 \;{\rm{Hz} },b = 2.78$(第20次地震)Harichandran和Vanmarcke(1986) 3 $\left| {\gamma (\omega,d)} \right| = \exp \left(- a\frac{ {\omega d} }{ {2\text{π} V} }\right)$ $a = 0.125$(第40次地震) ,$V$为波的视速度 Loh和Yeh(1990) 4 $\begin{array}{l} \left| {\gamma (\omega,d)} \right| = \exp (- a{d^2}) \\ \left| {\gamma (\omega,d)} \right| = \exp \{ (- a - b{\omega ^2})d\} \\ \left| {\gamma (\omega,d)} \right| = \exp \{ (- a - b\omega){d^c}\} \\ \end{array} $ $a = 2 \times {10^{ - 5}},b = 5 \times {10^{ - 6}}$
(第40次地震)
$a = 2 \times {10^{ - 5}},b = {\rm{2}}.5 \times {10^{ - 6}}$
(第45次地震)Loh和Lin(1988) 5 $\begin{array}{l} \left| {\gamma (\omega,d)} \right| = \exp \{ (- {a_1} - {b_1}{\omega ^2})\left| {d\cos \theta } \right|\}\cdot \\ \qquad\qquad\exp \{ (- {a_2} - {b_2}{\omega ^2})\left| {d\sin \theta } \right|\} \\ \end{array}$ $\theta $ 为地震波传播方向与两个台站连线的夹角 6 $\begin{array}{l} \left| {\gamma ({d_{\rm{L} } },{d_{\rm{T} } },\omega)} \right| = \exp (- {\beta _1}{d_{\rm{L} } } - {\beta _2}{d_{\rm{T} } })\cdot \\ {\rm{ } } \exp \{ - ({\alpha _1}(\omega)\sqrt { {d_{\rm{L} } } } + {\alpha _2}(\omega)\sqrt { {d_{\rm{T} } } }){\left(\frac{\omega }{ {2\text{π} } }\right)^2}\} \\ {\alpha _i}(\omega) = \frac{ {2\text{π} {a_i} } }{\omega } + \frac{ { {b_i}\omega } }{ {2\text{π} } } + {c_i}\begin{array}{*{20}{c} } \end{array}i = 1,2 \\ \end{array}$ ${d_{\rm{L}}}$-沿波传播方向的距离
${d_{\rm{T}}}$-垂直波传播方向的距离
(第24、30和45次地震)Hao等(1989) ; Oliveir等(1991) 7 $\begin{array}{l} \left| {\gamma (\omega,d)} \right| \\ = A\exp \left(- \frac{ {2d(1 - A)} }{ {ak} }\right){\left[1 + {\left(\frac{\omega }{ {2\text{π} {f_0} } }\right)^b}\right]^{1/2} } + (1 - A) \\ \end{array}$ 第20、24次地震记录进行分析,
建议对空间距离大于100 m的高频记录使用Harichandran(1991) 表 2 四次地震的相干拐点频率
${f_{{\rm{cc}}}}$ 取值Table 2. Values of coherent inflection point frequency
${f_{{\rm{cc}}}}$ of four earthquakes地震事件 汶川
(Ms 8.0)27th
(Ms5.9)San Simeon
(Ms 6.5)Parkfield
(Ms 6.0)台阵名称 ZGSA SMART-1 UPSAR UPSAR 震中距/km 226.6 135 55.6 11.6 ${f_{{\rm{cc}}}}$/Hz 1.5 1.0 0.75 0.50 -
[1] 阿布都瓦里斯, 俞瑞芳, 俞言祥. 基于汶川地震加速度记录的地震动相干函数变化特性[J], 振动与冲击, 2013, 32(16): 70-76.Abduvalis A, Yu R F, Yu Y X. 2013. Spatial variation of ground motions based on Wenchuan acceleration records[J]. Journal of Vibration and Shock, 32(16): 70-76 . [2] Abrahamson N A. Hard-rock coherency functions based on the Pinyon Flat array data[R]. Draft Report to EPRI, 2007, 5. [3] Abrahamson N A, Schneider J F, Stepp J C. Empirical spatial coherency functions for application to soil-structure interaction analyses[J]. Earthquake spectra, 1991a, 7(1): 1-27. doi: 10.1193/1.1585610 [4] Abrahamson N, Schneider J F, Stepp J C. Spatial coherency of shear waves from the Lotung, Taiwan large-scale seismic test[J]. Structural Safety, 1991b, 10(1-3): 145-162. doi: 10.1016/0167-4730(91)90011-W [5] Boissières H P, Vanmarcke E H. Estimation of lags for a seismograph array: wave propagation and composite correlation[J]. Soil Dynamics and Earthquake Engineering, 1995, 14(1): 5-22. doi: 10.1016/0267-7261(94)00026-D [6] Bolt B A, Loh C H, Penzien J, et al. 1982. Preliminary report on the SMART(strong motion array in Taiwan) 1[R]. Berkeley CA: University of California, Earthquake Engineering Research Center Report No. UCB/EERC-82/13. [7] Boore, D M. Effect of baseline correlation on displacement and response spectra for several recordings of the 1999 Chi-Chi, Taiwan, earthquake[J]. Bulletin of the Seismological Society of America, 2001, 91: 1199-1211. [8] Cranswick E. The information content of high-frequency seismograms and the near-surface geologic structure of “hard rock” recording sites[J]. Pure and Applied Geophysics, 1988: 128: 333-363. doi: 10.1007/BF01772604 [9] Der Kiureghian A. A coherency model for spatially varying ground motions[J]. Earthquake engineering & structural dynamics, 1996, 25(1): 99-111. [10] 丁海平, 刘启方, 金星, 袁一凡. 基岩地震动的一个相干函数模型—走滑断层情形[J]. 地震学报, 2004, 26(1): 62-67.Ding H P, Liu Q F, Jin X, Yuan Y F. 2004. A coherency function model of ground motion at base rock corresponding to strike-slipfault[J]. Acta Seismologica Sinica, 26(1): 62-67 . [11] 冯启民, 胡聿贤. 空间相关地面运动的数学模型[J]. 地震工程与工程振动, 1981, 1(2): 1-8.Feng Q M, Hu Y X. 1981. Mathematical model of spatially correlated ground motion[J]. Earthquake Engineering and Engineering Vibration, 1(2): 1-8 (in Chinese). [12] Hao H, Oliveira C S, Penzien J. Multiple-station ground motion processing and simulation based on SMART-1 array data[J]. Nuclear Engineering and Design, 1989, 111(3): 293-310. doi: 10.1016/0029-5493(89)90241-0 [13] Harichandran R S, Vanmarcke E H. Stochastic variation of earthquake ground motion in space and time[J]. Journal of engineering mechanics, 1986, 112(2): 154-174. doi: 10.1061/(ASCE)0733-9399(1986)112:2(154) [14] Harichandran R S. Estimating the spatial variation of earthquake ground motion from dense array recordings[J]. Structural Safety, 1991, 10(1-3): 219-233. doi: 10.1016/0167-4730(91)90016-3 [15] Hindy, A. and Novak, M. Response of pipelines to random ground motion[J]. Journal of Engineering Mechanics, 1980, 106: 339-360. [16] Hong H P, Liu T J. Assessment of coherency for bidirectional horizontal ground motions and its application for simulating records at multiple stations[J]. Bulletin of the Seismological Society of America, 2014, 104(5): 2491-2502. doi: 10.1785/0120130241 [17] Konakli K, Der Kiureghian A, Dreger D. Coherency analysis of accelerograms recorded by the UPSAR array during the 2004 Parkfield earthquake[J]. Earthquake engineering & structural dynamics, 2014, 43(5): 641-659. [18] 地震动参数及结构整体破坏相干性研究Li S, Xie L L, Hao M. Correlation between seismic ground motion parameters and their relationship with overall damage to structure[J]. Journal of Harbin Institute of Technology, 2007, 39(4): 505-560(in Chinese). [19] Liu T J, Hong H P. Assessment of spatial coherency using tri-directional ground motions[J]. Journal of Earthquake Engineering, 2016, 20(5): 773-794. doi: 10.1080/13632469.2015.1104760 [20] Loh C H. Analysis of the spatial variation of seismic waves and ground movements from smart-1 array data[J]. Earthquake engineering & structural dynamics, 1985, 13(5): 561-581. [21] Loh C H, Yeh Y T. Spatial variation and stochastic modelling of seismic differential ground movement[J]. Earthquake engineering & structural dynamics, 1988, 16(4): 583-596. [22] Loh C H, Lin S G. Directionality and simulation in spatial variation of seismic waves[J]. Engineering Structures, 1990, 12(2): 134-143. doi: 10.1016/0141-0296(90)90019-O [23] Lou C H. Analysis of the spatial variation of seismic waves and ground movements from SMART-1 data[J]. Earthquake Engineering and Structural Dynamic, 1985,13, 561-581 [24] Luco J E, Sotiropoulos D A. Local characterization of free-field ground motion and effects of wave passage[J]. Bulletin of the Seismological Society of America, 1980, 70(6): 2229-2244. [25] Luco J E, Wong H L. Response of a rigid foundation to a spatially random ground motion[J]. Earthquake Engineering & Structural Dynamics, 1986, 14(6): 891-908. [26] Menke W, Lerner-Lam A L, Dubendorff B, et al. Polarization and coherence of 5 to 30 Hz seismic wave fields at a hard-rock site and their relevance to velocity heterogeneities in the crust[J]. Bulletin of the Seismological Society of America, 1990, 80(2): 430-449. [27] Mohamed A S, Sunghyuk G, Sung G C, et al. Seismic responses of base-isolated nuclear power plant structures considering spatially varying ground motions[J]. Structural Engineering and Mechanics, 2015, 54(1): 169-188. doi: 10.12989/sem.2015.54.1.169 [28] Novak M, Hindy A. 1979. Seismic response of buried pipelines[C]// 3rd Canadian Conference on Earthquake Engineering. Montreal Canada. [29] Oliveira C S, Hao H, Penzien J. Ground motion modeling for multiple-input structural analysis[J]. Structural Safety, 1991, 10(1-3): 79-93. doi: 10.1016/0167-4730(91)90007-V [30] 屈铁军, 王君杰, 王前信. 空间变化的地震动功率谱的实用模型[J]. 地震学报, 1996, 18(1): 55-62.Qu T J, Wang J J, Wang Q X. A practical model of spatially varying ground motion power spectrum[J]. Acta Seismologica Sinica, 1996, 18(1): 55-62(in Chinese). [31] Rodda G K, Basu D. Coherency model for translational and rotational ground motions[J]. Bulletin of Earthquake Engineering, 2018, 16(7): 2687-2710. doi: 10.1007/s10518-017-0304-6 [32] Saxena V, Deodatis G, Shinozuka M. 2000. Effect of spatial variation of earthquake ground motion on the nonlinear dynamic response of highway bridges[C]// Proceedings of 12th World Conference on Earthquake Engineering, Auckland, New Zealand. [33] Schneider J F, Stepp J C, Abrahamson N A. 1992. The spatial variation of earthquake ground motion and effects of local site[C]// Proceedings of 10th World Conference on Earthquake Engineering. Madrid, Spain. [34] Somerville P G, McLaren J P, Saikia C K, et al. 1988. Site-specific estimation of spatial incoherence of strong ground motion[J]. Earthquake Engineering and Soil Dynamics, II-Recent Advances in Ground Motion Evaluation, Geotechnical Special Pub No 20, ASCE, New York. [35] Somerville P, Irikura K, Graves R, et al. Characterizing crustal earthquake slip models for the prediction of strong ground motion[J]. Seismological Research Letters, 1999, 70(1): 59-80. doi: 10.1785/gssrl.70.1.59 [36] Spudich P, Oppenheimer D.1986. Dense seismograph array observations of earthquake rupture dynamics[R]. Das S, Boatwright J, Scholz C H ed. Earthquake source mechanics, Geophysical Monograph 37, American Geophysical Union, Washington DC. [37] Spudich P. 1994. Recent seismological insights into the spatial variation of earthquake ground motions[R]. New Developments in Earthquake Ground Motion Estimation and Implications for Engineering Design Practice. ATC35-1. [38] Tian L, Gai X, Qu B, et al. Influence of spatial variation of ground motions on dynamic responses of supporting towers of overhead electricity transmission systems: An experimental study[J]. Engineering Structures, 2018, 128: 67-81. doi: 10.1016/j.engstruct.2016.09.010 [39] Todorovska M I, Trifunac M D, Ding H, et al. Coherency of dispersed synthetic earthquake ground motion at small separation distances: Dependence on site conditions[J]. Soil Dynamics and Earthquake Engineering, 2015, 79: 253-264. doi: 10.1016/j.soildyn.2015.08.004 [40] Toksöz M N, Dainty A M, Charrette III E E. Spatial variation of ground motion due to lateral heterogeneity[J]. Structural Safety, 1991, 10(1-3): 53-77. doi: 10.1016/0167-4730(91)90006-U [41] Wang D, Wang L, Xu J, et al. A directionally-dependent evolutionary lagged coherency model of nonstationary horizontal spatially variable seismic ground motions for engineering purposes[J]. Soil Dynamics and Earthquake Engineering, 2019, 117: 58-71. doi: 10.1016/j.soildyn.2018.11.021 [42] Wang G Q, Boore D M, Igel H, et al. Some observations on collocated and closely spaced strong ground-motion records of the 1999 Chi-Chi, Taiwan, earthquake[J]. Bulletin of the Seismological Society of America, 2003, 93(2): 674-693. doi: 10.1785/0120020045 [43] 王君杰, 陈虎. 面向设计应用的地震动空间相干函数模型[J]. 地震工程与工程振动, 2007, 27(1): 1-8.A spatial coherence model of seismic ground motion for design application [44] Watts D G, Jenkins G. 1968. Spectral Analysis and its Applications[M]. San Francisco. [45] Yang O S, Chen Y J. A practical coherency model for spatially varying ground motions[J]. Structural engineering and mechanics, 2000, 9(2): 141-152. doi: 10.12989/sem.2000.9.2.141 [46] Ye J, Pan J, Liu X. Vertical coherency function model of spatial ground motion[J]. Earthquake Engineering and Engineering Vibration, 2011, 10(3): 403-415. doi: 10.1007/s11803-011-0076-y [47] Yu R F, Yuan M Q, Yu Y X. Spatial coherency function of seismic ground motion based on UPSAR records[J]. Applied Mechanics and Materials, 2011, 90: 1586-1592. [48] Yu R F, Abduwaris A, Yu Y X. Practical coherency model suitable for near-and far-field earthquakes based on the effect of source-to-site distance on spatial variations in ground motions[J]. Structural Engineering and Mechanics, 2020, 73(6): 651-666. [49] Zerva A, Ang A H S, Wen Y K. Development of differential response spectra for lifeline seismic analysis[J]. Probabilistic engineering mechanics, 1986, 1(4): 208-218. doi: 10.1016/0266-8920(86)90014-7 [50] Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion coherence and strain estimates[J]. Soil Dynamics and Earthquake Engineering, 1997, 16(7-8): 445-457. doi: 10.1016/S0267-7261(97)00019-5 [51] Zerva A, Zervas V. Spatial variation of seismic ground motions: an overview[J]. Applied Mechanics Reviews, 2002, 55(3): 271-297. doi: 10.1115/1.1458013 [52] Zerva A. 2009. Spatial Variation of Seismic Ground Motions: Modeling and Engineering Applications[M]. Boca Raton, FL: Crc Press. [53] 钟菊芳, 胡晓, 屈铁军. 同一测点不同地震动分量空间相干性分析[J]. 地震研究, 2005, 28(4): 378-382.Spatial coherency analysis of different ground motion components at the same observational station -