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） 
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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
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