The influence of cut-off frequency on the statistical results of spatial coherency function of seismic ground motion
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摘要: 以SMART-1台阵第45号地震的南北分量加速度作为研究对象，计算截止频率fcut分别为8，16和24 Hz，台站间距分别为200，1 000和2 000 m这9种工况的相干系数，并进行曲线拟合，再根据9组拟合参数计算相应的相干系数并进行比较。结果表明：① 截止频率fcut的取值对相干函数统计模型参数和相干系数拟合曲线有很大影响；② 根据较小的截止频率fcut得到的拟合参数计算出的相干系数随频率的衰减很快，反之，根据较大的截止频率fcut得到的拟合参数计算出的相干系数随频率的衰减较慢；③ 当使用由确定的截止频率fcut得到的相干系数拟合参数计算超出截止频率fcut范围的相干系数时，将产生很大的误差。Abstract: Based on the north-south component of the 45th seismic record on the SMART-1 station, in this paper, we firstly calculated the lagged coherencies under nine conditions which were combined with the three cut-off frequencies （8, 16, 24 Hz） and three separation distances （200, 1 000, 2 000 m）. Then, we fitted the curves. Finally, the coherencies received according to nine groups of fitting parameters were compared. The conclusion can be drawn as follows: ① The selection of the cut-off frequency fcut has great effect on the parameters of coherence function model and lagged coherency fitting curves. ② The lagged coherency calculated by the fitting parameters which obtained by smaller cut-off frequency fcut decays rapidly with increasing frequency; on the other hand, the lagged coherency calculated by the fitting parameters which obtained by larger cut-off frequency fcut decays slowly with increasing frequency. ③ Great errors will be produced when using fitting parameters of the lagged coherency that got from the determined cut-off frequencyfcut to calculate the lagged coherency beyond that frequency range.
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图 2 第45号地震在台站C00处的NS向加速度时程及S波时间窗
Figure 2. Acceleration traces of NS and EW components for the 45th event recorded at station C00 and the S wave time window
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图 5 台站间距d＝2 000 m，截止频率fcut为8 Hz （a），16 Hz （b） 和24 Hz （c） 时的相干系数 |γ| （细实线）及其根据Abrahamson相干函数模型（黑粗线）和Loh相干函数模型（红粗线）拟合所得的拟合曲线
Figure 5. Lagged coherencies |γ| at station spacing d＝2 000 m （thin solid line） and the corresponding fitting curves based on Abrahamson （black rough line） and Loh （red rough line） coherence function model which obtained with different cut-off frequency values of 8 Hz （a），16 Hz （b） and 24 Hz （c）
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图 3 台站间距d＝200 m，截止频率fcut为8 Hz （a），16 Hz （b） 和24 Hz （c）时的相干系数 |γ| （细实线）及其根据Abrahamson相干函数模型（黑粗线）和Loh相干函数模型（红粗线） 拟合所得的拟合曲线
Figure 3. Lagged coherencies |γ| at station spacing d＝200 m （thin solid line） and the corresponding fitting curves based on Abrahamson （black rough line） and Loh （red rough line） coherence function model which obtained with different cut-off frequency values of 8 Hz （a），16 Hz （b） and 24 Hz （c）
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图 4 台站间距d＝1 000 m，截止频率fcut为8 Hz （a），16 Hz （b） 和24 Hz （c） 时的相干系数 |γ| （细实线）及其根据Abrahamson相干函数模型（黑粗线）和Loh相干函数模型（红粗线）拟合所得的拟合曲线
Figure 4. Lagged coherencies |γ| at station spacing d＝1 000 m （thin solid line） and the corresponding fitting curves based on Abrahamson （black rough line） and Loh （red rough line） coherence function model which obtained with different cut-off frequency values of 8 Hz （a），16 Hz （b） and 24 Hz （c）
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图 6 根据不同截止频率fcut所得的拟合参数计算得到的台站间距d＝200 m （a），1 000 m （b） 和2 000 m （c） 时的相干系数|γ|
左侧对应于Abr相干函数模型的结果，右侧对应于Loh相干函数模型的结果
Figure 6. Coherencies at separation distance d＝200 m （a），1 000 m （b），2 000 m （c） based on fitting parameters which obtained from different cut-off frequency values
The graphs on the left correspond to the result of the Abrahamson，and the graphs on the right correspond to the result of the Loh
表 2 基于Loh相干函数模型得到的拟合参数
Table 2 Fitting parameters based on Loh coherence function model
台站间距
d/m截止频率
fcut/Hz系数 a b 200 8 0.32 1.95×10−3 16 1.57 4.57×10−3 24 2.43 1.01×10−4 1 000 8 0.53 6.73×10−4 16 0.84 1.44×10−4 24 1.10 2.27×10−5 2 000 8 0.52 1.10×10−4 16 0.61 4.46×10−6 24 0.61 2.18×10−6 
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表 1 基于Abrahamson相干函数模型得到的拟合参数
Table 1 Fitting parameters based on Abrahamson coherence function model
台站间距
d/m截止频率
fcut/Hz系数 a1 a2 b1 b2 c n 200 8 −0.65 0.013 2.31 −1.34×10−2 −1.604 0.33 16 2.36 1.17×10−3 1.29 −9.39×10−3 −1.148 0.39 24 −0.16 9.74×10−3 −0.24 −1.71×10−3 −0.587 0.36 1 000 8 0.95 −2.70×10−4 0.62 1.16×10−3 −0.942 0.19 16 1.05 −3.76×10−4 −0.54 6.04×10−6 −0.988 0.20 24 0.62 −1.62×10−5 0.59 1.20×10−3 −1.778 0.27 2 000 8 −0.91 5.78×10−4 −0.02 −2.31×10−5 −1.570 0.10 16 1.06 −4.96×10−4 −3.10 1.53×10−3 −3.970 0.24 24 2.66 −1.27×10−3 −0.052 2.23×10−5 −3.203 0.18
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表 3 根据不同截止频率fcut拟合得到的相干系数在0—8 Hz频段内的变化
Table 3 Changes of the lagged coherencies obtained with different fcut in the 0—8 Hz frequency band
台站间距
d/m0—8 Hz内的相干系数 Abr相干函数模型 Loh相干函数模型 fcut＝8 Hz fcut＝16 Hz fcut＝24 Hz fcut＝8 Hz fcut＝16 Hz fcut＝24 Hz 200 0.99—0.39 0.99—0.45 0.99—0.47 0.93—0.33 0.73—0.58 0.61—0.58 1 000 0.78—0.22 0.78—0.23 0.79—0.24 0.58—0.11 0.43—0.29 0.33—0.31 2 000 0.62—0.24 0.60—0.28 0.60—0.28 0.35—0.20 0.29—0.28 0.30—0.29
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