Study on 2D in-plane HVSR simulation and application with transverse inhomogeneous body scattering
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摘要: 为分析场地的横向不均匀性对水平竖向谱比HVSR曲线产生的显著影响，本文基于Sánchez-Sesma等提出的扩散场方法，通过计算总波场格林函数虚部对二维沉积地形上的HVSR曲线进行模拟。格林函数虚部则通过刚度矩阵和平面内斜线格林函数采用间接边界元方法进行求解。对二维沉积地形和相应的一维层状半空间的HVSR曲线进行了参数分析，着重讨论了沉积地形的形状、相对计算点位置等因素对HVSR曲线的影响规律。结果表明：沉积地形内外材料阻抗比对HVSR曲线的影响最为显著；随着沉积地形内外材料阻抗差异和沉积侧界面坡度的增大，沉积地形上HVSR曲线的第一峰值点的频率显著增大至相应层状半空间结果的3.3倍，同时HVSR曲线的形态呈现出平台现象；随着计算点到沉积边界距离的减小，HVSR曲线高频段幅值相对较大。根据本文得到的局部地形对HVSR曲线的影响规律，在进行场地勘探时可采用HVSR方法初步确定局部地形的分布位置以降低勘探成本。Abstract: In order to analyze the significant influence of lateral inhomogeneity of site on horizontal-to-vertical spectral ratio (HVSR) curves, the diffuse field approach proposed by Sánchez-Sesma et alwas adopted to simulate the HVSR curves of 2-D sediment topography by calculating the imaginary part of Green’s functions of total wave field. The imaginary part of Green’s functions was solved by the dynamic stiffness matrix and in-plane inclined Green’s functions based on the indirect boundary element method (IBEM). The HVSR curves of 2-D sediment topographies and corresponding 1-D layered half-space were compared, the influences of sediment topography shapes and the relative position of calculation points on the HVSR curve were discussed in detail. The results show that the effect of impedance ratio between inside and outside materials of sediment topography on HVSR is the most significant; With the increase of the impedance ratio and the slopes of the interface on the sediment side, the frequencies of the first peak of HVSR curves increase significantly, which can be up to 3.3 times of the corresponding layered half-space results, simultaneously, platform emerges on HVSR curves; Amplitudes of HVSR curves in high frequency band increase with the decrease of distances from the calculation points to the sediment boundary. According to the results obtained in this study, the HVSR method can be used to preliminarily determine the place where local sediment topography exists. From this aspect, the cost of regional geophysical investigation can be reduced visibly via HVSR method.
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图 2 层状半空间中沉积内部中心点x＝0 m （a） 和非中心点x＝16 m （b）的方向能量密度随深度变化情况
Figure 2. Energy fluctuation of center point x＝0 m （a） and non-central point x＝16 m （b）in depth direction inside the alluvial canyon in layered half-space
图 3 采用本文方法得到的层状半空间HVSR曲线与Sánchez-Sesma等（2011）结果对比
Figure 3. Comparisons of the result of layered half-space HVSR curves in this method with Sánchez-Sesma et al （2011）
图 4 二维平面内采用本文方法得到的沉积地形上格林函数张量与Perton和Sánchez-Sesma （2016）的结果对比
Figure 4. Comparisons of the result of Green's function tensor on sedimentary topography in this method with Perton and Sánchez-Sesma （2016）
图 5 梯形沉积地形计算模型图
Figure 5. Calculation model graph of trape-zoidal sedimentary topography
图 6 层状半空间①—④ （a−d）计算模型图
Figure 6. Calculation model graph of layered half-spaces ①—④ （a−d）
图 8 沉积地形与相应层状半空间的HVSR曲线
Figure 8. HVSR curves of sedimentary topography and corresponding layered half-space
图 9 沉积地形（a）和其对应的层状半空间（b）内不同沉积内外阻抗比的HVSR曲线
Figure 9. HVSR curves of the sedimentary materials with different impedance ratios for sedimentarytopography (a) and its corresponding layered half space (b)
图 10 沉积内外材料阻抗比为1∶2 （a）和1∶4 （b）时不同横向不均匀界面坡度的HVSR曲线
Figure 10. HVSRs of different laterally inhomogeneous interface slopes with impedance ratios 1∶2 （a） and 1∶4 （b） between inside and outside materials of sedimental topography
图 11 沉积表面不同相对位置x/a上的HVSR曲线
（a） 沉积内外材料阻抗比1∶2；（b） 沉积内外材料阻抗比1∶4
Figure 11. HVSR curves of the sedimentary materials with different positions x/a
（a） The impedance ratio of internal and external material is 1∶2；（b） The impedance ratio of internal and external material is 1∶4
表 1 层状半空间计算参数
Table 1. Calculation parameters of layered half space
剪切波速
vS/（m·s−1）泊松比ν 土层密度
ρ/（kg·m−3）阻尼比ζ 土层 70 0.496 1 200 0.05 基岩 1 000 0.333 2 500 0.05 
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表 2 层状半空间不同阻抗比情况计算参数
Table 2. Parameters of different impedance ratios in layered half-space
剪切波速
vS/（m·s−1）泊松比
ν土层密度
ρ/（kg·m−3）阻尼比
ζ沉积土层① 140 0.496 1 200 0.05 沉积土层② 280 0.496 1 200 0.05 基岩半空间① 560 0.496 1 200 0.05 基岩半空间② 280 0.496 1 200 0.05
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表 3 不同沉积内外材料阻抗比情况计算参数
Table 3. Parameters of the alluvial canyon materials with different impedance ratios
剪切波速
vS/（m·s−1）泊松比
ν土层密度
ρ/（kg·m−3）阻尼比
ζ沉积内部土层 140 0.496 1 200 0.05 外部土层① 280 0.496 1 200 0.05 外部土层② 420 0.496 1 200 0.05 外部土层③ 560 0.496 1 200 0.05 外部土层④ 700 0.496 1 200 0.05 基岩半空间 1 000 0.333 2 500 0.05
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表 4 不同沉积形状情况沉积地形计算参数
Table 4. Parameters of the alluvial canyon materials with different topography shapes
剪切波速
vS/（m·s−1）泊松比
ν土层密度
ρ/（kg·m−3）阻尼比
ζ沉积内部土层 140 0.496 1 200 0.05 外部土层① 280 0.496 1 200 0.05 外部土层② 420 0.496 1 200 0.05 基岩半空间 1 000 0.333 2 500 0.05
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