浅壳与上地幔地震的水平加速度谱阻尼修正系数模型

苏开潍, 杨青松, 梁基冠

苏开潍, 杨青松, 梁基冠. 2019: 浅壳与上地幔地震的水平加速度谱阻尼修正系数模型. 地震学报, 41(6): 778-794. DOI: 10.11939/jass.20190052
引用本文: 苏开潍, 杨青松, 梁基冠. 2019: 浅壳与上地幔地震的水平加速度谱阻尼修正系数模型. 地震学报, 41(6): 778-794. DOI: 10.11939/jass.20190052
Su Kaiwei, Yang Qingsong, Liang Jiguan. 2019: Damping correction factor for horizontal acceleration response spectrum of shallow crustal and upper-mantle earthquakes. Acta Seismologica Sinica, 41(6): 778-794. DOI: 10.11939/jass.20190052
Citation: Su Kaiwei, Yang Qingsong, Liang Jiguan. 2019: Damping correction factor for horizontal acceleration response spectrum of shallow crustal and upper-mantle earthquakes. Acta Seismologica Sinica, 41(6): 778-794. DOI: 10.11939/jass.20190052

浅壳与上地幔地震的水平加速度谱阻尼修正系数模型

基金项目: 国家自然科学基金(51578470)资助
详细信息
    通讯作者:

    苏开潍: e-mail:sukaiweiim@gmail.com

  • 中图分类号: P315.9

Damping correction factor for horizontal acceleration response spectrum of shallow crustal and upper-mantle earthquakes

  • 摘要: 对日本K-net和KiK-net台网中6 466条浅壳与上地幔地震动记录进行了统计分析,考虑四种场地类别分别建立了浅壳与上地幔地震的加速度谱阻尼修正系数模型,并利用随机效应模型将模型总残差分离为事件间残差、事件内残差(分为场地间残差和场地内残差),计算其相应的标准差,探究地震震源、路径、场地等因素对模型误差的影响。研究结果显示:所提出的模型能较好地拟合实际计算值,其随机误差整体随谱周期和阻尼比的增大而增大;震源效应导致的随机误差在谱周期小于2.0 s时小于其它效应;在大部分谱周期上,场地效应导致的随机误差小于路径效应及其它效应;事件间残差与震源和震源深度显著相关,场地内残差与震源距显著相关。
    Abstract: In this paper, 6 466 strong-motion records from 76 shallow crustal and 47 upper-mantle earthquakes in Japan were compiled to develop a damping correction factor (DCF) model. This study is aimed to provide a DCF that can be used to scale a design spectrum without known source and path parameters, therefore the models presented in this study did not include any terms for earthquake magnitude or source distance. Site effect on DCF was found to be significant, and a separate DCF model was constructed for the records from each of the four site classes. The total residuals were approximately separated into within- and between-event residuals, and the within-event residuals were further separated into within- and between-site components using a random effect model. Then the corresponding standard deviations were calculated to illustrate the variabilities associated with the source, path and site effects for the DCF model presented in this study. The results showed that the random errors tend to increase with spectral periods and damping ratios increasing, and the random errors associated with source effect are smaller than those associated with path and site effect at spectral periods less than 2.0 s. At most spectral periods, the random errors associated with site effect were smaller than those associated with path and other effects not explicitly modelled in this study. The between-event residuals are significantly correlated with magnitude and focal depth, while the within-site residuals were significantly correlated with source distance. Therefore, source and path parameters should be included for a model that can be used to scale a design spectrum from an earthquake with a known magnitude and source distance.
  • 目前绝大多数抗震设计规范是给出5%的阻尼比加速度设计反应谱,把该阻尼比下的设计反应谱值与阻尼修正系数(damping correction factor,缩写为DCF)相乘可得到其它阻尼比下的设计反应谱,以便对多种阻尼水平下的建筑结构进行抗震设计。我国和欧美国家现行规范(International Conference of Building Officials,1997Building Seismic Safety Council,2000European Committee for Standardization,2005中华人民共和国住房和城乡建设部,中华人民共和国国家质量监督检验检疫总局,2010)均采用这一方法。随着减、隔震等技术的应用和发展,不同阻尼比的建筑结构日趋增多,根据现有规范计算得到的结果可能与实际需求有所差距,这对阻尼修正系数的研究提出了更高的要求,因此需要对设计反应谱的阻尼修正系数进行深入研究,以便获得更加准确的抗震设计参数。

    关于阻尼修正系数的研究始于Newmark和Hall (1982)对1973年之前美国地震动记录中阻尼比低于20%的修正系数进行统计分析并提出阻尼修正系数的表达式,该成果被应用于欧洲标准委员会指导现有结构抗震评估以及对安装有隔震和消震装置的结构进行抗震设计(Building Seismic Safety Council,2000)。Bommer等(2000)MS>5.5地震的地震动数据进行了分析统计,其阻尼修正系数公式被应用于EC8抗震设计规范(European Committee for Standardization,2005)。后期的深入广泛研究表明反应谱阻尼修正系数与结构的自振周期(Lin,Chang,2003)、场地因素(Lin,Chang,2004)、震级和震源距(Atkinson,Pierre,2004)等多种因素有关。随后,蒋建等(2009)研究了场地类别、阻尼比和谱周期对阻尼修正系数的影响;Castillo和Ruiz (2014)探究了震级和场地周期对阻尼修正系数的影响;卢皓(2017)的研究结果则表明各阻尼修正系数公式的计算精度明显受到震级和阻尼比大小的影响。虽然上述研究已对阻尼水平、自振周期、场地因素、震级和震源距对于阻尼修正系数的影响进行了深入研究,但少有研究考虑地震类别对其的影响,而且针对俯冲带地震建立的阻尼修正系数模型也少有。鉴于此,本文拟建立俯冲带地区浅壳与上地幔地震的加速度谱阻尼修正系数模型,并利用Abrahamson和Youngs (1992)提出的随机效应模型分析残差,探究震源效应、路径效应和场地效应对阻尼修正系数的影响,以期为俯冲带地区抗震规范的制定提供一定的参考。

    由于日本地区台站分布密集且强震动记录数据量大,对其研究能得到较为准确的结果并可应用于其它俯冲带地区。本文根据Zhao等(2015)提出的地震分类方法,基于日本K-net和KiK-net台网中123个浅壳与上地幔地震的6 466条强震记录,对36个谱周期(表1)、13个阻尼比(表2)的绝对加速度反应谱阻尼修正系数进行计算分析,研究不同场地类别下的DCF值的差异。

    表  1  谱周期点数量
    Table  1.  Number of spectral periods
    序号 周期/s 序号 周期/s 序号 周期/s 序号 周期/s 序号 周期/s
    2 0.01 10 0.09 18 0.25 26 0.80 34 3.50
    3 0.02 11 0.10 19 0.30 27 0.90 35 4.00
    4 0.03 12 0.12 20 0.35 28 1.00 36 4.50
    5 0.04 13 0.14 21 0.40 29 1.25 37 5.00
    6 0.05 14 0.15 22 0.45 30 1.50
    7 0.06 15 0.16 23 0.50 31 2.00
    8 0.07 16 0.18 24 0.60 32 2.50
    9 0.08 17 0.20 25 0.70 33 3.00
    下载: 导出CSV 
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    表  2  阻尼比数量
    Table  2.  Number of damping ratios
    序号阻尼比序号阻尼比序号阻尼比
    11% 6 6%1115%
    22% 7 7%1220%
    33% 8 8%1325%
    44% 9 9%1430%
    55%1010%
    下载: 导出CSV 
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    本研究所选取的强震记录的震级随震源距、震源深度的分布如图1所示。基于Hayes等(2012)建立的俯冲带几何模型Slab1.0,根据Zhao等(2015)提出的日本地震分类方法,浅壳地震指发生在俯冲板间之上,不属于俯冲板间且震源深度小于或等于25 km的地震,上地幔地震指发生在俯冲板之上但不属于浅壳地震的地震。由于这两类地震的数据量目前均较小,且据Zhao (2010)的研究,这两类地震具有相同的场地效应,故本文将浅壳与上地幔地震归为一组用于建立加速度反应谱阻尼修正系数模型。依据Zhao等(2006)的方法,根据场地特征周期将场地分为四类,具体标准列于表3

    图  1  震级随震源距(a)和震源深度(b)的分布
    Figure  1.  The distribution of MW with respect to source distance (a) and focal depth (b)
    表  3  场地类别分类标准
    Table  3.  Site class definitions used in this study
    场地类别土质类别场地特征周期Tg/s平均剪切波速v30/(km·s−1记录条数
    岩土Tg<0.2v30>6003 007
    硬土0.2≤Tg<0.4300<v30≤6001 828
    中硬土0.4≤Tg<0.6200<v30≤300 578
    软土Tg>0.6v30<2001 053
    下载: 导出CSV 
    | 显示表格

    计算每条记录的两个水平分量加速度反应谱的几何均值,并根据阻尼修正系数的定义

    ${B_{\rm a}} {\text{=}} \frac{{{S_{\rm a}}\left({T{\text{,}}\zeta } \right)}}{{{S_{\rm a}}\left({T{\text{,}}0.05} \right)}}$

    (1)

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    $|Z|{\text{=}} \frac{{\overline X - \overline Y}}{{\sqrt {\dfrac{{S_x^2}}{{{n_x}}} {\text{+}} \dfrac{{S_y^2}}{{{n_y}}}} }} {\text{~}} N(0{\text{,}}1){\text{,}}$

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    图2给出了四种场地类别间两两进行均值差异性检验的统计值| Z |的分布图。图2a表明 Ⅰ 类场地和 Ⅱ 类场地的DCF值在周期小于0.3 s时统计上显著不同,长周期上1%阻尼比时在很多周期上不显著; Ⅰ 类场地与 Ⅲ 类场地(图2b)和 Ⅰ 类场地与 Ⅳ 类场地(图2c)的对比表明| Z |在周期约为0.16 s较小范围内小于1.96; Ⅱ 类场地与 Ⅲ 类场地的对比(图2d)表明只有少数周期上| Z |<1.96,但| Z |值普遍小于前三组场地的对比; Ⅱ 类场地与 Ⅳ 类场地(图2e)以及 Ⅲ 类场地与 Ⅳ 类场地(图2f)的对比显示只在周期约为0.2 s的少数几个周期上| Z |<1.96。从这些结果可以看出:在大部分周期上| Z |>1.96,即四种场地类别的阻尼修正系数均值具有显著差异,表明了分场地类别建立阻尼修正系数模型的必要性。

    图  2  阻尼比为1%和30%时场地类别间的显著性检验统计值| Z |与谱周期的关系
    (a) Ⅰ类与Ⅱ类;(b) Ⅰ类与Ⅲ类;(c) Ⅰ类与Ⅳ类;(d) Ⅱ类与Ⅲ类;(e) Ⅱ类与Ⅳ类;(f) Ⅲ类与Ⅳ类
    Figure  2.  Variation of | Z | values with spectral periods for each pair of site classes with damping ratios 1% and 30%
    (a) Site classes Ⅰ and Ⅱ;(b) Site classes Ⅰ and Ⅲ;(c) Site classes Ⅰ and Ⅳ;(d) Site classes Ⅱ and Ⅲ;(e) Site classes Ⅱ and Ⅳ;(f) Site classes Ⅲ and Ⅳ

    图3给出了四种不同阻尼比水平下四种场地类别的DCF几何均值对数值与谱周期的关系曲线图,可以看出:任一阻尼比水平下,四种场地类别的DCF对数均值随谱周期值的变化趋势大概一致;阻尼比小于5%时,DCF对数均值随谱周期先增大后减小,阻尼比大于5%时,随谱周期先减小后增大,且阻尼比越大,整体上越小;任一阻尼比水平下,谱周期值小于0.03 s时,DCF对数均值均趋近于0,此时DCF对数均值与场地类型和阻尼比均无关,这是因为单自由度结构的地震反应放大系数在自频非常大时其动力反应与阻尼比无关;谱周期值处于0.03—0.16 s范围内时, Ⅰ 类场地的DCF值与其它三类场地差异较大,其它三类场地间的DCF值差异相对较小;当谱周期值大于0.32 s时, Ⅰ 类和 Ⅱ 类场地的DCF值较为相近,与Ⅲ类和Ⅳ类场地之间均有较大差异。

    图  3  不同阻尼比ζ时阻尼修正系数对数lnBa的几何均值与谱周期的关系曲线
    Figure  3.  Mean value of lnBa with spectral periods with different damping ratio ζ
    (a) ζ=3%;(b) ζ=8%;(c) ζ=15%;(d) ζ=30%

    上述结果表明,相同谱周期和阻尼水平下,不同场地的DCF几何均值的方差与平均数在大部分情况下存在显著差异,同样表明了分场地类别研究的必要性。

    为便于抗震设计直接应用,本文参考Stafford等(2008)Hatzigeorgiou (2010)的研究成果,给出日本浅壳与上地幔地震的阻尼修正系数仅包含谱周期和阻尼比的表达式为

    $\ln \left[ {\overline {{B}}_{\rm a} \left({T{\text{,}}\xi } \right)} \right] {\text{=}} a\left( {\ln \zeta {\text{-}} \ln 5 } \right) {\text{+}} b{\left( {\ln \zeta {\text{-}} \ln 5 } \right)^2} {\text{+}} c{\left( {\ln \zeta {\text{-}} \ln 5} \right)^3}{\rm{ {\text{+}} }}\xi _{i,j}^{\rm T}{\text{,}}$

    (3)

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    表  4  四类场地类别下阻尼修正系数模型的回归系数
    Table  4.  Regressive parameters for damping correction factor model for site classes Ⅰ to Ⅳ
    谱周期/s   Ⅰ类场地   Ⅱ类场地   
    a   b   c   a   b   c   
     0.03−0.507 8%   0.080 0%   −0.189 8%   −0.550 0%   0.100 0%   −0.049 9%   
     0.04−7.950 5%   1.218 2%   −0.170 9%   −4.200 9%   0.911 5%   −0.059 7%   
     0.05−15.022 4%   1.272 6%   −0.088 0%   −8.216 0%   1.228 8%   −0.039 6%   
     0.06−21.022 3%   0.843 4%   0.012 5%   −12.559 9%   1.209 6%   −0.000 2%   
     0.07−25.837 4%   0.285 8%   0.119 0%   −16.909 3%   1.025 3%   0.051 3%   
     0.08−29.366 4%   −0.255 8%   0.226 0%   −21.501 6%   0.758 2%   0.110 5%   
     0.09−31.472 6%   −0.700 0%   0.331 0%   −24.952 4%   0.451 1%   0.174 5%   
     0.10−32.970 3%   −0.992 1%   0.432 8%   −27.541 0%   0.127 5%   0.241 2%   
     0.12−34.812 4%   −1.326 8%   0.625 1%   −31.029 6%   −0.521 6%   0.378 0%   
     0.14−35.739 9%   −1.505 6%   0.801 9%   −33.120 8%   −1.137 0%   0.514 8%   
     0.15−35.996 1%   −1.556 0%   0.884 9%   −33.832 9%   −1.426 2%   0.582 1%   
     0.16−36.157 1%   −1.587 1%   0.964 3%   −34.385 3%   −1.674 3%   0.648 5%   
     0.18−36.275 6%   −1.605 1%   1.113 7%   −35.133 3%   −1.835 7%   0.777 7%   
     0.20−36.214 3%   −1.580 3%   1.251 4%   −35.545 6%   −1.910 6%   0.901 8%   
     0.25−35.668 4%   −1.406 8%   1.553 4%   −35.741 6%   −1.867 6%   1.189 2%   
     0.30−34.908 3%   −1.151 5%   1.807 0%   −35.345 2%   −1.646 4%   1.445 5%   
     0.35−34.112 2%   −0.862 3%   2.023 6%   −34.703 7%   −1.342 0%   1.674 2%   
     0.40−33.341 7%   −0.561 2%   2.211 1%   −33.964 9%   −0.999 6%   1.879 1%   
     0.45−32.616 5%   −0.259 4%   2.375 1%   −33.196 7%   −0.642 4%   2.063 3%   
     0.50−31.940 0%   0.037 6%   2.519 9%   −32.430 9%   −0.282 9%   2.229 6%   
     0.60−30.721 0%   0.607 2%   2.763 9%   −30.958 6%   0.418 8%   2.517 3%   
     0.70−29.647 9%   1.138 9%   2.961 2%   −29.589 9%   1.080 3%   2.756 2%   
     0.80−28.683 6%   1.632 6%   3.123 4%   −28.322 5%   1.695 7%   2.956 2%   
     0.90−27.799 0%   2.091 1%   3.258 3%   −27.143 3%   2.265 6%   3.124 9%   
     1.00−26.972 5%   2.517 6%   3.371 6%   −26.038 2%   2.793 1%   3.267 6%   
     1.25−25.066 9%   3.465 2%   3.584 4%   −23.522 0%   3.950 7%   3.536 5%   
     1.50−23.283 9%   4.275 7%   3.726 0%   −21.250 5%   4.920 1%   3.712 5%   
     2.00−19.830 8%   5.600 9%   3.878 3%   −17.126 9%   6.450 0%   3.883 8%   
     2.50−16.370 2%   6.650 6%   3.925 7%   −13.301 8%   7.601 0%   3.902 4%   
     3.00−12.845 6%   7.511 4%   3.911 4%   −9.628 3%   8.494 9%   3.826 4%   
     3.50−9.251 3%   8.235 0%   3.857 8%   −6.041 0%   9.204 9%   3.687 7%   
     4.00−5.596 0%   8.854 9%   3.778 0%   −2.508 1%   9.777 7%   3.505 7%   
     4.50−1.891 5%   9.393 8%   3.679 9%   0.986 4%   10.245 0%   3.292 7%   
     5.001.850 8%   9.867 6%   3.569 0%   4.450 9%   10.629 1%   3.057 1%   
    谱周期/s   Ⅲ类场地   Ⅳ类场地   
    a   b   c   a   b   c   
     0.03−0.267 5%   0.080 0%   −0.004 7%   −0.054 3%   0.100 0%   −0.059 8%   
     0.04−2.468 6%   0.935 0%   −0.004 5%   −1.320 6%   0.678 3%   −0.105 8%   
     0.05−6.151 3%   1.248 6%   0.010 4%   −3.806 6%   0.939 3%   −0.058 7%   
     0.06−10.524 5%   1.229 7%   0.037 4%   −7.517 3%   0.957 4%   −0.007 1%   
     0.07−14.976 8%   1.068 1%   0.072 7%   −11.806 0%   0.864 5%   0.046 1%   
     0.08−19.139 5%   0.847 7%   0.113 6%   −16.158 0%   0.720 7%   0.099 3%   
     0.09−22.600 0%   0.608 0%   0.158 2%   −20.231 6%   0.555 6%   0.151 8%   
     0.10−25.140 8%   0.368 6%   0.205 3%   −23.500 0%   0.384 4%   0.203 1%   
     0.12−28.559 5%   −0.075 5%   0.303 1%   −27.563 9%   0.052 6%   0.301 8%   
     0.14−30.831 7%   −0.455 2%   0.402 5%   −30.112 2%   −0.246 4%   0.395 1%   
     0.15−31.683 1%   −0.620 1%   0.451 9%   −31.053 7%   −0.381 4%   0.439 7%   
     0.16−32.393 3%   −0.769 2%   0.500 9%   −31.834 3%   −0.506 7%   0.483 0%   
     0.18−33.488 5%   −1.023 8%   0.597 2%   −33.032 2%   −0.729 7%   0.566 1%   
     0.20−34.264 2%   −1.227 1%   0.690 8%   −33.882 4%   −0.919 0%   0.644 7%   
     0.25−35.342 3%   −1.558 9%   0.911 0%   −35.110 2%   −1.271 0%   0.824 5%   
    下载: 导出CSV 
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    续表 4 
    谱周期/s   Ⅲ类场地   Ⅳ类场地   
    a   b   c   a   b   c   
     0.30−35.726 5%   −1.708 4%   1.111 7%   −35.653 5%   −1.490 8%   0.984 5%   
     0.35−35.751 4%   −1.736 2%   1.294 4%   −35.866 3%   −1.617 1%   1.128 7%   
     0.40−35.576 2%   −1.681 3%   1.461 1%   −35.906 1%   −1.675 7%   1.260 0%   
     0.45−35.284 2%   −1.569 3%   1.613 7%   −35.849 7%   −1.684 7%   1.380 5%   
     0.50−34.921 9%   −1.417 5%   1.753 9%   −35.737 5%   −1.656 5%   1.492 0%   
     0.60−34.086 0%   −1.039 1%   2.002 5%   −35.424 0%   −1.522 6%   1.692 8%   
     0.70−33.185 1%   −0.606 3%   2.216 3%   −35.051 8%   −1.321 2%   1.870 0%   
     0.80−32.266 7%   −0.151 1%   2.401 8%   −34.650 6%   −1.079 5%   2.028 5%   
     0.90−31.351 5%   0.308 9%   2.564 2%   −34.231 2%   −0.813 7%   2.172 0%   
     1.00−30.448 5%   0.763 8%   2.707 2%   −33.797 4%   −0.533 9%   2.303 2%   
     1.25−28.262 2%   1.846 2%   2.998 0%   −32.656 1%   0.188 2%   2.588 8%   
     1.50−26.178 2%   2.828 2%   3.217 3%   −31.435 4%   0.905 7%   2.829 1%   
     2.00−22.262 8%   4.486 3%   3.512 5%   −28.768 5%   2.255 8%   3.218 4%   
     2.50−18.599 1%   5.791 8%   3.682 8%   −25.840 9%   3.467 9%   3.526 7%   
     3.00−15.118 3%   6.816 0%   3.772 8%   −22.706 4%   4.548 9%   3.781 1%   
     3.50−11.777 9%   7.616 6%   3.807 6%   −19.411 9%   5.515 0%   3.996 9%   
     4.00−8.550 7%   8.237 1%   3.802 6%   −15.994 6%   6.382 3%   4.183 7%   
     4.50−5.418 1%   8.710 7%   3.768 1%   −12.483 9%   7.164 7%   4.348 0%   
     5.00−2.366 9%   9.062 7%   3.711 1%   −8.902 1%   7.873 7%   4.494 2%   
    下载: 导出CSV 
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    图4给出了DCF模型回归值与原始计算均值的拟合对比图,可以看出该模型对各场地类型、各阻尼比的绝对加速度反应谱均有较好的拟合效果,说明模型回归方程的形式及其考虑的影响因素基本合理。此外,在中长周期(约2 s以上)部分出现高阻尼比的DCF值反而高于较低阻尼比的DCF值且远高于1.0的现象,究其原因则是中长周期上,阻尼力与弹性力之比相对较大从而导致加速度随阻尼比的增大而增加。基于此,Naeim和Kelly (1999)建议:在进行隔震结构设计时,应当限制阻尼比在20%以内以减少隔震结构的加速度反应,这与Zhao (2004)以及Zhao和Zhang (2007)的结论一致。此外,Lin和Chang (2003)对这一现象也有所表述,即在所有中长周期上伪加速度反应谱值将随阻尼比的增大而减小,但加速度谱则不然,即在中长周期上阻尼水平较高的情况下,DCF将受影响,当谱周期在2 s以内时,DCF会随着阻尼比的增大而减小,但当谱周期超过2 s左右时,DCF将会随着阻尼比的增加而增大,甚至超过1.0。这是由于现用加速度反应谱是根据惯性力进行标定的,而阻尼比较大的结构在长周期条件下其阻尼力已不可忽略不计,导致在长周期高阻尼比情况下,用惯性力代替弹性力计算得到的反应谱值偏大,因而出现长周期部分阻尼修正系数反而较大的现象(罗开海,王亚勇,2011)。

    图  4  阻尼修正系数DCF模型的水平分量几何均值及拟合曲线
    (a) Ⅰ类场地;(b) Ⅱ类场地;(c) Ⅲ类场地;(d) Ⅳ类场地
    Figure  4.  Fitting curves of DCF model to the geometric mean of the two horizontal components
    (a) Site class Ⅰ;(b) Site class Ⅱ;(c) Site class Ⅲ;(d) Site class Ⅳ

    世界上几乎所有规范的阻尼修正系数均源于基于地震记录得到的经验模型,为便于使用,规范一般采用进一步简化的模型,而本文模型正是为了给规范制定提供参考。为便于计算与对比,选取场地特征周期Tg=0.2 s,将本文提出的DCF模型与现行抗震设计规范(中华人民共和国住房和城乡建设部,中华人民共和国国家质量监督检验检疫总局,2010)中的阻尼修正系数进行对比,结果显示,规范中场地分类为I0的情形对应于本文Ⅱ类场地模型。

    图5分别给出了阻尼比为3%,10%,20%和30%时本文模型与我国现行规范模型的对比图,虽然是源于不同地震的模型,但可以看出二者的大体趋势相近。另外本文使用了大量的地震动记录,采用了可靠的研究方法,能够确保该研究结果的可靠性。

    图  5  Ⅱ类场地下不同阻尼比ξ时DCF模型与规范模型的数值对比
    Figure  5.  Comparison of the predicted DCF from this study with designed specification for different damping ratios ξ for the class Ⅱ sites
    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    对阻尼修正系数DCF模型的标准差进行分析,以探究模型的其它影响参数引起的随机误差。根据Abrahamson和Youngs (1992)提出的随机效应模型,先将模型的总残差分离为事件间残差和事件内残差,即

    $\xi _{i,j}^{\rm T} {\text{=}} {\eta _i} {\text{+}} {\xi _{i,j}}{\text{,}}$

    (4)

    This page contains the following errors:

    error on line 1 at column 1: Start tag expected, '<' not found

    Below is a rendering of the page up to the first error.

    表  5  阻尼修正系数模型的总标准差
    Table  5.  Total standard deviations of DCF model
    谱周期
    /s
    总标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.014 50.006 90.004 10.002 00.001 90.003 70.005 50.007 20.008 90.015 90.021 60.026 40.030 5
    0.040.153 70.086 60.047 60.020 50.015 90.028 80.039 50.048 70.056 70.085 10.103 30.116 50.126 5
    0.050.177 50.103 10.058 10.025 60.020 70.037 70.052 00.064 40.075 20.114 30.139 70.158 20.172 3
    0.060.193 40.116 90.067 10.029 80.024 90.044 70.061 90.077 00.090 30.139 80.172 40.196 30.214 6
    0.070.199 30.123 30.073 90.032 50.026 70.050 10.071 80.087 20.102 70.160 30.198 80.226 60.247 8
    0.080.197 40.123 60.072 70.032 90.028 60.051 00.073 40.089 10.105 10.165 80.206 90.237 00.259 6
    0.090.187 50.119 20.071 00.032 40.028 10.050 50.070 60.088 50.104 60.165 80.207 80.238 50.261 5
    0.100.184 60.117 60.070 10.033 20.028 10.050 20.070 20.088 00.103 90.165 20.207 30.238 00.261 2
    0.120.174 90.112 10.067 00.030 80.026 10.049 60.067 80.087 00.100 40.159 20.199 00.227 80.249 5
    0.140.169 00.107 60.064 10.029 40.024 90.047 00.064 10.080 10.096 60.148 40.185 30.212 40.232 9
    0.150.164 80.105 90.062 00.029 10.024 90.044 60.062 10.079 70.093 90.143 30.179 50.206 00.226 5
    0.160.166 80.105 40.062 70.030 00.024 20.044 50.061 90.081 40.095 90.142 00.177 50.204 00.224 6
    0.180.160 50.102 00.060 40.028 10.023 20.042 40.059 00.074 70.086 50.136 40.171 10.197 50.218 3
    0.200.159 40.103 10.060 60.028 00.023 60.043 20.060 20.074 20.088 20.137 00.171 80.198 80.220 6
    0.250.161 40.102 90.060 90.028 10.023 50.043 00.059 70.073 40.087 40.136 00.172 50.202 10.227 3
    0.300.158 80.102 20.060 70.027 70.023 20.042 60.059 50.073 70.086 50.137 60.176 50.209 10.237 8
    0.350.158 00.102 60.061 60.028 20.023 40.043 40.059 70.074 60.089 50.141 50.182 80.218 20.250 4
    0.400.161 50.104 10.062 10.028 30.024 10.044 20.061 60.076 90.090 70.146 40.191 10.230 30.266 1
    0.450.160 90.104 40.062 50.028 60.024 10.044 30.061 80.077 20.091 60.149 30.196 80.238 90.278 0
    0.500.160 10.104 90.062 90.028 70.024 20.044 70.062 50.078 30.092 70.152 60.203 30.248 70.290 2
    0.600.165 70.107 50.064 50.029 60.025 10.046 40.065 10.082 00.097 50.164 10.219 70.269 30.314 4
    0.700.167 30.108 00.064 50.029 60.025 30.047 00.066 50.084 20.100 70.171 50.232 10.285 90.334 4
    0.800.170 70.111 00.066 90.030 80.026 40.049 30.069 80.088 80.106 50.183 30.248 60.305 90.356 7
    0.900.169 50.110 30.066 30.030 60.026 90.050 50.072 00.092 00.111 00.193 90.265 40.326 80.380 5
    1.000.171 20.112 80.068 70.032 10.028 20.053 30.076 10.097 30.117 20.205 30.280 00.344 00.399 2
    1.250.176 50.118 50.072 50.034 00.030 80.057 90.083 20.106 90.129 30.229 90.313 20.383 30.442 8
    1.500.189 80.128 50.079 20.037 20.033 60.064 20.092 80.119 80.145 50.257 90.350 00.424 80.487 0
    2.000.202 50.140 90.088 30.042 10.038 70.074 60.108 10.139 60.169 20.299 20.398 80.479 10.542 6
    2.500.218 10.154 50.097 60.047 10.043 80.084 20.122 30.158 10.191 20.331 60.438 50.521 10.584 2
    3.000.232 30.166 00.105 90.051 00.047 50.091 50.132 30.170 40.206 30.354 10.465 80.549 30.610 4
    3.500.240 90.175 10.112 90.054 60.051 20.098 80.143 00.184 50.222 60.379 70.492 70.574 60.636 0
    4.000.253 40.185 20.119 90.058 20.054 80.105 20.152 30.195 80.235 80.398 80.510 60.593 00.651 8
    4.500.266 10.195 90.127 10.061 90.057 70.110 70.159 40.204 40.246 10.411 30.525 20.606 30.663 9
    5.000.277 20.205 80.133 70.064 90.060 00.114 50.164 40.210 30.252 40.418 90.531 00.609 50.665 2
    下载: 导出CSV 
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    表  7  阻尼修正系数模型事件内标准差
    Table  7.  Within-event standard deviations of DCF model
    谱周期
    /s
    事件内标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.0080.0060.0040.0020.0020.0040.0050.0070.0080.0150.0200.0250.028
    0.040.1370.0780.0430.0190.0150.0260.0360.0440.0520.0770.0930.1050.114
    0.050.1600.0930.0530.0230.0190.0340.0470.0580.0680.1030.1250.1410.154
    0.060.1780.1070.0610.0270.0220.0410.0560.0700.0820.1260.1550.1760.192
    0.070.1860.1150.0660.0300.0250.0460.0640.0800.0940.1460.1800.2050.223
    0.080.1860.1160.0680.0310.0260.0480.0660.0830.0970.1520.1890.2160.236
    0.090.1780.1130.0670.0310.0260.0470.0660.0820.0970.1530.1900.2180.239
    0.100.1770.1120.0670.0300.0250.0470.0660.0820.0970.1530.1920.2200.241
    0.120.1680.1080.0650.0300.0250.0460.0650.0800.0950.1500.1870.2130.233
    0.140.1610.1030.0620.0280.0240.0440.0620.0770.0900.1420.1760.2020.221
    0.150.1590.1000.0600.0280.0240.0440.0600.0740.0870.1380.1720.1970.217
    0.160.1590.1010.0610.0280.0240.0430.0600.0740.0870.1360.1700.1950.215
    0.180.1530.0980.0580.0260.0220.0410.0570.0700.0830.1310.1650.1900.210
    0.200.1510.0970.0580.0270.0230.0410.0570.0720.0840.1330.1670.1930.214
    0.250.1540.0980.0590.0270.0220.0410.0570.0720.0830.1320.1670.1960.220
    0.300.1510.0970.0580.0270.0220.0410.0570.0710.0840.1330.1700.2010.227
    0.350.1480.0960.0580.0270.0220.0400.0570.0710.0830.1340.1710.2030.231
    0.400.1530.0970.0570.0260.0230.0410.0580.0720.0850.1360.1750.2080.237
    0.450.1510.0960.0570.0260.0220.0410.0570.0710.0850.1360.1770.2120.243
    0.500.1500.0960.0580.0270.0220.0410.0570.0720.0840.1370.1800.2170.250
    0.600.1550.0990.0590.0270.0230.0430.0600.0750.0890.1470.1930.2330.268
    0.700.1570.1010.0600.0280.0240.0440.0610.0770.0920.1530.2030.2470.285
    0.800.1590.1030.0620.0290.0240.0450.0640.0810.0960.1610.2150.2600.300
    0.900.1550.1000.0600.0280.0240.0450.0640.0820.0980.1670.2250.2730.315
    1.000.1570.1020.0620.0290.0250.0480.0680.0860.1030.1750.2340.2840.326
    1.250.1580.1040.0630.0300.0260.0500.0710.0900.1080.1870.2500.3020.345
    1.500.1660.1090.0670.0310.0280.0530.0760.0970.1170.2000.2670.3210.364
    2.000.1670.1120.0700.0330.0300.0570.0820.1040.1260.2140.2820.3350.377
    2.500.1650.1120.0710.0340.0310.0590.0850.1090.1310.2200.2870.3380.378
    3.000.1680.1180.0750.0360.0330.0630.0900.1150.1370.2300.2960.3460.384
    3.500.1660.1190.0770.0370.0350.0660.0940.1200.1440.2380.3040.3520.389
    4.000.1700.1230.0800.0390.0350.0670.0960.1230.1460.2400.3050.3520.388
    4.500.1710.1250.0810.0390.0360.0680.0980.1240.1480.2420.3060.3530.387
    5.000.1750.1290.0840.0410.0370.0690.0980.1250.1490.2420.3050.3500.383
    下载: 导出CSV 
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    表  6  阻尼修正系数模型的事件间标准差
    Table  6.  Between-event standard deviations of DCF model
    谱周期
    /s
    事件间标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.012 30.004 30.001 70.000 60.000 50.001 00.001 60.002 10.003 10.005 30.007 50.009 40.011 0
    0.040.069 40.037 80.020 10.008 40.006 30.011 60.016 10.019 90.023 30.035 70.044 10.050 60.055 8
    0.050.077 60.044 30.024 90.010 80.008 80.016 10.022 10.027 40.032 20.049 80.062 00.071 10.078 2
    0.060.076 00.046 70.027 10.012 00.011 60.018 50.025 80.032 40.038 60.061 30.076 40.087 90.096 7
    0.070.071 00.045 50.032 60.011 90.009 10.019 10.033 00.034 10.040 60.066 30.084 00.097 50.107 6
    0.080.066 90.042 60.024 80.011 30.012 70.018 70.033 10.033 30.040 10.066 30.084 40.098 30.108 7
    0.090.059 10.038 80.023 40.010 90.011 80.017 80.025 40.032 40.038 90.064 60.083 10.096 60.106 9
    0.100.054 10.034 70.021 20.013 70.012 00.017 20.024 30.031 00.037 20.061 90.079 00.091 90.101 7
    0.120.047 60.028 60.017 10.008 00.007 30.019 10.020 30.034 10.031 80.054 00.069 30.080 40.088 4
    0.140.051 00.030 40.017 40.007 60.006 50.017 00.017 50.022 40.036 20.044 50.057 40.066 60.072 9
    0.150.045 10.034 30.013 90.009 40.008 20.009 80.014 30.028 80.034 60.039 10.051 30.059 30.065 3
    0.160.050 20.028 90.015 90.011 90.005 60.010 60.015 20.034 30.041 10.040 10.051 30.059 30.064 9
    0.180.048 10.029 20.016 50.009 50.005 90.010 80.015 20.025 20.022 90.037 20.047 10.054 20.059 6
    0.200.051 10.035 80.016 20.009 00.007 10.012 90.018 10.017 20.026 50.032 60.041 30.047 10.052 0
    0.250.049 60.029 90.016 40.009 10.007 10.013 00.018 00.016 60.026 60.032 90.042 60.050 20.057 6
    0.300.050 40.033 30.018 40.007 70.006 00.010 70.017 90.018 20.021 20.034 50.046 30.058 40.071 1
    0.350.054 30.037 60.022 10.009 50.007 30.015 70.018 70.023 50.034 40.046 70.063 90.080 30.097 8
    0.400.053 10.038 80.023 70.010 80.008 60.015 80.021 90.027 50.032 40.055 40.077 30.098 80.120 6
    0.450.056 80.041 10.024 90.011 10.008 90.016 30.023 00.030 00.035 00.061 00.085 80.110 30.135 4
    0.500.056 50.042 00.024 90.011 10.009 50.017 80.025 30.032 10.038 50.067 00.094 20.121 30.148 2
    0.600.058 40.040 90.025 10.011 20.009 50.017 70.025 30.032 50.039 50.072 90.104 80.134 70.163 9
    0.700.057 40.039 20.023 50.010 90.009 30.017 60.025 60.033 40.041 00.077 50.112 10.144 40.175 7
    0.800.063 20.042 20.025 50.011 80.010 20.019 40.028 50.037 30.046 20.087 30.125 70.160 50.193 3
    0.900.068 00.046 90.028 60.013 10.011 50.022 00.032 30.042 20.052 10.098 20.141 40.179 00.214 0
    1.000.068 10.047 90.029 00.013 60.012 30.023 70.034 80.045 70.056 60.107 90.153 50.194 10.230 4
    1.250.077 80.057 00.035 50.016 70.016 40.029 70.043 70.057 30.070 50.134 00.189 00.236 50.277 6
    1.500.092 40.068 00.043 00.020 10.018 60.036 30.053 60.070 30.086 90.162 40.226 20.278 80.323 8
    2.000.114 80.085 10.054 00.025 90.024 40.048 10.070 90.092 70.113 60.209 10.282 40.342 90.390 1
    2.500.143 10.106 00.067 20.032 60.030 80.059 80.087 90.114 80.139 60.248 10.332 00.396 80.445 5
    3.000.160 30.117 10.074 80.036 00.034 00.066 50.097 20.126 10.154 10.269 60.359 60.426 70.474 3
    3.500.174 20.128 80.083 10.040 00.037 80.073 80.107 70.140 10.169 90.296 20.388 00.454 10.503 6
    4.000.187 50.138 40.089 70.043 60.041 90.081 00.118 20.152 70.184 90.318 30.409 30.477 10.523 9
    4.500.203 60.150 50.097 70.047 70.045 00.087 00.126 20.162 40.196 40.332 70.426 60.493 10.539 2
    5.000.214 80.160 30.104 00.050 50.047 30.091 10.131 80.169 30.204 00.342 10.435 10.499 30.544 0
    下载: 导出CSV 
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    由于DFC值在0.01 s和0.02 s谱周期上基本等于1.0,其参数值未列入表4中,其相应的标准差也未列入表57中。

    图6分别给出了3%,10%,20%和30%阻尼比水平下标准差随谱周期的分布图,可以看出:整个周期上,标准差大体呈增长趋势,除0.08 s处有一极值,模型随机误差大致随谱周期的增长而增大;谱周期约<2.0 s时,事件内标准差均大于事件间标准差,说明此时震源效应的影响较小,其它效应的影响较大;谱周期>1.28 s时,事件间标准差持续增大,而事件内标准差逐渐平缓,约2.0 s后,事件间标准差大于事件内标准差,这说明对于长周期部分,震源效应的影响所占比例更大。

    图  6  不同阻尼比ξ下标准差随谱周期的分布图
    Figure  6.  Distribution of standard deviations with spectral period
    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    事件内残差还可进一步分离为场地间残差和场地内残差,

    ${\xi _{l,n}} {\text{=}} \eta _l^{\rm{s}} {\text{+}} \xi _{l,n}^{\rm{s}}{\text{,}}$

    (5)

    This page contains the following errors:

    error on line 1 at column 1: Start tag expected, '<' not found

    Below is a rendering of the page up to the first error.

    场地间标准差反映场地效应相关参数的精确度,场地内标准差反映路径效应及其它效应相关参数的精确度。图7给出了 Ⅱ 类场地在不同阻尼比(3%,10%,20%,30%)情况下标准差随谱周期的分布图,可以看出大部分周期上模型的场地内标准差要大于场地间标准差,说明场地效应的影响整体上比路径效应和其它效应的影响要小。

    图  7  Ⅱ 类场地下不同阻尼比ξ时标准差随谱周期的分布图
    Figure  7.  Distribution of standard deviations with different damping ratio ξ for the class Ⅱ site
    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    图8a8b分别给出了30%阻尼比水平下四种场地类别的场地间标准差和场地内标准差的对比图,可以看出:四种场地类别的场地内标准差在0.07—0.35 s周期内变化相对小,整体上随着谱周期的增大而增大,表明路径效应和其它效应的影响随谱周期增大;四种场地类别的场地间标准差随谱周期整体上呈先增大后减小再增大的趋势,谱周期超过0.08 s时波动明显,反映了场地效应的影响在0.03—0.08 s之间随谱周期的增大而增大,在0.08—0.48 s之间场地效应的影响关系不明显,0.48 s之后随谱周期增大而呈增大的趋势。与图8b相比,图8a中不同场地类别的场地内标准差的差异在绝大多数周期上相对较小,特别是在长周期上,究其原因是场地内标准差主要来自于路径效应,当数据较多时,各类场地路径效应的模拟程度相当,而场地间标准差来自于场地效应的模拟误差,不同场地类别的模拟效果不同,从而导致场地间标准差在不同场地类别中有较大的差异。

    图  8  阻尼比为30%时四种场地类别的场地内标准差(a)和场地间标准差(b)的对比图
    Figure  8.  Comparison of standard deviations of between-site (a) and within-site (b) with damping ratio 30% for four-class sites

    随机选取几组数据,初步分析随机效应模型分离的残差,以探究震源效应、场地效应、路径效应等对随机误差的影响。

    图9ab给出了具有代表性的0.8 s谱周期下阻尼比为7%时事件间残差随震级、震源深度的分布图,图9cd给出了 Ⅱ 类场地下谱周期为0.08 s、阻尼比为3%和20%时的场地内残差随震源距的分布图,可以看出:事件间残差随震级、震源深度分布趋势线的斜率较明显,说明其与震级和震源深度有显著的相关性,DCF模型残差与震级、震源深度相关;场地内残差随震源距的分布趋势线的斜率也较大,这表明路径效应等影响因素也会对DCF模型产生影响,尚需进一步研究。

    图  9  阻尼修正系数模型事件间及场地内残差分布图
    (a) 事件间残差与震级关系分布图(谱周期T=0.08 s,阻尼比ξ=7%);(b) 事件间残差与震源深度关系的分布图(谱周期T=0.08 s,ξ=7%),(c) 场地内残差与震源距关系分布图(Ⅱ类场地,谱周期T=0.08 s,ξ=3%);(d) 场地内残差与震源距关系分布图(Ⅱ类场地,谱周期T=0.08 s,ξ=20%)
    Figure  9.  The variation of between-event and within-site residuals of DCF model
    (a) Between-event residuals with moment magnitude (T=0.08 s,ξ=7%); (b) Between-event residuals with focal depth (T=0.08 s,ξ=7%); (c) Within-site residuals with source distance (site class Ⅱ ,T=0.08 s,ξ=3%); (d) Within-site residuals with source distance (site class Ⅱ ,T=0.08 s,ξ=20%)

    本文对日本K-net和KiK-net台网中123个地震动事件的6 466条浅壳与上地幔地震动记录进行统计分析,根据场地平均剪切波速将场地分为四个类别,并分别计算了13个阻尼比36个谱周期对应的阻尼修正系数值,提出了关于阻尼比、谱周期的阻尼修正系数模型,并利用随机效应模型进行残差分析,探究了震源效应、路径效应、场地效应对阻尼修正系数的影响,得出的主要结论如下:

    1) 根据Z检验,不同场地的DCF均值存在显著差异,这表明区分场地类别研究DCF是必要的;

    2) 震源效应引起的随机误差在2.0 s谱周期之前小于其它效应引起的随机误差,但2.0 s后随机误差较大,其中场地效应引起的随机误差整体上要小于路径效应和其它效应引起的随机误差;

    3) 模型事件间残差与震级和震源深度显著相关,场地内残差与震源距显著相关,后续研究应考虑相关参数以建立标准误差更小的浅壳与上地幔加速度反应谱阻尼修正系数模型。

    本文建立的浅壳与上地幔地震的加速度设计反应谱阻尼修正系数模型,可为世界上其它俯冲带地区的阻尼修正系数模型的建立及完善提供参考。

  • 图  1   震级随震源距(a)和震源深度(b)的分布

    Figure  1.   The distribution of MW with respect to source distance (a) and focal depth (b)

    图  2   阻尼比为1%和30%时场地类别间的显著性检验统计值| Z |与谱周期的关系

    (a) Ⅰ类与Ⅱ类;(b) Ⅰ类与Ⅲ类;(c) Ⅰ类与Ⅳ类;(d) Ⅱ类与Ⅲ类;(e) Ⅱ类与Ⅳ类;(f) Ⅲ类与Ⅳ类

    Figure  2.   Variation of | Z | values with spectral periods for each pair of site classes with damping ratios 1% and 30%

    (a) Site classes Ⅰ and Ⅱ;(b) Site classes Ⅰ and Ⅲ;(c) Site classes Ⅰ and Ⅳ;(d) Site classes Ⅱ and Ⅲ;(e) Site classes Ⅱ and Ⅳ;(f) Site classes Ⅲ and Ⅳ

    图  3   不同阻尼比ζ时阻尼修正系数对数lnBa的几何均值与谱周期的关系曲线

    Figure  3.   Mean value of lnBa with spectral periods with different damping ratio ζ

    (a) ζ=3%;(b) ζ=8%;(c) ζ=15%;(d) ζ=30%

    图  4   阻尼修正系数DCF模型的水平分量几何均值及拟合曲线

    (a) Ⅰ类场地;(b) Ⅱ类场地;(c) Ⅲ类场地;(d) Ⅳ类场地

    Figure  4.   Fitting curves of DCF model to the geometric mean of the two horizontal components

    (a) Site class Ⅰ;(b) Site class Ⅱ;(c) Site class Ⅲ;(d) Site class Ⅳ

    图  5   Ⅱ类场地下不同阻尼比ξ时DCF模型与规范模型的数值对比

    Figure  5.   Comparison of the predicted DCF from this study with designed specification for different damping ratios ξ for the class Ⅱ sites

    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    图  6   不同阻尼比ξ下标准差随谱周期的分布图

    Figure  6.   Distribution of standard deviations with spectral period

    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    图  7   Ⅱ 类场地下不同阻尼比ξ时标准差随谱周期的分布图

    Figure  7.   Distribution of standard deviations with different damping ratio ξ for the class Ⅱ site

    (a) ξ=3%;(b) ξ=10%;(c) ξ=20%;(d) ξ=30%

    图  8   阻尼比为30%时四种场地类别的场地内标准差(a)和场地间标准差(b)的对比图

    Figure  8.   Comparison of standard deviations of between-site (a) and within-site (b) with damping ratio 30% for four-class sites

    图  9   阻尼修正系数模型事件间及场地内残差分布图

    (a) 事件间残差与震级关系分布图(谱周期T=0.08 s,阻尼比ξ=7%);(b) 事件间残差与震源深度关系的分布图(谱周期T=0.08 s,ξ=7%),(c) 场地内残差与震源距关系分布图(Ⅱ类场地,谱周期T=0.08 s,ξ=3%);(d) 场地内残差与震源距关系分布图(Ⅱ类场地,谱周期T=0.08 s,ξ=20%)

    Figure  9.   The variation of between-event and within-site residuals of DCF model

    (a) Between-event residuals with moment magnitude (T=0.08 s,ξ=7%); (b) Between-event residuals with focal depth (T=0.08 s,ξ=7%); (c) Within-site residuals with source distance (site class Ⅱ ,T=0.08 s,ξ=3%); (d) Within-site residuals with source distance (site class Ⅱ ,T=0.08 s,ξ=20%)

    表  1   谱周期点数量

    Table  1   Number of spectral periods

    序号 周期/s 序号 周期/s 序号 周期/s 序号 周期/s 序号 周期/s
    2 0.01 10 0.09 18 0.25 26 0.80 34 3.50
    3 0.02 11 0.10 19 0.30 27 0.90 35 4.00
    4 0.03 12 0.12 20 0.35 28 1.00 36 4.50
    5 0.04 13 0.14 21 0.40 29 1.25 37 5.00
    6 0.05 14 0.15 22 0.45 30 1.50
    7 0.06 15 0.16 23 0.50 31 2.00
    8 0.07 16 0.18 24 0.60 32 2.50
    9 0.08 17 0.20 25 0.70 33 3.00
    下载: 导出CSV

    表  2   阻尼比数量

    Table  2   Number of damping ratios

    序号阻尼比序号阻尼比序号阻尼比
    11% 6 6%1115%
    22% 7 7%1220%
    33% 8 8%1325%
    44% 9 9%1430%
    55%1010%
    下载: 导出CSV

    表  3   场地类别分类标准

    Table  3   Site class definitions used in this study

    场地类别土质类别场地特征周期Tg/s平均剪切波速v30/(km·s−1记录条数
    岩土Tg<0.2v30>6003 007
    硬土0.2≤Tg<0.4300<v30≤6001 828
    中硬土0.4≤Tg<0.6200<v30≤300 578
    软土Tg>0.6v30<2001 053
    下载: 导出CSV

    表  4   四类场地类别下阻尼修正系数模型的回归系数

    Table  4   Regressive parameters for damping correction factor model for site classes Ⅰ to Ⅳ

    谱周期/s   Ⅰ类场地   Ⅱ类场地   
    a   b   c   a   b   c   
     0.03−0.507 8%   0.080 0%   −0.189 8%   −0.550 0%   0.100 0%   −0.049 9%   
     0.04−7.950 5%   1.218 2%   −0.170 9%   −4.200 9%   0.911 5%   −0.059 7%   
     0.05−15.022 4%   1.272 6%   −0.088 0%   −8.216 0%   1.228 8%   −0.039 6%   
     0.06−21.022 3%   0.843 4%   0.012 5%   −12.559 9%   1.209 6%   −0.000 2%   
     0.07−25.837 4%   0.285 8%   0.119 0%   −16.909 3%   1.025 3%   0.051 3%   
     0.08−29.366 4%   −0.255 8%   0.226 0%   −21.501 6%   0.758 2%   0.110 5%   
     0.09−31.472 6%   −0.700 0%   0.331 0%   −24.952 4%   0.451 1%   0.174 5%   
     0.10−32.970 3%   −0.992 1%   0.432 8%   −27.541 0%   0.127 5%   0.241 2%   
     0.12−34.812 4%   −1.326 8%   0.625 1%   −31.029 6%   −0.521 6%   0.378 0%   
     0.14−35.739 9%   −1.505 6%   0.801 9%   −33.120 8%   −1.137 0%   0.514 8%   
     0.15−35.996 1%   −1.556 0%   0.884 9%   −33.832 9%   −1.426 2%   0.582 1%   
     0.16−36.157 1%   −1.587 1%   0.964 3%   −34.385 3%   −1.674 3%   0.648 5%   
     0.18−36.275 6%   −1.605 1%   1.113 7%   −35.133 3%   −1.835 7%   0.777 7%   
     0.20−36.214 3%   −1.580 3%   1.251 4%   −35.545 6%   −1.910 6%   0.901 8%   
     0.25−35.668 4%   −1.406 8%   1.553 4%   −35.741 6%   −1.867 6%   1.189 2%   
     0.30−34.908 3%   −1.151 5%   1.807 0%   −35.345 2%   −1.646 4%   1.445 5%   
     0.35−34.112 2%   −0.862 3%   2.023 6%   −34.703 7%   −1.342 0%   1.674 2%   
     0.40−33.341 7%   −0.561 2%   2.211 1%   −33.964 9%   −0.999 6%   1.879 1%   
     0.45−32.616 5%   −0.259 4%   2.375 1%   −33.196 7%   −0.642 4%   2.063 3%   
     0.50−31.940 0%   0.037 6%   2.519 9%   −32.430 9%   −0.282 9%   2.229 6%   
     0.60−30.721 0%   0.607 2%   2.763 9%   −30.958 6%   0.418 8%   2.517 3%   
     0.70−29.647 9%   1.138 9%   2.961 2%   −29.589 9%   1.080 3%   2.756 2%   
     0.80−28.683 6%   1.632 6%   3.123 4%   −28.322 5%   1.695 7%   2.956 2%   
     0.90−27.799 0%   2.091 1%   3.258 3%   −27.143 3%   2.265 6%   3.124 9%   
     1.00−26.972 5%   2.517 6%   3.371 6%   −26.038 2%   2.793 1%   3.267 6%   
     1.25−25.066 9%   3.465 2%   3.584 4%   −23.522 0%   3.950 7%   3.536 5%   
     1.50−23.283 9%   4.275 7%   3.726 0%   −21.250 5%   4.920 1%   3.712 5%   
     2.00−19.830 8%   5.600 9%   3.878 3%   −17.126 9%   6.450 0%   3.883 8%   
     2.50−16.370 2%   6.650 6%   3.925 7%   −13.301 8%   7.601 0%   3.902 4%   
     3.00−12.845 6%   7.511 4%   3.911 4%   −9.628 3%   8.494 9%   3.826 4%   
     3.50−9.251 3%   8.235 0%   3.857 8%   −6.041 0%   9.204 9%   3.687 7%   
     4.00−5.596 0%   8.854 9%   3.778 0%   −2.508 1%   9.777 7%   3.505 7%   
     4.50−1.891 5%   9.393 8%   3.679 9%   0.986 4%   10.245 0%   3.292 7%   
     5.001.850 8%   9.867 6%   3.569 0%   4.450 9%   10.629 1%   3.057 1%   
    谱周期/s   Ⅲ类场地   Ⅳ类场地   
    a   b   c   a   b   c   
     0.03−0.267 5%   0.080 0%   −0.004 7%   −0.054 3%   0.100 0%   −0.059 8%   
     0.04−2.468 6%   0.935 0%   −0.004 5%   −1.320 6%   0.678 3%   −0.105 8%   
     0.05−6.151 3%   1.248 6%   0.010 4%   −3.806 6%   0.939 3%   −0.058 7%   
     0.06−10.524 5%   1.229 7%   0.037 4%   −7.517 3%   0.957 4%   −0.007 1%   
     0.07−14.976 8%   1.068 1%   0.072 7%   −11.806 0%   0.864 5%   0.046 1%   
     0.08−19.139 5%   0.847 7%   0.113 6%   −16.158 0%   0.720 7%   0.099 3%   
     0.09−22.600 0%   0.608 0%   0.158 2%   −20.231 6%   0.555 6%   0.151 8%   
     0.10−25.140 8%   0.368 6%   0.205 3%   −23.500 0%   0.384 4%   0.203 1%   
     0.12−28.559 5%   −0.075 5%   0.303 1%   −27.563 9%   0.052 6%   0.301 8%   
     0.14−30.831 7%   −0.455 2%   0.402 5%   −30.112 2%   −0.246 4%   0.395 1%   
     0.15−31.683 1%   −0.620 1%   0.451 9%   −31.053 7%   −0.381 4%   0.439 7%   
     0.16−32.393 3%   −0.769 2%   0.500 9%   −31.834 3%   −0.506 7%   0.483 0%   
     0.18−33.488 5%   −1.023 8%   0.597 2%   −33.032 2%   −0.729 7%   0.566 1%   
     0.20−34.264 2%   −1.227 1%   0.690 8%   −33.882 4%   −0.919 0%   0.644 7%   
     0.25−35.342 3%   −1.558 9%   0.911 0%   −35.110 2%   −1.271 0%   0.824 5%   
    下载: 导出CSV
    续表 4 
    谱周期/s   Ⅲ类场地   Ⅳ类场地   
    a   b   c   a   b   c   
     0.30−35.726 5%   −1.708 4%   1.111 7%   −35.653 5%   −1.490 8%   0.984 5%   
     0.35−35.751 4%   −1.736 2%   1.294 4%   −35.866 3%   −1.617 1%   1.128 7%   
     0.40−35.576 2%   −1.681 3%   1.461 1%   −35.906 1%   −1.675 7%   1.260 0%   
     0.45−35.284 2%   −1.569 3%   1.613 7%   −35.849 7%   −1.684 7%   1.380 5%   
     0.50−34.921 9%   −1.417 5%   1.753 9%   −35.737 5%   −1.656 5%   1.492 0%   
     0.60−34.086 0%   −1.039 1%   2.002 5%   −35.424 0%   −1.522 6%   1.692 8%   
     0.70−33.185 1%   −0.606 3%   2.216 3%   −35.051 8%   −1.321 2%   1.870 0%   
     0.80−32.266 7%   −0.151 1%   2.401 8%   −34.650 6%   −1.079 5%   2.028 5%   
     0.90−31.351 5%   0.308 9%   2.564 2%   −34.231 2%   −0.813 7%   2.172 0%   
     1.00−30.448 5%   0.763 8%   2.707 2%   −33.797 4%   −0.533 9%   2.303 2%   
     1.25−28.262 2%   1.846 2%   2.998 0%   −32.656 1%   0.188 2%   2.588 8%   
     1.50−26.178 2%   2.828 2%   3.217 3%   −31.435 4%   0.905 7%   2.829 1%   
     2.00−22.262 8%   4.486 3%   3.512 5%   −28.768 5%   2.255 8%   3.218 4%   
     2.50−18.599 1%   5.791 8%   3.682 8%   −25.840 9%   3.467 9%   3.526 7%   
     3.00−15.118 3%   6.816 0%   3.772 8%   −22.706 4%   4.548 9%   3.781 1%   
     3.50−11.777 9%   7.616 6%   3.807 6%   −19.411 9%   5.515 0%   3.996 9%   
     4.00−8.550 7%   8.237 1%   3.802 6%   −15.994 6%   6.382 3%   4.183 7%   
     4.50−5.418 1%   8.710 7%   3.768 1%   −12.483 9%   7.164 7%   4.348 0%   
     5.00−2.366 9%   9.062 7%   3.711 1%   −8.902 1%   7.873 7%   4.494 2%   
    下载: 导出CSV

    表  5   阻尼修正系数模型的总标准差

    Table  5   Total standard deviations of DCF model

    谱周期
    /s
    总标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.014 50.006 90.004 10.002 00.001 90.003 70.005 50.007 20.008 90.015 90.021 60.026 40.030 5
    0.040.153 70.086 60.047 60.020 50.015 90.028 80.039 50.048 70.056 70.085 10.103 30.116 50.126 5
    0.050.177 50.103 10.058 10.025 60.020 70.037 70.052 00.064 40.075 20.114 30.139 70.158 20.172 3
    0.060.193 40.116 90.067 10.029 80.024 90.044 70.061 90.077 00.090 30.139 80.172 40.196 30.214 6
    0.070.199 30.123 30.073 90.032 50.026 70.050 10.071 80.087 20.102 70.160 30.198 80.226 60.247 8
    0.080.197 40.123 60.072 70.032 90.028 60.051 00.073 40.089 10.105 10.165 80.206 90.237 00.259 6
    0.090.187 50.119 20.071 00.032 40.028 10.050 50.070 60.088 50.104 60.165 80.207 80.238 50.261 5
    0.100.184 60.117 60.070 10.033 20.028 10.050 20.070 20.088 00.103 90.165 20.207 30.238 00.261 2
    0.120.174 90.112 10.067 00.030 80.026 10.049 60.067 80.087 00.100 40.159 20.199 00.227 80.249 5
    0.140.169 00.107 60.064 10.029 40.024 90.047 00.064 10.080 10.096 60.148 40.185 30.212 40.232 9
    0.150.164 80.105 90.062 00.029 10.024 90.044 60.062 10.079 70.093 90.143 30.179 50.206 00.226 5
    0.160.166 80.105 40.062 70.030 00.024 20.044 50.061 90.081 40.095 90.142 00.177 50.204 00.224 6
    0.180.160 50.102 00.060 40.028 10.023 20.042 40.059 00.074 70.086 50.136 40.171 10.197 50.218 3
    0.200.159 40.103 10.060 60.028 00.023 60.043 20.060 20.074 20.088 20.137 00.171 80.198 80.220 6
    0.250.161 40.102 90.060 90.028 10.023 50.043 00.059 70.073 40.087 40.136 00.172 50.202 10.227 3
    0.300.158 80.102 20.060 70.027 70.023 20.042 60.059 50.073 70.086 50.137 60.176 50.209 10.237 8
    0.350.158 00.102 60.061 60.028 20.023 40.043 40.059 70.074 60.089 50.141 50.182 80.218 20.250 4
    0.400.161 50.104 10.062 10.028 30.024 10.044 20.061 60.076 90.090 70.146 40.191 10.230 30.266 1
    0.450.160 90.104 40.062 50.028 60.024 10.044 30.061 80.077 20.091 60.149 30.196 80.238 90.278 0
    0.500.160 10.104 90.062 90.028 70.024 20.044 70.062 50.078 30.092 70.152 60.203 30.248 70.290 2
    0.600.165 70.107 50.064 50.029 60.025 10.046 40.065 10.082 00.097 50.164 10.219 70.269 30.314 4
    0.700.167 30.108 00.064 50.029 60.025 30.047 00.066 50.084 20.100 70.171 50.232 10.285 90.334 4
    0.800.170 70.111 00.066 90.030 80.026 40.049 30.069 80.088 80.106 50.183 30.248 60.305 90.356 7
    0.900.169 50.110 30.066 30.030 60.026 90.050 50.072 00.092 00.111 00.193 90.265 40.326 80.380 5
    1.000.171 20.112 80.068 70.032 10.028 20.053 30.076 10.097 30.117 20.205 30.280 00.344 00.399 2
    1.250.176 50.118 50.072 50.034 00.030 80.057 90.083 20.106 90.129 30.229 90.313 20.383 30.442 8
    1.500.189 80.128 50.079 20.037 20.033 60.064 20.092 80.119 80.145 50.257 90.350 00.424 80.487 0
    2.000.202 50.140 90.088 30.042 10.038 70.074 60.108 10.139 60.169 20.299 20.398 80.479 10.542 6
    2.500.218 10.154 50.097 60.047 10.043 80.084 20.122 30.158 10.191 20.331 60.438 50.521 10.584 2
    3.000.232 30.166 00.105 90.051 00.047 50.091 50.132 30.170 40.206 30.354 10.465 80.549 30.610 4
    3.500.240 90.175 10.112 90.054 60.051 20.098 80.143 00.184 50.222 60.379 70.492 70.574 60.636 0
    4.000.253 40.185 20.119 90.058 20.054 80.105 20.152 30.195 80.235 80.398 80.510 60.593 00.651 8
    4.500.266 10.195 90.127 10.061 90.057 70.110 70.159 40.204 40.246 10.411 30.525 20.606 30.663 9
    5.000.277 20.205 80.133 70.064 90.060 00.114 50.164 40.210 30.252 40.418 90.531 00.609 50.665 2
    下载: 导出CSV

    表  7   阻尼修正系数模型事件内标准差

    Table  7   Within-event standard deviations of DCF model

    谱周期
    /s
    事件内标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.0080.0060.0040.0020.0020.0040.0050.0070.0080.0150.0200.0250.028
    0.040.1370.0780.0430.0190.0150.0260.0360.0440.0520.0770.0930.1050.114
    0.050.1600.0930.0530.0230.0190.0340.0470.0580.0680.1030.1250.1410.154
    0.060.1780.1070.0610.0270.0220.0410.0560.0700.0820.1260.1550.1760.192
    0.070.1860.1150.0660.0300.0250.0460.0640.0800.0940.1460.1800.2050.223
    0.080.1860.1160.0680.0310.0260.0480.0660.0830.0970.1520.1890.2160.236
    0.090.1780.1130.0670.0310.0260.0470.0660.0820.0970.1530.1900.2180.239
    0.100.1770.1120.0670.0300.0250.0470.0660.0820.0970.1530.1920.2200.241
    0.120.1680.1080.0650.0300.0250.0460.0650.0800.0950.1500.1870.2130.233
    0.140.1610.1030.0620.0280.0240.0440.0620.0770.0900.1420.1760.2020.221
    0.150.1590.1000.0600.0280.0240.0440.0600.0740.0870.1380.1720.1970.217
    0.160.1590.1010.0610.0280.0240.0430.0600.0740.0870.1360.1700.1950.215
    0.180.1530.0980.0580.0260.0220.0410.0570.0700.0830.1310.1650.1900.210
    0.200.1510.0970.0580.0270.0230.0410.0570.0720.0840.1330.1670.1930.214
    0.250.1540.0980.0590.0270.0220.0410.0570.0720.0830.1320.1670.1960.220
    0.300.1510.0970.0580.0270.0220.0410.0570.0710.0840.1330.1700.2010.227
    0.350.1480.0960.0580.0270.0220.0400.0570.0710.0830.1340.1710.2030.231
    0.400.1530.0970.0570.0260.0230.0410.0580.0720.0850.1360.1750.2080.237
    0.450.1510.0960.0570.0260.0220.0410.0570.0710.0850.1360.1770.2120.243
    0.500.1500.0960.0580.0270.0220.0410.0570.0720.0840.1370.1800.2170.250
    0.600.1550.0990.0590.0270.0230.0430.0600.0750.0890.1470.1930.2330.268
    0.700.1570.1010.0600.0280.0240.0440.0610.0770.0920.1530.2030.2470.285
    0.800.1590.1030.0620.0290.0240.0450.0640.0810.0960.1610.2150.2600.300
    0.900.1550.1000.0600.0280.0240.0450.0640.0820.0980.1670.2250.2730.315
    1.000.1570.1020.0620.0290.0250.0480.0680.0860.1030.1750.2340.2840.326
    1.250.1580.1040.0630.0300.0260.0500.0710.0900.1080.1870.2500.3020.345
    1.500.1660.1090.0670.0310.0280.0530.0760.0970.1170.2000.2670.3210.364
    2.000.1670.1120.0700.0330.0300.0570.0820.1040.1260.2140.2820.3350.377
    2.500.1650.1120.0710.0340.0310.0590.0850.1090.1310.2200.2870.3380.378
    3.000.1680.1180.0750.0360.0330.0630.0900.1150.1370.2300.2960.3460.384
    3.500.1660.1190.0770.0370.0350.0660.0940.1200.1440.2380.3040.3520.389
    4.000.1700.1230.0800.0390.0350.0670.0960.1230.1460.2400.3050.3520.388
    4.500.1710.1250.0810.0390.0360.0680.0980.1240.1480.2420.3060.3530.387
    5.000.1750.1290.0840.0410.0370.0690.0980.1250.1490.2420.3050.3500.383
    下载: 导出CSV

    表  6   阻尼修正系数模型的事件间标准差

    Table  6   Between-event standard deviations of DCF model

    谱周期
    /s
    事件间标准差
    ξ=1%ξ=2%ξ=3%ξ=4%ξ=6%ξ=7%ξ=8%ξ=9%ξ=10%ξ=15%ξ=20%ξ=25%ξ=30%
    0.030.012 30.004 30.001 70.000 60.000 50.001 00.001 60.002 10.003 10.005 30.007 50.009 40.011 0
    0.040.069 40.037 80.020 10.008 40.006 30.011 60.016 10.019 90.023 30.035 70.044 10.050 60.055 8
    0.050.077 60.044 30.024 90.010 80.008 80.016 10.022 10.027 40.032 20.049 80.062 00.071 10.078 2
    0.060.076 00.046 70.027 10.012 00.011 60.018 50.025 80.032 40.038 60.061 30.076 40.087 90.096 7
    0.070.071 00.045 50.032 60.011 90.009 10.019 10.033 00.034 10.040 60.066 30.084 00.097 50.107 6
    0.080.066 90.042 60.024 80.011 30.012 70.018 70.033 10.033 30.040 10.066 30.084 40.098 30.108 7
    0.090.059 10.038 80.023 40.010 90.011 80.017 80.025 40.032 40.038 90.064 60.083 10.096 60.106 9
    0.100.054 10.034 70.021 20.013 70.012 00.017 20.024 30.031 00.037 20.061 90.079 00.091 90.101 7
    0.120.047 60.028 60.017 10.008 00.007 30.019 10.020 30.034 10.031 80.054 00.069 30.080 40.088 4
    0.140.051 00.030 40.017 40.007 60.006 50.017 00.017 50.022 40.036 20.044 50.057 40.066 60.072 9
    0.150.045 10.034 30.013 90.009 40.008 20.009 80.014 30.028 80.034 60.039 10.051 30.059 30.065 3
    0.160.050 20.028 90.015 90.011 90.005 60.010 60.015 20.034 30.041 10.040 10.051 30.059 30.064 9
    0.180.048 10.029 20.016 50.009 50.005 90.010 80.015 20.025 20.022 90.037 20.047 10.054 20.059 6
    0.200.051 10.035 80.016 20.009 00.007 10.012 90.018 10.017 20.026 50.032 60.041 30.047 10.052 0
    0.250.049 60.029 90.016 40.009 10.007 10.013 00.018 00.016 60.026 60.032 90.042 60.050 20.057 6
    0.300.050 40.033 30.018 40.007 70.006 00.010 70.017 90.018 20.021 20.034 50.046 30.058 40.071 1
    0.350.054 30.037 60.022 10.009 50.007 30.015 70.018 70.023 50.034 40.046 70.063 90.080 30.097 8
    0.400.053 10.038 80.023 70.010 80.008 60.015 80.021 90.027 50.032 40.055 40.077 30.098 80.120 6
    0.450.056 80.041 10.024 90.011 10.008 90.016 30.023 00.030 00.035 00.061 00.085 80.110 30.135 4
    0.500.056 50.042 00.024 90.011 10.009 50.017 80.025 30.032 10.038 50.067 00.094 20.121 30.148 2
    0.600.058 40.040 90.025 10.011 20.009 50.017 70.025 30.032 50.039 50.072 90.104 80.134 70.163 9
    0.700.057 40.039 20.023 50.010 90.009 30.017 60.025 60.033 40.041 00.077 50.112 10.144 40.175 7
    0.800.063 20.042 20.025 50.011 80.010 20.019 40.028 50.037 30.046 20.087 30.125 70.160 50.193 3
    0.900.068 00.046 90.028 60.013 10.011 50.022 00.032 30.042 20.052 10.098 20.141 40.179 00.214 0
    1.000.068 10.047 90.029 00.013 60.012 30.023 70.034 80.045 70.056 60.107 90.153 50.194 10.230 4
    1.250.077 80.057 00.035 50.016 70.016 40.029 70.043 70.057 30.070 50.134 00.189 00.236 50.277 6
    1.500.092 40.068 00.043 00.020 10.018 60.036 30.053 60.070 30.086 90.162 40.226 20.278 80.323 8
    2.000.114 80.085 10.054 00.025 90.024 40.048 10.070 90.092 70.113 60.209 10.282 40.342 90.390 1
    2.500.143 10.106 00.067 20.032 60.030 80.059 80.087 90.114 80.139 60.248 10.332 00.396 80.445 5
    3.000.160 30.117 10.074 80.036 00.034 00.066 50.097 20.126 10.154 10.269 60.359 60.426 70.474 3
    3.500.174 20.128 80.083 10.040 00.037 80.073 80.107 70.140 10.169 90.296 20.388 00.454 10.503 6
    4.000.187 50.138 40.089 70.043 60.041 90.081 00.118 20.152 70.184 90.318 30.409 30.477 10.523 9
    4.500.203 60.150 50.097 70.047 70.045 00.087 00.126 20.162 40.196 40.332 70.426 60.493 10.539 2
    5.000.214 80.160 30.104 00.050 50.047 30.091 10.131 80.169 30.204 00.342 10.435 10.499 30.544 0
    下载: 导出CSV
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  • 收稿日期:  2019-03-14
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  • 发布日期:  2019-10-31

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