Improved half-wavelength method for determining the velocity structure of site soil layer
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摘要: 地球物理反演浅地表土层波速剖面通常基于初始速度结构模型，为避免对地勘资料产生较多依赖，充分利用地震和背景噪声等被动源观测记录，快速简便地构建出土层反演的初始模型，本文基于场地土层的频散曲线和水平与竖向谱比提出了一种改进半波长法。该方法首先通过水平与竖向谱比确定场地土层的卓越频率，获取场地覆盖层厚度，确定反演所需的频带范围；其次，利用瑞雷波相速度对剪切波速的偏导数随土层深度的变化规律反演出场地的初始速度结构，并结合工程场地勘探中的半波长法，对常见的三类典型土层模型和日本Kushiro场地实测模型进行了实例分析；最后，将反演得到的速度结构与理论或实测速度结构进行误差对比分析。结果表明：改进半波长法获得的初始速度结构相对于理论或实测速度结构的最大误差不超过35%，可为利用地球物理方法反演工程场地波速剖面构造一个较小的搜索模型空间，进而提高反演计算的速度和结果的可靠性。Abstract: The initial velocity structure model is generally necessary in the study of geophysical inversion for shallow surface soil wave velocity profile. In order to avoid excessive reliance on the surveying data instead, and to make full use of observation records of passive sources such as earthquakes and ambient noise to quickly and easily construct the initial model of soil inversion, this paper proposes an improved half-wavelength method based on the dispersion curve and the horizontal-to-vertical spectral ratio （HVSR） of the soil layer. Firstly, the method deter-mines the predominant frequency of the soil layer by HVSR, and then obtains the depth of overlying soil, and determines the frequency band range required for the inversion. Next, invert initial velocity structure of the site by using the variation of the partial derivative of the Rayleigh wave velocity to the shear wave velocity with the depth of the soil layer. Combined with the half-wavelength method in engineering site exploration, three types of soil layer models frequently encountered and the measured model of Kushiro site in Japan were analyzed, and the error analysis of the velocity structure was carried out and compared with the theoretical or measured velocity structures. The research shows that the initial velocity structure obtained by the improved half-wavelength method has a maximum error not more than 35%, which can be used to reconstruct a small searching model space in inversion for the wave velocity profile of engineering site by using the geophysical method, so as to improve the speed of inversion calculations and the reliability of result.
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引言
浅地表剪切波速度结构是进行工程勘察和工程抗震的重要参数，了解地下的剪切波速度结构对地质灾害预测和防御具有重要意义（Strobbia，Cassiani，2011）。获取工程场地剪切波速结构剖面的常规方法分为跨（单）孔法和面波勘探法。跨（单）孔法虽然能获得较准确的速度值，但该方法经济耗费高、效率低，且对环境具有一定的破坏作用。面波勘探法分为人工面波和天然源面波勘探法，其中：人工源面波勘探法受场地条件限制较多，且勘探深度有限（Park et al，1999）；而天然源面波勘探法具有受场地条件限制小、可探测的深度范围大、对所测环境不产生破坏、高效、经济、适合人口密集地区等优点，因而受到越来越多的关注。天然源面波勘探法需从多点记录中提取面波频散曲线，提取面波频散曲线的常用方法主要有空间自相关法（spatial autocorrelation method，缩写为SPAC）（Aki，1957；Okada et al，1990）、频率-波数法（frequency-wavenumber spectral method，缩写为f-k）（Capon，1969；Wathelet et al，2008）。对于提取的频散曲线进行反演分析，则可获得试验场地速度结构。近年来，国内外很多学者利用更为简便的单台记录噪声水平与竖向谱比（noise horizontal to vertical spectral ratio，缩写为NHVSR）进行土层结构反演（王伟君等，2009；Bignardi et al，2016；García-Jerez et al，2016），Rong等（2017）和荣棉水等（2018）在利用测震或强震动观测数据反演浅地表土层速度结构方面也开展了研究，发展了全局优化反演实用方法。
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基于以上原因，本文借鉴崔建文等（1994）简化剥层法的思路，对半波长法进行改进，利用场地土层的频散曲线及水平与竖向谱比（horizontal-to-vertical spectral ratio，简写为HVSR）提出一种改进半波长反演方法。通过对理论和实测场地算例分析，得到改进半波长法对土层速度结构的反演误差范围，为采用其它地球物理手段进一步确定土层速度结构提供合理的参数约束区间。
1. 改进半波长法
利用面波频散资料反演土层波速的半波长法，是根据均匀半空间中面波相速度近似等于介质剪切波速，在成层弹性半空间中的面波相速度主要受控于一个波长深度的剪切波速，将与波长对应的等效相速度近似折算为相应深度处介质的剪切波速。由于该方法采用了过于简单的近似假定，其计算结果精度较低。而本文提出的改进半波长法侧重于反演工程场地的剪切波速剖面，首先通过场地土层的HVSR确定场地的卓越频率，结合频散曲线获取场地土层厚度；再根据半波长理论确定场地土层反演所需的频带范围，以提高实际资料反演结果的精度和可靠性；最后利用瑞雷面波相速度对剪切波速的偏导数随土层深度的变化规律得出权重，用加权法建立相速度与土层剪切波速的近似关系，从而反演出场地的初始速度结构。以下分别对实际反演过程中改进半波长法的两个步骤进行阐述。
1.1 基岩埋深及频带范围的确定
苏经宇和王广军（1985）通过对典型土层剖面的地震反应分析，指出不同场地的地震效应除受地震参数本身影响外，主要影响因素是场地基岩埋深、场地剖面深度以及土层刚度。由此可以看出，明确基岩埋深不仅可以确定场地土层反演所需的频带范围，而且对评估场地效应也必不可少。
依靠传统钻探技术获取土层厚度等参数由于花费高昂而受限制，充分利用地震和背景噪声等被动源观测记录推断场地土层参数受到了越来越多研究人员的关注。目前场地卓越频率的测定方法主要有依据数值计算的波速数值法和根据场地地震或背景噪声记录的直接测定法。近年来，国内外学者基于观测记录证实了运用HVSR可以准确地确定场地土层卓越频率（Wen et al，2006；王伟君等，2009，2011；Nagashima et al，2014；荣棉水等，2016）。理论HVSR的计算公式系Kawase等（2011）基于散射场理论提出，具体如下：
${\mathop{\rm HVSR}\nolimits} {\text{＝}}\sqrt {\frac{{2\alpha }}{\beta }} \frac{{\left| {{{{\mathop{\rm TF}\nolimits} }_{\mathop{\rm S}\nolimits} }\!\!\!{\text{（}}f{\text{）}}\!\!\!} \right|}}{{\left| {{{{\mathop{\rm TF}\nolimits} }_{\mathop{\rm P}\nolimits} }\!\!\!{\text{（}}f{\text{）}}\!\!\!} \right|}} {\text{，}}$
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${f_0} {\text{＝}} \frac{{{v_{\rm S}}}}{{4h}} {\text{，}}$
(2) 式中，h为盖土层厚度，vS为等效剪切波波速，f0为土层卓越频率。
当采用被动源台阵获取场地土层面波频散时，由单台三分量记录可额外获取场地的HVSR。基于此，本文提出一种简单、快速的基岩埋深获取方法。首先，根据HVSR方法与波速数值法获得的土层卓越频率相同，结合频散曲线可得出该卓越频率的相速度；然后，根据半波长理论，将该相速度等同于覆盖土层深度内介质的相速度，依此计算出覆盖土层厚度
$h {\text{＝}} \frac{{{v_{\rm S}}}}{{4{f_0}}} {\text{，}}$
(3) 式中，h是由观测记录所得的覆盖土层厚度，vS是由等效相速度折算而来的等效剪切波速。
由vR-f频散曲线可知，频率越高，波长越小，勘探深度越小。因此对于场地土层所需的频带范围，频率上限可取观测所得的最大值，频率下限则根据半波长理论确定为
${h_{\max }} {\text{＝}} \frac{{{\lambda _{\max }}}}{2} {\text{＝}}\frac{v_{\text{R}}\!\!{{{\!{\text{（}}\!f}_{\min } \!{\text{）}}}}}{{2{f_{\min }}}} {\text{，}}$
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1.2 场地土层剪切波速剖面的反演
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崔建文等（1994）![]()
图 1This page contains the following errors:
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在均匀弹性半空间中，剪切波速与瑞雷波相速度之间有如下近似关系（Harker，1988）：
${v_{\rm R}} {\text{≈}} \frac{{0.87 {\text{＋}} 1.12\mu }}{{1 {\text{＋}} \mu }}{v_{\rm S}} {\text{，}}$
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具体反演过程归纳如下：
1） 由实测瑞雷波频散曲线计算波长和等效剪切波速为
${\lambda _j} {\text{＝}} \frac{{{v_{{\mathop{\rm R}\nolimits} j}}}}{{{f_j}}} {\text{，}}$
(6) ${\overline v_{{\mathop{\rm S}\nolimits} j}} {\text{＝}} \frac{{1 {\text{＋}} \mu }}{{0.87 {\text{＋}} 1.12\mu }}{v_{{\mathop{\rm R}\nolimits} j}} {\text{，}}$
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${v_{{\mathop{\rm S}\nolimits} j}} {\text{＝}} \frac{{{{\overline v}_{{\mathop{\rm S}\nolimits} j}} {\text{－}} {P_{j {\text{－}} 1}}{{\overline v}_{j{\rm{ {\text{－}} 1}}}}}}{{{\rm{1 {\text{－}} }}{P_{j {\text{－}} 1}}}} {\text{，}}j{\text{＝}}1 {\text{，}}2 {\text{，}}3 {\text{，}}\cdots {\text{，}}n{\text{.}}$
(8) 由图1，式中权系数Pj按
${P_j} {\text{＝}} \dfrac{{\displaystyle \int_0^{{{{H_j}}}/{{{\lambda _{j + 1}}}}} {R\!\!\!\!{\text{（}}\!{\textit{z}}\!{\text{）}}\!\!\!\!} \rm d\xi }}{{\displaystyle \int_0^1 {R\!\!\!\!{\text{（}}\!{\textit{z}}\!{\text{）}}\!\!\!\!} \rm d\xi }} {\text{}}$
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${H_j} {\text{＝}} \left\{ {\begin{array}{*{20}{l}} {{h_1}{\text{，}}}&{j {\text{＝}} 1 {\text{，}}}\\ {{h_1} {\text{＋}} {h_2}{\text{，}}}&{j {\text{＝}} 2 {\text{，}}}\\ {\; \;\;\;\; \vdots}&{\;\;\;\vdots}\\ {{h_1} {\text{＋}} {h_2} {\text{＋}} \cdots {\text{＋}} {h_n}{\text{，}}}&{j {\text{＝}} n {\text{，}}} \end{array}} \right. $
(10) ${\overline v_j} {\text{＝}} \left\{ {\begin{array}{*{20}{l}} {{v_{\rm S1}}{\text{，}} }&{j {\text{＝}} 1{\text{，}}}\\ {\dfrac{{{h_1}{v_{\rm S1}} {\text{＋}} {h_2}{v_{\rm S2}}}}{{{h_1} {\text{＋}} {h_2}}}{\text{，}} }&{j {\text{＝}} 2{\text{，}}}\\ {\;\;\;\;\;\;\;\;\;\;\;\vdots}&{\;\;\;\vdots}\\ {\dfrac{{{h_1}{v_{\rm S1}} {\text{＋}} {h_2}{v_{\rm S2}} {\text{＋}} \cdots {\text{＋}} {h_{n {\text{－}} 1}}{v_{{\rm S}{\!\!\!\!{\text{（}}\!n {\text{－}} {\rm 1\!{\text{）}}\!\!\!\!}}}}}}{{{h_1} {\text{＋}} {h_2} {\text{＋}} ... {\text{＋}} {h_{n {\text{－}} 1}}}}{\text{，}} }&{j {\text{＝}} n {\text{－}} 1{\text{.}}} \end{array}} \right. $
(11) 2. 三类典型土层模型的数值试验及误差分析
2.1 三种典型的土层模型
为了验证改进半波长法的有效性，本文根据大量实际钻孔模型总结了三类典型土层模型对改进半波长法的计算程序进行验证。第一类为递增层模型，第二类为低速夹层模型，第三类为高速夹层模型，如表1所示。工程上一般将剪切波速vS＞500 m/s的层位定义为软基岩，本文将表1中理论模型的底层剪切波速取为600 m/s。
表 1 三类模型的具体属性参数Table 1. Specific attribute parameters of the three types of models模型 层数 层厚/m vP/（m·s−1） vS/（m·s−1） 密度/（g·cm−3） ${Q}_{\rm P}$ 
${Q}_{\rm S}$ 
泊松比μ 递增层 1 5 600 300 1.80 48 24 2 10 700 350 1.90 56 28 3 10 800 400 2.00 64 32 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 低速夹层 1 5 600 300 1.80 48 24 2 10 900 450 1.90 72 36 3 10 700 350 2.00 56 28 0.33 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 高速夹层 1 5 600 300 1.80 48 24 2 10 700 350 1.90 56 28 3 10 1100 550 2.00 88 44 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 This page contains the following errors:
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2.2 模型分层反演
基于频散曲线求取深度，需计算相速度对波长的偏导数，利用偏导数曲线判断介质的层界面，进而确定层数，并得到该界面深度对应的波长值（张恒山，王庆海，1998）。工程勘察中大多采用波长深度转换系数法，该方法认为，频散曲线的拐点对应于介质的层界面，从而可根据频散曲线的拐点反演出地层的层数，为土层分层提供参考（吴燕清，杨天春，2008）。半波长法就是波长深度转换系数取为0.5的一种半定量的波长解释法。
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图 2 三类模型的理论频散曲线（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型Figure 2. Theoretical dispersion curves of three types of models（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between![]()
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（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型Figure 3.This page contains the following errors:
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（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between2.3 模型波速反演及其误差分析
对于表1中三个典型土层模型，由式（1）计算得到三类模型的理论水平与竖向谱比HVSR曲线，如图4所示。根据HVSR曲线得出三类模型的卓越频率，结合三类模型的频散曲线，从而确定出三类模型的覆盖土层厚度，结果如表2所示。
表 2 三类模型的土层厚度Table 2. Soil thickness of three types of models模型名称 卓越频率/Hz 土层厚度/m 递增层模型 3.6 32 低速夹层模型 3.1 38 高速夹层模型 5.6 21 ![]()
图 4 三类模型的理论水平与竖向谱比HVSR曲线（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型Figure 4. Theoretical HVSR curves for three types of models（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between基于以上研究结果，根据递增层模型计算所得的32 m覆盖土层厚度，对频散曲线进行计算，频率上限取该场地频散曲线的最大值20 Hz，然后根据覆盖土层厚度，利用半波长法确定所需频散曲线的下限值为6 Hz，即当频散曲线选取在6—20 Hz时反演深度可达33 m。对其它两模型进行相同处理可知，低速夹层模型频散曲线选取在5.5—20 Hz时反演深度达到37.4 m，高速夹层模型频散曲线选取在9.5—20 Hz时反演深度达到20.6 m。对于高速夹层模型，由于深度在15—25 m之间有一层软基岩（vS＝550 m/s），运用卓越频率确定土层厚度时，则将此夹层作为该场地的基岩层。为了对本文方法反演剪切波速剖面进行合理的误差分析，在对该频散曲线进行计算时，覆盖土层厚度取为模型的厚度（35 m），则频率取值为6.2—20 Hz。对于三类模型分别运用传统半波长法以及改进半波长法对相应的频散曲线段进行反演，得到3个剪切波速度剖面，图5左显示出了三种理论模型及两种方法的计算结果。图5右是三种模型下两种方法计算结果相对于实际模型的百分比误差e随深度的变化，其中e计算如下：
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图 5 与反演厚度相关的三种模型的速度剖面（左）及传统、改进半波长法的相对百分比误差（右）（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型Figure 5. The velocity profiles of the three models related to inversion thickness （left） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （right）（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between${{e}} {\text{＝}} \frac{{{v_{\rm S}} {\text{－}} v_{\rm S}^*}}{{{v_{\rm S}}}} {\text{×}} {{100 {\text{%}}}}{\text{，}}$
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为了定量地分析反演模型与实际模型的剪切波速度误差，本文采用控制变量法，将三类模型反演所得的剪切波速按照实际模型的层厚分别计算出相应的层内平均剪切波速，得到3个剪切波速剖面，图6左为实际的模型和两种方法对应层厚的计算结果。图6右是两种方法计算结果相对于实际模型的百分比误差e随深度的变化。
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图 6 与实际厚度相关的三种模型的速度剖面（左）及传统、改进半波长法的相对百分比误差（右）（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型Figure 6. The velocity profiles of the three models related to actual thickness （left） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （right）（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in betweenThis page contains the following errors:
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${\overline v_{\rm{S}}} {\text{＝}} \frac{{\displaystyle \sum {{v_{\rm S {\textit{i}}}}{h_i}} }}{{\displaystyle \sum {{h_i}} }} {\text{，}}$
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${\overline v_{\rm{S}}^{*}} {\text{＝}} \frac{{\displaystyle \sum {{v_{\rm S{\textit{i}}}^*}h_{i}^{*}} }}{{\displaystyle \sum {h_{i}^{*}} }} {\text{，}}$
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(15) 反演平均波速与模型平均波速的相对误差列于表3。
表 3 理论模型与反演模型的平均波速及相对误差对比Table 3. Comparison of average wave velocity and relative error between theoretical model and inversion model反演方法 递增层模型 低速夹层模型 高速夹层模型 平均波速/（m·s−1） 相对误差 平均波速/（m·s−1） 相对误差 平均波速/（m·s−1） 相对误差 理论模型 400.0 414.3 442.9 传统半波长 429.4 −7.4% 437.1 −5.5% 464.6 −4.9% 改进半波长 421.3 −5.3% 427.9 −3.3% 472.4 −6.7% 对比以上计算结果，可得：
1） 两种反演方法所得平均速度比理论模型偏大，传统半波长法所得速度反演误差不超过7.4%，改进半波长法的速度反演误差不超过6.7%。
2） 两种反演方法的分层是人为给定的，是一种随深度渐变的结果，不代表实际分层，而利用拐点法分层也只能确定基岩界面和部分层面。
3） 传统半波长法在非递增层模型反演中获得的软基岩波速比改进半波长法更接近理论模型，但综合平均波速相对误差，计算结果相对于实际模型的误差，改进半波长法更具优势。
4） 在三个模型中：递增层模型的计算效果最好，两种方法的误差不超过21.7%；高速夹层模型的计算效果最差，两种反演方法的最大误差为35%。
3. 基于实际观测数据的反演算例
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图 7 日本钏路场地剪切波速度剖面、频散曲线与水平与竖向谱比（引自Arai，Tokimatsu，2005）（a） 由钻孔法所得剪切波速度剖面；（b） 由观测数据采用f−k法所得频散曲线；（c） 由三分量传感器观测所得HVSR曲线Figure 7. Shear wave velocity profile，dispersion curve and horizontal-to-vertical spectral ratio HVSR of Kushiro site in Japan （after Arai，Tokimatsu，2005）（a） Shear wave velocity profile obtained from drilling method；（b） The dispersion curve obtained by the f−k method from the observed data；（c） Observed HVSR curve from three-component sensor此外，A场地的钻孔延伸至工程基岩，其剪切波速为600—800 m/s，由钻孔法所得A场地剪切波速度剖面如图7a所示，可知A场地土层厚度为17 m。由此认为Arai和Tokimatsu（2005）对该场地观测台阵采用高分辨率f−k方法获得的频散曲线反演17 m深度内的土层速度结构是合理的。本文根据此频散曲线进行了简单计算，频率下限取该场地频散曲线最小值7.5 Hz，频率上限取为20 Hz，即当频散曲线选取在7.5—20 Hz时反演深度达到19 m，反演获得了基岩以上的土层场地初始速度结构，如图8a所示。从图中可以得出两种反演方法在平均波速方面，其反演误差不超过7.4%，与模型试验所得结果一致。实测模型反演中，在对应反演厚度所得的剪切波速剖面（图8a），改进半波长法所得剪切波速相较于钻孔模型，其相对误差不超过32%，而传统半波长法则误差范围很广，最大可达60%。在对应实测模型厚度所得的剪切波速剖面（图9a），改进半波长法的反演效果很好，与钻孔模型相比，其相对误差不超过10%，传统半波长法则不超过26%。
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图 8 与反演厚度相关的实测模型的速度剖面（a）及传统、改进半波长法的相对百分比误差（b）Figure 8. The velocity profiles of the measured models related to inversion thickness （a） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （b）![]()
图 9 与实际厚度相关的实测模型的速度剖面（a）及传统、改进半波长法的相对百分比误差（b）Figure 9. The velocity profiles of the measured models related to actual thickness （a） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （b）4. 结论
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中国地震局地壳应力研究所研究生王璞在软件编写中提供了帮助，审稿专家对本文提出了宝贵建议，本文理论频散曲线的正演计算采用了Geopsy软件，作者在此一并表示感谢。
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图 1
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Figure 1.
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图 2 三类模型的理论频散曲线
（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型
Figure 2. Theoretical dispersion curves of three types of models
（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between
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图 3
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（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型
Figure 3.
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$\partial {v_{\rm{R}}} / \partial {\lambda _{\rm{R}}}$（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between
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图 4 三类模型的理论水平与竖向谱比HVSR曲线
（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型
Figure 4. Theoretical HVSR curves for three types of models
（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between
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图 5 与反演厚度相关的三种模型的速度剖面（左）及传统、改进半波长法的相对百分比误差（右）
（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型
Figure 5. The velocity profiles of the three models related to inversion thickness （left） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （right）
（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between
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图 6 与实际厚度相关的三种模型的速度剖面（左）及传统、改进半波长法的相对百分比误差（右）
（a） 递增层模型；（b） 低速夹层模型；（c） 高速夹层模型
Figure 6. The velocity profiles of the three models related to actual thickness （left） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （right）
（a） The stratified media with S-wave velocity increasing from the top to the bottom；（b） The stratified media with the low-velocity layer in between；（c） The stratified media with the high-velocity layer in between
![]()
图 7 日本钏路场地剪切波速度剖面、频散曲线与水平与竖向谱比（引自Arai，Tokimatsu，2005）
（a） 由钻孔法所得剪切波速度剖面；（b） 由观测数据采用f−k法所得频散曲线；（c） 由三分量传感器观测所得HVSR曲线
Figure 7. Shear wave velocity profile，dispersion curve and horizontal-to-vertical spectral ratio HVSR of Kushiro site in Japan （after Arai，Tokimatsu，2005）
（a） Shear wave velocity profile obtained from drilling method；（b） The dispersion curve obtained by the f−k method from the observed data；（c） Observed HVSR curve from three-component sensor
![]()
图 8 与反演厚度相关的实测模型的速度剖面（a）及传统、改进半波长法的相对百分比误差（b）
Figure 8. The velocity profiles of the measured models related to inversion thickness （a） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （b）
![]()
图 9 与实际厚度相关的实测模型的速度剖面（a）及传统、改进半波长法的相对百分比误差（b）
Figure 9. The velocity profiles of the measured models related to actual thickness （a） and the relative percentage error variation by the traditional and improved half-wavelength methods with depth （b）
表 1 三类模型的具体属性参数
Table 1 Specific attribute parameters of the three types of models
模型 层数 层厚/m vP/（m·s−1） vS/（m·s−1） 密度/（g·cm−3） ${Q}_{\rm P}$ ${Q}_{\rm S}$ 泊松比μ 递增层 1 5 600 300 1.80 48 24 2 10 700 350 1.90 56 28 3 10 800 400 2.00 64 32 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 低速夹层 1 5 600 300 1.80 48 24 2 10 900 450 1.90 72 36 3 10 700 350 2.00 56 28 0.33 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 高速夹层 1 5 600 300 1.80 48 24 2 10 700 350 1.90 56 28 3 10 1100 550 2.00 88 44 4 10 1000 500 2.10 80 40 无限半空间 ∞ 1200 600 2.20 96 48 
下载: 导出CSV
表 2 三类模型的土层厚度
Table 2 Soil thickness of three types of models
模型名称 卓越频率/Hz 土层厚度/m 递增层模型 3.6 32 低速夹层模型 3.1 38 高速夹层模型 5.6 21
下载: 导出CSV
表 3 理论模型与反演模型的平均波速及相对误差对比
Table 3 Comparison of average wave velocity and relative error between theoretical model and inversion model
反演方法 递增层模型 低速夹层模型 高速夹层模型 平均波速/（m·s−1） 相对误差 平均波速/（m·s−1） 相对误差 平均波速/（m·s−1） 相对误差 理论模型 400.0 414.3 442.9 传统半波长 429.4 −7.4% 437.1 −5.5% 464.6 −4.9% 改进半波长 421.3 −5.3% 427.9 −3.3% 472.4 −6.7%
下载: 导出CSV
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