﻿ 分量式钻孔应变观测的最佳方案
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 大地测量与地球动力学  2022, Vol. 42 Issue (6): 650-654  DOI: 10.14075/j.jgg.2022.06.018

### 引用本文

ZHANG Yongqing, ZHAO Shuxian, QIU Zehua, et al. Optimal Solution for Component-Type Borehole Strain Observation[J]. Journal of Geodesy and Geodynamics, 2022, 42(6): 650-654.

### Foundation support

Special Fund for Basic Scientific Research of Central Public Research Institutes, No. ZDJ2018-17, ZDJ2017-10, ZDJ2012-11; National Natural Science Foundation of China, No. 41374051, 41974018.

### Corresponding author

ZHAO Shuxian, associate researcher, majors in borehole strain gauge and combined seismometer, E-mail: Shxzhaodqs@163.com.

### 文章历史

1. 应急管理部国家自然灾害防治研究院, 北京市安宁庄路1号, 100085

1 钻孔应变观测的一般化公式

 $\theta_{i}=\theta_{1}+\frac{\pi}{n}(i-1), i=1,2,3, \cdots, n$ (1)

i个测量单元的观测值为Si(i=1, 2, 3, …, n)，钻孔应变仪钢筒内壁的水平应变状态用N向正应变εN、E向正应变εE和N向剪应变γNE表示。根据平面应变转换方程[8]，第i个应变分量的理论值为：

 $\begin{gathered} \varepsilon_{i}=\frac{\varepsilon_{\mathrm{N}}+\varepsilon_{\mathrm{E}}}{2}+\frac{\varepsilon_{\mathrm{N}}-\varepsilon_{\mathrm{E}}}{2} \cos 2 \theta_{i}+\frac{\gamma_{\mathrm{NE}}}{2} \sin 2 \theta_{i}, \\ i=1,2,3, \cdots, n \end{gathered}$ (2)

εa=εN+εE(即水平面应变)，λNE=εNεE (即N向差应变)，则式(2)可简化为：

 $\begin{gathered} \varepsilon_{i}=\frac{\varepsilon_{a}}{2}+\frac{\lambda_{\mathrm{NE}}}{2} \cos 2 \theta_{i}+\frac{\gamma_{\mathrm{NE}}}{2} \sin 2 \theta_{i}, \\ i=1,2,3, \cdots, n \end{gathered}$ (3)

 $\begin{gathered} Q=\sum\limits_{i=1}^{n}\left(\varepsilon_{i}-S_{i}\right)^{2}= \\ \sum\limits_{i=1}^{n}\left(\frac{\varepsilon_{a}}{2}+\frac{\lambda_{\mathrm{NE}}}{2} \cos 2 \theta_{i}+\frac{\gamma_{\mathrm{NE}}}{2} \sin 2 \theta_{i}-S_{i}\right)^{2} \end{gathered}$ (4)

 $\begin{gathered} \frac{\partial Q}{\partial \varepsilon_{a}}=\frac{n}{2} \varepsilon_{a}+\frac{\lambda_{\mathrm{NE}}}{2} \sum\limits_{i=1}^{n} \cos 2 \theta_{i}+ \\ \frac{\gamma_{\mathrm{NE}}}{2} \sum\limits_{i=1}^{n} \sin 2 \theta_{i}-\sum\limits_{i=1}^{n} S_{i}=0 \end{gathered}$ (5)

 $\begin{gathered} S_{1}^{2}+S_{3}^{2}+S_{5}^{2}-S_{1} S_{3}-S_{1} S_{5}-S_{3} S_{5}= \\ S_{2}^{2}+S_{4}^{2}+S_{6}^{2}-S_{2} S_{4}-S_{2} S_{6}-S_{4} S_{6} \end{gathered}$ (16)

 青色竖线内区域为勒夫波显著的区域，红色竖线内区域为瑞利波显著的区域 图 2 S1+S3与S2+S4观测曲线、残差和相关系数曲线 Fig. 2 S1+S3 versus S2+S4 observation curves, residuals and correlation coefficient curves

 图 3 瑞利波显著区域放大图 Fig. 3 Enlarged view of the significant region of Rayleigh waves
3 结语

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Optimal Solution for Component-Type Borehole Strain Observation
ZHANG Yongqing1     ZHAO Shuxian1     QIU Zehua1     TANG Lei1
1. National Institute of Natural Hazards, MEM, 1 Anningzhuang, Beijing 100085, China
Abstract: In view of the incompleteness of the existing component-type borehole strainmeters and the future development of borehole strainmeters, we theoretically derive generalized formulas for surface strain, north-directional differential strain, and north-directional shear strain for the n≥3-component horizontally equally spaced borehole strain observation scheme. We qualitatively evaluate their accuracy self-verification capabilities. If only the accuracy self-verification of areal strain observation is considered, the existing four-component borehole strainmeter in China is the best solution; if the comprehensive self-verification of observation accuracy is considered, i.e., the accuracy self-verification of both areal strain and shear strain, then the six-component is the best solution. For the six-component scheme, we give formulas for accuracy self-verification of areal strain and shear strain.
Key words: borehole strainmeter; areal strain; shear strain; self-verification