## 2 Strain Transformation and Instrument Coupling  
The GTSM strainmeter can describe the strain state of an object under stress by calculating the data obtained from more than three transducers in different orientations [32], in which the extensile strain is set as a positive value and the compressive strain is set as a negative value in the original data. At a point in the body, let the Cartesian components of horizontal displacement be \(u\) in the x direction (west-east) and \(v\) in the y direction (south-north). In that case, the strain tensor can be expressed as three independent components, they are \(\varepsilon_{\alpha}=du/dx\), \(\varepsilon_{mn}=dv/dy\), and \(\varepsilon_{\alpha n}=\varepsilon_{ne}=1/2(du/dx+dv/dy)\), respectively. Based on the definitions above, the engineering strains \(\varepsilon_{\alpha}\), \(\varepsilon_{d}\), and \(\varepsilon_{\delta}\) describing the strain state can be further deduced. The areal strain \(\varepsilon_{\alpha}\) is equal to \(\varepsilon_{\alpha}+\varepsilon_{mn}\), which reflects the areal change; \(\varepsilon_{d}\) is named the differential shear strain or \(\gamma_{1}\) strain, equal to \(\varepsilon_{\alpha}-\varepsilon_{m}\), which are the west-east tensile behavior and south-north compressive behavior, respectively; \(\varepsilon_{\delta}\) is defined as the simple shear strain or \(\gamma_{2}\), which is equal to \(2\varepsilon_{\alpha n}\). The deformation behavior of a plane can be described using these three engineering strains [39; 40]. Although the borehole strainmeter is stacked with multiple transducers in different orientations, compared with the scale of the installation depth, we still regard it as a plane strain. Thus, instrument strains recorded in different orientations can be expressed as engineering strains by the following relation:  
\[e_{i}=1/2(\varepsilon_{a}+cos2\varphi_{i}\varepsilon_{d}+sin2\varphi_{i} \varepsilon_{s}) \tag{1}\]  
\(e_{i}\) is the instrument strain in the axial orientation of \(\varphi_{i}\), which shows the angle counter-clockwise from the x axis. However, due to such a small-scale geological inhomogeneity or installation situation, a cross-coupling terms is introduced into the expressions relating remote strains to instrument strains. For isotropic coupling, instrument and remote strain can be described as the following relation [41]:  
\[g_{i}e_{i}=C\varepsilon_{a}+Dcos2\varphi_{i}\varepsilon_{d}+Dsin2\varphi_{i} \varepsilon_{s} \tag{2}\]  
Although we generally do not consider the influences of vertical coupling, some cases were reported in some situations [40]. If vertical coupling is taken into account, the relation between the instrument and remote strain can be expressed as the following:  
\[g_{i}e_{i}=C\varepsilon_{a}+Dcos2\varphi_{i}\varepsilon_{d}+Hsin2\varphi_{i} \varepsilon_{s}+V\varepsilon_{zz} \tag{3}\]  
where \(g_{i}\) is the mechanical gain, \(\varphi_{i}\) is for the angle counterclockwise from the x axis; \(C\), \(D\), and \(H\) represents axial, perpendicular, and shearing strain deformational parameters in the \(i\)th channel, respectively; \(V\varepsilon_{zz}\) shows the effect of vertical strain where V is the vertical coupling parameter; and \(\varepsilon_{xz}\) represents the vertical strain. The responded strain by vertical stress such as rainfall and barometric has the relation with the horizontal areal strain as the following equation described:  
\[\varepsilon_{zz}=\frac{-\
u}{1-\
u}\varepsilon_{a} \tag{4}\]  
In the equation above, \(\
u\) is denoted as the Poisson ratio of the rock formation.