## 2 Three-Dimensional Cross-Hole Electrical Resistivity Tomography
The capacity to acquire extensive potential gradient information is a crucial feature of the 3D cross-hole ERT survey. It furnishes a rich data basis for high-resolution and detailed detection, contributing to mitigating inversion's multi-solution aspects. Its imaging quality and spatial resolution surpass those of typical ground resistivity detection methods, while its spatial accuracy exceeds that of conventional 2D surveys. Figure 1: Typical cross-hole ERT arrays: (**a**) dipole–dipole array (the current and potential bipoles are in different boreholes); (**b**) bipole–bipole array (electrodes of each bipole in different boreholes); and (**c**) pole–tripole array (the first current electrode moves along one borehole and the second current electrode with the potential bipole moves along the other borehole). The 3D Cartesian coordinate system is established for the drilling site coordinates. The adaptive grid is divided by the finite element numerical analysis method. The initial resistivity value \(\rho\) of the grid element is given, and the potential field data is calculated according to Formula (1), and the result is taken as the forward theoretical value. \[\frac{\partial}{\partial x}\bigg{(}\frac{1}{\rho}\frac{\partial U}{\partial x} \bigg{)}+\frac{\partial}{\partial y}\bigg{(}\frac{1}{\rho}\frac{\partial U}{ \partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}\frac{1}{\rho}\frac{ \partial U}{\partial z}\bigg{)}=-I\delta(x-x_{0})\delta\big{(}y-y_{0}\big{)} \delta(z-z_{0}), \tag{1}\]  
In the formula, \(\rho\) is the medium resistivity (\(\Omega\)-m); U is the potential at any point in space (V); I is the excitation current (A); \(\delta\) is the Dirac function; (x, y, z) is the space coordinate of the observation point; and (x\({}_{0}\), y\({}_{0}\), z\({}_{0}\)) is the space coordinate of the excitation source. To reconstruct the inversion image, the measured potential dots are fitted with the forward theoretical value g(m), and the objective function is  
\[S(m)=(d_{obs}-g(m))^{T}W_{d}(d_{obs}-g(m)), \tag{2}\]  
where m is the resistivity parameter matrix, and Wd is the weight coefficient matrix. The objective function has no model constraints, which may lead to inversion failure. Due to the large amount of potential field data, which is not conducive to the convergence of inversion data, the model parameter m needs to be modified many times during inversion, and the modification relationship is  
\[(J^{T}W_{d}J+\lambda I)\Delta m=J^{T}W_{d}(d_{obs}-g(m)), \tag{3}\]  
where J is a Jacobi matrix, and \(\lambda\) is the damping factor. When S(m) meets the given constraints, resistivity inversion is finished. Extract the resistivity value of each grid unit as the real value between the two boreholes to achieve resistivity imaging [23; 24]. Compared with the above three types of devices, the AM-BN quadrupole array device has more prominent advantages for 3D cross-hole ERT detection: From the perspective of electrode array distribution, the power supply electrodes A and B of the AM-BN quadrupole array are located in different boreholes. This distribution facilitates current penetration through the strata between the boreholes, thereby reducing the impact of the current channeling effect and better meeting the \"perspective cross-borehole\" observation requirements of cross-hole. Since the measurement electrodes M and N are closer to the power supply electrodes in the same borehole, stronger observation signals can be obtained. In metro shield zones, the resistivity of boulders is relatively high, making them high-resistivity  
Figure 2: Schematic of 3D cross-hole ERT for boulders. anomalies compared with the surrounding rock and soil media. The magnitude difference is significant (the resistivity of the surrounding rock is generally less than 500 \(\Omega\cdot\)m, while the resistivity of boulders is generally above 2000 \(\Omega\cdot\)m), allowing geophysical instruments to observe changes in the geoelectric field caused by this difference.