## 2 Three-Dimensional Cross-Hole Electrical Resistivity Tomography  
The ERT method involves injecting a steady current into the ground through two electrodes (one positive, one negative) placed on the surface or underground. This creates a stable current field. Due to the electrical differences between rock formations and geological anomalies, the observed current field distribution will vary. By recording the relevant parameters with a DC resistivity instrument, the apparent resistivity can be calculated. Changes in resistivity can then be used to infer geological anomalies underground. The cross-hole ERT detection method is a refined borehole detection technique that involves placing detection electrodes in boreholes to collect signals. By utilizing the resistivity differences between adverse geological bodies and the surrounding medium, it achieves the localization and identification of geological anomalies. In cross-hole ERT measurements, there are three observation configurations: dipole-dipole (AB-MN), bipole-bipole (AM-BN), and pole-tripole (A-BMN) [21]. In a dipole-dipole (AB-MN) configuration, the current and potential bipoles are placed in different boreholes (Figure 1a). In a bipole-bipole (AM-BN) configuration, the electrodes of each bipole are placed in different boreholes (Figure 1b). The electrode positions in one borehole are fixed with a constant electrode spacing. After measurements are taken, the electrode is moved to the other borehole for subsequent measurements. This process is repeated by selecting another fixed position with the same electrode spacing until all positions in the borehole are covered. Finally, the positions of the two drilling electrodes are interchanged, and the same process is repeated for the second measurement. The third array is known as the pole-tripole (A-BMN) configuration (Figure 1c), where the first current electrode moves along one borehole, and the second current electrode, along with the potential bipole, moves along the other borehole. In previous simulation studies on boulder detection, it was found that due to differences in the amount of data obtained from observations and the varying spatial distribution of electric fields, the spatial resolution of different device types can vary significantly. The A-M dipole array device can obtain strong observation signals but has relatively low spatial resolution. The spatial resolution of the A-MN tripole array device is greatly influenced by the distance between measurement electrodes (i.e., the MN spacing), making it difficult to determine scientific and reasonable parameters in advance for unknown strata, and it can only be controlled based on experience. The AB-MN quadrupole array device is easily affected by the current channeling effect, especially when there is a large range of low-resistivity layers near the electrodes, which influences the spatial distribution of the current and thus affects the detection results. In 2015, [PERSON] proposed a bipole-bipole electrode configuration that has high boulder-detection resolution and sensitivity [22]. The 3D cross-hole ERT method involves placing electrode arrays in three or more parallel boreholes to obtain potential gradient data through cross-borehole measurements (Figure 2). Through inversion interpretation methods, 3D resistivity imaging of the region between the boreholes is performed to achieve 3D localization and identification of adverse geological bodies hidden within the rock mass. In the engineering practice of boulder detection in metro shield zones, the typical observation model of the 3D cross-hole ERT is shown in Figure 1. It involves a \"perspective cross-borehole\" observation mode with four parallel boreholes. Well 1, Well 2, Well 3, and Well 4 are four boreholes drilled vertically into the strata. By inserting ERT cables into 4 boreholes, the detection electrodes are dispersed in three-dimensional space. Leveraging the mutual constraints among boreholes and dense sampling points improves the precision and spatial resolution of data collection.