## 2 Methodology
### Arbitrary Dipole-Dipole DC Measurements  
With the rapid development of GPS/GNSS (Global Navigation Satellite System) navigation and positioning technology [24] and the increasing maturity of China's BeiDou Satellite Navigation System [25], the positioning accuracy of handheld devices is continuously improving [26; 27]. This provides a new approach for setting up electrode grids in electrical exploration [28; 29]. When conducting regional geophysical surveys, a flexible three-dimensional electrode arrangement can be adapted to the terrain and environmental conditions of the work area without the need to strictly follow a predetermined grid order. In a dipole-dipole configuration, A and B are the current electrodes and M and N are the potential electrodes (Figure 1). The control equation for the direct current resistivity method with dual-electrode power supply can be expressed as:  
\[\
abla\cdot\left(\sigma\
abla\varphi\right)=-\mathrm{I}\left[\delta\left( \mathrm{r}-\mathrm{r}_{\mathrm{s}}^{+}\right)-\delta\left(\mathrm{r}-\mathrm{ r}_{\mathrm{s}}^{-}\right)\right] \tag{1}\]  
where \(\varphi\) represents the electrical potential, the supply current is I, the positive electrode position is \(\mathrm{r}_{s}^{+}\), the negative electrode position is \(\mathrm{r}_{s}^{-}\), and \(\sigma\) represents conductivity. The forward problem is to solve the distribution of electric potential in a known region, given the conductivity and source distribution. Using finite element discretization to solve the region, this equation can be written in matrix form:\[F(\sigma)u=q \tag{2}\]  
\(\mathbf{u}\) is a vector representing the potential distribution at the mesh nodes, F represents the forward Laplace operator, which is actually a symmetric and sparse matrix related to the mesh discretization and conductivity distribution, and \(\mathbf{q}\) is a vector describing the source distribution. We used the incomplete Gauss-Newton method [30] to realize inversion resistivity of the arbitrary dipole-dipole configuration. In any dipole-dipole configuration, selecting multiple sets of appropriate power supply current electrodes and corresponding receiving potential electrodes can greatly improve field observation efficiency, reduce observation costs, and enhance the resolution of the observation data. In the DC incomplete Gauss-Newton inversion, the observed data d is generally a subset of the potential u at the mesh nodes; observations are made only at a limited number of positions. This can be expressed as:  
\[\mathbf{d}=QF^{-1}(\mathbf{u}) \tag{3}\]  
In this context, Q is the mapping matrix from \(\mathbf{u}\) to \(\mathbf{d}\). The goal of the inversion is to obtain the subsurface conductivity \(\sigma\) (or resistivity) distribution from the potentials \(\mathbf{d}\) observed at these limited positions. In order to verify the advantages of using arbitrary dipole-dipole configuration in landslide detection, we conducted numerical model tests. The model for the numerical study has dimensions of 107 m (length) by 40 m (depth). The profile is divided into two layers. The first layer is an irregular strip-shaped soil layer 107 m long and 11 m thick, and its resistivity varies with relatively low resistivity values. The second layer is a trapezoidal bedrock layer 98 m long and 29 m thick, with a resistivity of 1000 \(\Omega\cdot\)m. A total of 24 electrodes are arranged along the length of the profile (Figure 2). Figure 1: Dipole–Dipole DC configuration. Figure 2: Schematic diagram of the numerical model. We used arbitrary dipole-dipole, Wenner, and Schlumberger arrays to model the sensitivity of the landslide body at different rainfall moments. The resistivity values for the overlying soil layers are set to 100 \(\Omega\) m, 50 \(\Omega\cdot\)m, and 10 \(\Omega\cdot\)m, respectively. The apparent resistivity profiles collected by the three arrays were subjected to inversion processing. The inversion results of the three arrays were then compared and analyzed (Figure 3). The root mean square (RMS = 1.2).