## 2 Methods
### Forward and Inversion Theory  
#### 2.2.1 Finite Element Method (FEM) Theory  
The boundary value problem that the potential of a point source in a uniform half-space [42] satisfies is presented below:  
\[ \begin{cases}\
abla\cdot(\sigma\
abla u)=-I\delta(A)&,\in\Omega\\ \partial u/\partial\mathbf{n}=0&,\in\Gamma_{s}\\ \partial u/\partial\mathbf{n}+cu=0&,\in\Gamma_{\infty}\end{cases} , \tag{1}\]  
where \(\sigma\) is the conductivity; \(u\) is the total potential; \(I\) is the point source current intensity; \(\delta(A)\) is the Dirac delta function centered at point source \(A\); \(\Omega\) is the overall region; \(\mathbf{n}\) is the unit normal vector outward from the boundary; \(\Gamma_{s}\) is the ground boundary; and \(c=\cos(\mathbf{r},\mathbf{n})/|\mathbf{r}|\), \(\mathbf{r}\) is the vector from point source \(A\) to any point on the infinite boundary \(\Gamma_{\infty}\). Figure 3: Schematic diagram of the “L”-shaped electrode array and measurement point distribution. (**a**) “L”-shaped electrode array (the green dots represent the electrodes, and the red line area indicates the anomalous body); (**b**) 3D spatial distribution and projection of measurement points. Figure 2: Schematic diagram of the “U”-shaped electrode array and measurement point distribution. (**a**) “U”-shaped electrode array (the green dots represent the electrodes, and the red line area indicates the anomalous body); (**b**) 3D spatial distribution and projection of measurement points. The boundary value problem of a point source in a steady-state current field can be transformed into the corresponding variational problem [42] as follows:  
\[\left\{ \begin{array}{l}F(u)=\int_{\Omega}\left[\frac{1}{2}\sigma(\
abla u)^{2}-I \delta(A)u\right]d\Omega+\frac{1}{2}\oint_{\Gamma_{\infty}}c\sigma u^{2}d \Gamma\\ \delta F(u)=0\end{array} \right., \tag{2}\]  
where \(F(u)\) represents the total energy function of the system. The steps for solving the variational problem of a point source steady-state current field using the FEM are as follows:  
Step 1: The study area is discretized by dividing the 3D electrical model into a finite number of hexahedral elements, each of which is further decomposed into five tetrahedral elements, as shown in Figure 4. Then, the integral of the variational problem is decomposed into the sum of the integrals over the tetrahedral elements \(e\) and the boundaries of each element \(\Gamma_{e}\). \[F(u)=\sum_{\Omega}\int_{e}\left[\frac{1}{2}\sigma(\
abla u)^{2}-I\delta(A)u \right]d\Omega+\sum_{\Gamma_{\infty}}\int_{\Gamma_{e}}\frac{1}{2}\sigma u^{2}d\Gamma, \tag{3}\]  
Step 2: Considering that the physical properties of underground media generally exhibit continuous variation with temperature and humidity, constructing a continuous electrical model is more consistent with real-world conditions. It is assumed that the vertices of the tetrahedral elements in Figure 4c are numbered as 1, 2, 3, and 4. Within the tetrahedral element, linear interpolation is applied to approximate the potential \(u\) and conductivity \(\sigma\) within the element, ensuring the continuous variation in both potential and conductivity inside the element, thus achieving continuous electrical properties:  
\[u=\sum_{i=1}^{4}N_{i}u_{i}\ \ \sigma=\sum_{i=1}^{4}N_{i}\sigma_{i}, \tag{4}\]  
where \(N_{i}\) is the shape function [43]; \(u_{i}\) is the potential value of the \(i\)-th vertex of the tetrahedral element; and \(\sigma_{i}\) is the conductivity value of the \(i\)-th vertex of the tetrahedral element.