## 4 Example Verification  
### D Isotropic Reservoir Model  
To validate the reliability and effectiveness of the 3D edge-based finite element method for borehole-to-surface electromagnetic modeling under isotropic resistivity conditions, a one-dimensional borehole-to-surface electromagnetic model was established for solutions. The results were compared with those obtained from the semi-analytical solution software DIPOLE1D (v3.13).  
As shown in Figure 3, a four-layer one-dimensional stratified reservoir model was established, consisting of an air layer with a resistivity of \(10^{8}\)\(\Omega\cdot\)m, an overburden layer with a thickness of 1000 m and a resistivity of 100 \(\Omega\cdot\)m, a high-resistivity reservoir layer with a thickness of 800 m and a resistivity of 1000 \(\Omega\cdot\)m, and a basement layer with a resistivity of 100 \(\Omega\cdot\)m. The model was simulated using both the 3D edge-based finite element method with anisotropic resistivity and the semi-analytical solution software DIPOLE1D. A vertical electric dipole transmitter was positioned at a depth of 500 m with transmitter coordinates at (0, 0, \(-\)500). The receiver array was deployed along the x-direction, spanning from \(-\)1000 m to 1000 m, with a receiver station placed every 100 m, resulting in a total of 21 stations. Given that the typical observation frequency range for borehole-to-surface electromagnetic surveys is 0.1 Hz to 1000 Hz, this study adopts a frequency of 2 Hz for simulations.  
Figure 4 shows the unstructured mesh generated using a mesh refinement strategy. The spatial dimensions of the layered model are \([-8000,8000]\times[-8000,8000]\times[-5000,\) 5000]. Figure 4 presents the meshing results generated by Gmsh, where the mesh consists of 334,489 elements, 681,379 faces, and 406,888 edges, with a total of 2,176,534 degrees of freedom.  
Figure 4: The y-z section of the mesh for the 1D reservoir model. (**a**) The schematic diagram of the mesh subdivision; (**b**) the schematic diagram of the mesh partitioning near the transmitter and receiver.  
Figure 3: One-dimensional layered model.  
To verify the performance of the proposed method, the one-dimensional controlled-transmitter electromagnetic forward algorithm proposed by [PERSON] was compared with the unstructured 3D edge-based finite element method used in this study, and the response curves were compared. As shown in Figure 5, the simulation results from the proposed algorithm agree well with the one-dimensional analytical solution. Figure 5a presents the amplitude of the electrical response \(\mid\!E_{x}\!\mid\) along the receiver measurement line, clearly indicating that the response results from the unstructured 3D edge-based finite element algorithm match very closely with those from DIPOLE1D. Additionally, the phase of the electrical response was compared, as shown in Figure 5b. Based on this, the relative errors of the amplitude (\(\mid\!E_{x}\!\mid\)) and phase (\(\Phi_{x}\)) were calculated, as illustrated in Figure 6. The relative errors of the magnitude and phase components of the electric field (Ex) are both less than 1%, proving the correctness of the algorithm in simulating isotropic conditions.