## 3 Accuracy Validation for 3-D Forward Modeling Algorithm  
To validate the accuracy of the forward modeling algorithm, we compared our 3-D forward modeling response with the analytical solution of a uniform half-space model and the layered model response calculated by an open-source 1-D forward program [20]. A 400 m \(\times\) 400 m loop transmitter was designed, and a unit step-off source current waveform was used. A measurement site was located at the center of the loop, with both source and receiver deployed on the surface. In addition, a 1-D layered model was designed for testing, as shown in Figure 1. The resistivity values from the top to bottom layers are \(10^{6}\)\(\Omega\cdot\)m, \(10\)\(\Omega\cdot\)m, \(100\)\(\Omega\cdot\)m, and the relative magnetic permeabilities are 1, 2, 1, respectively. The interfaces between the three layers are located at 0 m and 50 m. Figure 2a shows the comparison between our 3-D modeling response and the analytical solution of the uniform half-space model. Figure 2b shows a comparison between our 3-D forward modeling response and the 1-D numerical solution. It can be observed from Figure 2a that for the uniform half-space model, the relative differences between our 3-D modeling results and the 1-D analytical solution were less than 3%. In Figure 2b, when both resistivity and magnetic permeability anomalies existed, the relative differences between our 3-D response and the 1-D numerical solution were generally below 5%. Therefore, this example validated the computational accuracy of our 3-D forward algorithm presented in this work. Figure 1: Horizontal view of the 1-D layered model. Figure 2: _Cont._  
Calculation and Characteristic Analysis of 3-D Forward Modeling Response Considering Magnetic Permeability  
In this example, a 3-D model was designed to investigate the impact of magnetic permeability on the 3-D TEM response. The measurement configuration in this example was the same as that in the 1-D example described above. The uniform half-space model contained an anomaly block, located at \(x,y\in[-50,50]\) m and \(z\in[50,100]\) m. The background resistivity was 100 \(\Omega\cdot\)m, and the anomaly was located below the loop center. Figure 3 shows the modeling responses for the anomaly with (\(\rho\) = 100 \(\Omega\cdot\)m, \(\mu_{r}\) = 1), (\(\rho\) = 100 \(\Omega\cdot\)m, \(\mu_{r}\) = 1), (\(\rho\) = 100 \(\Omega\cdot\)m, \(\mu_{r}\) = 1), (\(\rho\) = 100 \(\Omega\cdot\)m, \(\mu_{r}\) = 1.5), and (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{r}\) = 1.5), respectively, along with the relative differences of these responses compared to the uniform half-space model response. As shown in Figure 3, the conductive block caused a distinct anomaly response; when the block included both resistivity and magnetic permeability anomalies, its modeling response had an obvious difference compared to the model response affected only by the  
Figure 3: Comparison of 3-D modeling responses for the anomaly with different parameters. Figure 2: Modeling accuracy validation: (**a**) Comparison with the 1-D analytical solution; (**b**) Comparison with the 1-D numerical solution. resistivity. Additionally, for the response at the middle time stage, the differences rapidly reached a maximum value (596%) and then faded out. Next, a 3-D model was designed to study the influence of a highly magnetizable anomaly (e.g., manhole cover) on the modeling response when it was present in shallow layers. The measurement configuration was also the same as in the previous 1-D example. A dodecahedron prism-shaped anomaly was placed below the center of the loop in the uniform half-space with a background resistivity of 100 \(\Omega\cdot\)m. The dodecahedron prism had an edge length of 10 m, a height of 1 m, and a burial depth of 10 m. Three cases were set with the anomaly holding different parameters: (\(\rho=10\)\(\Omega\cdot\)m, \(\mu_{r}=1\)), (\(\rho=100\)\(\Omega\cdot\)m, \(\mu_{r}=5\)), and (\(\rho=10\)\(\Omega\cdot\)m, \(\mu_{r}=5\)).