## 6 Synthetic Model Inversion Results and Analysis
### 3-D TEM Inversion Considering the Impact of Magnetic Permeability  
To analyze the effect of the magnetic permeability parameter on the 3-D TEM inversion, synthetic data was generated using a model that included both resistivity and magnetic permeability anomalies. This study tested the 3-D TEM inversions with and without considering magnetic permeability impact. The inversion results for the resistivity and magnetic permeability parameters are presented, as shown in Figure 9.  
Figure 8: Data inversion results: (**a**) Inversion result for anomaly with \(\mu_{\tau}\) = 2; (**b**) Inversion result for anomaly with \(\mu_{\tau}\) = 5. Figure 9: _Cont_. Figure 9: _Cont._  
Figure 9a,b display the resistivity inversion results for the anomalies with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 2) and (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 5), considering the impact of magnetic permeability. Figure 9c,d show the inversion results for the relative magnetic permeability. As it can be seen, the resistivity inversion results matched the true model (Figure 5) quite well, clearly reflecting the size, location, and resistivity of the subsurface target body. However, when the magnetic  
Figure 9: Data inversion results: (**a**) Resistivity inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 2), considering the impact of magnetic permeability; (**b**) Resistivity inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 5), considering the impact of magnetic permeability. (**c**) Relative magnetic permeability inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 2); (**d**) Relative magnetic permeability inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 5). (**e**) Resistivity inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 2), ignoring the impact of magnetic permeability; (**f**) Resistivity inversion result for anomaly with (\(\rho\) = 10 \(\Omega\cdot\)m, \(\mu_{\rm r}\) = 5), ignoring the impact of magnetic permeability. permeability value of the anomaly was large, a few spurious anomalies appeared in the inversion results. When the anomaly's relative magnetic permeability was set to 2, the inversion of the relative magnetic permeability parameter was not able to properly recover the underground target. In contrast, when the anomaly's relative magnetic permeability increased to 5, the position and size of the underground target were well-recovered, but the relative magnetic permeability value was only recovered to approximately 2, and a few spurious anomalies appeared near the anomaly. Comparing Figure 9a,b with Figure 9e,f, it can be observed that if the algorithm ignored the magnetic permeability and tried to invert the data that was influenced by magnetic permeability, although the general distribution of the underground target can be roughly obtained, large spurious anomalies would appear in the inversion results. These false anomalies were mostly concentrated around the anomaly. This example clearly demonstrates that when there is a significant magnetic permeability contrast in the underground target, ignoring magnetic permeability in the inversion algorithm (by assuming vacuum magnetic permeability) will result in poor inversion results, with large spurious anomalies near the target. This will complicate the subsequent interpretation for the inversion result. Figure 9c,d shows the inversion results for the magnetic permeability model when both resistivity and magnetic permeability were considered as inversion parameters. Because both physical parameters were inverted simultaneously, the solution space was larger compared to the inversion case only considering the resistivity variation. Moreover, since TEM has a higher sensitivity to resistivity compared to magnetic permeability, the inversion did not recover the magnetic permeability model well.