## 7 Conclusions  
We employed the edge-based finite element method using unstructured grids, combined with the second-order backward Euler method for spatial and time discretization, to develop a 3-D time-domain forward modeling algorithm for TEM that simultaneously considers the non-uniformity of resistivity and magnetic permeability. The accuracy of the forward modeling algorithm was verified through the comparisons with a uniform half-space analytical solution and a 1-D numerical solutions, and the influence of the magnetic permeability variations on the 3-D TEM forward response was demonstrated.  
Based on the forward modeling work, a 3-D TEM inversion algorithm was further implemented, which simultaneously inverts both resistivity and magnetic permeability parameters. Through tests on three types of model cases: resistivity model inversion, magnetic permeability model inversion, and simultaneous inversion for both resistivity and magnetic permeability, the positions of the anomalies were accurately recovered, validating the effectiveness of the inversion algorithm. Additionally, tests were conducted using synthetic data affected by magnetic permeability, where an inversion algorithm ignoring the magnetic permeability parameter was applied. The results showed that the inversion considering only the resistivity model led to significant spurious anomalies around the target body, whereas considering magnetic permeability greatly improved the recovery for the target body, highlighting the importance of accounting for magnetic permeability in 3-D TEM inversion.  
The 3D forward and inversion algorithm developed in this paper has certain limitations in its applicability. First, the algorithm only considers scenarios with a large-loop magnetic source and does not yet accommodate cases involving an electric field source. Second, the algorithm is designed specifically for isotropic resistivity models and does not yet account for the more complex anisotropic resistivity conditions typically encountered in real-world scenarios. Finally, the effects of induced polarization are not considered in the example models, suggesting that there is still potential for further enhancement of the approach presented in this paper.  
Our work significantly extends the flexibility and applicability of the 3-D TEM inversion algorithm, which is expected to aid in the quantitative interpretation of the TEM field data.  
C.W.: methodology; software; validation; writing--original draft preparation; visualization. Y.D.: software; writing--review and editing. J.G.: supervision; funding acquisition; writing--review and editing. H.T.: supervision; conceptualization; writing--review and editing. Y.W.: funding acquisition; writing--review and editing. W.D.: funding acquisition; writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China (Grant No. 42204081), National Key Research and Development Program of China (Grant No. 2023 YFB3907703), and Key Laboratory of Target Cognition and Application Technology (Grant No. 2023-CXPT-LC-005).  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.