## 2 Forward Modeling Principle
### Large-Scale SAI-Krylov Algorithm
Figure 9.2 Contour map of the horizontal electric field component in the _xoy_ plane for different transmitter positions: (**a–d**) source coordinates are (\(-250\) m, \(-250\) m, \(0\) m) and (\(-250\) m, \(250\) m, \(0\) m); (**e–h**) source coordinates are (\(-250\) m, \(-250\) m, \(0\) m) and (\(250\) m, \(-250\) m, \(0\) m); (**i–l**) source coordinates are (\(250\) m, \(-250\) m, \(0\) m) and (\(250\) m, \(250\) m, \(0\) m); (**m–p**) source coordinates are (\(-250\) m, \(250\) m, \(0\) m) and (\(250\) m, \(250\) m, \(0\) m). ## 4 Conclusions

In this study, we have developed an algorithm for simulating large-scale TEM responses based on the finite element and time-segmented SAI-Krylov subspace methods. This approach enables efficient simulation of large-scale TEM responses within an acceptable time frame while using less memory. The optimization parameters of the method, including the initial value, air conductivity \(\sigma_{air}\), tolerance \(\epsilon\), relaxation factor \(\omega\), and subspace order across different time intervals, were discussed in detail. The numerical simulation results show that the TEM response simulated by the proposed method is both accurate and reliable, laying the foundation for future research in large-scale inversion and TEM interpretation. While the proposed method achieves efficient large-scale 3D TEM forward modeling, there are still some limitations. Firstly, the selection of parameters in the algorithm is based on layered medium models, which may not be directly applicable to arbitrary 3D

Figure 9.3 Contour map of the \(\partial\mathbf{B}_{z}/\partial t\) in the \(xy\) plane for different transmitter positions: (**a–d**) source coordinates are (\(-250\) m, \(-250\) m, \(0\) m) and (\(-250\) m, \(250\) m, \(0\) m); (**e–h**) source coordinates are (\(-250\) m, \(-250\) m, \(0\) m) and (\(250\) m, \(-250\) m, \(0\) m); (**i–l**) source coordinates are (\(250\) m, \(-250\) m, \(0\) m) and (\(250\) m, \(250\) m, \(0\) m); (**m–p**) source coordinates are (\(-250\) m, \(250\) m, \(0\) m) and (\(250\) m, \(250\) m, \(0\) m). geological models. To address this limitation, future research will focus on optimizing parameter selection for random media. Secondly, the current method can benefit from the development of an adaptive SAI-Krylov subspace iterative algorithm, which could further improve computational efficiency. These aspects will be investigated in future studies to enhance the applicability and performance of the proposed method in more complex geological scenarios. Writing--original draft, validation, methodology, and formal analysis, [PERSON] and K.L. ; writing--review and editing, funding acquisition, and conceptualization: [PERSON], [PERSON], [PERSON] and T.F. ; review and editing, analysis, and interpretation: [PERSON], [PERSON], [PERSON] and T.F. All authors have read and agreed to the published version of the manuscript. This work was supported in part by the National Natural Science Foundation of China under Grant U2444220, 42404154 and 42477497; in part by the Postdoctoral Fellowship Program of CPSF under Grant GZC20233003; in part by the China Postdoctoral Science Foundation under Grant 2024M753507; in part by Basic Research Program of Jiangsu under Grant BK20241672 and BK20243024; and Jiangsu Funding Program for Excellent Postdoctoral Talent. Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. The authors would like to thank China University of Mining and Technology for providing the facilities required for this research. The authors also appreciate the constructive feedback from reviewers and colleagues, which helped improve the quality of this paper. The authors declare no conflicts of interest. ## References

* (1) [PERSON]; [PERSON]; [PERSON] Metalliferous mining geophysics--State of the art after a decade in the new millennium. _Geophysics_**2011**, _76_, W31-W50.