Abstract
With the increasing demand for both accuracy and efficiency in transient electromagnetic (TEM) simulations, conventional 3-D forward modeling methods face growing challenges. This study presents a high-accuracy and high-efficiency 3-D forward modeling approach that combines the spectral-element method (SEM) with a model order reduction (MOR) scheme. High-order orthogonal basis functions are employed, and the computational domain is discretized in a finite-element manner to improve modeling accuracy. During element-level analysis, a reduced-integration strategy is introduced to enhance the sparsity of the double-curl and conductivity matrices, thereby reducing the computational time and memory consumption required for matrix assembly. For temporal treatment, a shift-and-invert Krylov (SAI-Krylov) subspace algorithm is adopted: the basis and projection matrices are constructed using only one matrix factorization and tens of back-substitutions, after which low-dimensional matrix exponential functions are evaluated to efficiently obtain electromagnetic responses at arbitrary times. Comparisons with other numerical methods demonstrate the superior efficiency and accuracy of the proposed approach. Finally, simulations on a 3-D sulfide ore-body model are performed to investigate TEM field propagation for both galvanic and loop sources, confirming the capability of the method to model electromagnetic responses in complex geological settings.
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The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
References
Di, Q. et al. Summary of technology for a comprehensive geophysical exploration of gold mine in North China Craton. Sci. China Earth Sci. 64(9), 1524–1536 (2021).
Jiang, Z., Liu, L., Liu, S. & Yue, J. Surface-to-underground transient electromagnetic detection of water-bearing goaves. IEEE Trans. Geosci. Remote Sens. 57(8), 5303–5318 (2019).
Lu, K. et al. The application of multi-grounded source transient electromagnetic method in the detections of coal seam goafs in Gansu Province, China. J. Geophys. Eng. 18(4), 515–528 (2021).
Gottschalk, I., Knight, R., Asch, T., Abraham, J. & Cannia, J. Using an airborne electromagnetic method to map saltwater intrusion in the Northern Salinas Valley, CA. Geophysics 85(4), B119–B131 (2020).
Jahandari, H. & Bihlo, A. Forward modelling of geophysical electromagnetic data on unstructured grids using an adaptive mimetic finite difference method. Comput. Geosci. 25 (3), 1083–1104 (2021).
Commer, M. & Newman, G. A parallel finite-difference approach for 3D transient electromagnetic modeling with galvanic sources. Geophysics 69(4), 1192–1202 (2004).
Sun, H., Li, X. & Li, S. Three-dimensional FDTD modeling of TEM excited by a loop source considering ramp time. Chin. J. Geophys. 56 (3), 1049–1064 (2013).
Li, J., Farquharson, C. G. & Hu, X. Three effective inverse Laplace transform algorithms for computing time-domain electromagnetic responses. Geophysics 81(1), 113–128 (2016).
Lu, K. et al. 3D anisotropic TEM modeling with loop source using model reduction method. J. Geophys. Eng. 19(3), 403–417 (2022).
Haber, E. Computational Methods in Geophysical Electromagnetics (Society for Industrial and Applied Mathematics, 2014).
Zhou, J., Liu, W., Li, X. & Qi, Z. 3D transient electromagnetic modeling using a shift-and-invert Krylov subspace method. J. Geophys. Eng. 15 (4), 1341–1349 (2018).
Yin, C. et al. 3D airborne EM forward modeling based on time-domain spectral element method. Remote Sens. 13(4), 601 (2021).
Ren, Y., Chen, Y., Zhan, Q., Niu, J. & Liu, Q. A higher order hybrid SIE/FEM/SEM method for the flexible electromagnetic simulation in layered medium. IEEE Trans. Geosci. Remote Sens. 55(5), 2563–2574 (2017).
Huang, X. et al. 3D anisotropic modeling and identification for airborne EM systems based on the spectral-element method. Appl. Geophys. 14, 419–430 (2017).
Huang, X. et al. Spectral-element method with arbitrary hexahedron meshes for time-domain 3D airborne electromagnetic forward modeling. Geophysics 84(1), E37–E46 (2019).
Yin, C. et al. 3D frequency-domain airborne EM forward modelling using spectral element method with Gauss-Lobatto-Chebyshev polynomials. Explor. Geophys. 50(5), 461–471 (2019).
Botchev, M. A. & Knizhnerman, L. A. ART: Adaptive residual-time restarting for Krylov subspace matrix exponential evaluations. J. Comput. Appl. Math. 364, 112311 (2020).
Güttel, S., Kressner, D. & Lund, K. Limited memory polynomial methods for large scale matrix functions. GAMM-Mitt. 43(3), e202000019 (2020).
Eshof, V. & Hochbruck, M. Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27(4), 1438–1457 (2006).
Hochbruck, M., Paur, T., Schulz, A., Thawinan, E. & Wieners, C. Efficient time integration for discontinuous Galerkin approximations of linear wave equations. ZAMM J. Appl. Math. Mech. 95, 237–259 (2014).
Börner, R., Ernst, O. G. & Güttel, S. Three-dimensional transient electromagnetic modelling using rational Krylov methods. Geophys. J. Int. 202 (3), 2025–2043 (2015).
Zhou, J. et al. Source decoupling and model order reduction for 3-D full-time transient electromagnetic modeling. IEEE T. Geosci. Remote Sens. 61, 2002612 (2023).
Botchev, M. A. F., Grimm, V. & Hochbruck, M. Residual, restarting, and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35, 1376–1397 (2013).
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Heartfelt thanks extend to all the editors and reviewers for their valuable suggestions.
Funding
This work was supported by the National Natural Science Foundation of China (Grant numbers: 42404154, 42504143, 42230811) and Basic Research Program of Jiangsu (BK20241672, BK20243024).
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Conceptualization, Fan, Y. N. and Lu, K. L.; methodology, Fan, Y. N., Lu, K. L., Huang, Y. D. and Yang, Q. R.; software, Fan, Y. N., Lu, K. L. and Huang, Y. D.; validation, Lu, K. L., Yue, J. H. and Yang, Q. R.; investigation, Fan, Y. N. and Lu, K. L.; resources, Fan, Y. N. and Lu, K. L.; data curation, Lu, K. L. and Yang, Q., R.; writing—original draft preparation, Fan, Y. N. and Lu, K. L.; writing—review and editing, Fan, Y. N. and Lu, K. L.; visualization, Lu, K. L.; supervision, Fan, Y. N., Lu, K. L. and Yue, J. H.; project administration, Fan, Y. N. and Lu, K. L.; funding acquisition, Fan, Y. N., Lu, K. L. and Yue, J. H. All authors have read and agreed to the published version of the manuscript.
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Fan, Y., Lu, K., Huang, Y. et al. Model-order-reduced spectral-element method for high-accuracy and fast 3-D transient electromagnetic forward modeling with SAI-Krylov. Sci Rep (2026). https://doi.org/10.1038/s41598-026-44053-y
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DOI: https://doi.org/10.1038/s41598-026-44053-y
