{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T16:58:51Z","timestamp":1791565131816,"version":"4.3.4"},"reference-count":24,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T00:00:00Z","timestamp":1791417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T00:00:00Z","timestamp":1791417600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Aeromagnetic compensation mitigates the aircraft-induced magnetic interference in airborne magnetometer measurements, thereby providing the measurement accuracy required by geomagnetic navigation and aeromagnetic surveying. The most widely used compensation framework, the conventional Tolles\u2013Lawson (T-L) model, approximates the eddy-current magnetic field as an instantaneous linear response to the rate of change of the external magnetic field. Our analysis demonstrates that this assumption becomes invalid when the structural characteristic time scale is comparable to the aircraft maneuver time scale. We therefore generalize the eddy-current term into a first-order dynamic element, resulting in a dynamic eddy-current T-L model in which the eddy-current response is no longer instantaneous, while the 18-term design-matrix structure is preserved and no additional parameters are introduced. Utilizing a public flight dataset, we compare the two models on independent validation flight lines with a single calibration segment. Ridge regression on the Mag 4 channel yields a consistent and statistically significant improvement, where the median enhancement in the high-dynamic flight lines reaches +1.177 nT, approximately 70 times greater than that observed in regular survey lines. This benefit arises precisely under the high-dynamic flight conditions that are most critical for geomagnetic navigation, and its robustness is confirmed by dual-reference, vector-input, and regularization-parameter analyses.<\/jats:p>","DOI":"10.3390\/s26196338","type":"journal-article","created":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T16:43:38Z","timestamp":1791564218000},"page":"6338","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Newly Developed Tolles\u2013Lawson Compensation Model Accounting for Eddy-Current Dynamic Response"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-0287-2493","authenticated-orcid":false,"given":"Peng","family":"Gu","sequence":"first","affiliation":[{"name":"Key Laboratory of Electronics and Information Technology for Space Systems, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of the Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"}]},{"given":"Dong","family":"Gao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronics and Information Technology for Space Systems, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of the Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"project-administration","vocabulary":"credit"}]},{"given":"Chunjiang","family":"Bian","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronics and Information Technology for Space Systems, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of the Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Minghui","family":"Zhu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Electronics and Information Technology for Space Systems, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"University of the Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/navi.138","article-title":"Absolute Positioning Using the Earth\u2019s Magnetic Anomaly Field","volume":"63","author":"Canciani","year":"2016","journal-title":"Navig. J. Inst. Navig."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gnadt, A. (2022, January 3\u20137). Machine Learning-Enhanced Magnetic Calibration for Airborne Magnetic Anomaly Navigation. Proceedings of the AIAA SCITECH 2022 Forum, San Diego, CA, USA.","DOI":"10.2514\/6.2022-1760"},{"key":"ref_3","unstructured":"Tolles, W.E., and Lawson, J.D. (1950). Magnetic Compensation of MAD Equipped Aircraft, Airborne Instruments Laboratory Inc.. Report No. 201-1."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/TANE3.1961.4201799","article-title":"Identification and Evaluation of Magnetic-Field Sources of Magnetic Airborne Detector Equipped Aircraft","volume":"ANE-8","author":"Leliak","year":"1961","journal-title":"IRE Trans. Aerosp. Navig. Electron."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1109\/TAES.1979.308736","article-title":"Small Signal Compensation of Magnetic Fields Resulting from Aircraft Maneuvers","volume":"AES-15","author":"Bickel","year":"1979","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhao, G., Han, Q., Peng, X., Zou, P., Wang, H., Du, C., Wang, H., Tong, X., Li, Q., and Guo, H. (2019). An Aeromagnetic Compensation Method Based on a Multimodel for Mitigating Multicollinearity. Sensors, 19.","DOI":"10.3390\/s19132931"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"104791","DOI":"10.1016\/j.jappgeo.2022.104791","article-title":"Research on Aeromagnetic Compensation of a Multi-Rotor UAV Based on Robust Principal Component Analysis","volume":"206","author":"Qiao","year":"2022","journal-title":"J. Appl. Geophys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1049\/iet-smt.2018.5654","article-title":"Extended Aeromagnetic Compensation Modelling Including Non-Manoeuvring Interferences","volume":"13","author":"Du","year":"2019","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, C., Du, C., Peng, X., Han, Q., and Guo, H. (2022). An Aeromagnetic Compensation Method for Suppressing the Magnetic Interference Generated by Electric Current with Vector Magnetometer. Sensors, 22.","DOI":"10.3390\/s22166151"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Yu, Z., Ye, L., Ding, C., Chi, C., Liu, C., and Cheng, P. (2025). Dynamic Parameterization and Optimized Flight Paths for Enhanced Aeromagnetic Compensation in Large Unmanned Aerial Vehicles. Sensors, 25.","DOI":"10.3390\/s25092954"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, Z., Yu, Z., Liu, C., Wu, G., Li, J., Wang, D., Wang, Y., and Zhang, Y. (2025). Intelligent Dynamic-Enhanced Compensation for UAV Magnetic Interference. Sensors, 25.","DOI":"10.3390\/s25165059"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Li, Y., Wang, G., Han, Q., and Li, Q. (2026). High-Precision Aeromagnetic Compensation Method Under the Influence of the Geomagnetic Field. Sensors, 26.","DOI":"10.3390\/s26061867"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ye, L., Yu, Z., Zhang, Y., Chi, C., Cheng, P., and Chen, J. (2024). An Aeromagnetic Compensation Strategy for Large UAVs. Sensors, 24.","DOI":"10.3390\/s24123775"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1119\/1.2397095","article-title":"Electromagnetics from a Quasistatic Perspective","volume":"75","author":"Larsson","year":"2007","journal-title":"Am. J. Phys."},{"key":"ref_15","unstructured":"Jackson, J.D. (1999). Classical Electrodynamics, Wiley. [3rd ed.]."},{"key":"ref_16","unstructured":"Griffiths, D.J. (2008). Introduction to Electrodynamics, Pearson\/Benjamin Cummings. [3rd ed.]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.csda.2017.11.003","article-title":"A Note on the Validity of Cross-Validation for Evaluating Autoregressive Time Series Prediction","volume":"120","author":"Bergmeir","year":"2018","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_18","unstructured":"Gnadt, A.R., Belarge, J., Canciani, A., Conger, L., Curro, J., Edelman, A., Morales, P., O\u2019Keeffe, M.F., Taylor, J., and Jacobs, D. (2023). DAF-MIT AIA Open Flight Data for Magnetic Navigation Research. Zenodo."},{"key":"ref_19","unstructured":"Gnadt, A.R., Belarge, J., Canciani, A., Conger, L., Curro, J., Edelman, A., Morales, P., O\u2019Keeffe, M.F., Taylor, J., and Rackauckas, C. (2020). Signal Enhancement for Magnetic Navigation Challenge Problem. arXiv."},{"key":"ref_20","unstructured":"Sander Geophysics Ltd (2026, August 09). High Resolution Airborne Magnetometer Surveys. Available online: https:\/\/www.sgl.com\/magtf.html."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1080\/00401706.1970.10488634","article-title":"Ridge Regression: Biased Estimation for Nonorthogonal Problems","volume":"12","author":"Hoerl","year":"1970","journal-title":"Technometrics"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gu, B., Li, Q., and Liu, H. (2013, January 16\u201318). Aeromagnetic Compensation Based on Truncated Singular Value Decomposition with an Improved Parameter-Choice Algorithm. Proceedings of the 2013 6th International Congress on Image and Signal Processing (CISP), Hangzhou, China.","DOI":"10.1109\/CISP.2013.6743921"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wilcoxon, F. (1992). Individual Comparisons by Ranking Methods. Breakthroughs in Statistics: Methodology and Distribution, Springer.","DOI":"10.1007\/978-1-4612-4380-9_16"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R. (1993). An Introduction to the Bootstrap, Chapman & Hall. Monographs on Statistics and Applied Probability, No. 57.","DOI":"10.1007\/978-1-4899-4541-9"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6338\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T16:45:13Z","timestamp":1791564313000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/26\/19\/6338"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,8]]},"references-count":24,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2026,10]]}},"alternative-id":["s26196338"],"URL":"https:\/\/doi.org\/10.3390\/s26196338","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,8]]}}}