Supplementary material for "Magnetic dual-layer equivalent sources on the sphere"

About

The initial idea for this project emerged during a meeting between Arthur and Prof. Leo on January 22, 2024. In this meeting, Leo presented his preliminary thoughts on the topic, and Arthur immediately embraced the proposal. Shortly afterward, Arthur moved from Formosa, Goiás, to São Paulo to begin his master’s studies.

Abstract

The equivalent source method is widely used for processing and interpolating magnetic data, particularly in airborne surveys. However, implementations based on Cartesian coordinates present limitations at regional and global scales, where Earth curvature introduces geometric inconsistencies that affect data integration and modeling accuracy. To address this problem, this study proposes an adaptation of the magnetic equivalent source method to spherical coordinates, including revisions to its mathematical formulation to account for spherical geometry. The proposed framework enables consistent magnetic field modeling over large geographic areas. To improve the representation of magnetic sources, a dual-layer configuration is adopted to separate long- and short-wavelength components. Cross-validation is employed to determine optimal hyperparameters for each layer, ensuring stable and balanced inversions. To guarantee computational feasibility for large and high-resolution datasets, a gradient-boosting strategy is incorporated into the inversion process, significantly improving computational performance. Synthetic experiments demonstrate that the method remains stable and accurate for large-scale datasets, with tests conducted on synthetic data containing up to 500,000 observations and enables the reliable recovery of magnetic field components from total-field anomaly data. The approach was further applied to more than 1.5 million real observations, confirming its scalability and robustness. The recovered field amplitude provides additional constraints for data interpretation and enhances the geological analysis. The final implementation is released as open-source software to support reproducibility and broader adoption.

License

All Python source code (including .py and .ipynb files) is made available under the MIT license. You can freely use and modify the code, without warranty, so long as you provide attribution to the authors. See LICENSE-MIT.txt for the full license text.