by Arthur Siqueira-Macedo, Leonardo Uieda, India Uppal

This repository contains the data and source code used to produce the results presented in:

Siqueira-Macedo, A., Uieda, L., Uppal, I. (2026). Magnetic dual-layer equivalent sources on the sphere. EarthArXiv. doi:10.31223/X5M19R

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.

Since then, the project has represented a motivating challenge, as it marked Arthur’s first experience working in a research area that had not been part of his undergraduate research. Despite the initial difficulties, the transition proved to be highly enriching. Working closely with Leo and India throughout this project has been a valuable and rewarding experience, contributing significantly to Arthur’s academic and professional development.

Mapping the Earth's magnetic field using airplanes is a common technique, but traditional methods process this data as if the Earth were flat. Because our planet is round, projecting this information onto a flat surface introduces distortions, causing mapping errors over large areas. In this study, we updated the math behind these calculations to account for the Earth's actual curvature, allowing for accurate magnetic maps on a regional or global scale. To improve the results, we used a "dual-layer" model that captures both deep, broad magnetic features and shallow, detailed ones. Because mapping large areas involves massive amounts of data, we also added an advanced computational technique that allows computers to process the information much faster. We successfully tested our new method on over 1.5 million real observations, proving it is fast, reliable, and highly useful for understanding the geology of large regions. Finally, we released our software for free as open-source so anyone can use it.

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.

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.

The manuscript text (including all LaTeX files), figures, and data/models produced as part of this research are available under the Creative Commons Attribution 4.0 License (CC-BY). See LICENSE-CC-BY.txt for the full license text.