Inversion of Magnetic Data using Learned Dictionaries and Scale Space

TL;DR

Addresses the ill-posed problem of magnetic inversion by integrating learned dictionaries with a scale-space framework. The approach learns a convolutional dictionary from model ensembles and supports both a fixed shared dictionary and an iteration-dependent unrolled dictionary to enforce sparsity and progressive structure. In synthetic 3D experiments, the unrolled dictionary yields substantial gains in both model and data recovery over variational and fixed-dictionary baselines. The work demonstrates the potential of data-driven regularization for robust geophysical inversion and provides public code for reproducibility.

Abstract

Magnetic data inversion is an important tool in geophysics, used to infer subsurface magnetic susceptibility distributions from surface magnetic field measurements. This inverse problem is inherently ill-posed, characterized by non-unique solutions, depth ambiguity, and sensitivity to noise. Traditional inversion approaches rely on predefined regularization techniques to stabilize solutions, limiting their adaptability to complex or diverse geological scenarios. In this study, we propose an approach that integrates variable dictionary learning and scale-space methods to address these challenges. Our method employs learned dictionaries, allowing for adaptive representation of complex subsurface features that are difficult to capture with predefined bases. Additionally, we extend classical variational inversion by incorporating multi-scale representations through a scale-space framework, enabling the progressive introduction of structural detail while mitigating overfitting. We implement both fixed and dynamic dictionary learning techniques, with the latter introducing iteration-dependent dictionaries for enhanced flexibility. Using a synthetic dataset to simulate geological scenarios, we demonstrate significant improvements in reconstruction accuracy and robustness compared to conventional variational and dictionary-based methods. Our results highlight the potential of learned dictionaries, especially when coupled with scale-space dynamics, to improve model recovery and noise handling. These findings underscore the promise of our data-driven approach for advance magnetic data inversion and its applications in geophysical exploration, environmental assessment, and mineral prospecting. The code is publicly available at: https://github.com/ahxmeds/magnetic-inversion-dictionary.git.

Figures (2)

• Figure 1: Comparison of loss values obtained from 100 samples randomly drawn from the test set using different methods. The left panel compares the variational method with the unrolled ${\bm{\mathrm{\Psi}}}$ method, while the right panel compares the single shared ${\bm{\mathrm{\Psi}}}$ method with the unrolled ${\bm{\mathrm{\Psi}}}$ method. In both cases, the unrolled ${\bm{\mathrm{\Psi}}}$ (blue) consistently outperformed the competing models (orange) based on the nMSE loss for model recovery.
• Figure 2: Three sample models (01, 11, and 19) from the test set illustrating the quality improvement of unrolled dictionary learning for inversion compared to other methods. Each image highlights different features of the model reconstruction quality achieved using the proposed method. The 2D magnetic data from the 3D true model is inverted back to a 3D construction using four different methods. Views of the reconstruction are shown using contour potentials of the magnetic susceptibility.