3D inversion of magnetic gradient tensor data integrating analytical magnetization-vector estimation and Bayesian uncertainty quantification

Accurate 3D inversion of magnetic gradient tensor (MGT) data is challenging because target geometry and magnetization parameters are strongly coupled, and remanent magnetization may cause the total magnetization vector to deviate from the present geomagnetic field. To address this problem, this study proposes a physics-constrained and uncertainty-quantified deep-learning framework for 3D probabilistic inversion of six-component MGT data under unknown magnetization conditions. The network predicts the 3D target occupancy probability from 2D observation-plane tensor data, while the equivalent uniform magnetization vector is analytically estimated from the predicted geometry and the original observations through a stabilized least-squares procedure. A physics-consistency loss is then constructed by comparing the reconstructed forward response with the observed data, allowing the inversion result to be verified in the original observation space. To characterize inversion reliability, Bayesian posterior sampling is further introduced to quantify uncertainty in both the 3D occupancy probability and the estimated magnetization vector. Synthetic experiments show that the proposed method can recover the main 3D target geometry, estimate reasonable equivalent magnetization vectors, and maintain good observation-space consistency under different magnetization conditions. Additional robustness tests further demonstrate the stability of the method under varying burial depths and noise levels, while revealing its limitations for multi-target systems with strongly different magnetization directions. A controlled field experiment further demonstrates the feasibility of the method for measured MGT data. The results indicate that combining deep geometric prediction, analytical magnetization-vector estimation, physical consistency, and posterior uncertainty modeling provides a promising computational framework for MGT inversion under unknown magnetization parameters. Clinical trial registration Not applicable. This study is not a clinical trial.