An Optimized Multi-Layer Equivalent Source Method for Spatial Continuation of Magnetic Anomalies in the Geomagnetic Background

ObjectiveSpatial continuation of magnetic anomalies is a key technique in potential field data processing and supports geological interpretation and geomagnetic navigation. Existing methods remain limited: frequency-domain approaches are severely ill-posed and amplify high-frequency noise during downward continuation, whereas traditional single-layer equivalent source methods often fail to fit multi-scale anomalies generated by sources at different depths. Although the Multilayer Equivalent Source (MES) model improves depth resolution, its performance is constrained by subjective parameter selection and instability in large-scale inversion, which can lead to the loss of high-frequency structural information. This study proposes an optimized MES method for high-precision continuation in complex geological environments. The method establishes an objective parameterization scheme by combining Radially Averaged Power Spectrum (RAPS) analysis with Variational Mode Decomposition (VMD) to separate sources. It also introduces a collaborative inversion scheme based on the Fungal Growth Optimizer (FGO) and the Preconditioned Conjugate Gradient (PCG) method to adaptively optimize regularization parameters, suppress ill-posedness, and improve reconstruction robustness under noise.MethodsA four-step technical framework is developed. (1) Model construction: A Multi-layer Equivalent Source (MES) model is formed using uniformly magnetized rectangular prisms to represent subsurface sources. (2) Parameter configuration: An objective scheme combining RAPS and VMD is applied. RAPS estimates average source-layer depths from slope variations in the logarithmic power spectrum. VMD then decomposes the magnetic signal into intrinsic mode functions representing different depths, enabling calculation of layer thickness using the ratio of the Mean Total Horizontal Gradient (MTHD). (3) Collaborative inversion: A robust inversion strategy incorporates FGO into the PCG algorithm. Tikhonov regularization forms the objective function to mitigate ill-posedness, and FGO adaptively searches for optimal hyperparameters, including the regularization parameter, step-size scaling factor, and preconditioner weights, improving solution stability and convergence efficiency. (4) Comprehensive validation: Three evaluations are conducted. A five-prism theoretical model is used to benchmark performance against single-layer, double-layer, and frequency-domain methods. The global EMAG2 magnetic anomaly model with 5% Gaussian noise is applied to assess robustness. Finally, real aeromagnetic data from the Australian magnetic anomaly grid are tested in two sub-regions—a complex tectonic zone (Area A) and a sedimentary basin (Area B)—for downward continuation from 2 000 m to 0 m, using RMSE and GOF as indicators.Results and DiscussionsThe performance of the proposed method is validated in three stages. (1) Theoretical model verification: The radial average logarithmic power spectrum (Fig. 3) and VMD analysis (Fig. 4) identify three equivalent source layers, demonstrating the objectivity of the parameter configuration framework. The FGO-optimized inversion accelerates convergence by approximately 5～6 times and reduces the residual norm by 13% compared with the traditional Conjugate Gradient (CG) method (Fig. 7). In the 100 m upward continuation (Fig. 8, Table 4) and downward continuation (Fig. 9, Table 5) tests, the proposed method attains the lowest RMSE and highest GOF, addressing the ill-posedness of frequency-domain methods and the large fitting errors of single- and double-layer models. (2) Robustness analysis: Using the EMAG2 data (Fig. 10), the method demonstrates strong noise resistance. With 5% Gaussian noise added to the 1 000 m observation data, the downward continuation results remain stable and free of noticeable artifacts. Quantitative evaluation (Table 6) yields an RMSE of 7.36 nT and a GOF of 82.65%, confirming robustness in low signal-to-noise conditions. (3) Generalization verification: When applied to Australian magnetic anomaly grid data, two different geological regions are examined (Fig. 11, Fig. 12). In Area B (sedimentary basin), which has smooth gradients, the method achieves high-fidelity reconstruction with a GOF of 84.28% and an RMSE of 29.06 nT. In Area A (complex tectonic zone), despite the exponential decay of high-frequency signals, the method recovers key structural features (GOF = 76.14%), although localized residuals appear in high-gradient areas because of physical limits in field transformation. These findings support the method’s applicability across varied geological textures.ConclusionsThis study proposes a robust spatial continuation method for magnetic anomalies based on an optimized MES framework. By integrating RAPS analysis with VMD, the method establishes an objective parameterization scheme that reduces subjectivity in model construction. The incorporation of the FGO into the inversion algorithm improves convergence speed and stability, mitigating the ill-posedness inherent in downward continuation. Experimental results show that: (1) the method exhibits strong robustness, maintaining high signal fidelity under 5% Gaussian noise, as confirmed by the EMAG2 model tests; and (2) the method has broad geological applicability. In real Australian aeromagnetic grid data, it achieves high-precision reconstruction in deep sedimentary basins (Area B) and recovers major structural features in complex tectonic zones (Area A), outperforming traditional single-layer and frequency-domain methods. A remaining limitation is high memory demand due to storage of large dense kernel matrices. Future work will explore matrix compression or matrix-free inversion strategies to improve computational efficiency for large-scale geomagnetic data processing.