地磁背景下磁异常空间延拓的优化多层等效源方法

管宇; 张慧强

doi:10.11999/JEIT250958

地磁背景下磁异常空间延拓的优化多层等效源方法

doi: 10.11999/JEIT250958 cstr: 32379.14.JEIT250958

管宇^,,
张慧强

哈尔滨理工大学自动化学院哈尔滨 150006

详细信息

作者简介:
管宇：女，讲师，研究方向为容错控制与故障诊断

张慧强：男，硕士生，研究方向为地磁导航

通讯作者:
管宇　gy_hlg@163.com

中图分类号: TN911.7; O411.5
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- 文章访问数: 136
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- 被引次数: 0
出版历程
- 收稿日期: 2025-09-23
- 修回日期: 2026-01-22
- 录用日期: 2026-01-22
- 网络出版日期: 2026-02-11

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

GUAN Yu^,,
ZHANG Huiqiang

Institute of Automation, Harbin University of Science and Technology, Harbin 150006, China

摘要

摘要: 磁异常延拓是地磁空间信息获取与处理的重要技术手段。针对现有的频率域方法在向下延拓中具有不适定性，以及传统等效源方法难以兼顾多尺度场源拟合精度的问题，该文提出了一种磁异常空间延拓的优化多层等效源方法。该方法采用基于功率谱分析的深度估计与变分模态分解技术构建多层等效源参数设定框架，并引入真菌生长算法对等效源反演过程进行协同优化。理论模型与EMAG2仿真实验表明，该方法显著降低了模型构建的主观性，在5%高斯噪声干扰下仍能保持较高的信号保真度与抗噪鲁棒性。基于澳大利亚实测磁异常网格数据的应用验证，该方法在复杂构造区与平缓基底区均具有优异的普适性与延拓精度。
- 磁异常延拓 /
- 等效源 /
- 功率谱分析 /
- 地磁背景
Abstract: Objective Spatial 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. Methods A 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 Discussions The 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. Conclusions This 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.
- Magnetic anomaly continuation /
- Equivalent source /
- Power spectrum analysis /
- Geomagnetic background

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图 1 磁异常空间延拓的优化多层等效源方法流程图

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图 2 模型理论磁异常示意图

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图 3 径向平均功率谱曲线

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图 4 地表磁异常分离异常

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图 5 等效源分布范围实验结果

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图 6 关键参数敏感性曲线

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图 7 反演算法对比

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图 8 不同方法向上延拓100 m结果及对比残差

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图 9 不同方法向下延拓100 m结果及对比残差

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图 10 基于EMAG2区域数据的鲁棒性验证结果

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图 11 基于复杂构造区(A区)实测数据的向下延拓结果与残差分布

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图 12 基于平缓背景区(B区)实测数据的向下延拓结果与残差分布

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表 1 理论模型参数表

模型编号	中心深度(m)	水平坐标(m)	长a(m)	宽b(m)	高c(m)	磁化率(无量纲)
1	250	(1000, 1000)	200	200	300	0.2
2	450	(1600, 1600)	300	300	300	0.2
3	425	(1200, 800)	150	150	250	0.2
4	500	(500, 1500)	250	250	200	0.2
5	950	(1000, 1000)	1000	1000	100	0.2

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表 2 功率谱分段、谱斜率深度估计与VMD分解的参数设置

模块	参数	设定	说明
功率谱分段	分段区间(rad/m)	0～0.0024, 0.0024～0.0118, 0.0118～0.0168.	依据功率谱曲线斜率突变位置识别线性段区间
谱斜率深度	拟合方法	线性最小二乘法	拟合线性段
谱斜率深度	深度公式	式(9)	符合二维指数衰减模型
VMD 分解	模态数	5	控制分解模态数量
	带宽因子	1000	控制带宽和分解稳定性
	拉格朗日步长	0	采用无约束稳态更新形式
	收敛阈值	10^–6	控制迭代终止条件

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表 3 变分模态分解参数扰动敏感性分析

参数类型	设定值	频带表现	结果
模态数	5	中间层混叠	边界波动明显
	8	三层分量分离清晰	分解稳定可重复
	10	过度分解	分解稳定性降低
带宽因子	1000	频带重叠	收敛不稳定
	2000	结构层次明显	收敛快速稳定
	5000	过度压缩	高频成分被过度抑制

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表 4 不同方法向上100 m延拓结果的量化精度评估

方法	RMSE(nT)	MaxAbsError(nT)	GradRMS(nT/m)	GOF(%)
优化多层等效源	2.17	10.90	2.23	99.43
单层等效源	118.29	407.06	25.45	69.25
双层等效源	173.73	427.58	30.40	54.85
频率域	21.58	54.98	4.71	94.39

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表 5 不同方法向下100 m延拓结果的量化精度评估

方法	RMSE(nT)	MaxAbsError(nT)	GradRMS(nT/m)	GOF(%)
优化多层等效源	4.88	16.56	2.31	98.94
单层等效源	251.06	1574.95	226.55	7.76
双层等效源	280.05	1228.15	181.48	12.15
频率域	93.69	160.52	1.95	67.76
约束等效源	12.83	202.38	7.22	96.67
迭代补偿	12.38	218.11	7.15	96.79

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表 6 不同噪声水平下向下延拓结果的量化精度评估

噪声水平	RMSE(nT)	MaxAbsError(nT)	GradRMS(nT/km)	GOF(%)
无噪声	2.53	10.93	0.78	94.04
含1%噪声	5.83	25.46	1.40	86.25
含5%噪声	7.36	34.71	1.90	82.65

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表 7 不同地质区域参数

参数类别	实验区 A(浅部复杂构造区)	实验区 B(深部平缓基底区)
纬度范围	–30.9950°S～–29.0050°S	–25.9950°S ～–24.0050°S
经度范围	139.0050°E～141.9950°E	134.0050°E～136.9950°E
异常幅度范围	–2585～5112 nT	–538～2419 nT
地质特征	构造破碎、磁性梯度大、纹理复杂	构造平缓、磁性梯度弱、纹理简单

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表 8 不同地质区域向下延拓结果的量化精度评估

区域	RMSE(nT)	MaxAbsError(nT)	GradRMS(nT/km)	GOF(%)
A	90.60	637.28	127.76	76.14
B	29.06	231.93	5.26	84.28

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