Self-Structural Constraint Joint Inversion of Aeromagnetic and Gradient Data: Enhanced Imaging for Gold Deposits in Western Henan, China

Abstract

Innovative magnetic techniques are pivotal for advancing mineral exploration. This study presents a self-structural constraint (SSC) method that jointly inverts aeromagnetic and gradient data to resolve high-resolution magnetic susceptibility models for concealed ores. The SSC framework integrates gradient structures from multi-component data as mutual constraints, enhancing signal differentiation and noise suppression. Unstructured tetrahedral grids and Poisson-derived analytical expressions address complex terrains, enabling robust inversions. Synthetic tests show SSC improves resolution by 40%–60% over conventional methods and resists 10% Gaussian noise. Applied to gold exploration in western Henan, China, SSC delineated concealed ore bodies (300–2000 m depth) along NE- and NW-trending faults, correlating with andesite-hosted magnetic anomalies. Combined with volcanic facies analysis, magma migration through these faults provided metallogenic materials and structural traps. The SSC-derived 3D model identified new drill targets, bridging geophysical imaging with geological processes. This advancement enhances the detection of deep, structurally controlled mineralization, offering a transformative tool for resource discovery.

1. Introduction

Three-dimensional inversion of magnetic data has emerged as an indispensable methodology for characterizing subsurface magnetic susceptibility distributions, offering critical insights into the exploration of magnetic mineral resources [1,2,3,4,5,6,7]. Building upon the foundational framework established by Li and Oldenburg [8] for 3D magnetic inversion, subsequent advancements in computational efficiency, memory optimization, and resolution enhancement [9,10,11] have significantly broadened the scope of applications. These include, but are not limited to, mineral resource delineation [5,7], crustal architecture reconstruction [12], and detection of unexploded ordnance [13].

Magnetic gradient data exhibit inherently superior horizontal resolution [14], enabling enhanced delineation of near-surface magnetic sources. Building on this capability, recent advancements in joint inversion frameworks—integrating total-field and gradient magnetic data [14,15,16]—have significantly improved subsurface imaging accuracy by systematically leveraging multi-component datasets through unified computational matrices.

The inherent complexity of natural terrains—characterized by undulating topography and irregular subsurface geological bodies—challenges the efficacy of traditional inversion methods. To overcome these limitations, unstructured discretization techniques grounded in constrained Delaunay triangulation have emerged as robust alternatives [17]. In a seminal study, Abedi [18] developed a 2D focusing inversion method tailored for potential field data in rugged terrains, demonstrating enhanced stability for abrupt geological transitions. Meanwhile, Zuo et al. [19] advanced the field by proposing a 3D accelerated magnetic inversion algorithm that synergizes partial differential equations with unstructured tetrahedral meshing, effectively addressing computational bottlenecks in geometrically complex scenarios.

Recent advances in self-constrained inversion methodologies have demonstrated significant progress in addressing geophysical interpretation challenges. Paoletti et al. [20] pioneered a potential-field-constrained inversion framework utilizing a priori information derived exclusively from gravity and magnetic data analysis, establishing the foundational concept of self-constrained inversion. Building on this concept, Davide et al. [21] integrated microgravity surveys with a self-constrained inversion strategy to resolve the shallow geometry of the Irpinia Fault in Southern Italy, concurrently estimating its Holocene slip rate validated against independent geological constraints. Similarly, Sun and Chen [22] developed a magnetic gradient-driven self-constrained approach, where cross-correlation analyses between theoretical and observed data gradients were translated into spatial weighting functions to reinforce model geological plausibility. In parallel, Vitale and Fedi [23] introduced a two-step self-consistent inversion paradigm for potential fields, synergizing multiscale homogeneity analysis with 3D variable-exponent depth-weighting derived from field scaling properties, thereby enhancing the reconstruction fidelity of deep-seated complex sources. The integration of machine learning has further expanded methodological boundaries: Zhou et al. [24] devised a dual-network self-constrained architecture with adaptive fine-tuning to augment gravity inversion precision, effectively bridging data-driven learning with physical constraints, as evidenced by its successful application in geothermal reservoir delineation. Complementarily, Ming et al. [25] formulated a power-type structural self-constrained inversion (PTSS) scheme incorporating L2-norm regularization, where power-gradient self-constraints and cross-physics mutual constraints synergistically sharpened boundary resolution in joint gravity-magnetic inversions, a capability decisively validated in iron ore exploration scenarios.

In this study, we introduce a self-structural constraint (SSC) inversion method designed to significantly enhance the resolution of joint total-field and gradient magnetic data inversion. The SSC framework uniquely integrates gradient-derived structural attributes—extracted from multi-component inversion results—as physical constraints, thereby amplifying the differentiation of subsurface signals and suppressing noise interference. To address the challenges posed by complex terrains, we derive analytical expressions for tetrahedral magnetic gradient anomalies rooted in Poisson’s theory, enabling robust SSC inversion within an adaptive unstructured tetrahedral meshing framework. Theoretical model validations and real-world case studies demonstrate that the SSC method achieves 40%–60% higher spatial resolution in magnetic susceptibility imaging compared to conventional approaches while maintaining stability under noise levels up to 20% SNR. This advancement establishes SSC as a transformative tool for high-fidelity subsurface exploration in geologically intricate environments.

2. Methodology

Unstructured tetrahedral discretization provides an optimal framework for modeling undulating terrains and irregular geological geometries, enabling the derivation of tetrahedral magnetic gradient forward formulas based on Poisson’s theory to facilitate SSC inversion within an adaptive meshing framework.

2.1. Forward of Gradient Magnetic Anomaly of Tetrahedron

The expression of total-field magnetic anomaly (${\mathit{d}}_{\Delta T}$) is

$$\begin{array}{ll}{\mathit{d}}_{\Delta T}& ={\mathbf{G}}_{\Delta T}\cdot \mathcal{\kappa }\\ & ={\displaystyle \frac{T_0}{4\pi G\rho }}\left\{\left({\mathit{V}}_{xx}\cos i\cos d+{\mathit{V}}_{xy}\cos i\sin d+{\mathit{V}}_{xz}\sin i\right)\cdot \cos I\cos D+\right.\\ & \left({\mathit{V}}_{xy}\cos i\cos d+{\mathit{V}}_{yy}\cos i\sin d+{\mathit{V}}_{yz}\sin i\right)\cdot \cos I\sin D+\\ & \left({\mathit{V}}_{xz}\cos i\cos d+{\mathit{V}}_{yz}\cos i\sin d+{\mathit{V}}_{zz}\sin i\right)\cdot \sin I\}\cdot \mathcal{\kappa }\end{array}$$

(1)

where ${\mathit{G}}_{\Delta T}$ is the kernel matrix connecting the vector of total-field magnetic anomaly data (${\mathit{d}}_{\Delta T}$) and the vector of magnetic susceptibility ($\mathcal{\kappa }$), which is calculated using a forward analytical expression. $T_0$ is the geomagnetic field intensity, $\mathit{G}$ is the gravitational constant, and $\rho$ is the density of the geological body. $I$ and $D$ are the inclinations and declinations of the geomagnetic field, and $i$ and $d$ are the inclinations and declinations of the geological body, respectively. $x$, $y$, and $z$ axes indicate easting, northing, and vertically downward, respectively. Analytical expressions for the second-order derivatives of the gravity potential (${\mathit{V}}_{xx}$, ${\mathit{V}}_{xy}$, ${\mathit{V}}_{xz}$, ${\mathit{V}}_{yy}$, ${\mathit{V}}_{yz}$, and ${\mathit{V}}_{zz}$) of the tetrahedron are provided in Appendix A.

We derived the forward analytical expressions of magnetic gradient anomalies with unstructured tetrahedral grid meshing based on Poisson’s theory. The magnetic gradient anomalies in three directions (${\mathit{d}}_{\Delta T_x}$, ${\mathit{d}}_{\Delta T_y}$, and ${\mathit{d}}_{\Delta T_z}$) can be expressed as

$$\begin{array}{ll}{\mathit{d}}_{\Delta T_x}& ={\mathbf{G}}_{\Delta T_x}\cdot \mathcal{\kappa }\\ & ={\displaystyle \frac{T_0}{4\pi G\rho }}\left\{\left({\mathit{V}}_{xxx}\cos i\cos d+{\mathit{V}}_{xxy}\cos i\sin d+{\mathit{V}}_{xxz}\sin i\right)\right.\cdot \cos I\cos D+\\ & \left({\mathit{V}}_{yxx}\cos i\cos d+{\mathit{V}}_{yxy}\cos i\sin d+{\mathit{V}}_{yxz}\sin i\right)\cdot \cos I\sin D+\\ & \left.\left({\mathit{V}}_{zxx}\cos i\cos d+{\mathit{V}}_{zxy}\cos i\sin d+{\mathit{V}}_{zxz}\sin i\right)\cdot \sin I\right\}\cdot \mathcal{\kappa }\end{array}$$

(2)

$$\begin{array}{ll}{\mathit{d}}_{\Delta T_y}& ={\mathit{G}}_{\Delta T_y}\cdot \mathcal{\kappa }\\ & ={\displaystyle \frac{T_0}{4\pi G\rho }}\left\{\left({\mathit{V}}_{yxx}\cos i\cos d+{\mathit{V}}_{yxy}\cos i\sin d+{\mathit{V}}_{yxz}\sin i\right)\right.\cdot \cos I\cos D+\\ & \left({\mathit{V}}_{yxy}\cos i\cos d+{\mathit{V}}_{yyy}\cos i\sin d+{\mathit{V}}_{yyz}\sin i\right)\cdot \cos I\sin D+\\ & \left.\left({\mathit{V}}_{zxy}\cos i\cos d+{\mathit{V}}_{zyy}\cos i\sin d+{\mathit{V}}_{zyz}\sin i\right)\cdot \sin I\right\}\cdot \mathcal{\kappa }\end{array}$$

(3)

$$\begin{array}{ll}{\mathit{d}}_{\Delta T_z}& ={\mathit{G}}_{\Delta T_z}\cdot \mathcal{\kappa }\\ & =\left\{\left({\mathit{V}}_{zxx}\cos i\cos d+{\mathit{V}}_{zxy}\cos i\sin d+{\mathit{V}}_{zxz}\sin i\right)\right.\cdot \cos I\cos D+\\ & \left({\mathit{V}}_{zxy}\cos i\cos d+{\mathit{V}}_{zyy}\cos i\sin d+{\mathit{V}}_{zyz}\sin i\right)\cdot \cos IsinD+\\ & \left.\left({\mathit{V}}_{zxz}\cos i\cos d+{\mathit{V}}_{zyz}\cos i\sin d+{\mathit{V}}_{zzz}\sin i\right)\cdot \sin I\right\}\cdot \mathcal{\kappa }\end{array}$$

(4)

where ${\mathit{G}}_{\Delta T_x}$, ${\mathit{G}}_{\Delta T_y}$, and ${\mathit{G}}_{\Delta T_z}$ are the kernel matrices connecting the vectors of 3D magnetic gradient anomaly data (${\mathit{d}}_{\Delta T_x}$, ${\mathit{d}}_{\Delta T_y}$ and ${\mathit{d}}_{\Delta T_z}$) and the vector of magnetic susceptibility ($\mathcal{\kappa }$), which were calculated by forward analytical expressions. We used the coordinate transformation provided by Okabe [26] to derive the analytical expressions of the third-order derivative of the gravity potential (${\mathit{V}}_{xyz}$, ${\mathit{V}}_{zzy}$, ${\mathit{V}}_{zzz}$, ${\mathit{V}}_{zzx}$, ${\mathit{V}}_{xxy}$, ${\mathit{V}}_{xxz}$, ${\mathit{V}}_{yyx}$, ${\mathit{V}}_{yyz}$, ${\mathit{V}}_{xxx}$ and ${\mathit{V}}_{yyy}$) of the tetrahedron, which are provided in Appendix B.

To validate the accuracy of the derived forward analytical expressions for tetrahedral total-field and gradient magnetic anomalies, we constructed a regular hexahedral model with dimensions of 300 × 300 × 300 m (Figure 1a) and discretized it into 12 tetrahedral elements of varying sizes using Delaunay triangulation (Figure 1b₁–1b₁₂).

The forward analytical expressions were employed to compute the total-field and gradient magnetic anomalies for the 12 tetrahedral elements derived from the subdivision of the regular hexahedron (Figure 2a–d). These results were directly compared with those obtained using conventional hexahedral forward formulas (Figure 2e–h), with the geomagnetic field parameters set to an inclination of 45° and a declination of 60°. The subtraction of Figure 2a–d from Figure 2e–h, respectively, yields the anomalous residual diagrams shown in Figure 2i–l, where all residuals are close to zero. Subsequently, the root mean square errors (RMSEs) between the total-field and gradient magnetic anomalies calculated by the tetrahedral forward analytical expression and those directly computed using the hexahedral forward formula were evaluated. The RMSE between Figure 2a,e is 5.8395 × 10⁻⁴, between Figure 2b,f is 4.5219 × 10⁻⁶, between Figure 2c,g is 3.4496 × 10⁻⁶, and between Figure 2d,h is 6.7600 × 10⁻⁶. These results further validate the accuracy of the tetrahedral forward analytical expression, thereby laying the foundation for subsequent joint inversion of the magnetic total-field and its gradient under unstructured tetrahedral mesh discretization.

2.2. SSC Method

The conventional joint inversion of total-field and gradient magnetic data involves placing all data in a matrix, and the objective function ${\Phi }_{Ji}$ is

$$\begin{array}{ll}{\Phi }_{Ji}& ={\Phi }_{\mathit{d}}\left(\mathcal{\kappa }\right)+\delta {\Phi }_m\left(\mathcal{\kappa }\right)\\ & ={‖\left(\left[\begin{array}{l}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]\mathcal{\kappa }-\left[\begin{array}{l}{\mathit{d}}_{\Delta T}\\ {\mathit{d}}_{\Delta T_x}\\ {\mathit{d}}_{\Delta T_y}\\ {\mathit{d}}_{\Delta T_z}\end{array}\right]\right)‖}_2^2+\delta {‖{\mathit{W}}_{m_{Ji}}\mathcal{\kappa }‖}_2^2\to \min \end{array}$$

(5)

where ${\mathit{W}}_{m_{Ji}}=\mathit{diag}{\left({\left(\left[\begin{array}{c}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]\right)}^T\cdot \left[\begin{array}{c}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ is the model weighting matrix, and $\delta$ is the regularization parameter [8,15,27].

To enhance the resolution of the joint inversion of total-field and gradient magnetic data (Equation (5)), we propose a self-structural constraint (SSC) method. This approach leverages gradient-derived structural features from the inversion results of both total-field and gradient magnetic data as physical constraints. By integrating these constraints, the SSC method effectively combines the deep-source recovery capability of total-field magnetic data inversion with the high-resolution imaging advantages of magnetic gradient data inversion.

The objective function of the SSC method is

$${\Phi }_{SSC}={‖\left(\left[\begin{array}{l}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]{\mathcal{\kappa }}_3-\left[\begin{array}{l}{\mathit{d}}_{\Delta T}\\ {\mathit{d}}_{\Delta T_x}\\ {\mathit{d}}_{\Delta T_y}\\ {\mathit{d}}_{\Delta T_z}\end{array}\right]\right)‖}_2^2+\delta {‖{\mathit{W}}_{m_{SSC}}{\mathcal{\kappa }}_3‖}_2^2+{\mathsf{\gamma }}_{SSC}\left({‖{\mathit{t}}_1‖}_2^2+{‖{\mathit{t}}_2‖}_2^2\right)\to \min$$

(6)

The solution process differentiated Equation (6) with respect to ${\mathcal{\kappa }}_{3\mathit{W}}$ and can be expressed as

$$\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }{\Phi }_{\mathit{SSC}}}{\mathsf{\partial }{\mathcal{\kappa }}_{3w}}}=\left\{{{\mathit{G}}_{3w}}^T{\mathit{G}}_{3w}+\alpha \mathit{E}+{\mathsf{\gamma }}_{\mathit{SSC}}\left[{\left({\mathit{B}}_{\mathit{xw}}^{{\kappa }_1}\right)}^T{\mathit{B}}_{xw}^{{\kappa }_1}+{\left({\mathit{B}}_{yw}^{{\kappa }_1}\right)}^T{\mathit{B}}_{yw}^{{\kappa }_1}+{\left({\mathit{B}}_{zw}^{{\kappa }_1}\right)}^T{\mathit{B}}_{yw}^{{\kappa }_1}\right.\right.+\\ \left.\left.{\left({\mathit{B}}_{xw}^{{\kappa }_2}\right)}^T{\mathit{B}}_{xw}^{{\kappa }_2}+{\left({\mathit{B}}_{yw}^{{\kappa }_2}\right)}^T{\mathit{B}}_{yw}^{{\kappa }_2}+{\left({\mathit{B}}_{zw}^{{\kappa }_2}\right)}^T{\mathit{B}}_{yw}^{{\kappa }_2}\right]\right\}{\mathcal{\kappa }}_{3w}-{\mathit{G}}_{3w}^T{\mathit{d}}_3\end{array}$$

(7)

where ${\mathit{G}}_{3w}=\left[\begin{array}{l}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]\cdot {\mathit{W}}_{m_{SSC}}$, ${\mathit{d}}_3=\left[\begin{array}{l}{\mathit{d}}_{\Delta T}\\ {\mathit{d}}_{\Delta T_x}\\ {\mathit{d}}_{\Delta T_y}\\ {\mathit{d}}_{\Delta T_z}\end{array}\right]$. $\mathit{E}$ is an identity matrix and the optimal solution of the objective function transformed into Equation (7) is zero. ${\mathsf{\gamma }}_{SSC}$ is a regularization parameter. ${\mathcal{\kappa }}_{3w}$ is the weighting physical property, ${\mathcal{\kappa }}_{3w}={\mathcal{\kappa }}_3\cdot {\mathit{W}}_{m_{SSC}}$, ${\mathit{W}}_{m_{SSC}}$ is the weight of magnetic susceptibility,

$${\mathit{W}}_{m_{SSC}}={\left({\displaystyle \sum _{i=1}^Q{\left(\left[\begin{array}{l}{\mathit{G}}_{\Delta T}\\ {\mathit{G}}_{\Delta T_x}\\ {\mathit{G}}_{\Delta T_y}\\ {\mathit{G}}_{\Delta T_z}\end{array}\right]\right)}_{ij}^2}\right)}^{{\displaystyle \frac{\epsilon }{4}}}\cdot \mathit{diag}\left[{\displaystyle \raisebox{1ex}{$\left({\mathit{S}}_W-\min \left({\mathit{S}}_W\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\left(\max \left({\mathit{S}}_W\right)-\min \left({\mathit{S}}_W\right)\right)$}\right.}\right],j=1,\dots ,P,$$

$\mathit{Q}$ is the number of observation points, and $P$ is the number of tetrahedral units. ${\mathit{S}}_W={\displaystyle \raisebox{1ex}{$\left({\mathcal{\kappa }}_1+{\mathcal{\kappa }}_2\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, ${\mathcal{\kappa }}_1$, and ${\mathcal{\kappa }}_2$ are the inversion results of the total-field and gradient magnetic data, respectively. $\epsilon \left(0.5\le \epsilon \le 1.5\right)$ is the depth weighting factor, which is usually taken as 1. $\mathrm{diag}\left(\right)$ converts the vector in parentheses into a diagonal matrix.

In Equation (6), ${\mathit{t}}_1$ and ${\mathit{t}}_2$ are the gradient constraint terms of the inversion results for the total-field and gradient magnetic anomalies, respectively. The calculation formulas are as follows:

$${‖{\mathit{t}}_1‖}_2^2={\left({\mathit{B}}_{xw}^{{\mathcal{\kappa }}_1}{\mathcal{\kappa }}_{3W}\right)}^T\left({\mathit{B}}_{xW}^{{\mathcal{\kappa }}_1}{\mathcal{\kappa }}_{3W}\right)+{\left({\mathit{B}}_{yW}^{{\mathcal{\kappa }}_1}{\mathcal{\kappa }}_{3W}\right)}^T\left({\mathit{B}}_{yW}^{{\mathcal{\kappa }}_1}{\mathcal{\kappa }}_{3W}\right)+{\left({\mathit{B}}_{zW}^{k_1}{\mathit{k}}_{3W}\right)}^T\left({\mathit{B}}_{zW}^{k_1}{\mathit{k}}_{3W}\right)$$

(8)

$${‖{\mathit{t}}_2‖}_2^2={\left({\mathit{B}}_{xW}^{k_2}{\mathit{k}}_{3W}\right)}^T\left({\mathit{B}}_{xW}^{k_2}{\mathit{k}}_{3W}\right)+{\left({\mathit{B}}_{yW}^{k_2}{\mathit{k}}_{3W}\right)}^T\left({\mathit{B}}_{yW}^{k_2}{\mathit{k}}_{3W}\right)+{\left({\mathit{B}}_{zW}^{k_2}{\mathit{k}}_{3W}\right)}^T\left({\mathit{B}}_{zW}^{k_2}{\mathit{k}}_{3W}\right)$$

(9)

where ${\mathit{B}}_{xw}^{k_1}={\mathit{B}}_x^{k_1}\cdot {\mathit{W}}_{m_1}^{-1}$, ${\mathit{B}}_{xw}^{k_2}={\mathit{B}}_x^{k_2}\cdot {\mathit{W}}_{m_2}^{-1}$, ${\mathit{B}}_{yw}^{k_1}={\mathit{B}}_y^{k_1}\cdot {\mathit{W}}_{m_1}^{-1}$, ${\mathit{B}}_{yw}^{k_2}={\mathit{B}}_y^{k_2}\cdot {\mathit{W}}_{m_2}^{-1}$, ${\mathit{B}}_{zw}^{k_1}={\mathit{B}}_z^{k_1}\cdot {\mathit{W}}_{m_1}^{-1}$, ${\mathit{B}}_{z\mathit{W}}^{k_2}={\mathit{B}}_z^{k_2}\cdot {\mathit{W}}_{m_2}^{-1}$, ${\mathit{W}}_{m_i}=\mathit{diag}\left({\left(w_h^{\epsilon }\right)}^{-1}\cdot w_V^{\lambda }\right),i=1,2$, $w_h$ and $w_V$ are the depth and volume of each tetrahedron unit, respectively. $\lambda$ is a volume weight factor, which is generally 0.5. ${\mathit{B}}_x^{k_i}$, ${\mathit{B}}_y^{k_i}$, and ${\mathit{B}}_z^{k_i}$ are the gradient matrices of the physical properties (${\mathcal{\kappa }}_1$ and ${\mathcal{\kappa }}_2$) in the x, y, and z directions, respectively, and the gradient calculation with unstructured tetrahedral grid meshing was obtained using the second-order Taylor formula [28].

3. Theoretical Model Tests

To simulate realistic subsurface conditions, we designed two inclined prism models with varying burial depths, set within a geomagnetic field characterized by an inclination of 45° and a declination of 60°. The detailed model parameters are provided in

Table 1. Model parameters.

Model	Center Coordinates of the Top (m)	Center Coordinates of the Bottom (m)	Length of Top and Bottom (m)	Width of Top and Bottom (m)	Magnetic Susceptibility (SI)
Inclined prism 1	(1700, 1300, −300)	(2000, 1400, −1300)	600	500	0.025
Inclined prism 2	(2000, 2900, −500)	(1700, 2800, −1600)	850	750	0.025

.

Figure 3a shows the spatial distribution of the models and the total-field magnetic anomaly (${\mathit{d}}_{\Delta T}$). Figure 3b–d show the magnetic gradient anomalies in three directions. Inversion calculations of the total-field and gradient magnetic anomalies were carried out, and the sections of the 3-D inversion results with an easting of 2000 m were obtained. The white solid boxes are the true positions of the models with an easting of 2000 m (Figure 3e–h). The inversion result obtained by the 3D magnetic gradient anomalies (Figure 3f) better depicted the upper boundary of the geological body and the shallower geological body. Thus, we conducted joint inversion of the total-field magnetic anomaly and 3D magnetic gradient anomalies (Figure 3g) and compared it with the inversion result of the total-field magnetic anomaly (Figure 3e). It was inferred that the joint inversion strategy would provide more information on deeper geological bodies. The inversion result obtained by the SSC method (Figure 3h) had a higher resolution and clearer boundaries, indicating that the introduction of gradient structures of physical properties can significantly improve the recovery ability of the magnetic susceptibility distribution of subsurface magnetic bodies. Under identical conditions, the recovered magnetic susceptibility values are 2.6 times higher than those from the joint inversion of total-field magnetic data and its gradients, with a 1.6-fold improvement in resolution. Since the iteration-stopping criteria were met at the 100th iteration, the entire inversion process iterated 100 times, and the root mean square (RMS) fitting error was calculated for each iteration. The final RMS fitting error for the joint inversion of total-field magnetic data and its gradients was 0.15 nT, whereas the self-structural constrained joint inversion of total-field magnetic data and its gradients achieved a final RMS fitting error of 0.02 nT. These results further demonstrate that the proposed self-structural constrained joint inversion method significantly enhances the accuracy of subsurface source recovery, as evidenced by the markedly reduced fitting error.

To simulate realistic field conditions where measured total-field and gradient magnetic anomalies are typically contaminated by noise, we introduced Gaussian noise with a signal-to-noise ratio (SNR) of 10 to the anomalies shown in Figure 3a–d. The resulting noisy total-field magnetic anomaly and 3D magnetic gradient anomalies are presented in Figure 4a–d, respectively. Figure 4e,f display cross-sections of the 3D inversion results at an Easting of 2000 m, obtained using conventional joint inversion and the SSC method, with white solid boxes indicating the true model positions at this easting. Both methods demonstrated robust noise immunity, effectively recovering the true model distribution with accuracy comparable to noise-free inversion scenarios. In addition to the reported 10% Gaussian noise test, the SSC method was evaluated under varying noise levels (5%, 15%, and 20% SNR). At 5% noise, SSC maintained >90% recovery accuracy for shallow and mid-depth structures. At 15%–20% noise, resolution improvement remained significant (30%–50% over conventional methods), though minor artifacts emerged in deeper regions (>1500 m). The proposed method in this study recovers magnetic susceptibility values four times higher than those obtained from the joint inversion of total-field magnetic data and its gradients under identical conditions, with a threefold improvement in resolution. For comparative purposes, the inversion process was set to 100 iterations, and the root mean square (RMS) fitting error was calculated at each iteration. The final RMS fitting error for the joint inversion of total-field magnetic data and its gradients was 13.33 nT, while the self-structural constrained joint inversion achieved a final RMS fitting error of 0.34 nT. Due to the influence of noise on the anomalous data, the inversion results partially fit the noise, leading to an increase in the RMS fitting error compared to noise-free conditions. This further highlights the robustness and superior accuracy of the self-structural constrained joint inversion method in recovering subsurface sources under noisy scenarios.

In realistic geological scenarios, subsurface magnetic bodies often exhibit complex distributions. To evaluate the applicability of the SSC method under such conditions, we designed three inclined prism models with identical burial depths (Figure 5), whose specific parameters are detailed in

Table 2. Model parameters.

Model	Center Coordinates of the Top (m)	Center Coordinates of the Bottom (m)	Length of Top and Bottom (m)	Width of Top and Bottom (m)	Magnetic Susceptibility (SI)
Inclined prism 1	(1300, 1300, −300)	(1400, 1400, −1300)	1000	1000	0.25
Inclined prism 2	(2900, 1300, −300)	(2800, 1400, −1300)	1000	1000	0.25
Inclined prism 3	(2900, 2900, −300)	(2800, 2800, −1300)	1000	1000	0.25

.

Figure 5a illustrates the spatial distribution of the models and their corresponding total-field magnetic anomaly, while Figure 5b–d present the magnetic gradient anomalies along the three orthogonal directions (${\mathit{d}}_{\Delta T_x}$, ${\mathit{d}}_{\Delta T_y}$, and ${\mathit{d}}_{\Delta T_z}$). The 3-D inversion results for the region where easting > 2800 m or northing > 2800 m are shown in Figure 5e,f, with white solid boxes indicating the true positions of the models within this area. Compared to the conventional joint inversion results (Figure 5e), the SSC-based joint inversion (Figure 5f) demonstrated significantly improved resolution, with recovered magnetic susceptibility distributions more closely approximating the true model geometries. The proposed method in this study recovers magnetic susceptibility values 3.8 times higher than those from the joint inversion of total-field magnetic data and its gradients under identical conditions, with a 2.8-fold improvement in resolution. During the inversion process, the iteration count was set to 100, and the root mean square (RMS) fitting error was calculated at each iteration. The final RMS fitting error for the conventional joint inversion of total-field magnetic data and its gradients was 6.09 nT, whereas the self-structural constrained joint inversion achieved a final RMS fitting error of 0.76 nT. These results demonstrate that, for more complex subsurface scenarios, the proposed self-structural constrained joint inversion method yields significantly higher accuracy in inversion results and better recovers the magnetic structures of subsurface magnetic bodies.

4. Real Data Application

The study area is situated in the southern sector of the ore district within the western mountainous region of Songxian County, Henan Province, China. As illustrated on the right side of Figure 6, the geological and tectonic framework of the area is characterized by a predominance of fault structures, with limited fold development and NE-dipping monoclinic strata [29]. Multiphase tectonic activity involving stresses from varying directions has generated a complex fracture network throughout the region. This structural complexity is further manifested in the formation of mylonite belts, cataclastic rock zones, and altered cataclastic rock belts, which result from ductile–brittle deformation and metamorphic processes. These features exhibit a strong spatial correlation with gold mineralization. The gold deposits, labeled as I, II, III, and IV in Figure 6, are strategically positioned within this structurally controlled metallogenic system.

The left side of Figure 6 presents a detailed profile (corresponding to the solid white line on the right side of Figure 6) of gold deposit I within the study area. The exposed strata belong to the upper section of the Middle Proterozoic Great Wall System, predominantly comprising volcanic and tectonic breccia. Among the volcanic rocks, andesite is the most extensively distributed and exhibits strong magnetic properties. In contrast, tectonic breccia, being a sedimentary rock, is generally non-magnetic or weakly magnetic. The blue dotted circle indicates the approximate location and spatial extent of the gold orebody. Drill hole data, spanning depths of 21.22–759.57 m, confirm the presence of gold mineralization. The ore-bearing structures exhibit undulating geometries along their dip, with dip angles progressively varying from surface to depth. Notably, gold mineralization is predominantly concentrated in transitional zones where the dip angle shifts from steep to shallow, highlighting the structural control of ore localization.

Figure 7 displays the topographic map of the study area, which is characterized by a highly mountainous terrain. The region exhibits a distinct geomorphological pattern, with elevated northern and southern sectors and a relatively lower central zone, creating significant topographic variations. From south to north, the area is dominated by three primary geomorphological features: Xiong’er Mountain, Yihe River Valley, and Waifang Mountain. The Yihe River Valley serves as a natural boundary separating Xiong’er Mountain to the south from Waifang Mountain to the north. Topographic elevations within the study area range from 300 to 800 m, reflecting the rugged nature of the landscape. The presence of trenches in the western portion of the study area is manifested as linear features on the topographic map, adding to the complexity of the terrain.

In 2019, a comprehensive 1:10,000 low-altitude aeromagnetic survey was conducted in the study area using a multi-rotor unmanned aerial vehicle (M600). The survey maintained an average flight altitude of 96.7 m above ground level, employing an optical pump magnetometer (Ru/GSMP-35A) as the primary aeromagnetic surveying instrument. The geomagnetic field parameters during the survey were characterized by an inclination of 52.5° and a declination of −4.5°.

The acquired total-field magnetic anomaly (${\mathit{d}}_{\Delta T}$) and three-component magnetic gradient anomalies (${\mathit{d}}_{\Delta T_x}$, ${\mathit{d}}_{\Delta T_y}$, and ${\mathit{d}}_{\Delta T_z}$) are presented in Figure 8a–d. The observed anomalies exhibit a distinct banded distribution pattern, which aligns closely with the regional fault structure orientation, demonstrating a strong structural control on the magnetic signature of the study area.

To identify potential locations of undiscovered gold deposits, we applied the SSC method to invert the magnetic susceptibility structure of the study area. For comparative analysis, conventional joint inversion of total-field and gradient magnetic data was also performed, with both methods evaluated based on their ability to resolve known gold deposits. The inversion results are presented in Figure 9a,b, with the locations of four existing gold deposits (I–IV) annotated for reference. Using rock magnetic susceptibility data from known gold deposits, we established a cut-off threshold of 0.01 SI to delineate highly magnetic bodies (primarily andesite) in the subsurface. The conventional joint inversion results (Figure 9a) showed good correspondence with gold deposit I but failed to accurately locate gold deposit II. While gold deposits III and IV were partially resolved, their horizontal positions significantly deviated from actual locations. In contrast, the SSC-based joint inversion (Figure 9b) demonstrated superior performance, with recovered magnetic susceptibility distributions showing strong spatial correlation with all four gold deposits. These results indicate that the SSC method provides more reliable inversion outcomes, making it particularly suitable for delineating potential ore-forming areas in the study region.

The discovered gold deposits are structurally controlled, occurring within fault-related alteration zones and exhibiting a distribution pattern strongly influenced by the basement fault tectonic belt. Based on the SSC inversion results and the fault distribution in the study area, we have identified six potential ore bodies exhibiting banded (vein-type) distributions (Figure 9b). These ore bodies predominantly strike in NE and NW directions, with vertical extents ranging from approximately 300 m to 2000 m in depth.

Our analysis revealed the presence of volcanic zones in the western portion of the study area, characterized by distinct closed-ring magnetic anomalies in the total-field data and corresponding topographic highs. Figure 9c presents the magnetic susceptibility results following volcanic zone subdivision, along with the proposed metallogenic model. We interpret the deposits in the study area as typical magmatic–hydrothermal type mineralization. During the late stages of volcanic activity, intense exhalative and hydrothermal processes transported significant quantities of metallic compounds in gaseous and liquid phases. As magma ascended along regional structural pathways, these ore-bearing hydrothermal fluids interacted with surrounding rocks under specific geological and physicochemical conditions. This process facilitated the concentration and precipitation of gold from the hydrothermal fluids, ultimately forming vein-type magmatic–hydrothermal gold deposits. Such deposit types typically exhibit substantial mining potential due to their structural control and mineralization intensity.

5. Conclusions

The proposed self-structural constraint (SSC) method, integrating total-field and gradient magnetic data through unstructured tetrahedral meshing, significantly enhances the resolution of subsurface magnetic susceptibility imaging for mineral exploration. By leveraging gradient structures of multi-component inversion results as mutual constraints, the SSC framework overcomes limitations of conventional methods in resolving complex geological settings. The derived tetrahedral magnetic gradient formulas based on Poisson’s theory ensure computational accuracy, validated by synthetic hexahedral models. Theoretical tests demonstrate that SSC improves spatial resolution by 40%–60% and robustly locates magnetic sources under noise-contaminated scenarios.

Applied to a gold-bearing district in Henan Province, China, the SSC inversion delineated six concealed vein-type orebodies (300–2000 m depth) along NE- and NW-trending faults, aligning with high magnetic anomalies at andesite boundaries. Integration with the terrain and volcanic facies analysis revealed that NE-oriented faults acted as conduits for magma upwelling, supplying both metallogenic materials and structural traps for gold mineralization. These findings not only refine the ore-controlling fault model but also identify two high-priority drill targets. The SSC method bridges high-resolution geophysical imaging with critical metallogenic processes, offering a transformative approach for detecting deep-seated, structurally controlled mineral systems. This advancement underscores the pivotal role of innovative magnetic techniques in unlocking concealed resources and optimizing exploration efficiency.