Three-Dimensional Magnetic Inversion Based on Broad Learning: An Application to the Danzhukeng Pb-Zn-Ag Deposit in South China

Abstract

Three-dimensional (3-D) magnetic inversion is an essential technique for revealing the distribution of subsurface magnetization structures. Conventional methods are often time-consuming and suffer from ambiguity due to limited observations and non-uniqueness. To address these limitations, we propose a novel inversion method under the machine learning framework. First, we design a training sample generation space by extracting the horizontal positions of magnetic sources from the analytic signal amplitude and the reduced-to-the-pole anomalies of magnetic field data. We then employ coordinate transformation to achieve data augmentation within the designed space. Subsequently, we utilize a broad learning network to map the magnetic anomalies to 3-D magnetization structures, reducing the magnetic inversion time. The efficiency of the proposed method is validated through both synthetic and field data. Synthetic examples indicate that compared to the traditional inversion method, the proposed method approximates the true model more closely. It also outperforms traditional and deep learning methods in terms of computational efficiency. In the field example of the Danzhukeng Pb-Zn-Ag deposit in South China, the inversion result is consistent with drilling and controlled-source audio frequency magnetotelluric survey data, providing valuable insights for subsequent exploration. This study provides a new practical tool for processing and interpreting magnetic anomaly data.

1. Introduction

As an effective geophysical survey method, magnetic prospecting measures magnetic anomaly data on the Earth’s surface to investigate the underground magnetization structure. It has found widespread applications in mineral resource exploration [1,2,3], pollutant monitoring [4], and geological structure research [5,6]. The 3-D magnetic inversion constitutes a crucial step in processing magnetic anomaly data. It enables researchers to determine the shapes, burial depths, and magnitude of magnetic sources beneath the study area [7,8,9,10]. Consequently, 3-D magnetic inversion methods have garnered considerable attention and have seen significant advancements.

Over the past few decades, conventional 3-D inversion methods have aimed to achieve reasonable results by optimizing regularized objective functions. These functions balance data fitting and model constraints [11]. Various constraint conditions have been proposed to address the challenges posed by limited observations and non-uniqueness. Examples include smoothness constraints [12], focusing inversion [13], fuzzy c-means clustering [14], and sparsity constraints [15]. The inversion problem has been transformed into an optimization problem involving a large ill-posed matrix. Deterministic algorithms, such as the conjugate gradient [16] and Gauss–Newton [17] methods, have been adopted for inversion problems due to their advantages in solving large systems. Moreover, global optimization algorithms, like ant colony optimization [18] and particle swarm optimization [19], have also been introduced into magnetic inversion processes since they weakly depend on the initial model. These inversion methods have demonstrated their efficacy in field data applications, yielding reliable estimates of subsurface magnetization structures [20,21]. One limitation of conventional methods is the time-consuming iterative optimization process, particularly when addressing 3-D inversion problems [22]. Fast inversion algorithms based on 2-D FFT have been proposed to address this issue [23,24]. However, they struggle with undulating terrain and irregularly spaced data, limiting their applicability.

In recent years, deep learning (DL) has been increasingly applied to geophysical exploration due to the approximation ability of arbitrary functions [25]. By training on sample data, DL networks automatically learn and implicitly express the nonlinear relationships between geophysical parameters and their corresponding observational data. This has yielded significant results in various applications, including seismic data processing [26,27], magnetotelluric and electrical resistivity inversion [28,29,30], and fault identification [31]. For potential field data 3-D inversion, a deep convolutional neural network was developed for 3-D gravity inversion. Each layer of the density structure was treated as a separate channel output [32]. To enhance the inversion accuracy, a 3-D refiner was added to the encoder–decoder architecture. This allowed 2-D field data to be converted into a 3-D density model [33]. Furthermore, 3-D convolution was used to extract 3-D spatial information from multivariate magnetic data, enabling 3-D to 3-D mapping [34,35]. These successful applications reflect that the trained network can achieve real-time reflection of the field data without calculating the kernel matrix [12] and is independent of the initial model [36,37]. However, DL methods are hindered by time-consuming training processes due to increasingly complex architectures and numerous hyperparameters [38]. Additionally, the generation of adequate training samples is essential for ensuring the network’s generalization capability, but this process itself is also time-intensive [33,34,35,36,37]. As a result, DL training requires significant computational resources and frequently does not provide efficiency benefits over conventional methods in 3-D magnetic inversion.

As an alternative to DL, Chen and Liu proposed an efficient network structure known as broad learning (BL) [39]. Compared to DL, BL consists of a single hidden layer. It enhances mapping capabilities by increasing the complexity within that single layer, rather than by adding more hidden layers. This approach eliminates the need for layer-by-layer gradient function calculations of DL, thereby improving network training efficiency. Chen and Liu demonstrated a significant reduction in training time while maintaining classification accuracy comparable to other machine learning methods [39]. At present, the BL network has been successfully applied in time series prediction [40], electroencephalography [41], and geophysics inversion problems [42,43,44,45,46]. Recently, Xu et al. applied the BL approach to 3-D gravity data inversion and found that the BL network could promote the construction of density structures of subsurface [47]. Until now, although BL has made satisfactory advances in seismic, water resources, electrical resistivity, and gravity processing, it has not been used in the 3-D inversion of magnetic anomaly data.

Therefore, as a first attempt, we introduce the use of the BL technique to recover the distribution of magnetization structure from magnetic anomalies. Additionally, we also propose a strategy to quickly establish training samples, including magnetic anomaly data (input) and the corresponding magnetization structure (output). The combination of BL network and fast sample generation strategy facilitates rapid 3-D magnetic inversion. Finally, we applied the proposed method to both synthetic and field data to validate our approach.

2. Methods

In 3-D magnetic inversion, the subsurface space beneath the study area is typically discretized into rectangular grids, each characterized by unknown magnetization properties. A constant inclination (I) and declination (D) of magnetization within the study area and an absence of remanent magnetization are assumed. The total magnetic intensity (TMI) anomaly data on the observation surface, resulting from these magnetization grids, can be expressed as follows [12]:

d_i = G_ij·m_j

(1)

where d_i represents the TMI (nT) at the ith point on the observation surface, m_j denotes the magnetization magnitude (A/m) in the jth grid cell, and G_ij quantifies the contribution of magnetization in the jth grid cell to the ith point.

The BL inversion (BL–Inv) in this study followed the same grid meshing structure, utilizing the magnetization structure as the label and the corresponding TMI as input to learn the mapping relationship between the two:

m = ψ (d; Θ).

(2)

where ψ and Θ represent the objective function and the parameters of BL network, respectively. As shown in Figure 1, the general procedure of BL–Inv is summarized as follows:

• Step 1: Design the generation space of samples according to the distribution characteristics of TMI.
• Step 2: Establish training samples in the designed generation space. Train the BL network with the samples to obtain the parameter matrix Θ.
• Step 3: Input the field TMI data into the trained BL network to predict the underground magnetization structure.

2.1. Sample Generation Space Design

As a supervised learning network, the inversion performance of BL relies heavily on the training dataset. In this paper, we propose a method to enhance the quality of sample data and improve the efficiency of its establishment. As shown in Figure 2, we first performed reduced-to-the-pole (RTP) conversion on the TMI data requiring inversion (Figure 2a) to obtain the TMI distribution under vertical magnetization [48]. The high values of the RTP results (Figure 2b) reflect the horizontal positions of the subsurface magnetization structures, particularly their central locations. We then calculated the analytic signal amplitude (AS) (Figure 2c) of the RTP results as follows:

$$\mathrm{AS}=\sqrt{{\left({\displaystyle \frac{\partial {\mathrm{d}}_{\mathrm{rtp}}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial {\mathrm{d}}_{\mathrm{rtp}}}{\partial \mathrm{y}}}\right)}^2+{\left({\displaystyle \frac{\partial {\mathrm{d}}_{\mathrm{rtp}}}{\partial \mathrm{z}}}\right)}^2}$$

(3)

The high values of the AS indicate the boundaries of magnetization structures on the observation surface [49]. By integrating the significant RTP and AS values, we delineated the horizontal positions of the magnetization structures. To simplify the establishment process, the shape of the sample generation space was fixed as a cuboid (Figure 2d). Finally, we designed the 3-D space under the horizontal positions as the sample generation space (Figure 2e).

Following the above steps, we divided the study area into several independent regions, each corresponding to different magnetization structure. The training samples were specifically generated within these discrete spatial regions, rather than encompassing the entire study area. Prior information, including the number and horizontal positions of the magnetic sources, was embedded into the training samples to enhance their quality.

2.2. Sample Generation

Based on previous research [50,51], magnetic ore bodies typically form under specific temperature and pressure conditions, exhibiting significant magnetic susceptibility differences compared to the surrounding rocks [19,52]. To simulate this characteristic, this study employed sharp magnetization structures to generate the training samples. Specifically, within each independent designed space, we randomly assigned a magnetization value in the range 0 to 50 A/m to specific grids, forming a single magnetization structure. The length, width, height, and center burial depth of the magnetization structure can change randomly in a constrained space, with its shape remaining a rectangular cuboid with sharp boundaries [32,33,36]. We then performed coordinated rotations around the magnetization structure’s center point to generate complex structures oriented in different directions. The rotation method is as follows [47]:

$$\left[\begin{array}{c}\mathrm{x}\prime \\ \mathrm{y}\prime \\ \mathrm{z}\prime \end{array}\right]=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \beta & 0& -\sin \beta \\ 0& 1& 0\\ \sin \beta & 0& \cos \beta \end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \gamma & \sin \gamma \\ 0& -\sin \gamma & \cos \gamma \end{array}\right]·\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right].$$

(4)

where [x, y, z]^T represent the coordinates of the rectangular magnetization structure, [x’, y’, z’]^T are the corresponding coordinates after rotation; the angles α, β, and γ result from rotations along the Z, Y, and X axes, respectively.

Figure 3a illustrates a cuboid magnetization structure along a north–south trend with a rotation angle of (α = 0°, β = 0°, γ = 0°). As shown in Figure 3b, rotating it along the Z-axis (α = 45°) resulted in a horizontally inclined plate magnetization structure. Furthermore, the rotation of the cuboid along the Y-axis (β = 60°) enabled the facile generation of a complex dam-like magnetization structure (Figure 3c). The complexity and fast establishment of training samples were achieved through coordinate rotation. Figure 3d–f show three representative examples of magnetization structures generated by the proposed method within a designed space consisting of three spatial regions in Figure 2e. The corresponding TMI (Figure 3g–i) caused by these three forward models can be calculated using Equation (1).

2.3. Broad Learning

The number of samples used for BL training is denoted by L. Each sample pair comprised the TMI data, consisting of N observation points, and the corresponding magnetization structure, which consisted of M grids. The BL first generated feature nodes from input data as follows:

F_i = Φ_i(XΘ_Fi + K_Fi)  i = 1, 2, …, Q.

(5)

where X is an L × N dimensional matrix, representing the input samples; Φ_i is the mapping function, with Θ_Fi and K_Fi being the ith random weight and bias matrixes of sizes N × P and L × P, respectively; F_i is the ith feature node, where F_i≡ [f₁, f₂, …, f_P], P is the number of neurons for each feature node (Figure 1), and Q is the number of feature nodes. Second, the mapped features are enhanced by enhancement nodes:

E_j = Ψ_j(F^QΘ_Ej + K_Ej)  j = 1, 2, …, R.

(6)

where F^Q≡ [F₁, F₂, …, F_Q], Ψ_j is the hyperbolic tangent (tanh) function, with Θ_Ej and K_Ej being the jth randomly created weight and bias vectors of size (P × Q) × 1; E_j is the jth enhancement node, and R is the total number of enhancement nodes.

The output was obtained by combining the connections of all feature and enhancement nodes:

Y = [F^Q|E^R] Θ,

(7)

where Y is an L × M dimensional matrix representing the label samples; E^R≡ [E₁, E₂, …, E_R], and Θ is the weighting matrix for BL network. The objective function of the BL network can be defined using the following equation:

$$\arg \min : {‖\mathrm{A}\mathbf{\Theta }-\mathrm{Y}‖}^2+\mu {‖\mathbf{\Theta }‖}^2,$$

(8)

where A = [F^Q|E^R], μ is the regularization parameter. The desired weight Θ was determined through ridge regression using the pseudoinverse method [39]:

Θ = (A’A + μ·I)⁻¹A’Y.

(9)

where I is an identity matrix, A’ is the transposed matrix of A.

2.4. Parameter Tuning

As mentioned in Section 2.3, the BL network structure is controlled by three hyperparameters: P, Q, and R. P represents the number of neurons in each feature node, Q denotes the total number of feature nodes, and R indicates the number of enhancement nodes. Selecting an optimal set of hyperparameters is crucial to ensure the mapping capabilities of the BL network. Following previous studies [46,47], we conducted an exhaustive search within the range of [20:20:200] × [20:20:200] × [20:20:200] to identify the optimal hyperparameters.

We generated 30,000 samples and divided them into training and validation datasets using the k-fold cross-validation method, with k set to 5, to optimize the hyperparameters [53]. Two metrics were adopted to evaluate the inversion performance with different hyperparameters. The first metric was the relative error (B), which quantified the accuracy of the inversion results. The second metric was the coefficient of determination (R²), which assessed the similarity between the inversion result and the underground truth. The metrics are defined as follows [54]:

$$B={\displaystyle \frac{‖\widehat{\mathrm{m}}-\mathrm{m}‖}{‖\mathrm{m}‖}}\hspace{1em}\hspace{1em}{R}^2=1-{\displaystyle \frac{{‖\widehat{\mathrm{m}}-\mathrm{m}‖}^2}{{‖\mathrm{m}-\stackrel{\mathrm{-}}{\mathrm{m}}‖}^2}}.$$

(10)

where $\widehat{\mathrm{m}}$ and m represent the prediction and the true magnetization structure, respectively, and $\stackrel{\mathrm{-}}{\mathrm{m}}$ is the mean value of m. The value of B reaches its minimum, and R² reaches its maximum when P = 200, Q = 100, and R = 80.

3. Results

This section introduces two numerical models and field data to verify the proposed inversion method.

3.1. Synthetic Model I

The first test model, as displayed in Figure 4a, consisted of a rectangular anomaly body with a magnetization magnitude of 1 A/m. The magnetization structure was centered within the study area and had dimensions of 750 m (length), 750 m (width), and 450 m (height), with a center burial depth of 400 m. Figure 4b presents the original TMI generated by synthetic model I under the magnetization direction of I = 0° and D = 0°. The pseudo-inclination RTP algorithm [55] was applied to the original TMI, resulting in the RTP TMI shown in Figure 4c. The AS result derived from the RTP TMI is depicted in Figure 4d. The sample generation space was then designed based on the boundary (indicated by the black dashed lines in Figure 4c,d) and is represented by the light green block in Figure 4e.

To generate the training sample, we divided the underground 3-D space beneath the study area into 64 × 64 × 16 grids, with each grid measuring 50 × 50 × 50 m³. We respectively established 30,000 random samples within the entire study space and 30,000 designed samples within the restricted space shown in Figure 4e to verify the effectiveness of our proposed sample generation strategy. At the same time, we adopted the DL network structure (DL-Inv) published by Xu et al. [56] for a comparative analysis with the BL-Inv. Figure 5 shows the inversion results obtained by the two networks trained with different samples. A comparison between Figure 4a and Figure 5 reveals that our sample generation strategy simultaneously improved the inversion performance of both DL-Inv and BL-Inv. The metrics in

Table 1. Evaluation metrics and time consumption comparison of DL-Inv and BL-Inv methods.

Method	B	R2	Time (h)
DL-Inv (random)	0.6045	0.7757	2.33
BL-Inv (random)	0.5766	0.7006	0.18
DL-Inv (designed)	0.3966	0.8393	2.33
BL-Inv (designed)	0.4146	0.8380	0.18

demonstrate that the accuracy of our proposed inversion method is comparable to that of the DL network.

The entire process, including establishing 30,000 training samples, training the network, and predicting the magnetization structure, took the BL-Inv method a total of 0.18 h on a desktop computer (Intel 11th Gen Core i7-11700KF @ 3.60 GHz and 64.00 GB (3200 MHz) RAM) without parallel computing. In contrast, the DL-Inv method took 2.33 h with GPU acceleration (NVIDIA GeForce RTX 3080). Compared to DL-Inv, the BL-Inv approach achieves about a 92.3% reduction in time cost.

For enhanced visualization of the inversion results, we focus on the slice at X = 1850 m (Figure 6). Specifically, the inversion results obtained from DL-Inv exhibit sharp boundaries (Figure 6b). However, the magnitude of magnetization is higher than that of the true test model. In contrast, the inversion results of BL-Inv trained on random samples display blurred boundaries and lower magnitudes compared to the test model (Figure 6c). Figure 6d shows the inversion results of DL-Inv trained with designed samples. It is evident that the magnetization magnitude fluctuates within the structure, especially at the boundaries where high values are observed. The inversion results generated by BL-Inv trained with designed samples exhibit clear boundaries and a smooth internal structure (Figure 6e). Figure 6f further illustrates the comparison results on the horizontal profile, as indicated by the white dashed line in Figure 6a–e, with a depth of 300 m in the X = 1850 slice. The inversion results obtained from the machine learning methods trained with designed samples exhibit a closer approximation to the true magnetization magnitude (1 A/m). These results display a sharp boundary that is nearly consistent with the true magnetization structure.

3.2. Synthetic Model II

We designed the synthetic model II as a complex model (Figure 7a) to further test the proposed method. This model includes a rectangular magnetization structure situated at depths ranging from 300 to 600 m underground, a horizontally inclined plate located between 100 and 400 m underground, and a ladder structure positioned at depths between 150 and 450 m underground. Figure 7b,c display the original TMI generated by model II under the magnetization direction of I = 30°; D = 15° and the RTP TMI. The AS distribution of the RTP results is shown in Figure 7d. We obtained the horizontal positions based on the RTP and AS results and designed a sample generation space consisting of three independent cuboid areas, as shown in Figure 7e. We generated 30,000 designed samples to train the DL and BL network. In addition to these methods, we employed the smooth inversion method proposed by Li et al. [17], a traditional 3-D magnetic inversion method, as another comparative approach. The regularization parameter for the traditional inversion approach was determined by the L-curve method [11].

Figure 8a–c depict the 3-D magnetization structures obtained from three inversion methods. A false anomaly body [17] is found in the traditional inversion result at the bottom boundary (Figure 8a), whereas the DL-Inv and BL-Inv results exhibit no false anomalies. Figure 8d shows the slice at Y = 600 m for true model and three inversion results. We can see that the inversion results of the BL-Inv, trained with designed samples, reveal intricate ladder-like magnetization structures. In comparison, DL-Inv does not accurately depict the boundaries of the ladder structure (Figure 8b–d). Figure 8e presents comparison results along the vertical profiles at X = 500 m, X = 750 m, X = 2500 m, and X = 2800 m, as indicated by the white dashed line in Figure 8d. The inversion results of the smooth method extend deeper than the true magnetization structure, whereas the BL-Inv method more closely matches the true model. Furthermore, Figure 8f displays the slice at Y = 2500 m of the true model and inversion results. It can be seen that the machine learning methods recovered the deep regular magnetization structure. However, the DL-Inv method did not result in an accurate depiction of shallow horizontally inclined plates. Figure 8g also reflects that the inversion results of BL-Inv are closer to the true model. These test results demonstrate that BL-Inv is effective for recovering the shape and burial depth of multiple magnetic sources with varying sizes and magnitudes.

Table 2. Evaluation metrics and time consumption comparison of smooth inversion, DL-Inv, and BL-Inv methods in synthetic model II.

Method	B	R2	Time (h)
Smooth inversion	0.6157	0.6063	1.43
DL-Inv (designed)	0.5002	0.7357	2.33
BL-Inv (designed)	0.4988	0.7306	0.18

presents the evaluation metrics and total time consumption of the different methods, including the sample establishment, network training, and prediction. In terms of inversion accuracy, both DL-Inv and BL-Inv outperformed the traditional inversion method, with BL-Inv being very close to DL-Inv. Regarding computational efficiency, BL-Inv surpassed the other two methods, saving about 87.4% and 92.3% of the time consumption, respectively.

3.3. Field Data Application

We applied the BL-Inv method to field data derived from an aeromagnetic survey conducted over the Danzhukeng Pb-Zn-Ag deposit in Guangdong Province, China. Figure 9a illustrates the fault and strata information of the study area. The faults in this region are predominantly oriented in the northeast and northwest directions. Notably, the northwest-trending faults are mineralized zones, exhibiting intense mineralization at intersections with nappe structures [57]. The surface geology primarily consists of muddy siltstone, calcium–siliceous silty slate, quartz fine sandstone, metamorphic sandstone, slate, and siltstone. Surface samples and drilling data reveal that the principal economic minerals are Pb-Zn-Ag ores containing pyrrhotite. Magnetic parameter measurements (

Table 3. Statistical data of magnetic susceptibility for ores and rocks [57].

Type	Sample Number	Susceptibility κ (4π × 10−6 SI)
		Max	Min	Mean
Quartz sandstone	11	11.80	1.67	3.80
Siltstone	11	12.30	1.55	3.77
Pb-Zn-Ag ores	9	58.15	6.63	20.21
Silty slate	69	4414	431	1547
Pyrrhotite Pb-Zn	6	8374	1036	3112

) of the collected samples show that these ores have higher magnetic susceptibility than the surrounding rocks. The geomagnetic intensity of the study area is about 45,930 nT, with a magnetization direction of I = 35.5°; D = −3.7°. It was found that the induced magnetization magnitude produced by pyrrhotite Pb-Zn ores varies from 0.48 to 3.85 A/m [58]. Figure 9b displays the original TMI obtained from the aeromagnetic prospection data following standard protocols, ranging from −80 to 380 nT.

The RTP TMI for the study area was calculated using the pseudo-inclination RTP method, as shown in Figure 10a. The TMI distribution after RTP is observed to be largely consistent with the known mineralized zones, demonstrating the reliability of the RTP results. In view of this, we speculate that the study area is primarily characterized by induced magnetization [58]. Figure 10b shows the distribution of AS. Due to the high measurement altitude (100 m), the AS values are relatively low. Thus, we primarily designed the sample generation space based on the RTP results. We delineated four areas based on the relatively high values from the RTP and AS results, as indicated by the white dashed lines in Figure 10a,b. Finally, we designed a sample generation space consisting of four independent prismatic regions, as shown in Figure 10c.

We divided the underground space of the study area into 42 × 52 × 20 grids, each with a size of 100 × 100 × 50 m³. We established 30,000 input–output sample pairs within the designed space to train the BL network. For generating samples, we randomly selected rotation angles (α) at 15° intervals between −45° and 15°, considering that the mineralized zones predominantly trend north and northwest. Additionally, we set the rotation angles (β and γ) at 5° intervals within the range of −15° to 15°. Subsequently, we input the original TMI of the field data into the trained network to predict the inversion result. The BL-Inv method took only about 0.15 h for sample establishment, network training, and prediction. The high magnetization magnitude in the northern part of the study region aligns with drilling locations (Figure 11a), affirming the reliability of the inversion result. The magnitude of the BL-Inv inversion results ranges from 0 to 2.5 A/m, which is basically consistent with the physical property measurement. This is lower than the magnetization magnitude of ordinary iron mines [11,17,58], as the magnetic material in the study area is mainly pyrrhotite rather than magnetite.

We selected a slice (AA’ in Figure 9a, Figure 10a and Figure 11a) to provide detailed insights into the BL-Inv inversion results (Figure 11b). The AA’ slice successively crosses through mineralization faults F₈ and F₆. For fault F₈, a high magnetization magnitude region was found at depths ranging from 50 to 650 m, within an interval of approximately 1200 to 1700 m in the AA’ slice. ZK₁ drilling identified Pb-Zn-Ag ore and pyrrhotite within the high magnetization zone in fault F₈, occurring at depths of 50 to 400 m. This finding confirms the effectiveness of our method. Zhang and Li conducted a controlled-source audio frequency magnetotelluric (CSAMT) survey (indicated by the red line in Figure 9a) in this region, revealing a low-resistance area approximately 0 to 700 m underground, which aligns with our inversion results [57,59]. Both the CSAMT and BL-Inv magnetic 3-D inversion results highlight significant prospecting potential in the deeper portions of this region. For fault F₆, a relatively low magnetization structure exists at depths ranging from 150 to 550 m within an interval of approximately 2300 to 3500 m in the AA’ slice. Geochemical surveys indicate that elements such as Ag, Zn, As, Au, and Pb are enriched in this area [57], which is consistent with our inversion results around fault F₆.

4. Discussion

Prior studies [5,9,10,12,13,14,15,16,17,18] have demonstrated that 3-D inversion is a crucial technique for interpreting magnetic anomaly data and aiding mineral exploration. However, both traditional inversion and deep learning methods face time-consuming challenges. This study introduces the BL, a single-layer network architecture, into magnetic inversion, enhancing inversion efficiency. Compared to the traditional inversion method, the inversion results of our method are closer to the true model in synthetic data tests. Compared to deep learning, the inversion results from our approach are smoother. Studies [60,61] have indicated that adding a regularization term to the objective function of a deep network can improve stability. The lack of a regularization term in our DL-Inv structure [56] may be the reason for its structural fluctuations. In terms of computational efficiency, the BL-Inv method has significant advantages, reducing time consumption by at least 87%. The simplified network architecture enables BL-Inv to be flexibly applied to magnetic anomalies in various regions, including areas with undulating terrain.

In this study, we constructed a sample generation space using the RTP and AS transformation results of magnetic anomalies. Through this strategy, we implicitly embedded the number and horizontal coordinates of magnetic sources into BL-Inv, enhancing the network’s mapping capabilities. In field data measurement, noise interference may arise, affecting the estimation of the horizontal boundaries of magnetic sources. Furthermore, the establishment of the sample generation space requires manual delineation, which can lead to different results depending on the user. We tested the impact of noise and different sample generation spaces on BL-Inv using synthetic model I. Figure 12a displays the AS results of the noise-free original TMI generated by test model I. We designed a sample space based on the AS results that is consistent with the true model boundaries, as indicated by the black dashed lines in Figure 12a. We then added 5% Gaussian white noise to the original TMI [25,28,36], with the AS results shown in Figure 12b. In Figure 12b, we assumed that the establishment of the sample space was not affected by noise interference and remained consistent with the true model. Figure 12c,d show a sample space boundary smaller and larger than the true model, respectively, affected by noise and human influence, as indicated by the black dashed lines.

Figure 13a,b respectively show the inversion results for noise-free and noisy inputs within the same sample space. It can be observed that the proposed method was affected by noise, with the R² value dropping from 0.8599 to 0.8480. The results still accurately reflect the boundary and depth of model I, indicating that BL-Inv has a certain level of noise resistance. Figure 13c,d respectively show the inversion results within different sample spaces for noisy input. The R² values between the inversion results corresponding to different spaces and the true model are 0.8061 and 0.8098, respectively. Compared to Figure 13b, the R² values decrease due to inaccurate sample spaces. Yet, the R² values are all above 0.8, indicating that the proposed method is still applicable. In contrast, Liu et al. applied a U-Net structure to the conversion of magnetic field derivatives [62], and Zhang et al. used a fully convolutional network for delineating magnetic source boundaries [63]. These studies have demonstrated that machine learning methods can be directly used for the horizontal positioning of magnetic sources. In future studies, we can use these machine learning methods to determine the sample generation space objectively.

In our method, during the sample establishment process, the size, position, and rotation angle of the samples are randomly set within the designed space. This inherent randomness results in variations in the BL-Inv samples for each training session, potentially leading to different inversion results for the same field data, particularly under noise interference [28,32]. In order to estimate this uncertainty, we independently ran BL-Inv 25 times to achieve the inversion of the synthetic model II to verify its stability. The R² between the true magnetization structure and the 25 inversion results ranges from 0.7195 to 0.7396, demonstrating the method’s accuracy and stability. The mean value of the 25 inversion results is shown in Figure 14a. It can be seen that the mean result captures the 3-D distribution and magnitude of the magnetization structures in model II. Figure 14b presents the standard deviation of the 25 inversion results, and the maximum standard deviation is less than 0.2 A/m (10% of the true model).

The process of establishing training samples does not consider remanent magnetization, which may cause the magnetization direction to deviate from the direction of the geomagnetic field and become an unknown factor. The BL-Inv is generally more suitable for targets with strong induced magnetization. In the presence of significant remanent magnetization within the study area, the TMI data can be transformed into a magnitude magnetic anomaly data (MMA) [8,64]. The MMA training samples can then be established to train the BL network to achieve inversion. We employed RTP and AS results to delineate the generation space of training samples, which enhanced the mapping ability of the BL network. When the distribution of magnetic sources is highly complex, inaccurate delineation may lead to a decrease in the inversion accuracy. In the future, we plan to apply the BL network to the horizontal localization and 3-D inversion of complex geometric magnetic sources.

As described in Section 2.2, the magnetization structures within the training samples exhibited sharp boundaries. Therefore, the BL-Inv method is most effective in scenarios where magnetic sources display sharp boundaries and compact geometries, as these features were embedded into the training samples. For dispersed or gradational magnetization distributions, the method may not yield accurate results unless retrained with representative samples. An alternative approach is to use the inversion results of BL-Inv as the initial model for the traditional smooth inversion method. This strategy enhances the computational efficiency of traditional methods, as the BL-Inv result is closer to the true magnetization structure compared to a homogeneous initial model [46]. Our research has demonstrated that incorporating prior information into training samples can improve the accuracy of machine learning inversion. In the future, additional prior information, such as whether magnetic sources are dispersed and the upper limit of magnetization magnitude, derived from drilling and other geophysical exploration methods, can be incorporated to further constrain the inversion ambiguity.

5. Conclusions

In this study, we present a novel 3-D magnetic inversion method based on a BL network for reconstructing the 3-D magnetization structure from TMI data. It extracts the horizontal coordinates of the magnetic sources from the RTP and AS results of TMI to constrain the sample establishment, enhancing the network’s mapping ability. The simple network architecture allows for efficient magnetic inversion across various observation regions. The proposed BL-Inv approach is validated by both synthetic and field TMI data. Compared with conventional smoothing inversion, our results demonstrate clearer boundaries. This study provides a valuable sample generation strategy and a new 3-D magnetic inversion approach that may contribute to high-precision imaging of magnetization structures.