Inversion of Gravity Anomalies Based on U-Net Network

Abstract

The deep learning-based gravity anomaly inversion method addresses the complex challenge of deriving subsurface density variation models from surface gravity anomaly data. In order to generate various geological environments and their corresponding surface gravity anomaly datasets, three-dimensional density models considering different spatial locations and density variations are created in this paper. At the same time, the residual module and spatial attention mechanism are introduced into the U-Net architecture to improve the learning ability and inversion accuracy of complex geological structures. Experimental results demonstrate that the proposed method achieves the high-precision reconstruction of density variation models in complex anomaly environments, with a model residual error lower than 3%. Additionally, the inversion results of the density change and the gravity change in the Longshoushan fault zone show that the 2022 Menyuan MS6.9 earthquake is in the middle of the positive and negative density changes, which verifies the applicability of the U-Net network in the field of gravity change data, highlighting the method’s value in the real-world environment.

1. Introduction

Gravity inversion, a classical non-unique inverse problem, has seen significant progress through the development of various regularization techniques aimed at improving its stability and accuracy. Minimum volume constraints and their refinements have effectively restricted the solution space, resulting in more compact and interpretable inversion models [1,2,3,4,5,6,7]. Similarly, inversion methods that incorporate minimum gradient support constraints have greatly enhanced the sharpness and compactness of reconstructed structures [8,9,10]. The integration of depth-weighting functions into objective functions has helped mitigate the “skin effect”, thereby improving the vertical resolution of subsurface models [11,12]. Furthermore, the incorporation of the velocity model has significantly improved the reconstructions of density models by aligning inversion results with plausible geological environments [13].

Despite these advancements, challenges remain due to the reliance on initial models, susceptibility to local minima, and limitations in computational efficiency and stability. These issues are particularly pronounced in complex geological structures, where constructing accurate initial models is both technically challenging and computationally intensive [14].

Overcoming these challenges requires innovative methodologies to enhance the applicability of gravity inversion in complex subsurface systems. The integration of big data technologies and artificial intelligence has led to substantial advancements in geophysical data processing. Machine learning techniques, such as clustering [15,16], decision trees [17], support vector machines [18], Bayesian methods [19,20], and artificial neural networks (ANNs) [21], have gradually replaced the traditional reliance on initial models and sensitivity matrices. Its powerful global optimization capability makes machine learning one of the key research directions in geophysics [22].

Deep learning (DL), a specialized subset of machine learning, has tackled ill-posed inverse problems through hierarchical feature extraction [23,24]. This approach eliminates the need for manual feature engineering while improving the approximation of complex functions [25]. Studies by Kim and Nakata have demonstrated deep learning’s effectiveness in addressing intricate data challenges in large-scale inversion tasks [26]. In the geophysical data processing, deep learning algorithms have significantly improved the resolution and adaptability in reconstructing complex geological structures [27,28,29]. For instance, Wang et al. developed a novel method for magnetotelluric data inversion using genetic neural networks, significantly enhancing accuracy [30]. Pusyrev employed fully convolutional neural networks (FCNs) for the two-dimensional inversion of vertical magnetic dipole source data from boreholes [31]. Additionally, Pusyrev and Swidinsky applied deep learning to improve the precision and efficiency in the one-dimensional inversion of marine controlled-source electromagnetic data [32]. Noh et al. successfully integrated convolutional neural networks (CNNs) into airborne electromagnetic data inversion [33]. Li et al. utilized FCNs to reconstruct velocity models from seismic data, optimizing both resolution and efficiency [34]. Zhang et al. applied an optimized U-Net architecture to analyze gravity and gravity gradient data and achieved the accurate detection of density anomalies with improved efficiency [35].

The main challenge faced by deep learning in the inversion of gravity data is the difficulty in obtaining real underground spatial data, compounded by the potential for simulated data to deviate from real physical laws. To address this, this paper employs forward modeling to ensure that the results adhere to real physical laws. Additionally, deep learning models often exhibit a poor generalization to unseen data, necessitating transfer learning tailored to the specific requirements of different scenarios. Despite these challenges, the trained model demonstrates high efficiency, strong nonlinear mapping capabilities, and end-to-end automation.

This study employs a U-Net architecture to develop a nonlinear mapping model that directly links gravity anomaly inputs to density variation anomaly outputs. U-Net is a classical deep learning model, and its symmetry plays important roles, such as image segmentation and geophysical inversion. Two important mathematical properties—reflection symmetry and scale invariance—endow the model with an excellent robustness and generalization ability. Through supervised training and validation on a targeted dataset, this approach effectively inverts observed gravity data, realizes the time-varying gravity data inversion of three-dimensional density variation, completes accurate model reconstruction, and offers an advanced solution for exploring complex geological systems.

2. Dataset Construction

In this study, a sample dataset generator is developed to simulate three-dimensional structural anomalies and their corresponding gravity anomalies. This addresses the challenge of obtaining accurate and comprehensive real subsurface interface data. The generator can be configured with parameters that closely reflect subsurface conditions, enabling the creation of density variation anomalies with specific shapes, sizes, and spatial distributions, while simultaneously modeling the resulting surface gravity anomalies. Specifically designed for training deep learning networks, this tool facilitates the generation of realistic and representative training datasets, thereby enhancing model performance. The dataset includes both density variation anomalies and forward-simulated surface gravity anomalies, ensuring the validity and reliability of the simulation process.

2.1. Construction of 3D Density Variation Anomaly Models

A sample dataset representing a density variation anomaly model in a subsurface half-space is constructed. This initiative aims to mitigate overfitting and address potential issues arising from insufficient datasets during training. The generated data were then used to compute surface gravity anomalies through numerical simulations, ultimately constructing a comprehensive gravity anomaly dataset.

Model Space Construction

The subterranean half-space is systematically divided into multiple uniform cuboidal modeling units, as shown in Figure 1. The influence of each individual unit on surface observation points is calculated sequentially. The surface gravity anomaly is the cumulative contribution of all the grid units in the subsurface half-space to the surface.

When selecting the location of the abnormal body in the model, the crust above the Moho surface where shallow earthquakes usually occur is selected, and 50 km is selected here.

Based on the seismogenic range associated with pre-earthquake gravity anomalies, a horizontal extent of 100 km is selected. After several calculations and considering the computational load, a grid size of 1 km is chosen to ensure model accuracy. The subsurface half-space selected for this study is 100 km × 100 km × 50 km, and the size of each grid is 1 km × 1 km × 1 km. This configuration is chosen to accurately simulate the density variation anomaly of the subsurface region where the earthquakes occurred.

2.2. Forward Modeling of 3D Models

2.2.1. Model Generation

In the subsurface half-space of 100 km × 100 km × 50 km, a total of 8100 single density variation anomalies are systematically generated and used as the foundational dataset for training deep learning models, as shown in Figure 2. Using a 4090 graphics card, an Intel 19-13900K CPU, and 128 GB of RAM are used to generate the model with MATLAB 2022. The total processing time is 7 min and 6 s. Each density variation anomaly is set as 16 km × 16 km × 8 km. These anomalies are spaced 5 km in x and y directions, and 4 km vertically in z direction. At each spatial location, 10 distinct density anomalies, ranging from −0.5 g/cm³ to +0.5 g/cm³, are generated, ensuring an even distribution of density variation anomalies in the entire subsurface half-space.

2.2.2. Model Combination

In this study, a deep learning model training dataset is created. Standard models with varying numbers of anomalies are selected based on the characteristics of real gravity changes, rather than simply providing more shapes. This approach ensures that the model maintains a strong generalization ability and does not lose representativeness due to overfitting in real-world environments. A three-dimensional density variation anomaly model is constructed by integrating random selection with spatial constraints, the boundary test and random offset are used to ensure that the positions of anomalous bodies do not overlap and do not exceed the subsurface half-space, allowing for realistic and complex anomaly shape simulations. A diverse and intricate composite model is systematically developed by selecting multiple anomalies from a pre-generated independent outlier dataset and superimposing them onto a target composite grid. Using a 4090 graphics card, an Intel 19-13900K CPU, and 128 GB of RAM the single anomalous body model is combined into models with 2, 3, and 5 anomalous bodies, respectively. The total processing time is 1 h, 32 min, and 5 s.

Spatial consistency is maintained through the implementation of a placement algorithm that randomly selects offsets within the grid while verifying placement locations to prevent anomalies overlapping. If a conflict occurs, the algorithm dynamically recalibrates the positions until a valid configuration is achieved. Additionally, a filtering mechanism is employed to eliminate density variation anomalies that exceed the mesh size.

The generated anomaly bodies in the x, y, and z spaces exhibit varying characteristics including positive and negative values, orders of magnitude, spatial positions, and quantities. The anomalies include single-layer positive density anomalies at a depth of 10 km, as shown in Figure 3a; two density variation anomaly bodies at a depth of 40 km, as shown in Figure 3b; and different positive and negative density variation magnitudes, as shown in Figure 3c,d. Additionally, there are anomaly bodies located at depths of −20 km and −40 km, as well as more complex anomaly bodies characterized by simultaneous positive and negative densities, varying magnitudes, and deep differences in Figure 3d. These anomaly bodies are used as input for model training.

Adopting a batch processing strategy can effectively mitigate memory overflow issues that arise when generating large volumes of data rapidly. This approach addresses challenges related to computational efficiency and memory management. Furthermore, it facilitates the generation of combination exceptions within a controlled time frame, thereby enhancing overall system performance. By dynamically adjusting batch sizes (MATLAB provides memory commands (Windows only) to monitor available memory; these commands are used to dynamically resize a batch, set a minimum memory threshold of 2 GB, and release memory when the threshold is reached) and utilizing garbage collection mechanisms (when the memory usage is too large to run, the batch size is actively reduced to run), the method significantly reduces memory consumption and enhances computational stability, facilitating the efficient processing of large datasets. These strategies ensure the efficient handling of large datasets while maintaining computational stability.

The resulting composite anomaly model is stored in .npy format for seamless integration into subsequent analysis pipelines. This approach not only provides a comprehensive and diverse dataset for training and evaluating deep learning models but also establishes an advanced framework for constructing complex, high-fidelity data, advancing anomaly detection and geophysical interpretation.

2.2.3. Data Forward Modeling

The theoretical formulation for the forward computation of gravity anomalies at the observation point P(x,y,z), corresponding to a single cuboid model element, is presented as follows:

$$g=G{\displaystyle \sum _x{\displaystyle \sum _y{\displaystyle \sum _z\frac{\rho Vz}{r^3}}}}$$

(1)

$$r=\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2}$$

(2)

where r represents the distance between the observation point P(x,y,z) and the grid center of gravity (x₀,y₀,z₀), g is the surface gravity value, and G is the universal gravitation constant, which is approximately 6.67430 × 10⁻¹¹ m³/kg/s². Ρ is the density of the anomaly body, V is the volume of the anomaly body, and z is the depth perpendicular to the surface.

Figure 4 shows the gravitational effects of one, two, three, and five underground anomaly bodies as calculated using Equation (1), corresponding to Figure 3. We have established a vertical range for the anomaly bodies that extends 50 km below the surface in order to ensure the applicability of density variation anomalies within the real-space Moho surface and above. Using a 4090 graphics card, an Intel 19-13900K CPU, and 128 GB of RAM accelerates parallel computing using CUDA in python. The total processing time is 11 h, 5 min and 3 s.

When multiple anomaly bodies are present simultaneously with significant depth disparities, the surface gravity effect of a variation anomaly body characterized by low density variation may be obscured by the weight force effect from high-value density variation anomalies at the surface. This phenomenon, illustrated in Figure 4b, occurs when the distance between two positive density anomalies is small and the magnitude difference is significant. In this case, surface gravity anomalies generated by smaller anomalies can be masked by larger ones, which poses considerable challenges for future inversion work related to our model.

3. Inversion of Network Architecture

Compared to traditional gravity inversion methods, deep learning offers a direct approach to learning the complex nonlinear mapping between gravity anomalies and subsurface density distribution through end-to-end training. This approach eliminates the need for explicit mathematical modeling or prior assumptions. In this paper, a neural network structure based on U-Net is constructed, which adopts coding–decoding architecture to hierarchically extract and reconstruct underground density features. By training on a large dataset of high-precision theoretical models, the network can automatically learn density distribution patterns under various geological structures and effectively adapt to complex underground environments.

3.1. The Main Architecture of U-Net

Long et al. introduced the Fully Convolutional Network (FCN) method by transforming the fully connected layers of traditional Convolutional Neural Networks (CNNs) into convolutional layers, thus enabling end-to-end learning [36]. This innovation significantly reduces the number of hyper-parameters associated with nodes, enhancing computational efficiency. Furthermore, it mitigates the data loss typically linked to fully connected layers, preserving critical information necessary for optimizing output predictions and enabling accurate and efficient dense pixel prediction.

Building upon U-Net, a well-known convolutional neural network architecture, this study introduces a novel and streamlined network design that optimizes both computational efficiency and model performance. Specifically, the proposed architecture reduces the overall depth of the network and minimizes the number of convolutional channels, thereby decreasing memory consumption and accelerating training while maintaining high accuracy in the label classification at the output stage.

A key innovation of the architecture is the removal of traditional skip connections, which helps speed up the network training process. However, this enhancement comes at the expense of some fine-grained details, as the attention mechanism prioritizes dominant spatial features over local variations. Despite this trade-off, the overall performance of the model benefits from the improved feature representation, resulting in more robust and stable inversion results.

This study integrates a carefully designed residual module that compensates for the absence of skip connections by facilitating an effective gradient flow. This modification directly addresses the gradient vanishing issues that often hinder the training of deep networks, thereby accelerating convergence and improving stability during optimization.

Moreover, we introduce a spatial attention mechanism to further refine feature extraction. Unlike conventional convolutional layers that process all spatial features uniformly, the spatial attention module adaptively highlights important regions, enabling the network to focus on more relevant spatial structures in gravity field inversion. By doing so, the model can better capture large-scale patterns in the input data, which is crucial for improving the accuracy of spatial dimension reconstruction.

Additionally, the inclusion of residual modules enhances the network’s nonlinear approximation capabilities, allowing for a more precise reconstruction of spatial dimensions. As demonstrated in Figure 5, the proposed network outperforms conventional U-Net-based architectures, offering a more effective and efficient solution for geophysical applications.

The network is designed to train as much data as possible within a computing resource-constrained environment. For complex underground structural scenes, training focuses more on capturing the shapes and spatial locations of anomaly bodies rather than getting bogged down in intricate details. The incorporation of residual structures effectively addresses overfitting issues prevalent in small datasets, enhances model generalization capabilities, and produces smoother results.

The Conv2D layer in the shortcut path is used to resize the input layer to a different dimension; in order to achieve the role of the interlayer transfer, the dimension is still unified. Notably, the Conv2D layer within this shortcut path does not incorporate any nonlinear activation functions, ensuring that the dimensions align correctly when adding the values from the shortcut residual block back to the main pathway (similar to the effect achieved by matrix multiplication with WSW_SWS). This layer applies a linear transformation to reduce the size of the input, ensuring compatibility with the subsequent addition steps, as shown in Figure 6 [37].

Ultimately, this design aims to enhance training stability and efficiency by mitigating issues related to vanishing gradients. It seeks to improve training performance and streamline optimization processes for deep networks. Furthermore, it ensures that the characteristics of underground anomalies are sufficiently minimized for effective extraction.

Placing the spatial attention mechanism between the encoder and decoder offers distinct advantages. This design leverages high-dimensional semantic features extracted by the encoder to further optimize feature distribution through spatial attention mechanisms prior to entering the decoder [38]. This approach enhances feature representation in critical regions while mitigating interference from irrelevant areas. Given that the encoder has already captured rich contextual information, introducing a spatial attention mechanism at this stage allows for a more precise provision of sensitive features related to the target region, thereby facilitating detail recovery during the decoding phase. In the context of high-precision tasks, such as the 3D reconstruction addressed in this paper, implementing an attention mechanism before model dimensionality can significantly enhance model performance, preserve structural compactness, and maximize local feature retention while ensuring computational efficiency, as shown in Figure 7.

3.2. Regularization

In deep learning, the generalization ability of a U-Net network pertains to its predictive performance on previously unseen data [39,40]. Typically, datasets are divided into training and test sets, with network generalization assessed via test error metrics. To improve U-Net’s generalization performance, two key factors must be considered. First, minimizing loss function values; and second, reducing discrepancies between training errors and test errors. An excessively high loss function may result in under-fitting, a condition that can be rectified by adjusting either the U-Net architecture or its training algorithm. Conversely, a significant gap between the training error and test error indicates overfitting within the model. For environments involving ample data availability, regularization methods present an effective solution.

Dropout is a widely employed regularization technique in the training of U-Net models. It randomly deactivates a subset of neuronal connections during each training iteration, primarily within the encoders and decoders of the U-Net architecture. This method mitigates the model’s reliance on the co-adaptation between specific nodes, thereby simplifying the model structure and effectively reducing the risk of overfitting [41]. Additionally, Batch Normalization is frequently employed between convolutional layers in U-Net architectures. This regularization method serves to stabilize network performance and accelerate model convergence throughout the training process.

The study employs Dropout and Batch Normalization as regularization techniques, which are widely recognized for enhancing the generalization capability of neural networks. This enhancement improves the model’s adaptability to diverse datasets. Consequently, this approach ensures that our experiments can more accurately process complex surface gravity data in real geological environments, ultimately leading to improved inversion results for subsurface density variation anomaly bodies.

3.3. Loss Function

Mean Square Error (MSE) is employed as the loss function to align with the inherent continuity of underground spatial representations. This selection ensures smooth and continuous reconstruction by minimizing differences between predicted and true values. In this implementation, MSE functions as the primary loss metric for quantifying discrepancies between predicted 3D data (i.e., model output) and target 3D data. Specifically, MSELoss is computed as follows:

$$MSELoss={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(3)

where y_i represents the actual target value (ground truth), and ŷ_i denotes the target 3D data in this code. It signifies the predictive output of the model. N indicates the number of data samples.

This study introduces regional root-mean-square errors to address the disproportionate impact of high-error regions in traditional Root Mean Square Error (RMSE) assessments. The proposed method enhances the accuracy of the loss function within local contexts by independently evaluating the performance across sub-regions or subsets. It effectively mitigates the effects of spatial randomness in the distribution of subsurface anomaly bodies, where variations in size and location can result in significant errors in certain areas while causing only minor deviations elsewhere.

$${\mathrm{RMSE}}_{\mathrm{i},\mathrm{j},\mathrm{k}}={\displaystyle \frac{1}{{\mathrm{B}}_{\mathrm{H}}\times {\mathrm{B}}_{\mathrm{W}}\times {\mathrm{B}}_{\mathrm{D}}}}{{\displaystyle {\sum }_{x=0}^{{\mathrm{B}}_{\mathrm{H}}-1}{\displaystyle {\sum }_{y=0}^{{\mathrm{B}}_{\mathrm{w}}-1}{\displaystyle {\sum }_{z=0}^{{\mathrm{B}}_{\mathrm{D}}-1}\left(P_{\mathrm{i},\mathrm{j},\mathrm{k}}\left(x,y,z\right)-T_{\mathrm{i},\mathrm{j},\mathrm{k}}\left(x,y,z\right)\right)}}}}^2$$

(4)

where B represents the unit block of the RMSE calculation, x,y,z represents the length, width. And depth of each block, P_i,j,k represents the target value of the i,j,k block, and T_i,j,k represents the predicted output value of the i,j,k block model.

3.4. Algorithm Selection

This study adopts the Adaptive Moment Estimation (Adam) algorithm to enhance the training process of the 3D underground density reconstruction network. Traditional Stochastic Gradient Descent (SGD) with a globally uniform learning rate frequently encounters challenges [42]. Given that each variable’s dependency on the loss function varies, inappropriate learning rate settings can lead to a slow convergence for some variables and oscillations for others. To address this issue, the AdaGrad algorithm adjusts learning rates individually for each variable, allowing for more precise tuning. However, AdaGrad has a limitation: the rapid accumulation of squared gradients results in a progressively decreasing learning rate over time, which hinders the model’s ability to reach an optimal solution.

RMSProp improves upon AdaGrad by using an exponentially decaying average of squared gradients, thereby enabling continuous training and preventing the algorithm from getting stuck in a local minimum. Building on RMSProp, Adam further incorporates momentum through the calculation of first-order and second-order moment estimates of the gradients [43]. This combination provides more robust update directions and helps stabilize the training process. Thanks to its momentum-enhancing features, Adam not only stabilizes training but also accelerates the optimization process, making it particularly effective for large datasets requiring hyper-parameter tuning. With its high computational efficiency and fast convergence, Adam is especially well-suited for complex 3D network training tasks.

Simultaneously, the ReduceLROnPlateau method is employed as a dynamic learning rate adjustment strategy. This approach monitors specific indicators, such as the losses on validation sets, to determine if they have plateaued during the model training process. If these monitored metrics do not show significant improvement over a defined period, the ReduceLROnPlateau mechanism automatically decreases the learning rate, thereby facilitating the more stable convergence of our three-dimensional inversion model.

3.5. Network Configuration

In this study, the activation function employed is the widely adopted Rectified Linear Unit (ReLU), as presented in Equation (5). Additionally, an enhanced variant proposed by Xu et al. is utilized [44]. ReLU is favored for its capability to induce sparsity and alleviate the vanishing gradient problem, ensuring rapid convergence and minimizing computational cost.

$$f_{\mathrm{Re}Lu}(t)=\max (t,0)$$

(5)

$$f_{leaky\mathrm{Re}LU}(x)=\left\{\begin{array}{l}x\begin{array}{ccc}& & \end{array}if x\ge 0\\ \alpha x\begin{array}{ccc}& & \end{array}if x<0\end{array}\right.$$

(6)

For the output layer, the Leaky ReLU function (depicted in Equation (6)) is implemented. This activation function effectively mitigates the dying ReLU issue, facilitating smooth gradient propagation and enhancing flexibility in capturing nonlinear features, making it particularly suitable for three-dimensional reconstruction tasks.

To extract features, this study adopts maximum pooling as the pooling strategy. Compared with average pooling, maximum pooling more effectively highlights salient features [45], which is essential for retaining critical information during feature extraction. This method significantly contributes to the construction of underground three-dimensional anomaly weights.

4. Inversion and Discussions

4.1. Single Anomaly Body Inversion

During each training epoch, the input data were processed by the U-net architecture, which is based on convolutional neural networks (CNNs). The input images and target body data were loaded in batches, with each batch containing 32 samples. RMSE losses were computed for smaller blocks consisting of 10 × 10 × 5 voxels. This block-by-block comparison with the target data allowed for the quantification of the difference between the predicted output and the target using the RMSE. This approach reduced computational requirements while maintaining the accuracy of local data.

4.1.1. Classification Results

A total of 50 epochs were conducted throughout the training process. After each epoch, training losses, validation losses, and the current learning rate were calculated and documented. Initially, a modest learning rate of 10⁻⁵ was established alongside the ReduceLROnPlateau learning rate scheduler; this approach entails halving the learning rate when no further reduction in validation loss occurs. This strategy aims to enhance training stability while mitigating overfitting risks. Additionally, an early stopping criterion was implemented during the training process: if there was no significant improvement in the validation loss for five consecutive epochs, training would be terminated prematurely to prevent overfitting.

In the training of a single anomaly body, the utilization of the Adam optimizer markedly accelerates convergence, mitigates fluctuations during training, and substantially reduces the overall convergence time, as shown in Figure 8a. The model exhibits commendable characteristics throughout the training process: it achieves rapid convergence within the first five epochs with a significant reduction in the loss function. As training progresses, by the 10th epoch, the rate of loss declines, plateaus, and the model stabilizes. By the 30th epoch, the training process is essentially complete, with successful convergence. Ultimately, the residual error is maintained below 2%, which underscores the model’s superior fitting capability and provides robust assurance for subsequent applications, as shown in Figure 8b.

4.1.2. Forecast Results

Under the inversion framework based on deep learning, the results obtained from inverting a single anomaly body demonstrate remarkable effectiveness. In comparison to traditional inversion methods, deep learning-based approaches exhibit significant advantages. Deep learning models possess the capability to learn complex nonlinear mapping relationships from extensive training datasets, which enhances their flexibility when addressing anomaly bodies situated within heterogeneous or intricate backgrounds.

Particularly in environments involving a solitary anomaly body, as shown in Figure 9, these models can accurately capture the density distribution characteristics of the anomaly by optimizing the loss function and even detect subtle variations. On the horizontal slice, there is no diffusion at all, the model size is good, and there is some upward diffusion on the z direction, but the overall shape and size are intact. This proficiency allows the model not only to reconstruct the overall structure of the anomaly body but also to maintain a high level of precision regarding its finer details.

The model demonstrates a commendable performance in both the recovery of form and the spatial position reconstruction of the anomaly body. It presents a clear and complete geometric structure, with calculated values closely approximating real values and exhibiting minimal errors. This outcome validates that the model can accurately invert anomalous bodies under known conditions while maintaining high stability and accuracy, irrespective of whether the density of these bodies is positive or negative. The reconstruction results for both positive and negative density anomalies indicate that the model effectively captures the density distribution characteristics associated with different types of anomaly bodies, thereby circumventing the reliance on specific density intervals typical of traditional methods. In contrast to conventional gravity inversion techniques, deep learning models do not depend on prior simplified assumptions; instead, they adaptively learn complex feature distributions related to both positive and negative anomalies through training. This inherent flexibility not only enhances their efficacy in detecting individual positive and negative anomaly bodies but also lays a robust foundation for addressing intricate geological challenges involving abnormal bodies with diverse properties in future research. Overall, the deep learning-based inversion model exhibits significant flexibility and adaptability. It efficiently identifies both the nature and spatial distribution of singular abnormal entities while simultaneously offering theoretical possibilities for subsequent inversions involving multi-density structures as well as actual surface gravity data.

4.2. Inversion of Multiple Classified Anomaly Bodies

In this study, datasets containing varying numbers of anomalies (1, 2, 3, and 5) were combined to generate a total of 32,000 forward and inverse models with different anomaly counts (8000 of each type). The approach was employed to evaluate the robustness of the network across diverse model configurations and to assess the performance of complex models. The block RMSE loss function was utilized during network training. The primary objective was to enable the model to accurately predict anomalies within three-dimensional data.

During each training epoch, the input data were processed by the U-Net architecture, which is based on convolutional neural networks (CNNs). The input images and target body data were loaded in batches, with each batch containing 96 samples. RMSE losses were computed for smaller blocks consisting of 10 × 10 × 5 voxels. This block-by-block comparison with the target data allowed for the quantification of the difference between the predicted output and the target using the RMSE. This research seeks to enhance the predictive accuracy for anomalies present in three-dimensional datasets while managing computational efficiency through localized error calculations. The inversion of the standard model for complex subsurface anomaly bodies was achieved.

4.2.1. Classification Result

A total of 50 epochs were conducted during the multiple anomaly bodies training process. After each epoch, the training losses, validation losses, and current learning rate were calculated and recorded. Initially, a small learning rate of 10⁻⁵ was set, and the ReduceLROnPlateau learning rate scheduler was employed. When the validation loss ceased to decrease, the learning rate was halved. This strategy aimed to enhance training stability and mitigate overfitting. Additionally, an early stopping criterion was implemented during the training process: if there was no significant improvement in the validation loss for five consecutive epochs, training would be terminated prematurely to prevent overfitting.

During the training process, the Adam optimizer significantly enhances the convergence rate of the model, minimizes fluctuations during training, and substantially reduces the time required for convergence, as shown in Figure 10a. The model exhibits favorable training characteristics: it achieves rapid convergence within the first 10 epochs with a marked decrease in the loss function. As training progresses, the rate of decline in the loss function stabilizes, and the model reaches stability by the 30th epoch. By the 40th epoch, the training process is essentially complete, and the model successfully converges. The final residual error of the model is maintained below 5%, demonstrating its superior fitting performance and establishing a robust foundation for subsequent applications, as shown in Figure 10b.

4.2.2. Forecast Results

1.

Two anomalies

A two anomaly body characterized by significant differences in burial depth is employed to evaluate the model. This design aims to simulate the features of geological anomalies at varying depths, thereby allowing for a comprehensive assessment of the model’s adaptability in addressing complex variations in burial depth. The combination of high and low buried depth differences effectively verifies the sensitivity and inversion accuracy of the model, providing a solid foundation for the further optimization of the algorithm.

The inversion results of the model are commendable, as shown in Figure 11. As it accurately restores the primary geometric features, the two anomalous bodies have no diffusion, size, and range inversion effects in the horizontal direction and the z direction is good. Regarding positional accuracy, the inversion model effectively captures both the horizontal and vertical positions of these anomalies, with depth errors maintained within a minimal range. This demonstrates the model’s superiority in spatial analytical capabilities.

In terms of density distribution characteristics, the inversion results provide detailed insights into the internal structure and boundary changes in both anomalies. The transition between density values for these anomalies and their surrounding background is remarkably smooth. Furthermore, the characterization of boundaries for each anomaly body is distinctly clear, indicating that the input data have been well restored by the model.

It is noteworthy that not only do these inversion results accurately recover both the shape and relative positioning of the two anomaly bodies, they also reveal potential patterns regarding their underground interaction characteristics. For instance, a gradual change in relative density gradients between these two bodies suggests that the model can sensitively detect anomaly signal characteristics even within complex environments. Additionally, a satisfactory resolution at depth ensures the preservation of fine structures within both anomaly bodies in a vertical context—providing an effective reference for multi-layer geological body inversions in intricate settings.

In the process of verifying the inversion results, it was observed that the residuals remain stable within ±5 μGal when comparing ground gravity observations with values generated by the inversion. Furthermore, the distribution of these residuals is uniform, exhibiting no significant systematic bias or concentrated areas of error. This strongly indicates that the model’s inversion results possess a robust capability to accurately represent the ground gravity signal, demonstrating a high degree of consistency between observation and simulation data. Such findings further validate both the physical binding force and stability inherent in the model. Additionally, the smooth distribution observed in the residual region suggests that the inversion process effectively mitigates the unnecessary amplification of noise, thereby minimizing its impact on the result accuracy.

2.

Three anomaly bodies

When testing complex model tests involving three anomaly bodies, a scenario was designed that incorporated both positive and negative anomalies while preserving the depth variations among them. The design not only increased the data diversity but also introduced more challenging targets, thereby substantially enhancing the complexity of the inversion process. This approach was intended to comprehensively evaluate the model’s adaptability to complex geological structures.

The model exhibits a strong capacity for reverse signal capture, as shown in Figure 12. The abnormal body with density variations of −0.2 g/cm³ and +0.2 g/cm³ in the shallow layer has no diffusion and deformation in its contour boundary and its value in the horizontal direction and depth direction. The abnormal body with a density variation of +0.3 g/cm³ in the deep layer has a certain diffusion boundary deformation ambiguity due to the influence of the abnormal body with a relatively close horizontal distance in the shallow layer, but the overall shape and value are correct. The model accurately restores the characteristics of both positive and negative anomaly bodies while effectively representing the main geometric features, depth distribution, and density variation anomalies within the subsurface model over time. Regarding positional accuracy, the inversion model successfully captures both horizontal and vertical positions of the three anomalies; furthermore, the depth errors are maintained within a minimal range—highlighting the superiority of this model in spatial analytical capabilities.

Concerning the density distribution characteristics, the inversion results provide detailed insights into the internal structures and boundary changes associated with the three anomalies. The transition between the density values among these anomalies and their surrounding background is smooth without any abrupt shifts or abnormal high/low density readings. Additionally, while there is a minor overflow observed locally at certain points the overall shape stability is preserved, thus ensuring that input data models are restored effectively.

In the process of verifying the inversion results, the residuals remain stable within ±2 μGal when comparing ground gravity observations with the values generated by the inversion. The distribution is uniform, exhibiting no significant systematic bias or concentrated areas of error. The model demonstrates an enhanced performance in environments where both positive and negative density signals are present, resulting in a surface residual that is lower than that observed for the two anomaly bodies.

The random introduction of three outliers adds complexity to the verification process by incorporating both positive and negative values along with deep differences. This effectively showcases the model’s superior capability in handling complex environments. The findings indicate that the model not only sensitively captures subtle features within intricate scenes but also maintains an excellent performance amidst variations in depth and noise suppression. These attributes establish a robust foundation for the extensive application of this model in practical geological exploration, further validating its effectiveness and applicability for multi-objective and multi-level geological body inversion tasks.

3.

Five anomaly bodies

In the complex density variation model test involving five anomaly bodies, a carefully designed scenario was established in which both positive and negative anomaly bodies coexisted, thereby comprehensively assessing the model’s adaptability to intricate and realistic geological structures. This design not only enhances data diversity but also presents a more challenging target for analysis. Particularly, when confronted with both positive and negative anomalies alongside varying signal gradients within the region, the complexity of inversion is significantly heightened.

The test results indicate that the model’s recognition capability remains exceptional even in complex situations, as shown in Figure 13. From −5 km near the surface to −40 km below the deep layer, the lowest density variation in the anomalous body is −0.5 g/cm³, and the largest density variation is 0.4 g/cm³, and the density variation is not as significant as −0.1 g/cm³, which achieves the inversion effect. The shape boundary of the deepest anomalous body is fuzzy, but the shape integrity is still accurate. For positive anomalies, the density distribution effectively highlights high-value regions, while for negative anomalies, the inversion results accurately restore both the depth and extent of low-value areas. This demonstrates that the model can adeptly manage differences between positive and negative densities while maintaining high accuracy and preventing diffusion or the blurring of edge values. The precise inversion of both positive and negative density characteristics further substantiates its performance potential when dealing with multi-source heterogeneous signals in intricate geological contexts.

Moreover, it is noteworthy that the model exhibits a commendable resolution in vertically inverted high and low anomaly bodies. Particularly, within environments involving deep structural anomalies, variations in high and low-density gradients are meticulously restored, with corresponding transition zones exhibiting a remarkable degree of continuity. This level of detail underscores the model’s ability to sensitively capture minor structural changes along the vertical axis while providing an extensive characterization across multiple geological layers.

A performance worthy of in-depth analysis is observed in the intersection region of the anomaly bodies characterized by drastic density changes. In this context, the model demonstrates remarkable resilience against signal mixing, effectively delineating the boundaries of each independent anomaly body with a high degree of stability. Despite these complex conditions, the model maintains an impressive level of attention to local details, indicating that its structural design is adept at managing various regional disturbances. Moreover, the model exhibits both stability and adaptability across different testing environments, including environments involving random arrangements of positive and negative anomalies as well as unique distributions characterized by varying elevations. This flexibility enables it to address inversion requirements in straightforward situations while consistently delivering reliable results under more intricate conditions. Additionally, these experiments unveil potential expansion capabilities within the model framework, thereby supporting future practical applications in multi-domain geological inversion.

In summary, this study validated the model through randomly generated five anomaly bodies featuring unequal positives and negatives alongside alternating high and low levels. The inversion outcomes reaffirmed both the adaptability and accuracy inherent in this methodology; it exhibited exceptional sensitivity in feature detection while maintaining stability and precision when addressing complex geological structures. Collectively, these test results establish a robust foundation for practical applications in geological inversion and further substantiate the efficacy of this approach when confronted with challenging environments.

5. Model Application and Verification

The gravity measurement data around the Longshoushan fault zone, collected from 2020 to 2021 by the Gravity Branch of the China Earthquake Administration, were subjected to inversion analysis, by integrating these data with the tectonic dynamic background of the northeastern margin of the Qinghai–Tibet Plateau [46]. We conducted a comprehensive analysis of density fluctuations and subsurface attributes preceding seismic events, thereby substantiating the practical applicability and intrinsic value of our model. According to information from the USGS, the Menyuan M_S 6.9 earthquake in 2022 occurred in a region characterized by long-term, left-lateral, and strike-slip shear and vertical uplift, with a focal depth of 15 km. The inversion results indicated significant density variation anomalies at various depths within the study area. The observation area spans 100 km × 100 km, while the surface gravity data interpolation is conducted at a resolution of 1 km × 1 km, aligning with the input parameters for our model. The subsurface was divided into cuboid units measuring 100 km × 100 km × 50 km, each unit being sized at 1 km × 1 km × 1 km. High-precision inversion research was conducted utilizing the U-Net network framework. The distribution pattern of residual errors within experimental data indicates that the peak values for surface gravity residual errors reach up to 15 μGal but progressively decrease from peripheral areas towards the center of the investigation. In proximity to the epicenter, residual errors fall below 5 μGal—well within acceptable limits—and exert minimal influence on the interpretations regarding underground structures in this region. This observation underscores that our model possesses strong regional focusing capabilities which effectively diminish secondary interference while minimizing impacts from external density variation anomalies on local inversions within our study area. However, in practical applications, the influence of density anomalies outside the region introduce interferences, leading to a poor spatial edge effect in the trained model and a slight reduction in the overall accuracy. Meanwhile, the limitations of single gravity inversion, such as poor shape, also exist in deep learning inversion. Overall, this network demonstrates high accuracy and adaptability when detecting large-scale anomalies as well as complex subterranean features.

Based on the available data, the U-Net network employed in this study was utilized to conduct the inversion analysis of the target region. A significant positive density anomaly of 0.2 g/cm³ was identified at depths of 20–30 km northwest of the Tolai Mountain fault zone. In contrast, a negative density anomaly of 0.3 g/cm³ was detected at depths ranging from 15 to 25 km southeast of the Lenglongling fault zone. These anomalies are horizontally distributed and parallel along their respective fault zones, with stress effects aligning with the orientations of these faults. This alignment exhibits a strong consistency with the left-lateral strike-slip characteristics observed by the USGS during the Menyuan MS 6.9 earthquake event at a focal depth of 15 km. Figure 14d illustrates the distribution of 20 km underground sections, faults, and earthquake aftershocks in the epicentral area. The distribution of earthquake aftershocks was consistent with the inversion results. By relying on the spatial relationship between fault zones and density changes, the U-Net-based inversion results effectively explain pre-earthquake gravity variations. Additionally, the distribution of density changes correlates with the strike of the Lenglongling fault zone, showing an increase on the northwest side and a decrease on the southeast side of the epicenter. These findings align with the focal mechanism solution and epicenter shift reported by Dai et al., as well as the spatial extent of major aftershock distributions [47]. Furthermore, regional gravity anomalies reveal fluctuating characteristics in the Moho surface, indicating substantial changes at approximately 50 km of depth. The results demonstrate that the enhanced U-Net network successfully captures complex gravity signals while significantly reducing secondary interference, showcasing robust generalization capabilities across diverse geological contexts. This provides reliable technical support and valuable tools for investigating density variations and intricate subsurface structures.

In summary, this study demonstrates the feasibility of applying the U-Net-based gravity anomaly inversion approach in real-world geological investigations. The improved network effectively captures complex gravitational signatures, mitigates secondary influences, and effectively generalizes across diverse subsurface environments, providing robust support for practical geological challenges.

6. Conclusions

In this study, the U-Net network model is employed for the inversion of underground anomaly bodies. Compared with traditional seismic inversion methods, this model can automatically extract features from a broad range of gravity anomaly data, thereby eliminating the reliance on manual feature extraction and prior assumptions inherent in conventional approaches. This capability significantly enhances both the efficiency and flexibility of the inversion process, making it particularly well suited for complex geological environments.

• During the theoretical model testing phase, a RMSE loss function based on partition calculations is introduced. This method takes into account the spatial distribution characteristics of underground anomaly bodies and effectively optimizes inversion accuracy.
• In model tests, not only are the shape, size, and boundaries of subsurface anomalies accurately constrained, but also the residual gravity from the surface flow is maintained below 5 μGal. These results further validate both the feasibility and superiority of using U-Net networks for gravity anomaly inversion.
• In practical applications, U-Net networks have successfully addressed inversion challenges in geological contexts, as demonstrated by their ability to identify underground density variation changes preceding the Menyuan M_S 6.9 earthquake. The focal depth matches the one produced by the USGS while keeping surface gravity residuals within 15 μGal. Additionally, the distribution and focal mechanisms of aftershocks are consistent with those obtained using other researchers’ GNSS methods, thereby demonstrating U-Net’s capability in addressing non-unique gravity inversion issues.

Whether in theoretical testing or real-world earthquake case studies, U-Net networks have proven to be accurate and robust for gravity anomaly inversion, offering innovative methodologies for exploring density changes in complex geological environments.