An important approach to obtain the underground structure is the inversion of potential field data, which is an ill-posed problem. Regularization inversion is a commonly used method for addressing such ill-posed problems. The selection of regularization parameters determines the accuracy and stability of the inversion. However, conventional methods are computationally complex, and the calculation process may introduce errors in parameter selection. We discuss the shortcomings of existing methods and propose a regularization parameter selection neural network (RPSNN) to calculate regularization parameters for potential field data inversion. This network is designed to more effectively capture the relationship between potential field data and regularization parameters. Comparative analysis reveals that our method can directly obtain the optimal regularization parameter (ORP) and lead to more accurate inversion results. Therefore, our method is more effective in selecting regularization parameters for potential field data inversion. Numerical experiments demonstrate that trained RPSNN outperforms existing methods in predicting regularization parameters, leading to improved solutions for inverse problems. The proposed method was applied to the Lu-Zong ore concentration area in the middle and lower reaches of the Yangtze River region to obtain a 3-D distribution model of magnetite within 5 km underground. The results showed that the magnetite locations corresponded well with the shallow minerals being mined and were verified with drilling data, confirming the existence of usable deep resources in the ore concentration area. The extent of deep minerals was circled to delineate favorable mineralization areas.

The inversion of potential field data dispersed by nonstructured tetrahedron mesh is better applied for undulating terrain area, but it has a poor computational efficiency owing to the irregular distribution of the nodes of the mesh, which makes it challenging to potential field data fine inversion over a large area. We propose an equivalent geometric-discrete function fitting (EG-DFF) method for the fast inversion of potential field data with nonstructured tetrahedral mesh, and we prove that this mesh yields equivalent property of adjacent surface, which is used to accelerate the calculation of kernel function. The efficiency of inversion solution is improved by using a discrete function to fit the physical parameters of the grid cells to reduce the number of solution parameters. The test results showed that the EG-DFF method did not lose the inversion when 500 mesh cells are fit with a third-order function and 200 mesh cells are fit with a second-order function and can improve the computational efficiency by more than four times and reducing memory consumption by eight times compared to the traditional method. The proposed method also exhibited good resistance to noise. We applied the EG-DFF method to airborne gravity and magnetic data to analyze the origin mechanism of Kirin Lake, which is located in the subglacial rift valley of Princess Elizabeth Land. The density and magnetization structure showed that the upwelling of the mantle with the action of the rift accounted for the formation of Kirin Lake.

The observation surface of gravity data is often undulating, and the density inversion efficiency of gravity data is seriously reduced due to the low computational efficiency of the kernel function when the undulating surface is considered, which makes it impossible to apply to the detailed inversion of large-scale data. We proposed a multivariate geometric equivalent transformation (MGET) method to rapidly compute the kernel matrix with an undulating observation surface for improving density inversion efficiency. We first build the equivalence relation of the kernel function for different grid nodes, grid cells, and adjacent layers with a structured hexahedron mesh, which reduces the computation time and the number of kernel function, and can significantly improve the density inversion efficiency of gravity data. Tests show that the kernel matrices computed by the MGET method are accurate and efficient, and the computational efficiency of the kernel function is improved by more than ten times, and the density inversion can be performed efficiently and quickly. Finally, we apply the MGET method to real gravity data from Harbin City, China, to obtain the fault distribution and trends. This will contribute to future city construction planning.

The three-direction magnetization intensities of a source can be obtained by the magnetization vector inversion (MVI) of magnetic data, and therefore, MVI can be well applied to a magnetic anomaly with remanence. However, MVI exhibits poor applicability attributed to its low computational efficiency for a data with large area and undulating terrain. We proposed a discrete function (DF) method to improve the efficiency of MVI, which is suitable for uniform hexahedron mesh and unstructured tetrahedral mesh. In this method, DF was used to fit the magnetization vectors of a given number of cells, and the computation of the physical properties of the cells was replaced by the computation of function coefficients, and the memory and calculation time were reduced, as the number of function coefficients was less than the number of cells. The Gramian constraint item in MVI could be directly computed by function coefficients without increasing the computational process. For uniform hexahedron mesh, the integration of DF with the block-Toeplitz-Toeplitz-block-fast Fourier transform (DF-BTTB-FFT) method can further improve the inversion efficiency. The model tests revealed that the DF method could reduce the storage and computation time by approximately five times without precision loss, and DF-BTTB-FFT method could achieve approximately four times improvement compared with the DF method. Finally, the proposed DF method was used to accomplish the MVI of real magnetic anomaly and its gradient data in Shaanxi Province to delineate the range of the subsurface burned zone, and the distribution was ascertained by combining the nuclear magnetic water measurements.

The inversion of gravity and magnetic data with unstructured tetrahedral mesh is the main means to obtain subsurface information for exploring regions with undulating terrain. However, carrying out these inversions with unstructured tetrahedral meshes is time-consuming. In this work, we propose a novel method for fast inversion with an unstructured tetrahedral mesh, which we call the equivalent surface and Gaussian quadrature with a nonuniform fast Fourier transform (ES-GNF) method. The ES-GNF method can improve the kernel matrix computing efficiency in the inversion process. We use the equivalent surface element characteristics of the common facets of adjacent tetrahedral cells to reduce the kernel matrix calculation process and apply Gaussian quadrature with a nonuniform fast Fourier transform (NUFFT) method to efficiently calculate the function of a triangular facet in the kernel matrix. Calculation analysis and synthetic model tests verify that the computing efficiency of the ES-GNF method is nearly$10 \times $that of the traditional method, with almost no reduction in precision and no additional memory requirements. Finally, we apply the ES-GNF method to obtain physical property distributions in order to analyze deep magma and the source of ore-forming material in the southern Greater Khingan Mountains and the northern margin of the North China Craton. The inversion results indicate that mineral resources are under the proximal control of volcanic activity. Furthermore, magma has migrated through large faults to the shallow crust and formed metallic ores.

The subsurface density and magnetization distribution obtained from joint inversion of gravity and magnetic data could map ore distribution. In order to consider irregular terrain, irregular geological body and remanent magnetization, the gravity and magnetic joint inversion scheme developed here employs tetrahedral unstructured grid. We also employ a volume weighting function in order to improve the convergence of the scheme. Model comparison showed that the developed method with volume weighting function can yield accurate density and magnetization inversion results. The tetrahedral unstructured grid with small volume differences has a higher inversion resolution than that of prismatic structured grid. The method has been applied to gravity and magnetic data from the Huanggangliang-Ganzhuermiao metallogenic belt from mineral exploration. We predicted the favorable area for mineralization in the survey area according to the characteristics of the known deposits, and the favorable mineralization areas are mainly distributed along the faults. Combined with the analysis of deep inversion results, we found that the Baiyinnuoer and Shuangjianzishan mineral deposits have the same metallogenic material source, and the magma migrated upward along the Y-shaped magma channel. Therefore, it was speculated that the difference of these two deposits was caused by the different migration structural environments.

In the 3-D density inversion calculation of gravity data, the entire subsurface space is discretized into rectangular prisms, and the density value of each prism is calculated. However, the method of dividing the entire space often results in invalid calculations. In this study, we proposed a highly efficient density inversion method using a locally adaptive unstructured mesh. In this method, the inversion scope is reduced by using the feature that the zero value of the tilt angle method corresponds to the edge of the field source. In addition, the local inversion area is divided by unstructured meshes, which can better represent irregular geological bodies and undulating terrain. In the proposed method, the size of the grid cells is changed according to the value of the tilt angle to reduce the amount of mesh, which reduces the computational complexity and improves the computational efficiency. In addition, by introducing a volume weighting function, the sensitivity of grid cells of different sizes can be balanced. Through synthetic modeling experiments, we verified that the locally adaptive unstructured mesh method can improve the efficiency and accuracy of inversion, flexibly deal with undulating terrain, and obtain the distribution features of irregular bodies. We applied this method to gravimetric data of the North Qinling region of the Shaanxi province, which clearly shows the spatial distribution information of five high-density geological bodies, and the depth range of ore bodies formed is from 240 to 1481 m according to our inversion results.

Gravity measurement is an important geophysical prospecting method for mineral exploration. With most of the shallow ore deposits in China being exploited, the future targets for exploration will be aimed at the deep mineral resources about 4-km underground. To improve the resolving ability in the vertical direction of the 3-D density inversion to find deep-source minerals, we propose to achieve the inversion of gravity anomalies by combining power-spectrum derived adaptive weighting and cross-gradient regularization. First, spectral analysis is utilized to conduct the source-depth separation, and the adaptive weighting function is designed depending on the slope of the radial logarithmic power spectrum of the gravity anomaly, which can enhance the correspondence for the field sources with different depths to improve the vertical resolution relatively. Thus, each separated source with different depths could be given an adaptive weighting coefficient in each separated inversion. Second, the cross-gradient technique is introduced as a structural constraint in the objective function to further constrain the separated inversion process. By performing two-cuboid-source cases, it can be seen that this combination calculated approach has great potential to improve the quality of the inversion results. When examining the inversion of the actual gravity data from certain ore district in the west of Zhen’an, South Qinling, we speculate that a potential deep ore body would exist in the deposit via utilizing the recovered density model obtained by the proposed method.

The cross-gradient joint inversion of gravity and magnetic data is a commonly used way, which requires dividing the subsurface into closely arranged cells with structured or unstructured grids to solve the physical properties of the discrete subsurface. Unstructured grids can more effectively fit undulating terrain and irregular geological bodies than structured grids. However, the existing cross-gradient joint inversion method with structured grids cannot be used to complete the inversion with unstructured grids. We proved that the second-order Taylor formula is more suitable for calculating the cross-gradients of unstructured grids; therefore, we first proposed and implemented the second-order Taylor formula cross-gradient (2-TCG) method for the cross-gradient joint inversion of gravity and magnetic data with unstructured grids, which can calculate the cross-gradient item more easily than other ways because the first few coefficients of Taylor formula are gradient items. The model tests showed that the 2-TCG joint inversion method can accurately achieve the 3-D joint inversion of gravity and magnetic data with unstructured grids, and this method can better describe the detailed information of irregular geological bodies in complex terrain. To explore the minerals and analyze the metallogenic model in the Huanggang–Ganzhuermiao metallogenic belt of the southern Greater Khingan Range, Inner Mongolia, we measured gravity and magnetic data on a scale of 1:10000. We estimated the mineral distributions with 150–280 m depth by 2-TCG joint inversion. We analyzed the relationship between ore bodies and faults, and we established a mixed metallogenic model of the skarns and hydrothermal veins in the survey area.

The airborne gravimetry measurement can better highlight the anomalous response of deeper resources than ground gravity data, so an integrated survey of airborne and ground gravity data is an effective strategy to obtain the obvious response of shallow and deep mineral resources simultaneously. In order to obtain a more reliable and higher resolution 3-D density distribution model through the joint inversion of airborne and ground gravity data to support deeper source exploration, we propose a high-resolution joint inversion method with a cross-constraint technique. We use structural constraints to establish an objective function for joint inversion of airborne and surface gravity data, and add density-weighting constraints through cross-calculation to make full use of the sensitivity of different data and geological targets. Synthetic model tests prove that the cross-constraint joint inversion method of airborne and ground data improves the model resolution effectively, and can delineate the distribution of deeper targets more clearly compared to the existing joint inversion way. In order to predict the spatial distribution of deep ore-bearing rock bodies in a north Qinling area of Shaanxi province, we obtained regional measured airborne and surface gravity data. High-resolution density joint cross-constraint inversion method was used for real data interpretation, and the distribution feature of possible deep ore-forming rock bodies.