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.

Localized water-rich areas in aquifers can cause severe accidents during coal mining, including property damage and casualties. Traditional ground-based geophysical methods often struggle in the mountainous terrains where coal mines are typically located. To address this, we applied a high-resolution exploration and interpretation strategy based on advancements in semi-airborne transient electromagnetic (SATEM) technology, integrating drilling and logging to detect water-rich areas in the Huangling coal mine. Our approach involved acquiring high signal-to-noise ratio electromagnetic (EM) data near the transmitting line source, using unstructured tetrahedral meshes to simulate complex topography, and employing a quasi-Newton optimization algorithm for detailed 3-D inversion. Synthetic tests demonstrate the accuracy of this method in resolving underground conductivity structures, even in challenging terrains. Field data inversion showed a good fit with observed data and good agreement with resistivity logging, confirming the reliability of the results. This study not only addresses the critical need for efficient water hazard detection in coal mining but also offers significant potential for mineral exploration, geological surveying, and disaster prevention.

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.

Unstructured grid generation is more suitable for finishing the joint inversion of gravity and magnetic anomalies in the case of undulating observation surfaces and irregular field sources. To realize the joint inversion with unstructured grid of gravity and magnetic anomalies corrupted by remanence, we proposed a joint inversion method of gravity and magnetic analytic signal (MAS) with unstructured grid, and an adaptive subdivision method based on anomaly feature is introduced to improve the computation efficiency. The least-squares method was used to perform the computation of cross-gradient item in joint inversion for non-homologous gravity and magnetic source. Synthetic tests on a dipping-slab model show that the proposed method can complete the joint inversion, and can effectively recover the distribution characteristics of the field source without precision loss, and can reduce calculation time and storage space. Finally, we applied our new method to real gravity and magnetic data to ascertain the distribution of magnetite in Shandong Province, China. We obtained the range of favorably mineralized regions based on the inversion results, which provides an important basis for further exploration.

Cross-gradient joint inversion of gravity and seismic data can more accurately provide the characteristics of subsurface density and velocity structure. The subsurface is usually divided into triangular grids to accurately simulate the undulating terrain and irregularity of geological bodies in the inversion, and the existing cross-gradient inversion with triangular grid is finished by the linear trend method. However, the density and velocity changes are not all linear. To better describe both nonlinear and linear variation feature of the physical property, we propose the second-order finite-difference cross-gradient joint inversion method of gravity and seismic data with triangular grid, which can obtain high-resolution results and effectively reflect the linear and nonlinear physical property changes. We also compare the effect of cross-gradient inversion results computed by different order finite-difference method and verify that the second-order finite-difference method is more reasonable according to the computational efficiency and accuracy. To reveal the distribution of the polymetallic minerals in the Lu (Lujiang)-Zong (Zongyang) ore concentration area, we carry out regional gravity and profile seismic measurements in this area. We first obtain the 2-D density and velocity results, and then use the 2-D density as a constraint to compute the 3-D density distribution. The results reveal the distribution of six minerals and that the burial depth of high-density polymetallic ores range from 567 to 959 m, which provides reliable information for subsequent exploitation.

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.

We develop a fast and stable inversion method to solve the underdetermined linear sparsity gravity inversion problem. A new sparsity constraint based on$L0$norm is studied to improve the accuracy of the recovered block density. A projected space algorithm avoids the update of forward matrix in the general reweighted iterative method, whose operations are complex and time consuming. Another advantage of the proposed projected algorithm is that it is compatible with other fast gravity methods. A general iterative algorithm is applied to solve the general gravity inversion. This space algorithm projects the recovered density on the sparsity constraint. With the application of space algorithm, the large forward matrix is calculated once. A test of synthetic example demonstrates the advantages of inversion efficiency and accuracy. The application of real gravity obtained from Mobrun ore, northeast of Noranda, QC, Canada shows the practical value of the proposed inversion method. This novel study can be used in other geophysical inversions.

Airborne gravity gradiometry is an effective tool for its high efficiency and sensitivity to the interesting buried targets, which has been increasingly introduced in mineral and petroleum exploration. When compared to gravimetry, full tensor gradient (FTG) measurements provide higher resolution and signal-to-noise ratio. In this article, we build and test models with different sampling intervals and compare the performances of individual components or their combinations in minimum structure inversion to evaluate the benefits of FTG combinations. As part of a theoretical study, the singular value spectrum (SVS) and depth-resolution plot (DRP) are examined to determine how much information and resolution different datasets provide in the underdetermined inverse problem. The synthetic examples show that integrating more components in inversion improves the resolution of the recovered model, and the inversion result of gravity or individual component cannot accurately restore the distribution of anomalous bodies. With the increasing sampling interval, joint inversion of FTG data still provides enough information corresponding to the source density distribution, and the limit survey interval is tested through models. In addition, we use a real survey over the Vinton dome as a case study and discover that$V_{\mathrm {zz}}$component fails to reconstruct the caprock model at coarse line spacing, whereas FTG combinations recover the acceptable geometry flawlessly. It also reflects the benefit of FTG combinations for inversion processing, which reduces the interference of measured noise on inversion results and allows for longer sampling intervals during the survey, effectively lowering the cost of an airborne survey and improving exploration efficiency

The gravity gradient data have higher horizontal resolution and highlight the shallow sources relative to gravity data, therefore, the comprehensive measurement of gravity and its vertical gradient anomalies is commonly used to reveal the density structure of planets. To obtain higher resolution density results by the joint inversion of large-scale gravity and its gradient, we propose a density-weighted and cross-gradient constrained (DWCG) joint inversion method in the sphere coordinate, which improves the inversion accuracy by introducing a density structural constraint provided by the gravity inversion result. Furthermore, the Block-Toeplitz Toeplitz-Block (BTTB) structure is used to finish the DWCG joint inversion, which effectively accomplishes a high-efficiency calculation, and is more practical for performing the inversion of large-scale data. Theoretical modeling experiments show that the DWCG joint inversion method can effectively improve the resolution, and identify the distribution of deep and shallow field sources, and the accuracy is improved by nearly 40% compared to the regularization inversion method of gravity data. Simultaneously, the DWCG joint inversion method has certain noise immunity. Finally, we apply the DWCG joint inversion method to the lunar data to obtain the density distribution and further revealed the Moho relief by nonlinear inversion based on the density results. The results show that the high-density structure corresponds to the uplift of the mantle, and a large number of meteorite impacts caused mantle flow, which is shown as a clear impact basin and ring anomaly on the topography and anomaly map.