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 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.

Since polymetallic ores show higher anomalies in gravity exploration methods, we usually obtain the position and range of ore bodies by density inversion of gravity data. The three-dimensional (3D) gravity focusing density inversion is a common interpretation method in mineral exploration, which can directly and quantitatively obtain the density distribution of subsurface targets. However, in actual cases, it is computation inefficient. We proposed the preconditioned Jacobian-free Newton-Krylov (JFNK) method to accomplish the focusing inversion. The JFNK method is an efficient algorithm in solving large sparse systems of nonlinear equations, and we further accelerate the inversion process by the preconditioned technique. In the actual area, the gravity anomalies are distributed on the naturally undulating surface. Nowadays, the gravity inversion under undulating terrain was mainly achieved by discretizing the ground into unstructured meshes, but it is complicated and time-consuming. To improve the practicality, we presented an equivalent-dimensional method that incorporates unstructured meshes with structured meshes in gravity inversion, and the horizontal size is determined by the gradient of observed gravity and terrain data. The small size meshes are adopted at the position where the terrain or gravity gradient is large. We used synthetic data with undulating-terrain to test our new method. The results indicated that the recovered model obtained by this method was similar to the inversion method of unstructured meshes, and the new method computes faster. We also applied the method to field data in Huayangchuan, Shaanxi Province. The survey area has complicated terrain conditions and contains multiple polymetallic ores. Based on the high-density characteristics of polymetallic ore bodies in the area, we calculate the field data into 3D density models of the subsurface by the preconditioned JFNK method and infer six polymetallic ores.

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.

It is of great significance to clarify the ranges and states of subsurface coalfield spontaneous combustion areas for coal mining and disaster management. Since the spontaneous combustion of coal seams produces highly magnetic burnt rocks and high temperatures, magnetic and infrared remote sensing measurements are commonly used for detection. To infer the accurate ranges of highly magnetic burnt rocks, we propose a three-dimensional constrained magnetization vector inversion method based on coal seam information, which considers highly magnetic burnt rocks to be produced via the combustion of a coal seam and to have thermal remanence, and this method can more accurately obtain the ranges of magnetic source for deducing coalfield spontaneous combustion areas. Combined with infrared remote sensing temperature measurement data, we analyze the range, state, and future spread direction of coalfield spontaneous combustion areas in Liaoning Province, China, according to the relative positions of high-temperature areas and highly magnetic burnt rocks. Based on the inversion results, we divided the survey area into nine blocks and obtained corresponding interpretation results. The accuracy of the interpretation was verified through drilling. This provides comprehensive spontaneous combustion area information for coal mining and disaster management.

The inversion of gravity and magnetic data can obtain the density and magnetic structure of underground space, which provide important information for resource exploration and geological structure division. The most commonly used inversion method is smooth inversion in which the objective function is built with L2-norm, which has good stability, but it produces non-focused results that make subsequent interpretation difficult. The power-type structural self-constrained inversion (PTSS) method with L2-norm is proposed to improve the resolution of smooth inversion. A self-constraint term based on the power gradient of the results is introduced, which takes advantage of the structural feature that the power gradient can better focus on the model boundary to improve the resolution. For the joint inversion of gravity and magnetic data, the power-type mutual-constrained term between different physical structures and the self-constrained term can be simultaneously used to obtain higher-resolution results. The modeling tests demonstrated that the PTSS method can produce converged high-resolution results with good noise immunity in both the respective inversions and the joint inversion. Then, the PTSS joint inversion was applied to the airborne gravity and magnetic data of the iron ore district in Shandong, revealing the shape and location of the mineralized rock mass, which are crucial information for subsequent detailed exploration.