# Three-Dimensional Magnetic Inversion Based on an Adaptive Quadtree Data Compression
## 1 Introduction  
The magnetic survey is widely used in geological mapping, geological structure surveys, mineral resources exploration, sedimentary basin basement detection and other fields [1]. Three-dimensional magnetic inversion refers to the computation of the spatial distribution and properties of underground sources from magnetic anomalies of an observation surface. According to the different inversion algorithms, magnetic inversion can be summarized into two categories [2], one is to obtain the equivalent magnetism through inversion calculations to outline the distribution of magnetic anomaly field sources, called the probability tomography [3; 4; 5]. This is a semi-quantitative inversion interpretation method that is simple, stable, and fast to compute. It can be used to locate the geometric position of the anomaly and solve complex geological problems. Another type of inversion method is to directly calculate the magnetic distribution in the imaging space through linear or nonlinear algorithms to obtain information about the shape, volume, magnetic parameter distribution and properties of anomalies. This is called the physical property inversion method, and it can quantitatively interpret the geometric and physical properties of the target geologic map and provide a strong basis for subsequent geologicalinterpretation. In this kind of inversion algorithm, nonlinear methods such as simulated annealing algorithms, genetic algorithms, and neural network algorithms are global stochastic optimization algorithms [6; 7], which can effectively reduce the possibility of falling into local extremes during the search process, but generally suffer from the problem of high computational volume. In this paper, we studied the linear physical property inversion algorithm, which belongs to the second type of method. It has been successfully applied in mineral resource exploration, oil and gas exploration, and identification of geological structural units, etc. [8; 9; 10]. The linear physical property inversion method is now relatively mature after decades of development. In 1975, [PERSON] [11] introduced the \"linear inversion\" method by adding the reference model and prior information into the inversion method, and they used the Backus-Gilbert method to perform two-dimensional gravity inversion. By introducing the minimum volume constraint into the inversion, [PERSON] and [PERSON] [12] proposed a compact inversion method to make the parameter distribution centralized. [PERSON] and [PERSON] [13; 14] introduced the minimum constraint of a spatial derivative and depth weighting to realize the three-dimensional inversion of gravity and magnetic data. [PERSON] and [PERSON] introduced the minimum support function and the minimum gradient support function to clarify the boundary of the inversion result, thus obtaining a focused result [15; 16; 17; 18]. By adding petrophysical data to the inversion, [PERSON] and [PERSON] [19; 20] proposed an inversion method based on fuzzy c-clustering, which improved the resolution of the inversion results. [PERSON] et al. [21] discussed various constraints and noted that the application of constraints in the inversion should be combined with actual geological problems. In addition, many scholars have improved the constraints and inversion algorithms to varying degrees [22; 23; 24; 25]. The inversion methods reviewed above, all discretize the underground into numerous prism elements and estimate the three-dimensional distribution characteristics of physical parameters [26]. In the three-dimensional inversion, the number of elements in the kernel function matrix is huge. Whether it is calculating the kernel function matrix in the forward algorithm or multiplying the kernel function matrix with its transpose in the inversion algorithm, it is quite time-consuming.