# Three-Dimensional Magnetic Inversion Based on an Adaptive Quadtree Data Compression
## 1 Introduction
New sensing technologies and acquisition platforms, such as aeromagnetic surveys, unmanned aerial surveys, satellite magnetic surveys, ground-based high-precision sampling, etc., can collect dense data over large exploration areas, which requires more computer memory and longer computation time to process using traditional inversion methods. Therefore, how to reduce storage requirements and improve computational efficiency is a popular research topic in the field of three-dimensional inversion (3D inversion). [PERSON] and [PERSON] [27] performed a wavelet transform on the sensitivity matrix. By setting a threshold value on the wavelet coefficients, the sensitivity matrix was transformed into a sparse matrix for storage and calculation, to reduce the required memory and the amount of calculation. However, this method causes the sensitivity matrix to have more empty elements, and the faster the inversion, the greater the accuracy loss. Considering the symmetry and finiteness of a forward kernel matrix, [PERSON] et al. [28] proposed the concept of a geometric lattice to reduce the computing time and memory space of the kernel matrix. By using highly parallel Graphics Processing Unit (GPU) acceleration techniques, forward and inversion calculations can be accelerated tens of times [29; 30; 31], but, this method requires high computer performance and does not reduce the computation of the algorithm. To solve the problem of a large amount of computation, local grid refinement technology [32] and adaptive grid generation technology [33; 34] have been proposed for model space. [PERSON] et al. [35] optimized a local mesh based on edge and center anomaly detection. [PERSON] et al. [36] developed an adaptive down-sampling method to reduce the number of observed data points and achieve data space compression. In addition, the fast gravity inversion method based on singular value decomposition [37] and Lanczos double diagonalization [38; 39; 40] have also had beneficial effects on large-scale data calculations. To reduce the amount of calculation, increase the calculation speed, and reduce the memory footprint, the main aim of this paper was to propose an adaptive quadtree decomposition compression algorithm that compresses the collected magnetic data and reduces the amount of data, thus reducing the size of the kernel function matrix. A quadtree is a tree-like data structure, with four sub-blocks on each node. Quadtree decomposition is the process of dividing the data into four quadrants and then recursively dividing them until each quadrant meets the precision requirement. The concept of the image quadtree was proposed by [PERSON] and [PERSON] in the early 1970s [41]. The quadtree is an important method in image processing. It has the advantages of fast decomposition, low time complexity, and flexibility. It is widely used in computer graphics, image segmentation, image fusion, remote sensing image processing, geographic information system, geological modeling, and other fields [42; 43; 44]. This paper first reviews the development of 3D inversion and then analyzes the solutions proposed by different scholars for large-scale data. Finally, an adaptive quadtree decomposition technology is proposed, which can compress the data while preserving the anomalous information of the observation data, thereby greatly improving the inversion efficiency. The effectiveness of the method is demonstrated by experiments on synthetic models and measured data.