# Three-Dimension Inversion of Magnetic Data Based on Multi-Constraint UNet++
## 1 Introduction  
Magnetic exploration is an important branch of applied geophysics, which principally measures various geological bodies and magnetic fields, achieving the purpose of studying underground structures [1]. The 3D inversion of magnetic data is an effective method for restoring the magnetic susceptibility distribution in an underground space using magnetic anomaly data [2]. There are two main indexes for evaluating the advantages and disadvantages of the 3D inversion method: one is the model-fitting effect, and the other is the data-fitting effect after the forward modeling of the inversion model [3]. Conventional regularization inversion is a commonly used, traditional 3D inversion method, which has a good data-fitting effect; however, it has problems such as a low depth resolution and insufficient model-fitting ability, so the study of new physical property inversion methods has always been a popular research direction [4]. In recent years, 3D inversion methods based on deep learning have an improved depth resolution and physical property distribution, and also have a better data-fitting effect. With enhancements in the exploration ability of instruments, it is of great significance to try to develop a new high-precision 3D physical property inversion method to improve the application effect of potential field exploration and the interpretation ability of geological problems. Neural networks (NNs) are one of the most popular machine learning methods, and they have proven to be very effectivein a large number of various data processing and analysis tasks, including in the field of geophysics. In addition, deep learning methods are the product of the development of artificial neural networks. In the past five years, the number of papers on the inversion of geophysical data using deep learning methods has gradually increased, which is due to both the strong learning and excellent generalization abilities of deep learning methods. Similarly, deep learning performs well in the field of the three-dimensional inversion of field data. For example, [PERSON] et al. implemented a 3D sparse gravity inversion based on a U-Net neural network, obtained inversion results with a high depth resolution, and discussed the anti-noise capability of U-Net [5]. [PERSON] et al. proposed a three-dimensional magnetic gradient tensor (MGT) inversion method based on convolutional neural networks (CNNs), which could automatically predict physical parameters from the two-dimensional images of the MGT. The geometric form, depth, magnetic inclination (I), magnetic declination (D), magnetic susceptibility, and other parameters of the magnetic anomaly were extracted, and the three-dimensional model of the magnetic anomaly was obtained through comprehensive analysis [6]. [PERSON] et al. proposed a 3D geological structure inversion method using convolutional neural networks (CNNs) to train their respective classification and regression models, which was able to quickly predict the parameters of geological structures for the construction of 3D models. The classification model was used to classify and judge geological structures, the regression model predicted the occurrence of geological structures with a classification accuracy of about 100%, and the regression accuracy of the different structures was mostly between 80% and 97% [7]. [PERSON] et al. proposed an optimized DL network for gravity and magnetic joint inversion, which was able to extract structural similarity features and the mapping relationship between original potential field data and physical parameters on multiple scales, and which adopted different inversion strategies (independent or combined) in different regions to improve the accuracy of the joint inversion results [8]. [PERSON] et al. proposed a deep learning three-dimensional inversion strategy based on multivariate magnetic data to reduce the non-uniqueness of potential field inversion.