## 1 Introduction  
Magnetic exploration is a kind of geophysical exploration method that uses the magnetic field changes (magnetic anomalies) caused by the magnetic differences between various rocks (ore) in the crust to find useful mineral resources and study the underground structure [1; 2; 3]. Magnetic prospecting is widely used to search for useful mineral resources [4; 5; 6]. In the late 1950s and early 1960s, the Soviet Union and the United States began ocean magnetic surveys, and continental drift theory entered a new historical stage. Since the 1980s in China, the rapid development of high-precision magnetometers has significantly advanced magnetic exploration [7].  
The regularized inversion algorithm based on the logarithmic barrier method [8; 9] proposed by [PERSON] and [PERSON] in 1996 and 2003 is mostly used in the three-dimensional inversion of magnetic methods [10]. The logarithmic barrier term increases the complexity of retrieving the objective function. In the direct current inversion method, the logarithm of the model resistivity is usually adopted to avoid negative values of the model parameters. Based on the same idea, the logarithm of the magnetic susceptibility is used, and the inversion can then be carried out. At the same time, the efficiency of the three-dimensional inversion of magnetic methods based on the logarithmic barrier method is low. Currently,the limited-memory BFGS (L-BFGS) method has been mostly applied to the inversion of various geophysical methods [11; 12]. The limited-memory BFGS (L-BFGS) method has good stability, requires less memory, performs only one quasi-forward simulation and is fast and efficient.  
In the past, the objective function used in the inversion of magnetic methods was generally an additive objective function. One problem faced by this type of objective function in the inversion is that regularization factors need to be selected, and the selection of regularization factors can only be carried out according to experience or repeated inversion trial calculations, resulting in a low inversion efficiency and requiring more inversion time. With the further development of inversion at home and abroad, the objective function has been further developed. In 2002 and 2009, [PERSON] et al. proposed a contrast source inversion method based on the multiplicative objective function [13; 14; 15]. This method has been applied to large-scale electromagnetic inversion and log inversion [16]. The multiplicative objective function has been applied to three-dimensional inversion via the resistivity method, and good inversion results have been obtained. [17] The advantage of this kind of objective function is that it automatically adjusts the weight between the data fitting difference term and the regularization term in the inversion iteration process, and it does not need to repeatedly select regularization factors, which provides great convenience for inversion. At present, the three-dimensional inversion of the resistivity method based on the multiplicative objective function has been achieved in China, but the multiplicative objective function in other geophysical methods has not been applied.  
In this paper, the logarithmic magnetic susceptibility method, which eliminates the logarithmic barrier in the objective function and simplifies the objective function, is adopted. The limited-memory BFGS (L-BFGS) method is used to achieve three-dimensional magnetic inversion based on the multiplicative objective function. In this paper, the synthetic data of single prism and oblique prism anomalous body models are used for inversion, and the measured data are also used for inversion. The feasibility and effectiveness of magnetic inversion based on multiplicative objective function are verified, and a new idea is provided for selecting the weights between data items and model items.