## 1 Introduction
Consequently, the iterative process of converting the magnetic field gradient to apparent resistivity variation becomes more complex. The advancement of computer algorithms and the widespread use of machine learning has provided new approaches to the solution process [11; 12; 13]. Machine learning (ML) fits a certain non-linear mapping relationship based on big data, thereby avoiding the challenges associated with data analysis and mathematical modeling encountered in traditional mathematical-physical methods during data processing. Hence, ML offers distinct advantages for solving non-linear and complex problems [14; 15]. In geophysical exploration, most geological models are characterized by non-linearities, so machine learning has been widely used in geophysical data processing and interpretation, such as lithology identification, curve fitting, and geophysical inversion, due to their unique non-linear fitting ability [16; 17; 18; 19]. Support vector machine (SVM) is a classical machine learning method with a solid theoretical foundation and strong generalization performance [20]. Compared to deep neural networks, SVM requires fewer training samples, resulting in lower computing costs [21]. Many researchers have successfully applied SVM to geophysical data processing [22; 23; 24]. LSSVMs (least square support vector machines) are an extension of SVM that offers superior approximations for non-linear problems, making them a valuable tool for modeling and forecasting non-linear systems [25; 26]. It requires fewer parameters and faster convergence to the optimal solution. However, the effectiveness of predictions heavily relies on the selection of kernel function parameters and regularization parameters. Therefore, when employing an LSSVM in computational optimization, careful consideration must be given to choosing the appropriate type of kernel function, as well as determining suitable values for its associated parameters. Generally, the Gaussian kernel function exhibits excellent anti-interference ability and is widely adopted. However, it is highly sensitive to parameter values which makes it necessary to optimize parameters when utilizing the Gaussian function. In this paper, the Gaussian kernel function is adopted to analyze the problem. Due to the parametric sensitivity of Gaussian kernel functions, it may not be appropriate to select parameters based on common experience. Therefore, an optimization algorithm is needed to optimize the two main parameters of the Gaussian kernel function, the kernel parameter, and the regularization parameter [27; 28; 29]. NGO (Northern Goshawk optimization) is a metaheuristic optimization algorithm designed for global optimization, which efficiently explores solution spaces [30; 31; 32; 33]. Considering the non-linear relationship between the magnetic field gradient and apparent resistivity variation, this paper introduces NGO-LSSVM in apparent resistivity variation imaging for GAFEM, where NGO optimizes LSSVM parameters. First, the basic electromagnetic theory of magnetic field gradient response was studied to determine the characteristic parameters that affect apparent resistivity variation calculation. On this basis, the basic theory of LSSVM models and NGO algorithms was studied, and an optimized NGO-LSSVM model was developed. Finally, the prediction effect of the NGO-LSSVM model was verified by predicting the apparent resistivity variation of the theoretical models and field data.