## 1 Introduction  
Magnetic dipole source detection and parameter estimation, including position parameters, magnetic moment magnitude, and direction parameters, are critical issues in magnetic anomaly detection [1, 2]. This technology has been extensively applied in underwater target detection [3, 4], unexploded ordnance (UXO) detection [5, 6], naval mine detection [7], geomagnetic assisted matching navigation [8, 9], archaeology [10], and biomedical fields [11, 12]. Research has demonstrated that when the distance between the target source and the measurement point exceeds 2.5 times the size of the target source, the source can be modeled as a magnetic dipole. This approximation is suitable for most \"far-field\" detection scenarios, such as underwater target detection and UXO detection [13, 14]. With this understanding, the detection, localization, and parameter estimation of magnetic sources can be simplified to those of magnetic dipole sources. Despite magnetic anomaly detection being largely unaffected by environmental media, challenges persist, including weak target anomaly responses and complex background magnetic noise. These challenges contribute to low precision in target magnetic source localization and identification and reduce the convergence performance of optimization algorithms [15]. Addressing how to achieve high-precision localization and magnetic moment parameter inversion of target magnetic dipoles under strong noise interference remains vital in enhancing the accuracy and robustness of magnetic dipole parameter estimation. Currently, magnetic dipole source parameter estimation is primarily divided into two parts: estimation of source position and estimation of source magnetic moment parameters. [PERSON] et al. [16] utilized the magnetic tensor field, independently measured by a superconducting magnetic gradiometer and an array magnetic measurement system, to determine the position of the dipole target using the eigenvector method, achieving dipole source localization. [PERSON] et al. [17] derived a closed-form formula for the precise localization of dipole sources based on the measured magnetic gradient tensor field of the target source without the need to know the orientation of the target source. [PERSON] et al. [18] introduced the Scalar Triangulation And Ranging (STAR) method, which employs a cubic magnetic gradient tensor array for multi-point contraction to locate ferromagnetic targets. The STAR method facilitates immediate target localization and is unaffected by fluctuations in the Earth's magnetic field; however, the spherical assumption of the tensor invariant introduces inherent non-spherical errors. Researchers have proposed various methods to address these errors. [PERSON] et al. [19] proposed a STAR method that compensates for real-time direction and distance errors. However, the iterative process of the STAR method still encounters convergence difficulties in specific directions and numerical errors. [PERSON] et al. [20] proposed a robust calibration and localization method for magnetic gradient tensor measurement systems, which enhances the calibration accuracy of the magnetometer array through multi-invariant constraint calibration technology and improves the localization robustness for non-cooperative targets by adaptively selecting multiple pairs of magnetic gradient tensors combined with the Levenberg-Marquardt algorithm optimization. [PERSON] et al. [21] introduced a source localization technique called Normalized Source Strength (NSS), based on the eigenvalues of magnetic gradient tensor data. This technique allows the localization results to be independent of the target source's magnetization direction, enabling the unique determination of the target source's position. Methods based on NSS have been extensively used for localizing magnetic dipole targets [22; 23]. With advances in computer technology and artificial intelligence, neural networks and deep learning techniques have increasingly been applied to the localization and identification of magnetic dipole targets.