## 1 Introduction  
In recent years, electromagnetic (EM) and magnetic methods have been essential in geophysical exploration, widely applied in subsurface detection for identifying targets with varying electrical and magnetic properties [1, 2, 3, 4]. However, traditional inversion techniques face significant limitations, including susceptibility to local minima and dependence on initial values, which can lead to inaccurate results. These challenges are further compounded by the non-uniqueness of solutions, a common issue when using single-type geophysical data [5, 6, 7]. Specifically, transient electromagnetic (TEM) methods effectively capture conductivity variations but are limited in depth penetration, while magnetic methods excel in detecting deeply buried ferromagnetic targets but are often hindered by geological noise [7, 8, 9]. To overcome these challenges, joint inversion approaches, which integrate data from multiple geophysical modalities, have been developed [10; 11; 12; 13; 14]. These methodologies capitalize on the complementary nature of EM and magnetic responses, thereby enabling the derivation of more constrained and reliable subsurface models. Joint inversion techniques can be broadly categorized into two frameworks: those based on empirical relationships between physical properties and those relying on spatial distribution coupling. The former assumes predefined relationships, such as the correlation between resistivity and density, and has been successfully applied to the joint inversion of electromagnetic, seismic, and gravity data [15]. However, these methods are often limited by the variability of such relationships across different geological contexts. The latter, on the other hand, eliminates the need for direct correlation by employing structural coupling, such as cross-gradient functions, to achieve spatially coherent inversions. This approach has been effectively demonstrated in the integration of gravity and magnetic datasets [9]. Joint inversion methodologies, which integrate data from multiple geophysical sources, have emerged to overcome these limitations. By leveraging the complementary nature of EM and magnetic responses, joint inversion approaches provide a more constrained and reliable subsurface model, improving both the accuracy and robustness of inversion results. [PERSON] et al. [8] demonstrated that combining magnetic and TEM data enables more effective target discrimination, especially in applications such as unexploded ordnance (UXO) detection, where traditional methods may struggle due to environmental noise and complex subsurface interactions. [PERSON] and [PERSON] [9] further highlighted that integrating gravity and magnetic data using cross-gradient joint inversion significantly reduces the ill-posedness inherent in single-method inversions, offering a more stable solution for complex geological settings. [PERSON] et al. [5] showed that combining EM with elastic data enhances the inversion's robustness, allowing for a more comprehensive understanding of the subsurface by incorporating both electrical and mechanical properties. The effectiveness of joint inversion has been further validated in a range of geophysical applications, showing improved robustness and stability compared to traditional methods [16]. Among various applications of magnetic data inversion, unexploded ordnance (UXO) detection is a prominent example. The following studies provide insights into advancements and limitations in this domain: Magnetic data inversion has been extensively studied in the context of UXO detection. For example, [PERSON] et al. [17] highlighted the importance of magnetic moments in UXO clearance, while [PERSON] and [PERSON] [18] proposed a Kaczmarz regularization-based approach to address magnetic anomaly inversion. Furthermore, [PERSON] [19] and [PERSON] et al. [20] investigated the impact of magnetic backgrounds and multisensor methods for UXO detection, providing valuable insights into the challenges of signal discrimination and data fusion.