# Regional Geomagnetic Field Modeling Based on Associated Legendre Polynomials
## 1 Introduction  
The geomagnetic field--an intrinsic vector field of the Earth--is generated through the interplay of the liquid outer core, magnetic materials in the mantle and crust, and the combined effects of internal and external current systems [1, 2]. The presence of a geomagnetic field can be confirmed through geomagnetic measurements and observations of auroral phenomena in high-latitude regions. Furthermore, the geomagnetic field plays a crucial role in supporting biological activity in the Earth's ecosystem and acts as a natural barrier to protect the Earth's surface from direct exposure to energetic particle radiation [3]. Geomagnetic fields have significant application potential in areas such as target detection, motion carrier navigation, and missile guidance [4, 5, 6, 7]. A critical challenge in these applications lies in generating high-precision reference maps, which require extensive high-resolution geomagnetic measurements and the support of efficient and accurate modeling technologies. Geomagnetic field modeling techniques can be broadly categorized into global and regional modeling. Global modeling provides magnetic field distributions on a world-wide scale, making it suitable for large-scale geomagnetic studies; however, when thefocus shifts to the magnetic field structure of localized regions, when only regional geomagnetic data are available, or when higher-resolution magnetic field characterization is required, the significance of regional modeling becomes particularly pronounced. Regional geomagnetic field modeling is typically performed using numerical fitting techniques and mathematical-physical modeling approaches. These latter approaches are based on mathematical-physical theories, including rectangular harmonic analysis (RHA) [8; 9], spherical cap harmonic analysis (SCHA) [10; 11], revised spherical cap harmonic analysis (R-SCHA) [12], and inversion of equivalent sources [13]. These approaches are often constrained by truncation levels, extension effects, and boundary effects. In contrast, numerical fitting approaches do not depend on potential theory. Instead, they utilize a set of functions or polynomials to model the detailed structure of a geomagnetic field in a localized region. Although Taylor polynomials use basis functions in exponential form to characterize physical quantities, they are not suitable for constructing higher-degree models due to their non-orthogonality, which significantly increases computational complexity during inversion. Legendre polynomials, with orthogonal properties, help to simplify modeling processes and enhance the truncation level of the model, allowing for a more accurate representation of the magnetic field at smaller scales. For instance, the 50 th-degree small-scale regional Legendre polynomial model developed by [PERSON] et al. can capture magnetic anomaly information within a scale range of approximately 0.4 km, with the average error in fitting experimental data maintained within 30 nT. [14]. The Legendre polynomial method for regional geomagnetic field modeling has several limitations. First, the existing Legendre polynomials are usually simplified versions of the associated Legendre functions [15], which limits the accuracy of the model. Second, boundary effects are typically addressed by refining the boundary data, which enhances the accuracy of local data but also increases the computational complexity and workload. In addition, there may be inconsistencies in the data collected at different time points, affecting the overall accuracy of the model. Furthermore, the truncation level of the model remains insufficient, resulting in significant fitting errors that restrict its applicability to magnetic fields with small-scale distributions. In this study, global geomagnetic anomaly field grid data derived from the Earth Magnetic Anomaly Grid (EMAG) model are used to create a basic model of localized regional geomagnetic anomalies--referred to as the R\(-\)ALPOLM--by employing associated Legendre polynomials.