Exploring the spatial distribution of sea surface height involves two primary methodologies: utilizing gridded reanalysis data post-secondary processing or conducting direct fitting along-track data. While processing gridded reanalysis data may entail information loss, existing direct fitting methods have limitations. Therefore, there is a pressing need for novel direct fitting approaches to enhance efficiency and accuracy in sea surface height fitting. This study demonstrates the viability of Legendre polynomial surface fitting, benchmarked against bicubic quasi-uniform B-spline surface fitting, which has been proven to be a well-established direct fitting method. Despite slightly superior accuracy exhibited by bicubic quasi-uniform B-spline surface fitting under identical order combinations, Legendre polynomial surface fitting offers a simpler structure and enhanced controllability. However, it is pertinent to note that significant expansion of the spatial scope of fitting often results in decreased fitting efficacy. To address this, the current research achieves the precise fitting of sea surface height across expansive spatial ranges through a regional stitching methodology.

The direct fitting of sea level anomaly (SLA) using satellite along-track data provides a critical approach for monitoring mesoscale ocean dynamics. While bicubic quasi-uniform B-spline surface fitting has demonstrated feasibility in localized sea areas, its applicability to basin-scale regions remains underexplored. This study focuses on the northern Atlantic Ocean, employing B-spline surface fitting to derive SLA fields from satellite along-track data. The results show strong agreement with in situ measurements, yielding a mean absolute error (MAE) of 1.89 cm and a root mean square error (RMSE) of 3.02 cm. Comparative analysis against the Copernicus Marine Environment Monitoring Service (CMEMS) Level-4 gridded SSH data reveals nearly equivalent accuracy (MAE: 1.95 cm; RMSE: 3.06 cm). The relationship between the order of fitting and the spatial extent of the fitting domain is also examined. Furthermore, the influence of the coastline on the fitting results is investigated in detail. As the coastline area expanded, the MAE and RMSE for the entire region increased. But the maximum increase in MAE was only 1.20 cm, and the maximum increase in RMSE was only 2.49 cm. Notably, there was no upward trend in MAE and RMSE in the mesoscale vortex dense area, which highlights the advantage of B-spline’s local support. Geostrophic flow and vertical component of relative vorticity are computed from the satellite along-track SLA data, with results showing agreement with Level-4 gridded geostrophic flow and vertical component of relative vorticity data.

Satellite altimetry technology provides along-track sea level anomaly (SLA) data for studying mesoscale eddies. However, accurately reconstructing their spatial structures from discrete and non-uniform along-track observations remains a significant challenge. This study systematically evaluates the performance of bi-quadratic, bi-cubic, and bi-quartic quasi-uniform B-spline surface fitting methods for mesoscale eddy reconstruction in the South Indian Ocean (60°S–30°S, 75°E–105°E). By combining idealized experiments with real satellite data, a comprehensive comparison is conducted across several dimensions, including fitting accuracy, computational efficiency, parameter robustness, error distribution, and the physical plausibility of derived vorticity fields. For SLA surface fitting, all three methods achieve comparable accuracy, but the bi-quadratic B-spline demonstrates marked advantages in computational efficiency. Its single-fit time is only 53% and 27% of that of the bi-cubic and bi-quartic methods, respectively, and it shows insensitivity to node configuration, highlighting its practicality. In contrast, vorticity field inversion, which relies on the second derivative of the fitted surface, requires higher-order continuity. Only the bi-quartic B-spline, with C³ continuity, produces physically credible and smooth vorticity fields, whereas lower-degree methods result in discontinuous or non-smooth fields. Based on these findings, this study proposes an application-oriented selection principle: the bi-quadratic B-spline is recommended for efficiency-focused tasks, such as eddy detection, while the bi-quartic B-spline is necessary for dynamic analyses involving vorticity.