Physics-informed neural network for gridded SSH from SWOT observations considering the next-order balanced model

The Surface Water and Ocean Topography (SWOT) satellite mission, launched in December 2022, provides revolutionary measurements of the sea surface height (SSH) variations with unprecedented spatial resolution down to Ο(1 km). As a result, SWOT products have significant potential in monitoring ocean dynamics down to the submesoscale. However, the repeat cycle of 21 days introduces a barrier to fully capture these dynamics as they vary on the order of days. To fully exploit the potential of the satellite mission and simplify processing requirements for potential users, we propose a physics-informed neural network (PINN) to generate gridded SSH products from SWOT L3 along-track snapshots. The neural network has a U-Net-like architecture combined with residual learning to consider the spatial variations of the SSH field, and takes time, geolocations, and gridded SSH from conventional altimetry missions as input features, while the SWOT observations serve as ground truth. In addition to the classical data loss, the PINN model applies direct constraints on the model's trainable parameters by forcing them to fulfill the next-order correction of the quasi-geostrophic theory (SQG⁺¹), which has been demonstrated to be able to capture cyclogeostrophic balance and frontogenesis attributed to submesoscale dynamics. To this end, the high resolution of SWOT observations is kept, while the velocities and pressure fields associated with the SQG⁺¹ theory are predicted. We conducted experiments using both simulated data and real-world data. Both experiments demonstrate the benefits of incorporating physical loss to achieve higher generalizability, thereby filling the gaps between SWOT tracks reasonably. Based on the real-world data, 2-km gridded SSH products with a temporal resolution of five days are achieved. The proposed method shows promising potential for generating high-resolution gridded products while considering physical constraints. The product will be beneficial for the community to analyze mesoscale to submesoscale ocean dynamics, and compare with other sources of surface and in-situ data in the upper ocean.