This paper presents a physics-constrained neural differential equation framework for parameterization, and employs it to model the time evolution of seasonal snow depth given hydrometeorological forcings. When trained on data from multiple SNOTEL sites, the parameterization predicts daily snow depth with under 9% median error and Nash Sutcliffe Efficiencies over 0.94 across a wide variety of snow climates. The parameterization also generalizes to new sites not seen during training, which is not often true for calibrated snow models. Requiring the parameterization to predict snow water equivalent in addition to snow depth only increases error to ~12%. The structure of the approach guarantees the satisfaction of physical constraints, enables these constraints during model training, and allows modeling at different temporal resolutions without additional retraining of the parameterization. These benefits hold potential in climate modeling, and could extend to other dynamical systems with physical constraints.

Estimating the variability of seasonal snow cover, in particular snow depth in remote areas, poses significant challenges due to limited spatial and temporal data availability. This study uses snow depth measurements from the ICESat-2 satellite laser altimeter, which are sparse in both space and time, and incorporates them with climate reanalysis data into a downscaling-calibration scheme to produce monthly gridded snow depth maps at microscale (10 m). Snow surface elevation measurements from ICESat-2 along profiles are compared to a digital elevation model to determine snow depth at each point. To efficiently turn sparse measurements into snow depth maps, a regression model is fitted to establish a relationship between the retrieved snow depth and the corresponding ERA5 Land snow depth. This relationship, referred to as subgrid variability, is then applied to downscale the monthly ERA5 Land snow depth data. The method can provide timeseries of monthly snow depth maps for the entire ERA5 time range (since 1950). The validation of downscaled snow depth data was performed at an intermediate scale (100 m x 500 m) using datasets from airborne laser scanning (ALS) in the Hardangervidda region of southern Norway. Results show that snow depth prediction achieved R2 values ranging from 0.74 to 0.88 (post-calibration). The method relies on globally available data and is applicable to other snow regions above the treeline. Though requiring area-specific calibration, our approach has the potential to provide snow depth maps in areas where no such data exist and can be used to extrapolate existing snow surveys in time and over larger areas. With this, it can offer valuable input data for hydrological, ecological or permafrost modeling tasks.