A gradient-enhanced sequential nonparametric data assimilation framework for soil moisture flow

Soil water content (SWC) is a vital variable in the hydrological cycle, while simulation of it often relies on resolving the soil water flow equation. To cope with the unavailability or poor quality of physical models, various nonparametric data assimilation (DA) schemes have been established. However, there tends to be two significant common challenges in such methods: (1) the difficulty in capturing locally changing behaviors of time series, especially the peaks and troughs, and (2) poor statistical interpolation capability in time and space. These two challenges are no doubt attributed to the complete renunciation of physical constraints. Unlike previous physics-informed approaches that incorporated physical governing equations and engineering control into the loss functions, this study attempts to introduce additional physical constraints from data gradient into the model-free DA framework. As a follow-up study of Wang et al. (2021), a gradient-enhanced version of nonparametric DA schemes (i.e., GE-EnKFGP) is proposed. The temporal (daily) and spatial (vertical) gradients of the SWC are merged into the construction of the unsaturated flow dynamical models based on the Gaussian process (GP), while the Kalman update formulation is used to reconcile real-time observations. With the aid of a series of real-world cases, the performance of the GE-EnKFGP was compared with the original EnKFGP and its gradient-based version (GB-EnKFGP), where the temporal gradients of the SWC were used as the proxy for the SWC as the GP output. The results showed that the enhancement of the gradient information in the GE-EnKFGP led to a better estimation than the initial EnKFGP due to its more accurate identification of multiple local extrema. This should be attributed to the mass conservation constraint hidden within the temporal gradients and the implicit constraint of the driving force (or upper boundary) from the spatial gradients. Spatial gradients of the SWC outperformed temporal ones under various application scenarios. The GB-EnKFGP and GE-EnKFGP exhibited superior performances in retrieving surface SWC than that in the deeper layer. Hence, an enhancement scheme using only the temporal gradients of the surface layer was recommended. In the context of spatial extrapolation, the assistance of spatial gradients yielded an improved estimate of the deeper SWC quite robustly through GP training and assimilation of easy-to-access surface data. However, the implementation of the GB-EnKFGP and temporal gradient-enhanced EnKFGP [i.e., GE-EnKFGP (t)] run the risk of triggering a performance collapse due to the delayed response of SWC profiles to rainfall events.