Reduced order rainfall-discharge model for hydro-climatic data assimilation: a data-driven approach

Hydro-climatic variables such as precipitation (P), evapotranspiration (ET), terrestrial water storage (TWS) or river discharge define the terrestrial water cycle at local and global scales. The robust detection and quantification of steady trends in these variables require analysing sufficiently long time series of observations. Yet, historical discharge records may suffer from long data gaps or simply be too short; different reanalyses or data-driven models of P and ET often show large discrepancies and the associated uncertainty is not systematically provided. Finally, TWS has been observed only since the launch of GRACE in 2002 and also suffers from dozens of missing epochs.

Here, we present a 3-step approach to consistently reconstruct the historical time series of TWS and discharge at the catchment scale. In the first step, we use in-situ discharge observations and TWS anomaly derived from GRACE(-FO) observations to identify a reduced-order and mass-conserving rainfall-discharge model of the catchment. In the second step, the model is run with different precipitation and evapotranspiration data sets to select the pair P and ET reproducing most accurately the observed discharge and TWS. If necessary, the resulting net water flux (P-ET) is adjusted with a bias to improve the simulation accuracy. lastly, we apply a Bayesian smoother such as the Rauch–Tung–Striebel smoother to estimate TWS and discharge along with their respective uncertainty over the period covered by the P-ET time series. Critical to the proposed approach is the rainfall-discharge model identification. Here, we assume that the observed monthly-averaged discharge at the outlet is primarily driven by the TWS in the upstream catchment. As a consequence, we first estimate a storage-discharge model in the form of a continuous-time differential equation. This equation is subsequently coupled with the water mass balance equation to form the rainfall-discharge model. Remarkably, this final model is estimated independently of any P and ET models.

Finally, we apply the proposed approach to Amazonian and Siberian catchments for a period spanning from 1980 to 2020. In the first case, linear and time-invariant models capture with reasonable accuracy the observed drainage dynamics. In contrast, non-linear or linear and time-variable models are necessary to take correctly into account the temperature-dependent snow and ice accumulation and thaw in the case of Siberian catchments.