A data-driven framework for forward and inverse hydrology in large basins: insights from GRACE(-FO) and discharge observation

Since 2002, the Gravity Recovery and Climate Experiment (GRACE) and its successor, GRACE Follow-On (GRACE-FO), has enabled the estimation of Terrestrial Water Storage (TWS) anomalies (ΔS), significantly improving our ability to constrain water balance components at regional scales. These observations also provide new insights into large-scale drainage dynamics by allowing direct examination of the storage–discharge (S–Q) relationship, a fundamental element to conceive lumped rainfall–runoff models.

In many catchments, including those in the Amazon basin, the empirical S–Q relationship can be reasonably approximated by a deterministic function Q=ƒ(ΔS, t), where t denotes time. Substituting this function into the water balance equation yields a mass-conserving rainfall–runoff model expressed as a nonlinear first-order differential equation in ΔS . This formulation supports forward simulation of discharge and storage given precipitation and evapotranspiration (P, ET) and is amenable to assimilation of discharge observations using techniques such as Bayesian smoothing. More importantly, the model can be rearranged to perform inverse estimation — also known as “hydrology backward”— to infer net recharge (P-ET) from observed discharge Q.

In this study, we examine 50 catchments of varying size within the Amazon basin and estimate for each of them the function ƒ using two approaches: (1) by fitting a spline, which can incorporate time dependence, and (2) a Single-Hidden-Layer Feedforward Neural Network (SLFN) trained via the Extreme Learning Machine (ELM), a lightweight learning algorithm which does not require iterative backpropagation. Forward simulations (P, ET → ΔS, Q) demonstrate good skill in reconstructing hydrographs, independently of the catchment size, with some limitations in reproducing TWS anomalies during high-flow periods. For inverse modeling, we focus on reconstructing evapotranspiration (P, Q → ET) using for the precipitation a combination of various products. We show that although the estimated uncertainty on ET remains substantial, the resulting estimates are broadly consistent with existing independent ET datasets.