An Inverse Predictor-Corrector Data Assimilation Method for One-Dimensional Discharge and Bathymetry Estimation from Surface Flow Measurements

Abstract

This study presents the Predictor-Corrector Data Assimilation (IPCDA) method, a numerical algorithm for estimating discharge and bed topography in one-dimensional steady subcritical open-channel flows when boundary conditions and bathymetry are unavailable. The governing equation is derived from the onedimensional Saint-Venant equation under steady-state conditions. The Working Grid (WG) is constructed from a subset of observational nodes selected according to a minimum water surface elevation drop criterion grounded in the diffusion-wave approximation and motivated by practical instrument resolution constraints. To address the inherent underdetermination arising from incomplete information, the algorithm proceeds in two sequential phases: in the Predictor-phase, a local inverse problem is solved independently within each cell of the WG, providing initial discharge estimates through closed-form algebraic expressions derived from the discretized energy balance equation; in the Corrector-phase, the global discharge estimate is refined through the iterative solution of a quadratic programming problem obtained via a derivative-free linearization of the nonlinear friction slope term. The methodology was assessed using two synthetic test cases generated with HEC-RAS. The first case corresponds to a 2.6 [km] trapezoidal channel with variable bed slope and uniform roughness coefficient, considering reference discharges of 1, 10, and 100 [m3/s]. The second case consists of a 7.5 [km] reach of the Biobío river in Chile under a flood discharge condition of 2000 [m3/s] and variable Manning’s n. In both experiments, the estimated discharges generally exhibited relative errors below 5%, specifically when the tolerance for the creation of the WG uses the higher resolutions. In the trapezoidal channel, bed elevation profiles were successfully reconstructed with normalized errors below 2.3% for low to-moderate flow conditions, increasing at high discharge where a relatively small-parameter hypothesis of the geometry assumption is less satisfied. For the Biobío river case, the flow area was recovered within ±6%
at a representative cross section, despite the use of a rectangular approximation for a highly irregular natural geometry. A sensitivity analysis combining variations in the objective function weights and the Manning roughness coefficient n provides a practical framework for constructing a plausible discharge interval from IPCDA runs considering independent segments of the same river: by pooling the estimates across parameters configurations and assumptions, the range of estimated discharges serves as a data-driven uncertainty band. The width of this band is controlled primarily by the uncertainty in n. For the two cases tested, the median estimate remains within 4% of the reference discharge across all configurations, suggesting that the interval is approximately centered on the true discharge even under significant parametric uncertainty.