Development of the consider cubature Kalman filter for state estimation of hydrological models with parameter uncertainty

Abstract

Filtering methods are popular for improving the forecast performances of hydrological models. Parameter uncertainty can result in poor state estimates; hence, augmented state-parameter estimation methods have gained much attention in the hydrologic literature. However, researchers have not treated the robustness of augmented methods in much detail. Another powerful approach to include the parameter uncertainty in the state estimation process is the consider Kalman filter (consider KF), which accounts for the parameter uncertainty without updating the parameters. The consider KF has received little to no attention in the hydrologic context. This paper investigates the usefulness of the consider KF by updating the states of a hydrological model (without updating the parameters), with the augmented state-parameter estimation method as the benchmark. Two nonlinear extensions for the augmented state-parameter estimation method (augmented cubature Kalman filter, ACKF) and consider Kalman filter (consider cubature Kalman filter, CCKF) are developed since hydrological models are mostly nonlinear. Both filters are tested with a lumped hydrological model. The results of a state update synthetic case show that both filters can improve the accuracy of the updated states by including the parameter uncertainty, while the ACKF may deteriorate the original performance when given perfect parameters. The results of a reforecast experiment show that both filters can improve the forecast performance by updating the model states based on the observed discharge, while the ACKF may suffer from potential divergence. Either the ACKF or CCKF can be considered for state estimation in the presence of parameter uncertainty in hydrological forecasting, with the CCKF being the more robust option.