State and parameter estimation of hydrologic models using the constrained ensemble Kalman filter

Data assimilation techniques, such as the Kalman filter (KF) and its extensions, update state variables based on the KF type of algorithms, but state‐space models usually have relevant physical laws or settings. The updated states by the KF may violate some physical constraints, which can be equality or inequality, and linear or nonlinear constraints. This paper presents three methods, the naive method, the projection method and the accept/reject method, to incorporate constraints into the ensemble Kalman filter (EnKF). Essentially, the projection method projects the updated ensemble members from the unconstrained EnKF to the feasible space characterized by the constraints. The accept/reject method tries to enforce the updated states to obey the constraints by resampling the observation error and model error. The three methods are applied to a conceptual hydrologic model for state estimation and sequential parameter learning, with specific treatment of inequality constraints. Both the reject/accept and projection methods perform better than the naive method which treats the hard constraint in a simple way and ignores other constraints. It is easy to implement the accept/reject method, the performance of which is comparable to the projection method regarding both the estimation quality and the computational time with the case study.