State and parameter estimation with the extended Kalman filter: an alternative formulation of the model error dynamics

An alternative formulation of the extended Kalman filter for state and parameter estimation is presented, referred to as Short‐Time Augmented Extended Kalman Filter (ST‐AEKF). In this algorithm, the evolution of the model error generated by the uncertain parameters is described using a truncated short‐time Taylor expansion within the assimilation interval. This allows for a simplification of the forward propagation of the augmented error covariance matrix with respect to the classical state augmented approach. The algorithm is illustrated in the case of a scalar unstable dynamics and is then more extensively analyzed in the context of the Lorenz 36‐variable model. The results demonstrate the ability of the ST‐AEKF to provide accurate estimate of both the system's state and parameters with a skill comparable to that of the full state augmented approach and in some cases close to the EKF in a perfect model scenario. The performance of the filter is analyzed for different initial parametric errors and assimilation intervals, and for the estimates of one or more model parameters. The filter accuracy is sensitive to the nature of the estimated parameter but more importantly to the assimilation interval, a feature connected to the short‐time approximation on which the filter formulation relies. The conditions and the context of applications of the present approach are also discussed