Approximating Optimal State Estimation

Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for linear systems and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time dependent error covariance matrix which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model which suggests the use of reduced order error models to obtain near optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced order approximation of the forecast error system. This reduced order system is obtained by balanced truncation of the Hankel operator representation of the full error system. As an example application a reduced order Kalman filter is constructed for a time-dependent quasi-geostrophic storm track model. The accuracy of the state identification by the reduced order Kalman filter is assessed and comparison made to the state estimate obtained by the full Kalman filter and to the estimate obtained using an approximation to 4D-Var. The accuracy assessment is facilitated by formulating the state estimation methods as observer systems. A practical approximation to the reduced order Kalman filter that utilizes 4D-Var algorithms is examined.