Abstract:

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 a wide class of 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 and is used to construct a reduced-order Kalman 
filter for the purpose of state identification in a time-dependent quasigeostrophic storm track model. The accuracy 
of the state identification by the reduced-order Kalman filter is assessed by comparison to the true state, to the 
state estimate obtained by the full Kalman filter, and to the state estimate obtained by direct insertion.