Abstract:

Minimising 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 sys-
tems and experience shows that the extended Kalman filter also performs well in 
non-linear 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 repre-
sentation 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 with the state estimate obtained by the full Kalman filter and 
with the estimate obtained using an approximation to 4D-Var. The accuracy assess-
ment is facilitated by formulating the state estimation methods as observer systems. 
A practical approximation to the reduced order Kalman filter that utilises 4D-Var 
algorithms is examined