Ioannou, P. J., & Farrell, B. F. (2006).
Application of Generalized Stability Theory to Deterministic and Statistical Prediction., Predictability of Weather and Climate. In
T. Palmer & R. Hagedorn (Ed.), (pp. 181-216) . Cambridge University Press, Cambridge.
AbstractUnderstanding of the stability of deterministic and stochastic dynamical systems
has evolved recently from a traditional grounding in the system’s normal modes
to a more comprehensive foundation in the system’s propagator and especially in
an appreciation of the role of non-normality of the dynamical operator in deter-
mining the system’s stability as revealed through the propagator. This set of ideas,
which approach stability analysis from a non-modal perspective, will be referred
to as generalised stability theory (GST). Some applications of GST to determinis-
tic and statistical forecast are discussed in this review. Perhaps the most familiar
of these applications is identifying initial perturbations resulting in greatest error
in deterministic error systems, which is in use for ensemble and targeting appli-
cations. But of increasing importance is elucidating the role of temporally dis-
tributed forcing along the forecast trajectory and obtaining a more comprehensive
understanding of the prediction of statistical quantities beyond the horizon of deter-
ministic prediction. The optimal growth concept can be extended to address error
growth in non-autonomous systems in which the fundamental mechanism produc-
ing error growth can be identified with the necessary non-normality of the sys-
tem. The influence of model error in both the forcing and the system is examined
using the methods of stochastic dynamical systems theory. In this review determin-
istic and statistical prediction, i.e. forecast and climate prediction, are separately
discussed.
.pdf Farrell, B. F., & Ioannou, P. J. (2006).
Approximating Optimal State Estimation., Predictability of Weather and Climate. In
T. Palmer & R. Hagedorn (Ed.), (pp. 181-216) . Cambridge University Press, Cambridge.
AbstractMinimising 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
.pdf