  Balanced truncation is a well-established model order reduction method which
has been applied to a variety of problems. Recently, a connection between
linear Gaussian Bayesian inference problems and the system-theoretic concept of
balanced truncation has been drawn. Although this connection is new, the
application of balanced truncation to data assimilation is not a novel idea: it
has already been used in four-dimensional variational data assimilation
(4D-Var). This paper discusses the application of balanced truncation to linear
Gaussian Bayesian inference, and, in particular, the 4D-Var method, thereby
strengthening the link between systems theory and data assimilation further.
Similarities between both types of data assimilation problems enable a
generalisation of the state-of-the-art approach to the use of arbitrary prior
covariances as reachability Gramians. Furthermore, we propose an enhanced
approach using time-limited balanced truncation that allows to balance Bayesian
inference for unstable systems and in addition improves the numerical results
for short observation periods.
