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

Time-limited Balanced Truncation for Data Assimilation Problems