Low-dimensional representation of error covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensionalapproximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause thesteady-state analysis error covariance to admit a low-dimensional representation. A usefulmeasure of forecast/analysis cycle stability is thebound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-stateanalysis error covariance matrix eigenvalues are derived from the bound matrix. The estimatesgeneralize to time-dependent systems. If much of the steady-state analysis error variance is dueto a few dominant modes, the leading eigenvectors of the bound matrix approximate those ofthe steady-state analysis error covariance matrix. The analytical results are illustrated in twonumerical examples where the Kalman filter is carried to steady state. The first example usesthe dynamics of a generalized advection equation exhibiting non-modal transient growth.Failure to observe growing modes leads to increased steady-state analysis error variances.Leading eigenvectors of the steady-state analysis error covariance matrix are well approximatedby leading eigenvectors of the bound matrix. The second example uses the dynamics of adamped baroclinic wave model. The leading eigenvectors of a lowest-order approximation ofthe bound matrix are shown to approximate well the leading eigenvectors of the steady-stateanalysis error covariance matrix.