Low-dimensional representation of error covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here s tability properties of the forecast/analysis cycle for linear, time-indepen dent systems are used to identify factors that cause the steady-state analy sis error covariance to admit a low-dimensional representation. A useful me asure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lo wer estimates for the steady-state analysis error covariance matrix eigenva lues are derived from the bound matrix. The estimates generalize to time-de pendent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix appr oximate those of the steady-state analysis error covariance matrix. The ana lytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting non-modal transient growth. Fail ure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covaria nce matrix are well approximated by leading eigenvectors of the bound matri x. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matri x are shown to approximate well the leading eigenvectors of the steady-stat e analysis error covariance matrix.