Low-dimensional representation of error covariance

Publication Status is "Submitted" Or "In Press: 
LDEO Publication: 
Publication Type: 
Year of Publication: 
2000
Editor: 
Journal Title: 
Tellus Series a-Dynamic Meteorology and Oceanography
Journal Date: 
Oct
Place Published: 
Tertiary Title: 
Volume: 
52
Issue: 
5
Pages: 
533-553
Section / Start page: 
Publisher: 
ISBN Number: 
0280-6495
ISSN Number: 
Edition: 
Short Title: 
Accession Number: 
ISI:000089782400006
LDEO Publication Number: 
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Abstract: 

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations 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 the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical 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. 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 approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.

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