Analysis error covariance versus posterior covariance in variational data assimilation

I. Yu. Gejadze, V. Shutyaev, F.X. Le Dimet

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22 Citations (Scopus)
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The problem of variational data assimilation for a nonlinear evolution model is
formulated as an optimal control problem to find the initial condition function
(analysis). The data contain errors (observation and background errors); hence
there is an error in the analysis. For mildly nonlinear dynamics the analysis error
covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, and for stronger nonlinearity by the ‘effective’ inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its ‘effective’ counterpart is introduced. An approach for computing the mentioned estimates in the matrix free environment using the Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term.
Original languageEnglish
Pages (from-to)1826–1841
Number of pages16
JournalQuarterly Journal of the Royal Meteorological Society
Issue number676
Early online date21 Dec 2012
Publication statusPublished - 2013


  • large-scale flow models
  • hessian
  • Bayesian posterior covariance
  • analysis error covariance
  • optimal control
  • data assimilation
  • nonlinear dynamics


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