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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 8, issue 6
Nonlin. Processes Geophys., 8, 331–340, 2001
https://doi.org/10.5194/npg-8-331-2001
© Author(s) 2001. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

Special issue: Quantifying Predictability

Nonlin. Processes Geophys., 8, 331–340, 2001
https://doi.org/10.5194/npg-8-331-2001
© Author(s) 2001. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  31 Dec 2001

31 Dec 2001

Adjoints and low-rank covariance representation

M. K. Tippett1 and S. E. Cohn2 M. K. Tippett and S. E. Cohn
  • 1International Research Institute for climate prediction, LDEO, Palisades, NY, USA
  • 2Data Assimilation Office, NASA/GSFC, Greenbelt, MD, USA

Abstract. Quantitative measures of the uncertainty of Earth system estimates can be as important as the estimates themselves. Direct calculation of second moments of estimation errors, as described by the covariance matrix, is impractical when the number of degrees of freedom of the system state is large and the sources of uncertainty are not completely known. Theoretical analysis of covariance equations can help guide the formulation of low-rank covariance approximations, such as those used in ensemble and reduced-state approaches for prediction and data assimilation. We use the singular value decomposition and recently developed positive map techniques to analyze a family of covariance equations that includes stochastically forced linear systems. We obtain covariance estimates given imperfect knowledge of the sources of uncertainty and we obtain necessary conditions for low-rank approximations to be appropriate. The results are illustrated in a stochastically forced system with time-invariant linear dynamics.

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