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13 - Stochastic parametrized extended Kalman filter for filtering turbulent signals with model error

from Part III - Filtering turbulent nonlinear dynamical systems

Published online by Cambridge University Press:  05 March 2012

Andrew J. Majda
Affiliation:
New York University
John Harlim
Affiliation:
North Carolina State University
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Summary

Throughout the book, we have stressed that a central issue in practical filtering of turbulent signals is model error. Naively, one might think that model errors always have a negative effect on filter performance and indeed this was illustrated in Chapters 2 and 3 in simple examples with simple time differencing methods like backward or forward Euler with associated natural time discrete noise. However, a central issue of this book is to emphasize that judicious model errors in the forward operator, guided by mathematical theory, can both ameliorate the effect of the curse of ensemble size for turbulent dynamical systems and retain high filtering skill, remarkably, often exceeding that with the perfect model! In particular, we have illustrated these principles with various model errors arising from using approximate numerical solvers (Chapter 2), reduced strategies for filtering dynamical systems with instability (Chapters 3 and 8), reduced strategies for filtering sparsely observed signals (Chapter 7) and simple linear stochastic models for filtering turbulent signals from nonlinear dynamical systems (Chapters 5, 10, and 12). In our earlier discussion (see Chapters 8, 10, and 12), we demonstrated that the off-line strategy accounting for model errors, the mean stochastic model (MSM), under some circumstances produces reasonably accurate filtered solutions. However, this off-line strategy often has limited skill for estimating real-time prediction problems with rapid fluctuations that are often observed in nature since the off-line strategy (MSM) relies heavily on a fixed parameter set that is extracted from long-time or climatological statistical quantities such as the energy spectrum and correlation time.

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Publisher: Cambridge University Press
Print publication year: 2012

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