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Discussion papers | Copyright
https://doi.org/10.5194/npg-2018-4
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 08 Feb 2018

Research article | 08 Feb 2018

Review status
This discussion paper is a preprint. A revision of this manuscript was accepted for the journal Nonlinear Processes in Geophysics (NPG) and is expected to appear here in due course.

Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

Colin Grudzien1, Alberto Carrassi1, and Marc Bocquet2 Colin Grudzien et al.
  • 1Nansen Environmental and Remote Sensing Center, Bergen, Norway
  • 2CEREA, joint laboratory École des Ponts ParisTech and EDF R&D, Université Paris-Est, Champs-sur-Marne, France

Abstract. The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. A reduced rank representation of the estimated covariance, however, leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, describing the intrinsic role of covariance inflation in reduced rank, ensemble based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.

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Colin Grudzien et al.
Colin Grudzien et al.
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Latest update: 20 Aug 2018
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While the butterfly effect renders the forecasting problem in chaotic physical applications inherently volatile, chaotic dynamics also put strong constraints on the evolution of prediction errors. Using the framework of chaotic dynamical systems, we analyze the asymptotic properties of ensemble based Kalman filters, and how these are influenced by the dynamical constraints in the model, especially in the context of random model errors and small ensemble sizes.
While the butterfly effect renders the forecasting problem in chaotic physical applications...
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