Articles | Volume 25, issue 3
Nonlin. Processes Geophys., 25, 633–648, 2018

Special issue: Numerical modeling, predictability and data assimilation in...

Nonlin. Processes Geophys., 25, 633–648, 2018

Research article 04 Sep 2018

Research article | 04 Sep 2018

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. However, a reduced rank representation of the estimated covariance 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, providing a novel theoretical interpretation for the role of covariance inflation in 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.

Short summary
Using the framework Lyapunov vectors, we analyze the asymptotic properties of ensemble based Kalman filters and how these are influenced by dynamical chaos, especially in the context of random model errors and small ensemble sizes. Particularly, we show a novel derivation of the evolution of forecast uncertainty for ensemble-based Kalman filters with weakly-nonlinear error growth, and discuss its impact for filter design in geophysical models.