Vol. 16, No. 1, 2021

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Balanced data assimilation for highly oscillatory mechanical systems

Gottfried Hastermann, Maria Reinhardt, Rupert Klein and Sebastian Reich

Vol. 16 (2021), No. 1, 119–154
Abstract

Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems.

Keywords
data assimilation, ensemble Kalman filter, balanced dynamics, highly oscillatory systems, Hamiltonian dynamics, geophysics
Mathematical Subject Classification
Primary: 65C05, 62M20, 93E11, 62F15, 86A22
Milestones
Received: 9 September 2020
Revised: 30 March 2021
Accepted: 14 April 2021
Published: 22 June 2021
Authors
Gottfried Hastermann
Institut für Mathematik
Freie Universität Berlin
Berlin
Germany
Maria Reinhardt
Institut für Mathematik
Universität Potsdam
Potsdam
Germany
Rupert Klein
Institut für Mathematik
Freie Universität Berlin
Berlin
Germany
Sebastian Reich
Institut für Mathematik
Universität Potsdam
Potsdam
Germany