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Nonlinear error dynamics for cycled data assimilation methods

Moodey, A. J. F., Lawless, A. S., Potthast, R. W. E. and Van Leeuwen, P. J. (2013) Nonlinear error dynamics for cycled data assimilation methods. Inverse Problems, 29 (2). 025002. ISSN 0266-5611

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To link to this article DOI: 10.1088/0266-5611/29/2/025002

Abstract/Summary

We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ..., with a first guess given by the state propagated via a dynamical system model from time tk − 1 to time tk. In particular, for nonlinear dynamical systems that are Lipschitz continuous with respect to their initial states, we provide deterministic estimates for the development of the error ||ek|| := ||x(a)k − x(t)k|| between the estimated state x(a) and the true state x(t) over time. Clearly, observation error of size δ > 0 leads to an estimation error in every assimilation step. These errors can accumulate, if they are not (a) controlled in the reconstruction and (b) damped by the dynamical system under consideration. A data assimilation method is called stable, if the error in the estimate is bounded in time by some constant C. The key task of this work is to provide estimates for the error ||ek||, depending on the size δ of the observation error, the reconstruction operator Rα, the observation operator H and the Lipschitz constants K(1) and K(2) on the lower and higher modes of controlling the damping behaviour of the dynamics. We show that systems can be stabilized by choosing α sufficiently small, but the bound C will then depend on the data error δ in the form c||Rα||δ with some constant c. Since ||Rα|| → ∞ for α → 0, the constant might be large. Numerical examples for this behaviour in the nonlinear case are provided using a (low-dimensional) Lorenz '63 system.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical and Physical Sciences > National Centre for Earth Observation (NCEO)
Faculty of Science > School of Mathematical and Physical Sciences > Department of Mathematics and Statistics
Faculty of Science > School of Mathematical and Physical Sciences > Department of Meteorology
ID Code:30559
Publisher:Institute of Physics

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