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Conditioning and preconditioning of the variational data assimilation problem

Haben, S. A., Lawless, A. S. and Nichols, N. K. (2011) Conditioning and preconditioning of the variational data assimilation problem. Computers & Fluids, 46 (1). pp. 252-256. ISSN 0045-7930

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To link to this article DOI: 10.1016/j.compfluid.2010.11.025

Abstract/Summary

Numerical weather prediction (NWP) centres use numerical models of the atmospheric flow to forecast future weather states from an estimate of the current state. Variational data assimilation (VAR) is used commonly to determine an optimal state estimate that miminizes the errors between observations of the dynamical system and model predictions of the flow. The rate of convergence of the VAR scheme and the sensitivity of the solution to errors in the data are dependent on the condition number of the Hessian of the variational least-squares objective function. The traditional formulation of VAR is ill-conditioned and hence leads to slow convergence and an inaccurate solution. In practice, operational NWP centres precondition the system via a control variable transform to reduce the condition number of the Hessian. In this paper we investigate the conditioning of VAR for a single, periodic, spatially-distributed state variable. We present theoretical bounds on the condition number of the original and preconditioned Hessians and hence demonstrate the improvement produced by the preconditioning. We also investigate theoretically the effect of observation position and error variance on the preconditioned system and show that the problem becomes more ill-conditioned with increasingly dense and accurate observations. Finally, we confirm the theoretical results in an operational setting by giving experimental results from the Met Office variational 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:24145
Uncontrolled Keywords:Variational data assimilation; Error covariances; Least-squares optimization; Condition number; Hessian
Additional Information:Special issue: 10th ICFD Conference Series on Numerical Methods for Fluid Dynamics (ICFD 2010)
Publisher:Elsevier

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