Conditioning and preconditioning of the variational data assimilation problem

Haben, S. A., Lawless, A. S. ORCID: https://orcid.org/0000-0002-3016-6568 and Nichols, N. K. ORCID: https://orcid.org/0000-0003-1133-5220 (2011) Conditioning and preconditioning of the variational data assimilation problem. Computers & Fluids, 46 (1). pp. 252-256. ISSN 0045-7930 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.

Additional Information	Special issue: 10th ICFD Conference Series on Numerical Methods for Fluid Dynamics (ICFD 2010)
Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/24145
Item Type	Article
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > National Centre for Earth Observation (NCEO) Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
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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