Analysis of cycled 4D‐Var with model error. (12th November 2012)

• Record Type: Journal Article
• Authors: Cullen, M. J. P.
• Is Part Of: Quarterly journal of the Royal Meteorological Society. Volume 139:Number 675(2013:Jul.)
• Page Start: 1473
• Page End: 1480
• Publication Date: 2012-11-12
• Subjects: Meteorology -- Periodicals; 551.5
• DOI: 10.1002/qj.2045
• Languages: English
• ISSNs: 0035-9009

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>In a previous study the author showed, using a toy model, that forecast performance using four‐dimensional variational analysis (4D‐Var) and an imperfect forecast model was dramatically improved by using a regularization matrix instead of a matrix estimating the true background‐error covariance. This article carries out a theoretical analysis of this behaviour. The analysis first highlights the interpretation of short‐window 4D‐Var as a fixed lag smoother, which means that the control over model error growth exerted by observations late in the assimilation window is taken into account. It is then shown that, in a scalar case, the solution will converge to that of infinitely long‐window weak‐constraint 4D‐Var where the same regularization constraint is applied at each subwindow. The optimum choice of regularization constraint is analyzed for the converged solution and is such that the regularization evolved by the model to the observation time equals the true error evolved to the observation time. The true error is controlled by the observation error regardless of the amount of model error provided that the optimum regularization is used. The same calculation is done for 3D‐Var, showing that the true error is not so controlled and is typically larger. It is shown that the feedback between forecast‐error growth and the choice of regularization constraint can have a strong effect on the optimal choice, as observed in the experiments. The results are consistent with setting the regularization equal to the background error calculated relative to the truth projected on to the model attractor, rather than the truth itself. In this context, the model attractor can be regarded as the model state closest to the truth that gives the correct time derivatives.</p> </abstract>