On extending the limits of variational assimilation in nonlinear chaotic systems

Abstract

A study is made of the limits imposed on variational assimilation of observations by the chaotic character of the atmospheric flow. The primary goal of the study is to determine to which degree, and how, the knowledge of past noisy observations can improve the knowledge of the present state of a chaotic system. The study is made under the hypothesis of a perfect model. Theoretical results are illustrated by numerical experiments performed with the classical three-variable system introduced by Lorenz. Both theoretical and numerical results show that, even in the chaotic regime, appropriate use of past observations improves the accuracy on the estimate of the present state of the flow. However, the resulting estimation error mostly projects onto the unstable modes of the system, and the corresponding gain in predictability is limited. Theoretical considerations provide explicit estimates of the statistics of the assimilation error. The error depends on the state of the flow over the assimilation period. It is largest when there has been a period of strong instability in the very recent past. In the limit of infinitely long assimilation periods, the behaviour of the cost-function of variational assimilation is singular: it tends to fold into deep narrow “valleys” parallel to the sheets of the unstable manifold of the system. An unbounded number of secondary minima appear, where solutions of minimization algorithms can be trapped. The absolute minimum of the cost-function always lies on the sheet of the unstable manifold containing the exact state of the flow. But the error along the unstable manifold saturates to a finite value, and the absolute minimum of the cost function does not, in general, converge to the exact state of the flow. Even so, the absolute minimum of the cost function is the best estimate that can be obtained of the state of the flow. An algorithm is proposed, the quasi-static variational assimilation, for determining the absolute minimum, based on successive small increments of the assimilation period and quasi-static adjustments of the minimizing solution. Finally, the impact of assimilation on predictability is assessed by forecast experiments with that system. The ability of the present paper lies mainly in the qualitative results it presents. Qualitative estimates relevant for the atmosphere call for further studies.