ON EXTENDING THE LIMITS OF VARIATIONAL ASSIMILATION IN NONLINEAR CHAOTIC SYSTEMS

A study is made of the limits imposed on variational assimilation of o bservations by the chaotic character of the atmospheric flow. The prim ary goal of the study is to determine to which degree, and how, the kn owledge of past noisy observations can improve the knowledge of the pr esent state of a chaotic system. The study is made under the hypothesi s of a perfect model. Theoretical results are illustrated by numerical experiments performed with the classical three-variable system introd uced 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 mod es of the system, and the corresponding gain in predictability is limi ted. Theoretical considerations provide explicit estimates of the stat istics of the assimilation error. The error depends on the state of th e 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-fun ction of variational assimilation is singular: it tends to fold into d eep 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 mini mum of the cost-function always lies on the sheet of the unstable mani fold containing the exact state of the flow. But the error along the u nstable manifold saturates to a finite value, and the absolute minimum of the cost function does not, in general, converge to the exact stat e of the flow. Even so, the absolute minimum of the cost function is t he best estimate that can be obtained of the state of the flow. An alg orithm is proposed, the quasi-static variational assimilation, for det ermining the absolute minimum, based on successive small increments of the assimilation period and quasi-static adjustments of the minimizin g solution. Finally, the impact of assimilation on predictability is a ssessed by forecast experiments with that system. The ability of the p resent paper lies mainly in the qualitative results it presents. Quali tative estimates relevant for the atmosphere call for Further studies.