ADVANCED DATA ASSIMILATION IN STRONGLY NONLINEAR DYNAMICAL-SYSTEMS

Advanced data assimilation methods are applied to simple but highly no nlinear problems. The dynamical systems studied hem are the stochastic ally forced double well and the Lorenz model. In both systems, linear approximation of the dynamics about the critical points near which reg ime transitions occur is not always sufficient to track their occurren ce or nonoccurrence. Straightforward application of the extended Kalma n filter yields mixed results. The ability of the extended Kalman filt er to track transitions of the double-well system from one stable crit ical point to the other depends on the frequency and accuracy of the o bservations relative to the mean-square amplitude of the stochastic fo rcing. The ability of the filter to track the chaotic trajectories of the Lorenz model is limited to short times, as is the ability of stron g-constraint variational methods. Examples are given to illustrate the difficulties involved, and qualitative explanations for these difficu lties are provided. Three generalizations of the extended Kalman filte r are described. The first is based on inspection of the innovation se quence, that is, the successive differences between observations and f orecasts; it works very well for the double-well problem. The second, an extension to fourth-order moments, yields excellent results for the Lorenz model but will be unwieldy when applied to models with high-di mensional state spaces. A third, more practical method-based on an emp irical statistical model derived from a Monte Carlo simulation-is form ulated, and shown to work very well. Weak-constraint methods can be ma de to perform satisfactorily in the context of these simple models, bu t such methods do not seem to generalize easily to practical models of the atmosphere and ocean. In particular, it is shown that the equatio ns derived in the weak variational formulation are difficult to solve conveniently for large systems.