State Estimation using model order reduction for unstable systems

State Estimation using model order reduction for unstable systems

In this talk the state estimation problem as it occurs within data assimilation is discussed. The main target of data assimilation for numerical weather prediction is to find the best estimate of the true state of the atmosphere by using measured observations. This is an ill-posed inverse problem caused by the fact that the number of available observations is at least two orders of magnitude smaller than the dimension of the state vector. The focus of this talk is on the well-known and approved method of four-dimensional variational data assimilation. It can be interpreted as a Tikhonov regularization of the ill-posed data assimilation problem. The methodrequires the minimization of a series of simplified cost functions. These simplified functions are usually derived from a spatial or spectral truncation of the full system being approximated. In this talk a new method for deriving these simplified problems is proposed, based on control theoretic model reduction methods. The models used for numerical weather prediction often possess inherent instabilities. But most standard model reduction methods are designed for asymptotically stable systems only. A new approach for model reduction of unstable systems is proposed. It is shown that this performs well within the state estimation problem. To illustrate the theoretical results numerical experiments are performed using a two dimensional Eady model - a simple model of baroclinic instability, which is the dominant mechanism for the growth of storms at mid-latitudes. It is a suitable test model to show the benefit that may be obtained by using model reduction techniques to approximate the unstable systems within the state estimation problem.