Approximate Kalman Filters for Unstable Dynamics
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1. INTRODUCTION A number of recent studies have made clear that rapid error growth in regions of baroclinic and barotropic instability is nonmodal, and therefore can generally be explained by the singular values and singular vectors, rather than the eigenvalues and eigenvectors, of the tangent linear dynamics (Farrell 1989; Trefethen et al. 1993). Modern four-dimensional (4D) data assimilation methods, such as 4D variational algorithms and nonlinear Kalman lter (KF) schemes, ooer the potential to exploit this fact through their direct use of the tangent linear dynamics. Such methods would assign more weight to observations in unstable regions than those in more quiescent regions, all else being equal. It is well known that practical implementation of Kalman lter schemes requires sensible approximation. Evaluation of several possible approximations, known as suboptimal schemes (SOS's), was carried out by Todling and Cohn (1994; TC94 hereafter) for a stable 2D linear barotropic model. Results were encouraging for this highly idealized model. However, when these SOS's were evaluated recently for a barotropically unstable version of that model, we found that all these schemes failed to provide reliable error bars. On the other hand, Todling and Ghil (1994) have shown that the complete KF is well{behaved in the presence of instability, even when the number of observations is very limited. This motivates the need for additional approximations to the KF.
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