APPROXIMATE DATA ASSIMILATION SCHEMES FOR STABLE AND UNSTABLE DYNAMICS

Two suboptimal data assimilation schemes for stable and unstable dynam ics are introduced. The first scheme, the partial singular value decom position filter, is based on the most dominant singular modes of the t angent linear propagator. The second scheme, the partial eigendecompos ition filter, is based on the most dominant eigenmodes of the propagat ed analysis error covariance matrix. Both schemes rely on iterative pr ocedures like the Lanczos algorithm to compute the relevant modes. The performance of these schemes is evaluated for a shallow-water model l inearized about an unstable Bickley jet. The results are contrasted ag ainst those of a reduced resolution filter, in which the gains used to update the state vector are calculated from a lower-dimensional dynam ics than the dynamics that evolve the state itself. The results are al so contrasted against the exact results given by the Kalman filter. Th ese schemes are validated for the case of stable dynamics as well. The two new approximate assimilation schemes are shown to perform well wi th relatively few modes computed. Adaptive tuning of a modeled trailin g error covariance for all three of these low-rank approximate schemes enhances performance and compensates for the approximation employed.