Two suboptimal data assimilation schemes for stable and unstable dynamics are introduced. The first scheme, the partial singular value decomposition filter, is based on the most dominant singular modes of the tangent linear propagator. The second scheme, the partial eigendecomposition filter, is based on the most dominant eigenmodes of the propagated analysis error covariance matrix. Both schemes rely on iterative procedures like the Lanczos algorithm to compute the relevant modes.
The performance of these schemes is evaluated for a shallow-water model linearized about an unstable Bickley jet. The results are contrasted against those of a reduced resolution filter, in which the gains used to update the state vector are calculated from a lower-dimensional dynamics than the dynamics that evolve the state itself. The results are also contrasted against the exact results given by the Kalman filter. These schemes are validated for the case of stable dynamics as well.
The two new approximate assimilation schemes are shown to perform well with relatively few modes computed. Adaptive tuning of a modeled trailing error covariance for all three of these low-rank approximate schemes enhances performance and compensates for the approximation employed.