A new filter: The Error Subspace Transform Kalman Filter - ESTKF

Recently, we have developed a new filter formulation, the Error Subspace Transform Kalman Filter. Details on this filter will be available in the paper "A unification of ensemble square-root filters" by L. Nerger, T. Janjic, J. Schroeter, and W. Hiller (Monthly Weather Review, 140, 2335-2345, ​doi:10.1175/MWR-D-11-00102.1).

While studying the detailed relationship of the SEIK filter and the Ensemble Transform Kalman Filter (ETKF), we found a possibility to obtain the same minimum transformation of the ensemble (i.e. the distance of the ensemble transformation matrix is minimal in the Frobenius norm) at a slightly lower computational cost than that of the ETKF. The new filter results from a modification of the SEIK filter to use consistent projections between the state space and the error subspace represented by the ensemble of model states. (The SEIK filter itself already projects onto the error subspace. However, this projection is not fully consistent and will lead to small differences in the analysis ensemble that depend on the order of the states on the ensemble matrix.) As the new formulation is very similar to the ETKF, but operates directly in the error subspace instead of the ensemble-representation of it, the new filter was termed "Error-subspace Transform Kalman Filter" ESTKF.

In the paper, we also tested the use of the symmetric square root in the SEIK filter instead of the Cholesky decomposition that is commonly applied. The symmetric square root is commonly used in the ETKF and will also be the default for the ESTKF. For the SEIK filter, the symmetric square-root improved state estimates in experiments with the Lorenz-96 model, for the case that the ensemble transformation was deterministic.

The ESTKF, its localized variant LESTKF, as well as a formulation the SEIK filter with symmetric square-root and explicit ensemble transformation are available with the release of Version 1.8 of PDAF.