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Derivation and characterization of a new filter for nonlinear high-dimensional data assimilation
(2015)
Data assimilation (DA) combines model forecasts with real-world observations to achieve an optimal estimate of the state of a dynamical system. The quality of predictions in nonlinear and chaotic systems such as atmospheric or oceanic circulation is strongly sensitive to the initial conditions. Therefore, beyond the consistent reconstruction of past states, a primary relevance of advanced DA methods concerns the proper model initialization. The ensemble Kalman filter (EnKF) and its deterministic variants, mostly square root filters such as the ensemble transform Kalman filter (ETKF), represent a popular alternative to variational DA schemes. They are applied in a wide range of research and operations. Their forecast step employs an ensemble integration that fully respects the nonlinear nature of the analyzed system. In the analysis step, they implicitly assume the prior state and observation errors to be Gaussian. Consequently, in nonlinear systems, the mean and covariance of the analysis ensemble are biased and these filters remain suboptimal. In contrast, the fully nonlinear, non-Gaussian particle filter (PF) relies on Bayes' theorem without further assumptions, which guarantees an exact asymptotic behavior. However, it is exposed to weight collapse, particularly in higher-dimensional settings, known as the curse of dimensionality.
This work presents a new method to obtain an analysis ensemble with mean and covariance that exactly match the corresponding Bayesian estimates. This is achieved by a deterministic matrix square root transformation of the forecast ensemble, and subsequently a suitable random rotation that significantly contributes to filter stability while preserving the required second-order statistics. The forecast step remains as in the ETKF. The algorithm, which is fairly easy to implement and computationally efficient, is referred to as the nonlinear ensemble transform filter (NETF). The limitation with respect to fully-nonlinear filtering is that the NETF only considers the mean and covariance of the Bayesian analysis density, neglecting higher-order moments.
The properties and performance of the proposed algorithm are investigated via a set of experiments. The results indicate that such a filter formulation can increase the analysis quality, even for relatively small ensemble sizes, compared to other ensemble filters in nonlinear, non-Gaussian scenarios. They also confirm that localization enhances the applicability of this PF-inspired scheme in larger-dimensional systems. Finally, the novel filter is coupled to a large-scale ocean general circulation model with a realistic observation scenario. The NETF remains stable with a small ensemble size and shows a consistent behavior. Additionally, its analyses exhibit low estimation errors, as revealed by a comparison with a free ensemble integration and the ETKF. The results confirm that, in principle, the filter can be applied successfully and as simple as the ETKF in high-dimensional problems. No further modifications are needed, even though the algorithm is only based on the particle weights. Thus, it is able to overcome the curse of dimensionality, even in deterministic systems. This proves that the NETF constitutes a promising and user-friendly method for nonlinear high-dimensional DA.