Reduced-rank sigma-point Kalman filter for geophysical data assimilation

The main goal of my research was to develop a practical scheme for the sigma-point Kalman filter (SPKF) for its application in a realistic climate model. Large computational expense has been an obstacle to applying the SPKF to a high-dimensional system. I addressed this issue by developing an advanced SPKF data-assimilation system. My work also addressed several other factors related to the practical implementation of SPKF. The main objectives of this research were to: (i) investigate two methods to construct a reduced-rank sigma-point unscented Kalman filers (RRSPUKF) (ii) propose a localization scheme for the SPKF and (iii) implement RRSPUKF in a realistic climate model. I present two methods to approximate the error covariance by a reduced-rank approximation. In the first method, truncated singular-value decomposition (TSVD) is applied on the error-covariance matrix calculated in the data space (RRSPUKF(D)) while in the second method TSVD is applied on the error-covariance matrix calculated in the ensemble space (RRSPUKF(E)). The new algorithms are first tested on the Lorenz-96 model, a one-dimensional atmospheric '~toy' model. The performance of both rank-reduction methods are close to that of the full-rank SPKF. I propose a localization method for RRSPUKF(E). The results from numerical experiments on the Lorenz-96 model showed that when the localization and inflation were implemented, the optimal estimate was achieved with a finite number of sigma points. The realistic model I used in this study was the Zebiak-Cane (ZC) model, an intermediate complexity coupled El Niño Southern Oscillation (ENSO) prediction model. The RRSPUKFs are implemented for the ZC model with the assimilation of sea surface temperature anomalies. The results showed that both RRSPUKF(D and E) were able to correctly analyze the phase and intensity of all major ENSO events during the study period with relatively similar estimation accuracy. Furthermore, the RRSPUKF was compared against ensemble square-root filter (EnSR