Decomposition of likelihoods and techniques for multi-scale data assimilation

Approaches are developed for splitting of data assimilation (DA) problems into manageable pieces so that different DA methods may be applied. A block matrix generalized inverse result allows for the decomposition of Gaussian likelihoods based upon projected data models. A sophisticated implementation of a prototype method is developed and promising results are obtained for different model problems. Numerical techniques are focused on a combined particle filter/Kalman filter (PF-EnKf) using low-dimensional projected optimal proposal-particle filter (OP-PF) and ensemble Kalman filter (EnKf) after stratified resampling. The technique is evaluated using a modified 'storm track' Lorenz '96 model in which the chaotic/laminar solution structure changes in space and time. Results obtained are shown to be competitive when compared with different EnKF methods and unprojected OP-PF.