  The understanding of nonlinear, high dimensional flows, e.g, atmospheric and
ocean flows, is critical to address the impacts of global climate change. Data
Assimilation techniques combine physical models and observational data, often
in a Bayesian framework, to predict the future state of the model and the
uncertainty in this prediction. Inherent in these systems are noise (Gaussian
and non-Gaussian), nonlinearity, and high dimensionality that pose challenges
to making accurate predictions. To address these issues we investigate the use
of both model and data dimension reduction based on techniques including
Assimilation in Unstable Subspaces, Proper Orthogonal Decomposition, and
Dynamic Mode Decomposition. Algorithms that take advantage of projected
physical and data models may be combined with Data Analysis techniques such as
Ensemble Kalman Filter and Particle Filter variants. The projected Data
Assimilation techniques are developed for the optimal proposal particle filter
and applied to the Lorenz'96 and Shallow Water Equations to test the efficacy
of our techniques in high dimensional, nonlinear systems.
