  We introduce a framework for Data Assimilation (DA) in which the data is
split into multiple sets corresponding to low-rank projections of the state
space. Algorithms are developed that assimilate some or all of the projected
data, including an algorithm compatible with any generic DA method. The major
application explored here is PROJ-PF, a projected Particle Filter. The PROJ-PF
implementation assimilates highly informative but low-dimensional observations.
The implementation considered here is based upon using projections
corresponding to Assimilation in the Unstable Subspace (AUS). In the context of
particle filtering, the projected approach mitigates the collapse of particle
ensembles in high dimensional DA problems while preserving as much relevant
information as possible, as the unstable and neutral modes correspond to the
most uncertain model predictions. In particular we formulate and numerically
implement a projected Optimal Proposal Particle Filter (PROJ-OP-PF) and compare
to the standard optimal proposal and to the Ensemble Transform Kalman Filter.
