  Prediction of the state evolution of complex high-dimensional nonlinear
systems is challenging due to the nonlinear sensitivity of the evolution to
small inaccuracies in the model. Data Assimilation (DA) techniques improve
state estimates by combining model simulations with real-time data. Few DA
techniques can simultaneously handle nonlinear evolution, non-Gaussian
uncertainty, and the high dimension of the state. We recently proposed
addressing these challenges using a Proper Orthogonal Decomposition (POD)
technique that projects the physical and data models into a reduced-dimensional
subspace. POD is a tool to extract spatiotemporal patterns (modes) that
dominate the observed data. We combined the POD-based projection operator,
computed in an offline fashion, with a DA scheme that models non-Gaussian
uncertainty in lower dimensional subspace. If the model parameters change
significantly during time evolution, the offline computation of the projection
operators ceases to be useful. We address this challenge using a sliding window
POD (SWPOD), which recomputes the projection operator based on a sliding subset
of snapshots from the entire evolution. The physical model projection is
updated dynamically in terms of modes and number of modes, and the data model
projection is also chosen to promote a sparse approximation. We test the
efficacy of this technique on a modified Lorenz'96 model with a time-varying
forcing and compare it with the time-invariant offline projected algorithm. In
particular, dynamically determined physical and data model projections decrease
the Root Mean Squared Error and the resampling rate.
