  In this article we develop algorithms for data assimilation based upon a
computational time dependent stable/unstable splitting. Our particular method
is based upon shadowing refinement and synchronization techniques and is
motivated by work on Assimilation in the Unstable Subspace (AUS) and
Pseudo-orbit Data Assimilation (PDA). The algorithm utilizes time dependent
projections onto the non-stable subspace determined by employing computational
techniques for Lyapunov exponents/vectors. The method is extended to parameter
estimation without changing the problem dynamics and we address techniques for
adapting the method when (as is commonly the case) observations are not
available in the full model state space. We use a combination of analysis and
numerical experiments (with the Lorenz 63 and Lorenz 96 models) to illustrate
the efficacy of the techniques and show that the results compare favorably with
other variational techniques.
