Chaotic systems, such as oceanic and atmospheric circulations, are characterized by extreme sensitivity of the state to perturbations in the initial condition and/or forcing. This creates difficulties for data assimilation, especially for suboptimal estimation algorithms that may not be able to properly account for the effects of small uncertainties.

In this paper, we focus on the filtering problem, where only past observations are available to estimate the state at a given time. We use geometric properties of the attractor of a chaotic system in the neighborhood of the analysis to identify a subspace that can give an acceptable low rank local representation of the attractor. We use Floquet Theory to build a dynamic basis for this subspace and to construct a reduced rank filter that resolves the effects of uncertainty. Resolving uncertainty in this subspace stabilizes the locally unstable modes of the tangent linear model.

We have tested our low rank Floquet-based algorithm on a chaotic Lorenz95 system with 144 state variables, comparing its analysis RMSE to a known true state, to the analysis obtained from a full rank, large sample Ensemble Kalman Filter (assumed to approximate the optimal linear filter), and to the analysis obtained from a low rank filter based on singular vectors of the tangent linear model (which maximizes the spread of the forecast ensemble). Results show that our filter is robust and converges to the optimal linear filter. Additionally, it is scaleable to large systems and only needs the forward model to compute the basis for the tangent subspace. This is in contrast to the singular vector filter, which requires an adjoint model for efficient computation of the singular vectors.