Deterministic Linear Bayesian Updating of State and Model Parameters for a Chaotic Model

We present a sampling-free implementation of a linear Bayesian filter. It is based on spectral series expansions of the involved random variables, one such example being Wiener's polynomial chaos. The method is applied to a combined state and parameter estimation problem for a chaotic system, the well-known Lorenz-63 model. We compare it to the ensemble Kalman filter (EnKF), which is essentially a stochastic implementation of the same underlying estimator---a fact which is demonstrated in the paper. The spectral method is found to be more reliable for the same computational load, especially for the variance estimation. This is to be expected due to the fully deterministic implementation.