Filtering nonlinear spatio-temporal chaos with autoregressive linear stochastic models

Fundamental barriers in practical filtering of nonlinear spatio-temporal chaotic systems are model errors attributed to the stiffness in resolving multiscale features. Recently, reduced stochastic filters based on linear stochastic models have been introduced to overcome such stiffness; one of them is the Mean Stochastic Model (MSM) based on a diagonal Ornstein-Uhlenbeck process in Fourier space. Despite model errors, the MSM shows very encouraging filtering skill, especially when the hidden signal of interest is strongly chaotic. In this regime, the dynamical system statistical properties resemble to those of the energy-conserving equilibrium statistical mechanics with Gaussian invariant measure; therefore, the Ornstein-Uhlenbeck process with appropriate parameters is sufficient to produce reasonable statistical estimates for the filter model. In this paper, we consider a generalization of the MSM with a diagonal autoregressive linear stochastic model in Fourier space as a filter model for chaotic signals with long memory depth. With this generalization, the filter prior model becomes slightly more expensive than the MSM, but it is still less expensive relative to integrating the perfect model which is typically unknown in real problems. Furthermore, the associated Kalman filter on each Fourier mode is computationally as cheap as inverting a matrix of size D, where D is the number of observed variables on each Fourier mode (in our numerical example, D=1). Using the Lorenz 96 (L-96) model as a testbed, we show that the non-Markovian nature of this autoregressive model is an important feature in capturing the highly oscillatory modes with long memory depth. Second, we show that the filtering skill with autoregressive models supersedes that with MSM in weakly chaotic regime where the memory depth is longer. In strongly chaotic regime, the performance of the AR(p) filter is still better or at least comparable to that of the MSM. Most importantly, we find that this reduced filtering strategy is not as sensitive as standard ensemble filtering strategies to additional intrinsic model errors that are often encountered when model parameters are incorrectly specified.