Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demonstrate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catastrophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations.
With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy, A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007, E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008, J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008 is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.
"Catastrophic filter divergence in filtering nonlinear dissipative systems." Commun. Math. Sci. 8 (1) 27 - 43, March 2010.