| We consider the application of the Extended Kalman Filter to a chaotic system with stable, neutral and unstable manifolds of dimension given by the number of positive, null and negative Lyapunov exponents, respectively. We show that the rank of the EKF covariance matrix, initially equal to the total number of degrees of freedom of the system, asymptotically reduces to the dimension of the unstable and neutral subspace. In a reduced form of the algorithm (Extended Kalman Filter with Assimilation in the Unstable Subspace, EKF-AUS), the assimilation is confined to the unstable and neutral directions of the tangent space. In the limits of validity of the assumptions, that is when the observations are sufficiently dense and accurate that filter divergence does not occur, we show that the EKF and its reduced form EKF-AUS converge to the same solution. |
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