Abstract
The performance of (ensemble) Kalman filters used for data assimilation in the geosciences critically depends on the dynamical properties of the evolution model. A key aspect is that the error covariance matrix is asymptotically supported by the unstable–neutral subspace only, i.e. it is spanned by the backward Lyapunov vectors with non-negative exponents. The analytic proof of such a property for the Kalman filter error covariance has been recently given, and in particular that of its confinement to the unstable–neutral subspace. In this paper, we first generalize those results to the case of the Kalman smoother in a linear, Gaussian and perfect model scenario. We also provide square-root formulae for the filter and smoother that make the connection with ensemble formulations of the Kalman filter and smoother, where the span of the error covariance is described in terms of the ensemble deviations from the mean. We then discuss how this neat picture is modified when the dynamics are nonlinear and chaotic, and for which analytic results are precluded or difficult to obtain. A numerical investigation is carried out to study the approximate confinement of the anomalies for both a deterministic ensemble Kalman filter (EnKF) and a four-dimensional ensemble variational method, the iterative ensemble Kalman smoother (IEnKS), in a perfect model scenario. The confinement is characterized using geometrical angles that determine the relative position of the anomalies with respect to the unstable–neutral subspace. The alignment of the anomalies and of the unstable–neutral subspace is more pronounced when observation precision or frequency, as well as the data assimilation window length for the IEnKS, are increased. These results also suggest that the IEnKS and the deterministic EnKF realize in practice (albeit implicitly) the paradigm behind the approach of Anna Trevisan and co-authors known as the assimilation in the unstable subspace.
Highlights
The unstable–neutral subspace of a dynamical system is defined as the vector space generated by the backward Lyapunov vectors with non-negative Lyapunov exponents (Legras and Vautard, 1996), i.e. the space of the small perturbations that are not growing exponentially under the action of the backward dynamics
We have provided an upper bound for the rate of such convergence, derived the explicit covariance dependence on the initial condition and on the information matrix, and proved the existence of a universal sequence of covariance matrices to which the solutions converge if the system is sufficiently observed and the initial error covariances have non-zero projections onto each of the unstable–neutral forward Lyapunov vectors
data assimilation (DA) in this case is usually carried out using ensemble Kalman filter (EnKF) or smoothers (Evensen, 2009), and the reduced-rank covariance description is implemented using an ensemble of model realizations
Summary
The unstable–neutral subspace of a dynamical system is defined as the vector space generated by the backward Lyapunov vectors with non-negative Lyapunov exponents (Legras and Vautard, 1996), i.e. the space of the small perturbations that are not growing exponentially under the action of the backward dynamics. Applications to atmospheric and oceanic models (Uboldi and Trevisan, 2006; Carrassi et al, 2008b) showed that even for high-dimensional models, an efficient error control is achieved by monitoring only the unstable directions, making AUS a computationally efficient alternative to standard procedures in this. Elementary results about backward and covariant Lyapunov vectors are recalled They are put to use in our subsequent numerical investigation of the approximate convergence of the ensemble of the EnKF and IEnKS onto the unstable–neutral subspace using a low-order chaotic model. This convergence is characterized geometrically in regimes that deviate to different extents from the linear and Gaussian assumptions on which the analytic results are built.
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