Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace

Abstract

International audienceThe characteristics of the model dynamics are critical in the performance of (ensemble) Kalmanfilters. In particular, as emphasized in the seminal work of Anna Trevisan and coauthors, theerror covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., itis spanned by the backward Lyapunov vectors with nonnegative exponents. This behavior is at thecore of algorithms known as assimilation in the unstable subspace, although a formal proof was stillmissing. This paper provides the analytical proof of the convergence of the Kalman filter covariancematrix onto the unstable-neutral subspace when the dynamics and the observation operator are linearand when the dynamical model is error free, for any, possibly rank-deficient, initial error covariancematrix. The rate of convergence is provided as well. The derivation is based on an expression thatexplicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown thatif the unstable and neutral directions of the model are sufficiently observed and if the column spaceof the initial covariance matrix has a nonzero projection onto all of the forward Lyapunov vectorsassociated with the unstable and neutral directions of the dynamics, the covariance matrix of theKalman filter collapses onto an asymptotic sequence which is independent of the initial covariances.Numerical results are also shown to illustrate and support the theoretical findings

Asymptotic Forecast Uncertainty and the Unstable Subspace in the Presence of Additive Model Error

“…Proposition 1 is similar results in perfect models [9,27,8], but with some key differences. Once again that the estimation errors are dissipated by the dynamics in the span of the stable BLVs, but the recurrent injection of model error prevents the total collapse of the covariance to the unstable-neutral subspace.…”

“…al. [8], where in perfect models, the stable dynamics alone are sufficient to dissipate forecast error in the span of the stable BLVs. With α = 0, the uniform bound in Corollary 3 may be understood by the two components which equation (65) is composed of, the bound on Υ k:k−Ni, and q sup e 2(λi+ )Ni, +1 1 − e 2(λi+ ) .…”

“…al. [8], proving the asymptotic equivalence of reduced rank initializations of the Kalman filter with the full rank Kalman filter: as the number of assimilations increases towards infinity, the covariance of the full rank Kalman filter converges to a sequence of low rank covariance matrices initialized only in the unstable-neutral BLVs. The convergence of the Kalman smoother error covariances onto the span of the unstable-neutral BLVs, and stability of low rank initializations, was established by Bocquet & Carrassi [7]; this latter work also numerically extended this relationship to weakly nonlinear dynamics and ensemble-variational methods.…”

Four-dimensional ensemble variational data assimilation and the unstable subspace

“…The AUS approach has recently motivated a research effort that has led to a proper mathematical formulation and assessment of its driving idea, i.e. the span of the estimation error covariance matrices asymptotically (in time) tends to the subspace spanned by the unstable and neutral backward Lyapunov vectors (Bocquet et al, 2017;Gurumoorthy et al, 2017).…”

“…In Section 2, we introduce an alternative derivation of some of the results of the degenerate (or rank-deficient) KF onto the unstable and neutral subspace, obtained in Bocquet et al (2017) (later Boc17). The filter is presented in an ensemble square-root form to make the link with the ensemble square-root EnKF.…”