Insensitivety to initial condition/prior in data assimilation for the case of the optimal filter and deterministic model

Abstract

<p>Data assimilation is a term used to describe efforts to improve our knowledge<br>of a system by combining incomplete observations with imperfect models.<br>This is more generally known as filtering, which is ’optimal’ estimation of<br>the state of a system as it evolves over time, in the mean square sense. In<br>a Bayesian framework, the optimal filter is therefore naturally a sequence of<br>conditional probabilities of a signal given the observations and can be up-<br>dated recursively with new observations with Bayes’ formula. When, the<br>dynamics and observations errors are linear, this is equivalent to the Kalman<br>filter. In the nonlinear case, deriving an explicit form for the posterior dis-<br>tribution is in general not possible.<br>One of the important difficulties with applying the nonlinear filter in practice<br>is that the initial condition, the prior, is required to initialise the filtering.<br>However we are unlikely to know the correct initial distribution accurately<br>or at all. A filter is called stable if it is insensitive with respect to the<br>prior, that is, it converges to the same distribution, regardless of the initial<br>condition.<br>A body of work exists showing stability of the filter which rely on the stochas-<br>ticity of the underlying dynamics. In contrast, we show stability of the op-<br>timal filter for a class of nonlinear and deterministic dynamical systems and<br>our result relies on the intrinsic chaotic properties of the dynamics. We build<br>on the considerable knowledge that exists on the existence of SRB measures<br>in uniformly hyperbolic dynamical systems and we view the conditional prob-<br>abilities as SRB measures ‘conditional on the observation’ which are shown<br>to be absolutely continuous along the unstable manifold. This is in line with<br>the result of Bouquet, Carrassi et al [1] regarding data assimilation in the<br>“unstable subspace”, where they show stability of the filter if the unstable<br>and neutral subspaces are uniformly observed.</p><p>[1] M. Bocquet et al. “Degenerate Kalman Filter Error Covariances and<br>Their Convergence onto the Unstable Subspace”. In: SIAM/ASA Jour-<br>nal on Uncertainty Quantification 5.1 (2017), pp. 304–333. url: https:<br>//doi.org/10.1137/16M1068712.</p>