Quantifying Bayesian Filter Performance for Turbulent Dynamical Systems through Information Theory

Incomplete knowledge of the true dynamics and its partial observations pose a notoriously difficult problem in many contemporary scientific applications which require predictions of high-dimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. This information-theoretic framework for an off-line assessment of filter performance is an important complement to the path-wise approach, and it has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this framework is utilized to study the performance of imperfect, reduced-order filters with Gaussian models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advectiondiffusion equation. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance. Important effects of biases due to model error in the filter estimates of the mean dynamics are quantified through appropriate information measures, including a new information-theoretic measure of barriers to imperfect filter performance.