Improved variational methods in statistical data assimilation

Data assimilation transfers information from an observed system to a physically based model system with state variables x(t). The observations are typically noisy, the model has errors, and the initial state x(t(0)) is uncertain: the data assimilation is statistical. One can ask about expected values of functions h < G(X)> on the path X = {x(t(0)), ..., x(t(m))} of the model state through the observation window t(n) = {t(0),..., t(m)). The conditional (on the measurements) probability distribution P(X) = exp[-A(0)(X)] determines these expected values. Variational methods using saddle points of the "action" A(0)(X), known as 4DVar (Talagrand and Courtier, 1987; Evensen, 2009), are utilized for estimating < G(X)>. In a path integral formulation of statistical data assimilation, we consider variational approximations in a realization of the action where measurement errors and model errors are Gaussian. We (a) discuss an annealing method for locating the path X-0 giving a consistent minimum of the action A(0)(X-0), (b) consider the explicit role of the number of measurements at each t n in determining A(0)(X-0), and (c) identify a parameter regime for the scale of model errors, which allows X-0 to give a precise estimate of < G(X-0)> with computable, small higher-order corrections.