Improved variational methods in statistical data assimilation

Abstract

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(t0) is uncertain: the data assimilation is statistical. One can ask about expected values of functions ⟨G(X)⟩ on the path X = {x(t0), ..., x(tm)} of the model state through the observation window tn = {t0, ..., tm}. The conditional (on the measurements) probability distribution P(X) = exp[−A0(X)] determines these expected values. Variational methods using saddle points of the "action" A0(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 X0 giving a consistent minimum of the action A0(X0), (b) consider the explicit role of the number of measurements at each tn in determining A0(X0), and (c) identify a parameter regime for the scale of model errors, which allows X0 to give a precise estimate of ⟨G(X0)⟩ with computable, small higher-order corrections.