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(t_m)} of the model state through the observation window t_n = {t₀, ..., t_m}. 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 t_n 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.