Articles | Volume 22, issue 2
Research article
07 Apr 2015
Research article |  | 07 Apr 2015

Improved variational methods in statistical data assimilation

J. Ye, N. Kadakia, P. J. Rozdeba, H. D. I. Abarbanel, and J. C. Quinn

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.

Short summary
We propose an improved method of data assimilation, in which measured data are incorporated into a physically based model. In data assimilation, one typically seeks to minimize some cost function; here, we discuss a variational approximation in which model and measurement errors are Gaussian, combined with an annealing method, to consistently identify a global minimum of this cost function. We illustrate this procedure with archetypal chaotic systems, and discuss higher-order corrections.