Articles | Volume 22, issue 2
Nonlin. Processes Geophys., 22, 205–213, 2015
Nonlin. Processes Geophys., 22, 205–213, 2015

Research article 07 Apr 2015

Research article | 07 Apr 2015

Improved variational methods in statistical data assimilation

J. Ye1, N. Kadakia1, P. J. Rozdeba1, H. D. I. Abarbanel1,2, and J. C. Quinn3 J. Ye et al.
  • 1Department of Physics, University of California, San Diego, La Jolla, CA 92093-0374, USA
  • 2Marine Physical Laboratory (Scripps Institution of Oceanography), University of California, San Diego, La Jolla, CA 92093-0374, USA
  • 3Intellisis Corporation, 10350 Science Center Drive, Suite 140, San Diego, CA 92121, USA

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.