Representation of correlation functions in variational assimilation using an implicit diffusion operator

Correlation models are required in data assimilation to characterize the error structures of variables defined on a numerical grid. Previous studies have shown that the diffusion equation can provide a flexible and computationally efficient framework for representing grid-point correlation functions for problems of large dimension such as those encountered in atmospheric or ocean variational data assimilation. In this article, an implicit formulation of the diffusion-based correlation model is presented as an alternative to the traditional explicit formulation. The implicit formulation is analyzed in detail for the one-dimensional (1D) problem and shown to be closely related to the first-order recursive filter. Integrating a 1D implicit diffusion equation, with constant coefficient, over M steps is shown to be equivalent to convolving the initial condition with an Mth order auto-regressive (AR) function. Expressions for both the length-scale of the AR function and the normalization factor required to generate unit-amplitude (correlation) functions are given in terms of M and the diffusion coefficient. For a fixed length-scale the Gaussian function, which is the only function that can be represented using an explicit formulation of the constant-coefficient diffusion equation, is the limiting case as M -> infinity of the AR functions generated by the implicit diffusion equation. Generalizations of the diffusion model are discussed to allow for different shapes in the correlation function and spatial variations in the length-scale. An important consequence of employing spatially varying length-scales is that the normalization factors are no longer constant. Approximate expressions for the normalization factors are evaluated in terms of their effectiveness to provide viable alternatives to estimates produced using expensive algorithms such as randomization. Boundary conditions can distort the correlation functions near the boundaries and significantly degrade the accuracy of the analytical expressions for the normalization factors. These problems can be avoided through a straightforward extension of the diffusion model that makes the boundaries effectively transparent, although the solution comes at the expense of an extra application of the diffusion equation. Extensions of the method to construct two-and three-dimensional correlation models are discussed. Copyright (C) 2010 Royal Meteorological Society