Correlation modelling on the sphere using a generalized diffusion equation



An important element of a data assimilation system is the statistical model used for representing the correlations of background error. This paper describes a practical algorithm that can be used to model a large class of two- and three-dimensional, univariate correlation functions on the sphere. Application of the algorithm involves a numerical integration of a generalized diffusion-type equation (GDE). The GDE is formed by replacing the Laplacian operator in the classical diffusion equation by a polynomial in the Laplacian. The integral solution of the GDE defines, after appropriate normalization, a correlation operator on the sphere. The kernel of the correlation operator is an isotropic correlation function. The free parameters controlling the shape and length-scale of the correlation function are the products kpT, p = 1, 2, …, where (-1)pkp is a weighting (‘diffusion’) coefficient (kp > 0) attached to the Laplacian with exponent p, and T is the total integration ‘time’. For the classical diffusion equation (a special case of the GDE with kp = 0 for all p > 1) the correlation function is shown to be well approximated by a Gaussian with length-scale equal to (2k1T)1/2.

The Laplacian-based correlation model is particularly well suited for ocean models as it can be easily generalized to account for complex boundaries imposed by coastlines. Furthermore, a one-dimensional analogue of the GDE can be used to model a family of vertical correlation functions, which in combination with the two-dimensional GDE forms the basis of a three-dimensional, (generally) non-separable correlation model. Generalizations to account for anisotropic correlations are also possible by stretching and/or rotating the computational coordinates via a ‘diffusion’ tensor. Examples are presented from a variational assimilation system currently under development for the OPA ocean general-circulation model of the Laboratoire d'Oceanographie Dynamique et de Climatologie.