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.

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