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Correlation modelling on the sphere using a generalized diffusion equation
Author(s) -
Weaver Anthony,
Courtier Philippe
Publication year - 2001
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.49712757518
Subject(s) - mathematics , laplace operator , correlation function (quantum field theory) , mathematical analysis , normalization (sociology) , operator (biology) , diffusion equation , statistics , biochemistry , spectral density , chemistry , economy , repressor , sociology , anthropology , transcription factor , economics , gene , service (business)
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) p kp 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.