On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation

Differential operators derived from the explicit or implicit solution of a diffusion equation are widely used for modelling background‐error correlations in geophysical applications of variational data assimilation. Key theoretical results underpinning the diffusion method are reviewed. Solutions to the isotropic diffusion problem on both the spherical space \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb S}^2$ \end{document} and the d ‐dimensional Euclidean space \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb R}^d$ \end{document} are considered first. In \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb R}^d$ \end{document} the correlation functions implied by explicit diffusion are approximately Gaussian, whereas those implied by implicit diffusion belong to the larger class of Matérn functions which contains the Gaussian function as a limiting case. The Daley length‐scale, defined as \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$D=\sqrt{-\! \left.d/\nabla^{2} c(r)\right|_{r=0}}$ \end{document} where ∇ 2 is the d ‐dimensional Laplacian operator and r = | r | is Euclidean distance, is used as a standard parameter for comparing the different isotropic functions c ( r ). Diffusion on \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb S}^2$ \end{document} is shown to be well approximated by diffusion on \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb R}^2$ \end{document} for length‐scales of interest. As a result, fundamental parameters that define the correlation model on \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb S}^2$ \end{document} can be specified using more convenient expressions available on \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb R}^2$ \end{document} . Anisotropic Gaussian or Matérn correlation functions on \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\mathbb R}^d$ \end{document} can be represented by a diffusion operator furnished with a symmetric and positive‐definite diffusion tensor. For anisotropic functions c ( r ), the tensor \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}${\bf D}=-\! \left( \nabla \nabla^{\rm T} c({\bf r}) |_{{\bf r}={\bf 0}} \right)^{-1}$ \end{document} where ∇ is the d ‐dimensional gradient operator, is a natural generalization of the (square of) the Daley length‐scale for characterizing the spatial scales of the function. Relationships between this tensor, which we call the Daley tensor, and the diffusion tensor of the explicit and implicit diffusion operators are established. Methods to estimate the elements of the local Daley tensor from a sample of simulated background errors are presented and compared in an idealized experiment with spatially varying covariance parameters. Since the number of independent parameters needed to specify the local diffusion tensor is of the order of the total number of grid points N , sampling errors are inherently much smaller than those involved in the order N 2 estimation problem of the full correlation function. While the correlation models presented in this paper are general, the discussion is slanted to their application to background‐error correlation modelling in ocean data assimilation. Copyright © 2012 Royal Meteorological Society