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

Authors: Anthony Weaver (CERFACS, Toulouse)
Isabelle Mirouze (CERFACS, Toulouse)

Differential operators derived from the explicit or implicit solution of a diffusion equation are widely used for modelling background-error correlations in variational assimilation. In the isotropic case, the correlation functions implied by explicit diffusion are approximately Gaussian, whereas those implied by implicit diffusion belong to the larger class of Whittle-Matern functions which contains the Gaussian function as a limiting case. Anisotropic Gaussian and Whittle-Matern correlation functions can be constructed from their isotropic counterparts by replacing the standard normalized Euclidean distance measure with a Mahalanobis distance measure involving the inverse of an aspect tensor that accounts for directionality. Likewise, a `Daley tensor' can be defined that generalizes the standard definition of the Daley length-scale to the anisotropic case. Relationships between the aspect tensor and Daley tensor of the Gaussian and Whittle-Matern functions are established. These tensors are in turn related to the parameters of anisotropic formulations of the explicit and implicit diffusion operators. Methods to estimate the elements of the Daley tensor from an ensemble are presented and compared in idealized and real-data experiments. Since the number of independent parameters needed to specify the local Daley 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 N2 estimation problem of the full correlation matrix. The approach can be viewed as a hybrid technique for combining variational and ensemble methods.

GMAO Head: Michele Rienecker Global Modeling and Assimilation Office NASA Goddard Space Flight Center

Curator: Nikki Privé Last Updated: May 27 2011