Estimation of the local diffusion tensor and normalization for heterogeneous correlation modelling using a diffusion equation

Abstract As the background‐error covariance matrix is a key component of any assimilation system, its modelling is an important step. Usually, this matrix is decomposed into correlations and standard deviations matrices. An interesting method for modelling the correlation matrix of the background error for complex geometry, like an ocean grid, consists in computing correlation functions using a diffusion operator. The background‐error correlation functions can be estimated for example from an ensemble of perturbed forecasts. The diffusion operator is able to represent heterogeneous correlation functions at a reasonable numerical cost. But a first challenge resides in the determination of the local diffusion tensor corresponding to the local correlation function. Then the second challenge resides in the determination of the normalization to make sure that the matrix modelled through the diffusion operator is a correlation matrix. In this article, we propose to build a background‐error correlation matrix using a diffusion operator based on a local diffusion tensor. The estimation of this local tensor is performed using an ensemble of perturbed forecasts. A validation within a randomization method illustrates the feasibility and the accuracy of the proposed method. In particular, it is shown that the local geographical variations of diagnosed correlation functions (through an ensemble of perturbed forecast) are well represented. This is first illustrated in an analytical one‐dimensional framework. In that context, the diffusion field and the normalization field are deduced from a given correlation length‐scale field. The resulting length‐scales are shown to correspond to the initial length‐scale when the given length‐scale field spectrum is red. The approximate normalization, computed from the local length‐scale, is close to the true normalization under the same condition of a red spectrum. Then, the method is illustrated in a real context using an ensemble of perturbed forecasts from the MOCAGE–PALM assimilation system. In that case, length‐scale and anisotropy diagnosis reveal the complexity of the correlation of stratospheric ozone forecast errors. The local diffusion tensor deduced from these diagnoses is shown be able to represent such an existing heterogeneity and anisotropy. As in the one‐dimensional case, the approximate normalization, based on the local diffusion tensor, appears to be a really good approximation of the true normalization. Copyright © 2008 Royal Meteorological Society