Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. II) pp 177-203 | Cite as

# Background Error Correlation Modeling with Diffusion Operators

## Abstract

Many background error correlation (BEC) models in data assimilation are formulated in terms of a positive-definite smoothing operator B that is employed to simulate the action of correlation matrix on a vector in state space. In this chapter, a general procedure for constructing a BEC model as a rational function of the diffusion operator D is presented and analytic expressions for the respective correlation functions in the homogeneous case are obtained. It is shown that this class of BEC models can describe multi-scale stochastic fields whose characteristic scales can be expressed in terms of the polynomial coefficients of the model. In particular, the connection between the inverse binomial model and the well-known Gaussian model \(\mathbf{\mathsf{B}}_{g} =\exp \mathbf{\mathsf{D}}\) is established and the relationships between the respective decorrelation scales are derived.By its definition, the BEC operator has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of B) is a computationally expensive procedure, therefore an efficient numerical approximation is needed. Under the assumption of local homogeneity of D, a heuristic method for computing the diagonal elements of B is proposed. It is shown that the method is sufficiently accurate for realistic applications, and requires 10^{2} times less computational resources than other methods of diagonal estimation that do not take into account prior information on the structure of B.

## Keywords

Diagonal Element Diffusion Tensor Polynomial Coefficient Hadamard Matrix Diffusion Operator## Notes

### Acknowledgements

This study was supported by the Office of Naval Research (Program element 0602435N) as part of the project “Exploring error covariances in variational data assimilation”.

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