Background Error Correlation Modeling with Diffusion Operators

  • Max YaremchukEmail author
  • Matthew Carrier
  • Scott Smith
  • Gregg Jacobs


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 102 times less computational resources than other methods of diagonal estimation that do not take into account prior information on the structure of B.


Diagonal Element Diffusion Tensor Polynomial Coefficient Hadamard Matrix Diffusion Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Max Yaremchuk
    • 1
    Email author
  • Matthew Carrier
    • 1
  • Scott Smith
    • 1
  • Gregg Jacobs
    • 1
  1. 1.Naval Research Laboratory, Stennis Space CenterHancockUSA

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