Notes and Correspondence On the correlation functions associated with polynomials of the diffusion operator

Correlation functions (CFs) associated with the inverse background-error correlations (iBECs) represented by polynomials of the diffusion operator D are obtained analytically for the binomial approximations of the Gaussian BEC and in the general case of a quadratic polynomial of D. The respective analytical expressions for one-, twoand three-dimensional cases have two tuning parameters, which provide enough freedom in adjusting the CFs’ shape to experimental data. The polynomial coefficients of the corresponding iBEC operator are obtained in terms of these tuning parameters and may be useful in the design of the BEC models for variational data assimilation. Published in 2011 by John Wiley & Sons Ltd.