Use of discrete Fourier transforms in the 1D-Var retrieval problem

For some purposes, observation error covariance matrices having a symmetric Toeplitz form can be well approximated by circulant matrices. This amounts to modifying correlations so that they are periodic on the scale of the observing domain. The inverse of a circulant matrix can be evaluated efficiently with a discrete Fourier transform, as can circulant matrix-vector and matrix-matrix products. This could be useful for 1D-Var retrievals of measurements made with high-spectral-resolution infrared sounders, when the observation vector is large and the errors are correlated. Two 1D-Var simulation studies indicate that symmetric Toeplitz observation error covariance matrices can in this context be accurately approximated with circulant matrices, although some care is required when the correlations have a Gaussian fall-off. The simulation studies also show that assuming the observation errors are uncorrelated, when they are in fact correlated, can give misleading 'information content' estimates. This may be important for channel selection calculations when the errors are assumed to be uncorrelated.