Sampling and misspecification errors in the estimation of observation‐error covariance matrices using observation‐minus‐background and observation‐minus‐analysis statistics

Specification of the observation‐error covariance matrix for data assimilation systems affects the observation information content retained by the analysis, particularly for observations known to have correlated observation errors (e.g., geostationary satellite and Doppler radar data). A widely adopted approach for estimating observation‐error covariance matrices uses observation‐minus‐background and observation‐minus‐analysis residuals, which are routinely produced by most data assimilation systems. Although this approach is known to produce biased and noisy estimates, due to sampling and misspecification errors, there has been no systematic study of sampling errors with this approach to date. Furthermore, the eigenspectrum of the estimated observation‐error covariance matrix is known to influence the analysis information content and numerical convergence of variational assimilation schemes. In this work, we provide new theorems for the sampling error and eigenvalues of the estimated observation‐error covariance matrices with this approach. We also conduct numerical experiments to illustrate our theoretical results. We find that this method produces large sampling errors if the true observation‐error standard deviation is large, while the other error characteristics, including the true background‐error standard deviation and observation‐ and background‐error correlation length‐scales, have a relatively small effect. We also find that the smallest eigenvalues of the estimated matrices may be smaller or larger than the true eigenvalues, depending on the assumed and true observation‐ and background‐error statistics. These results may provide insights for practical applications: for example, in deciding on appropriate sample sizes and choosing parameters for matrix reconditioning techniques.