Convergence Issues in the Estimation of Interchannel Correlated Observation Errors in Infrared Radiance Data

A posteriori consistency diagnostics have been used in recent years to estimate correlated observation error. These diagnostics provide an estimate of what the observation error covariances should be and could in turn be introduced in the assimilation to improve the statistical consistency between the error statistics used in the assimilation and those obtained from observation departures with respect to the background and the analysis. To estimate the observation error covariances, it is often assumed that the background error statistics are optimal, an assumption that is open to criticism. The consequence is that if the background error covariances are in error, then the estimated observation error statistics will adjust accordingly to fit the {\em innovation} error covariances. In this paper, a Jacobian obtained from the RTTOV radiative transfer model is used as the observation operator. Using controlled experiments, the background error being considered to be fixed, it is shown that the iterative application of the statistical consistency diagnostics may require more than one iteration to reach convergence. It is also shown that the underlying matrix equation being solved can be factorized and the exact solution can be obtained. If the true background error covariances are used in the assimilation, the estimated observation error covariances are then obtained by subtracting the background error covariances from those of the innovations. This is simpler and can be applied to the full set of assimilated observations. Using the Environment Canada assimilation system, the results for several types of observations indicate that the background error estimation would deserve additional attention.