Diagnosis of background errors for radiances and other observable quantities in a variational data assimilation scheme, and the explanation of a caseof poor convergence

In any statistical data assimilation scheme the ratio between the observati on and background errors fundamentally determines the weight given to the o bservations. The observation errors are specified directly in terms of the observable quantities, in variational data assimilation schemes these can i nclude satellite-measured radiances as well as conventional observations. T he background errors, on the other hand, are specified in terms of those qu antities that lead to a compact formulation of the background term (the J(b ) of the variational analysis), viz. balanced vorticity, unbalanced tempera ture, divergence and surface pressure, and specific humidity. It is not obv ious how the magnitudes of these background errors can be compared with the various observation errors. Within the variational analysis, the backgroun d errors are implied in terms of observed quantities, i.e. not normally com puted explicitly. They depend, in general, on the J(b) formulation and on t he observation operators. In the case of radiance observations this involve s the Jacobian of the radiative-transfer model which, in turn, depends on t he atmospheric state. By applying the observation operators of a variational data assimilation sc heme to a set of random vectors, drawn from a population whose probability density function is given by the assumed background-error covariance matrix , we obtain grid-point fields of background-error standard deviations for a ny observed quantity. These are valuable for diagnosing the response of the data assimilation system to observational data, and for tuning the specifi ed observation and background errors in general. The calculated error stand ard deviations can be compared with those obtained from studies of innovati on statistics (i.e. observed departures from the background). The technique has been applied to a range of observable quantities, including the radian ce data from both the infrared and microwave instruments of the TIROS opera tional vertical sounder (TOVS). We used the results for some of the higher- peaking channels to verify that the specified background errors in the rece ntly introduced 50-level version of the ECMWF model are also reasonable in the upper stratosphere, where there are few conventional data. We also foun d that the operational background errors for humidity were set unrealistica lly large in some dry subtropical areas. A case of poor convergence of the variational analysis was found to be due to unrealistically high background errors in terms of one of the humidity-s ensitive radiance channels (the Meteosat water-vapour channel, similar to T OVS channel 12). Excessively large ratios between background and observatio n errors locally led to larger than normal eigenvalues of the analysis Hess ian-thus increasing the condition number of the minimization problem, with an associated decrease in the rate of convergence of the minimization. The mis-specification of background errors was confined to relatively small are as in the subtropics, but it affected the minimization globally.