Diagnosis of background errors for radiances and other observable quantities in a variational data assimilation scheme, and the explanation of a case of poor convergence
Authors: Anderson Erik , Fisher Mike , Munro Rosemary , McNally Anthony 
Published: July 2010

In any statistical data assimilation scheme the ratio between the observation and background errors fundamentally determines the weight given to the observations. The observation errors are specified directly in terms of the observable quantities. In variational data assimilation schemes these can include satellite‐measured radiances as well as conventional observations. The background errors, on the other hand, are specified in terms of those quantities that lead to a compact formulation of the background term (the Jb of the variational analysis), viz. balanced vorticity, unbalanced temperature, divergence and surface pressure, and specific humidity. It is not obvious how the magnitudes of these background errors can be compared with the various observation errors. Within the variational analysis, the background errors are implied in terms of observed quantities, i.e. not normally computed explicitly. They depend, in general, on the Jb formulation and on the observation operators. In the case of radiance observations this involves the Jacobian of the radiative‐transfer model which, in turn, depends on the atmospheric state.

By applying the observation operators of a variational data assimilation scheme 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 any observed quantity. These are valuable for diagnosing the response of the data assimilation system to observational data, and for tuning the specified observation and background errors in general. The calculated error standard deviations can be compared with those obtained from studies of innovation statistics (i.e. observed departures from the background). The technique has been applied to a range of observable quantities, including the radiance data from both the infrared and microwave instruments of the TIROS operational vertical sounder (TOVS). We used the results for some of the higher‐peaking channels to verify that the specified background errors in the recently introduced 50‐level version of the ECMWF model are also reasonable in the upper stratosphere, where there are few conventional data. We also found that the operational background errors for humidity were set unrealistically 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‐sensitive radiance channels (the Meteosat water‐vapour channel, similar to TOVS channel 12). Excessively large ratios between background and observation errors locally led to larger than normal eigenvalues of the analysis Hessian—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 areas in the subtropics, but it affected the minimization globally.