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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

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.