Modern data assimilation schemes basically rely on linear estimation theory, or on an extension of this theory. In such an approach, each observation is given a weight proportional to the inverse of its specified error variance, measuring the confidence or the precision given to this particular observation. Because of the poor precision of certain observations or their sparse density in some areas, practical implementations of operational analysis schemes are based on the use of proper observation sets but also of background fields, given by a short-range forecast. These background fields can be seen as another source of observations with a given confidence that corresponds to the forecast error covariances. Since the final analysis is very dependent on the specification of the relative weights given to each source of observations, through the error covariances, and since these errors are not perfectly known, a large potential for improvement on analyses is offered by methods producing a posteriori diagnostics of a misspecification of a priori errors, or by procedures allowing an adaptive tuning of these parameters.

On the other hand large operational centers are now using, or have planned to use, assimilation schemes based on a 3D or 4D variational approach, that especially allows the use of a wider range of observations. Diagnostics based on statistics between observations (including the background) and the minimizing solution have been proposed, that can be applied in a variational framework. In particular, it has been shown that a simple diagnostic is the value at the minimum of the cost function that measures the distance between observations and analysis.

We present a method based on diagnostics of observations-minus-analysis differences but that aims at performing an adaptive tuning of observation error parameters from a single batch of observations and background fields. The sum of these squared differences is computed for each subset of observations and compared to its statistical expectation: it is shown that the ratio of these two quantities should be close to the value of the error parameter for this given subset. The computation of the statistical expectation of a term of the cost function is shown to be feasible by performing a second analysis with perturbed observations. The principle of such a method is first shown on a simplified 1D-Var assimilation scheme, where one tries to recover the correct ratio between background and observation terms.

The method is then applied in the framework of the 3D/4D-Var assimilation scheme developed at Météo-France. The possibility to tune the statistics of observation errors for TOVS radiances is in particular investigated, as well as the impact of such a tuning for the analysis and the forecast of cyclones observed during the FASTEX experiment.