Following the a posteriori diagnosis approach proposed by some authors, a practical computation of the expectation of sub-parts of the value of a cost function at the minimum is shown to be feasible by using a randomization technique based on a perturbation of observations or background fields. These computations allow the tuning of observation-error weighting parameters by applying a simple iterative fixed-point procedure.The procedure is first tested in a simplified variational scheme on a circular domain and then in a similar scheme but with the addition of the vertical coordinate. The relationship between the proposed approach and the Generalized Cross Validation is also shown. A test in the French Action de Recherche Petite Echelle Grande Echelle (ARPEGE) three-dimensional variational framework with both simulated observations and background fields is finally performed. It shows that a complete description of observation-error parameters can be retrieved with only a few iterations and, thus, at a reasonable cost. = hTr(R7' [I'jE(caeyT)r; + E ( c y~y )~ -E{(l'jxa)cYT} -E[cy(rjxa)T}]), since r j x ay? = r j x a + rjxTr j x Ty? = rjea -€ 9 , and using the fact that E(Tr(.)} = Tr{E(.)} and the linearity properties of the expectation operator E. J J J
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