Estimation of the representativeness error caused by the incremental formulation of variational data assimilation

A formalization of the representativeness error caused by the use of a simplified increment in certain implementations of the variational formalism is introduced. The incremental representativeness-error can be determined by summing up the part of the background-error power-spectrum that is beyond the truncation of the simplified increment. Thus, this error will be more important for fields, such as humidity, whose forecast errors have large components at small scales. The length scale and the oscillating structure of this incremental representativeness-error are explained as the consequences of the application of a square window in spectral space to the corresponding background-error covariances. A computation of the covariances of this error has been performed in a realistic framework given by the French numerical weather prediction system. In particular, it shows that this error is not negligible, and that its size is of the same order as that of observational instrument error which occurs when using a T63 spectral resolution of the analysis increment.