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 simp lified 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 t he truncation of the simplified increment. Thus, this error will be more im portant for fields, such as humidity, whose forecast errors have large comp onents at small scales. The length scale and the oscillating structure of t his incremental representativeness-error are explained as the consequences of the application of a square window in spectral space to the correspondin g background-error covariances. A computation of the covariances of this er ror has been performed in a realistic framework given by the French numeric al weather prediction system. In particular, it shows that this error is no t negligible, and that its size is of the same order as that of observation al instrument error which occurs when using a T63 spectral resolution of th e analysis increment.