First-order and higher-order approximations of observation impact

In studies by Errico (2007) and Gelaro et al. (2007), it was found that first order approximations of the reduction of forecast error by the analysis were systematically close to twice the correct value, while higher order approximations lead to relatively accurate estimations. We show that this factor of 2 is due to the quadratic nature of the forecast error measure and to the fact that the analysis point is near the minimum of this quadratic measure.