<p>This contribution adresses the characterization of the model-error covariance matrix from the new theoretical perspective provided by the parametric Kalman filter method which approximates the covariance dynamics from the parametric evolution of a covariance model. The classical approach to obtain the modified equation of a dynamics is revisited to formulate a parametric diagnosis of the model-error covariance matrix. As an illustration, the particular case of the advection equation is considered as a simple test bed. After the theoretical derivation of both the forecast-error and the predictability-error covariance matrices, a numerical simulation is proposed which demonstrates the skill of the parametric methodology in reproducing the model-error covariance matrix information.</p>