Model-error estimation in 4D-Var

Current operational implementations of 4D-Var rely on the assumption that the numerical model representing the evolution of the atmospheric flow is perfect, or at least that model errors are small enough (relative to other errors in the system) to be neglected. This paper describes a formulation of weak-constraint 4D-Var that removes that assumption by explicitly representing model error as part of the 4D-Var control variable. The consequences of this choice of control variable for the implementation of incremental 4D-Var are discussed. It is shown that the background-error covariance matrix cannot be used as an approximate model-error covariance matrix. Another model-error covariance matrix, based on statistics of model tendencies, is proposed. Experimental results are presented. They show that this approach to accounting for and estimating model error captures some known model errors and improves the fit to observations, both in the analysis and in the background, but that it also captures part of the observation bias. We show that the model error estimated in this approach varies rapidly, and cannot be used to correct medium- or long-range forecasts. We also show that, because it relies on the tangent linear assumption for the entire assimilation window, the incremental formulation of weak-constraint 4D-Var is not the most suitable formulation for long assimilation windows. Copyright (C) 2007 Royal Meteorological Society.