Optimal solution error quantification in variational data assimilation involving imperfect models

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimalcontrol problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) errorrises because of the presence of the input data errors (background and observation errors). Then, this erroris quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliarycontrol problem. If the model is not perfect, the optimal solution error includes an additional componentbecause of the presence of the model error. In this paper, we study the influence of the model error on theoptimal solution error covariance, considering strong and weak constraint data assimilation approaches. Forthe latter, an additional equation describing the model error dynamics is involved. Numerical experimentsfor the 1D Burgers equation illustrate the presented theory