Sensitivity of the model error parameter specification in weak-constraint four-dimensional variational data assimilation

• Model error parametrization is considered in weak-constraint data assimilation.
• Equations are derived for sensitivity to bias, standard deviation, and correlation.
• A feedback mechanism is proposed for adaptive tuning of the model error covariance.
• The tuning procedure is based on a variable step-size gradient descent algorithm.
• Preliminary results show the potential to significantly improve the state estimates.

This article presents the mathematical framework to evaluate the sensitivity of a forecast error aspect to the input parameters of a weak-constraint four-dimensional variational data assimilation system (w4D-Var DAS), extending the established theory from strong-constraint 4D-Var. Emphasis is placed on the derivation of the equations for evaluating the forecast sensitivity to parameters in the DAS representation of the model error statistics, including bias, standard deviation, and correlation structure. A novel adjoint-based procedure for adaptive tuning of the specified model error covariance matrix is introduced. Results from numerical convergence tests establish the validity of the model error sensitivity equations. Preliminary experiments providing a proof-of-concept are performed using the Lorenz multi-scale model to illustrate the theoretical concepts and potential benefits for practical applications.

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