On discretization error and its control in variational data assimilation

In four-dimensional variational data assimilation (4D-Var), the model equations are treated as strong constraints on an optimization problem. In reality, the model does not represent the system behaviour exactly and errors arise due to physical approximations, discretization, variability of physical parameters, and inaccuracy of initial and boundary conditions. Errors are also inherent in observation due to inaccuracies in the direct measurement and mapping of the state (model) space onto the observational space or vice versa. The purpose of this work is to define these errors, in particular the discretization and projection errors, and to formulate a canonical problem to study their impact on the quality of the data assimilation process and resulting predictions.