Accounting formodel error in strong-constraint 4D-Var data assimilation

The strong-constraint formulation of four-dimensional variational data assimilation (4D-Var) assumes that the model used in the process perfectly describes the true dynamics of the system. However, this assumption often does not hold and the use of an erroneous model in strong-constraint 4D-Var can lead to a sub-optimal estimation of the initial conditions. We show how the presence of model error can be correctly accounted for in strong constraint 4D-Var by allowing for errors in both the observations and the model when considering the statistics of the innovation vector. We demonstrate that, when these combined model error and observation-error statistics are used in place of the standard observation error statistics in the strong-constraint formulation of 4D-Var, a statistically more accurate estimate of the initial state is obtained. The calculation of the combined model error and observation-error statistics requires the specification of model error covariances, which in practice are often unknown. We present a method to estimate the combined statistics from innovation data that does not require explicit specification of the model error covariances. Numerical experiments using the linear advection equation and a simple nonlinear coupled model demonstrate the success of the new methods in reducing the error in the estimate of the initial state, even in the case when only the uncorrelated part of the model error is accounted for.