Systematic comparison of four-dimensional data assimilation methods with and without the tangent linear model using hybrid background error covariance: E4DVar versus 4DEnVar

Two ensemble formulations of the four-dimensional variational (4DVar) data assimilation technique are examined for a low-dimensional dynamical system. The first method, denoted E4DVar, uses tangent linear and adjoint model operators to minimize a cost function in the same manner as the traditional 4DVar data assimilation system. The second method, denoted 4DEnVar, uses an ensemble of nonlinear model trajectories to replace the function of linearized models in 4DVar, thus improving the parallelization of the data assimilation. Background errors for each algorithm are represented using a hybrid error covariance, which includes climatological errors as well as ensemble-estimated errors from an ensemble Kalman filter (EnKF). Numerical experiments performed over a range of scenarios suggest that both methods provide similar analysis accuracy for dense observation networks, and in perfect model experiments with large ensembles. Nevertheless, E4DVar has clear benefits over 4DEnVar when substantial covariance localization is required to treat sampling error. The greatest advantage of the tangent-linear approach is that it implicitly propagates a localized, full-rank ensemble covariance in time, thus avoiding the need to localize a time-dependent ensemble covariance. The tangent linear and adjoint model operators also provide a means of evolving flow-dependent information from the climate-based error component, which is found to be beneficial for treating model error. Challenges that need to be overcome before adopting a pure ensemble framework are illustrated through experiments estimating time covariances with four-dimensional ensembles and comparing results with those estimated with a tangent linear model.