Comparison of ensemble-based and variational-based data assimilation schemes in a quasi-geostrophic model

In this study, data assimilation schemes related to variational and ensemble methods are implemented for a quasi-geostrophic model. Four different schemes are discussed: 3D-Var, 4D-Var, 3D-Var hybridized with bred vectors, and LEKF. The goal of this study is not only to compare individual performances but also to try to understand advantages and disadvantages for practical implementation in operational systems, as we know that 4D-Var has already been implemented in several operational centers, and Ensemble Kalman Filters are regarded as the possible next phase data assimilation systems. Our results are discussed in terms of the rms error in potential vorticity (which is a model variable) for all the data assimilation experiments. Given the same rawinsonde observations, the LEKF scheme outperformed 4D-Var in both accuracy and computational cost. This scheme converged to the lowest rms error level, and the 4D-Var with a long assimilation window of 36 hour reached the second lowest error level. However, the slow convergence rate of the LEKF suggests that that an initial guess sufficiently close to the true state of the atmosphere (such as that obtained from a 3D-Var assimilation cycle) is a key factor for fast convergence. Results from 4D-Var experiments suggest that the amplitude of the background error covariance and the length of the assimilation window greatly influence its performance. However, for a high-resolution model with full physics, a long assimilation window is too costly. The minimization process requires special care with long windows by gradually lengthening the window length in order to deal with multiple minima (Pires et al., 1996). Also, the minimization problem is more difficult if the cost function is ill-conditioned and the chosen preconditioner cannot effectively reduce the condition number, so that the minimizer cannot converge, resulting into too many iterations. We also found that the hybrid system with BVs can effectively improve the regular 3D-Var system by suppressing the spurious error spikes. With 20 bred vectors (global approach), the hybrid system is competitive with 12-hour window 4D-Var, and has a very low computational cost. A local hybrid system with 10 bred vectors competes even with the 4D-Var with an optimal 36-hour window. The success of the hybrid scheme using a rather small ensemble suggests that including the background error covariance from 3D-Var provides an important base for the representation of the average behavior of background error. This may be also helpful for methods using ensemble Kalman filters that suffer from rank-deficiency or sampling problems (Hamill and Snyder, 2002). Our results also indicate that a small amount of random perturbations can be helpful for stabilizing the scheme and reducing the error level for all ensemble-related schemes. Variational schemes, including the hybrid one, have larger vertical dependence of the error than the LEKF. More experiments have to be performed, in particular in order to take into account more sophisticated features in the schemes and, most importantly, in order to abandon a perfect model approach. As far as the LEKF is concerned, it is important to understand the potential improvement that can arise from the introduction of observation localization (Miyoshi, 2005) and the impact due to a vertical localization of the model domain. In fact, for the QG-model this latter feature was of minor importance; though we expect that for a real model, including boundary layer description and more sophisticated physics, vertical localization can have a major impact on the performance of the LEKF.