Multivariate correlations: Applying a dynamic constraint and variable localization in an ensemble context

The structure of the background error covariance matrix is critical for performance in both variational and sequential data assimilation schemes, determining both the form and the magnitude of the analysis increment. Multivariate correlations are provided by the dynamic constraint for variational systems, whereas for ensemble systems, these cross-correlations are provided by the ensemble itself, which is subject to sampling error. Potentially, the variationally-derived covariances can provide more useful information than the ensemble if the sample size is not large enough, especially given the computational burden of computing many ensemble members. Therefore, we implement the dynamic constraint in an Ensemble Kalman Filter to determine its impact. This same methodology is also applied to a hybrid system, with the constraint being applied to both the static and the ensemble contributions simultaneously.

Sampling error can be mitigated through the use of localization, eliminating covariances that are likely unphysical. Typically used in a spatial or temporal sense, this concept can also be applied between different variable types, eliminating correlations between variables that are not physically related. This method is complementary to the dynamic constraint within the ensemble context given that the constraint removes the balanced portions of the variables, leaving only portions that should be uncorrelated. Variable localization can ensure that there are no spurious correlations remaining due to limited sample size.

We use a global atmospheric model of intermediate complexity (SPEEDY, Molteni 2003) to investigate the impacts of the dynamic constraint and variable localization in an Ensemble Kalman Filter, LETKF (Hunt et al 2007), and then in a hybrid 3DEnVar system based on the work of Lorenc (2003) and Buehner (2005). Positive impacts are seen from applying the dynamic constraint to the ensemble portion of the hybrid system in addition to the static portion. Further benefits are seen in both systems by applying both the dynamic constraint and variable localization in conjunction with each other.