Multivariate Correlations: Balance Operators and Variable Localization in Ensemble Data Assimilation
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Localization is performed in ensemble data assimilation schemes to eliminate correlations that are contaminated by sampling error. This method is frequently necessary within numerical weather prediction (NWP) applications due to the computational constraints present, limiting the number of ensemble members to a size much smaller than the dimension of the system. The most common form of localization occurs in the spatial dimensions, reducing the correlations for points that are distant and likely dominated by sampling error. Spatial localization can introduce imbalance in the system due to the disruption of physical relationships that are dictated by gradients or column integrated quantities, which produce fast-moving gravity waves within NWP models and degrade the forecast. The first part of this dissertation explores the impact of including a balance operator within ensemble data assimilation schemes and how the type of spatial localization interacts with it. The inclusion of a balance operator allows the localization to be performed on the unbalanced portion of the correlation, preserving the balanced correlation. Two data assimilation schemes are explored: a hybrid 4D ensemble-variational (4DEnVar) scheme and a Local Ensemble Transform Kalman Filter (LETKF). Observing system simulation experiments are performed using an intermediate complexity model, SPEEDY. It is shown that localizing on the background error as in the hybrid 4DEnVar is more effective than localizing on the observation error as in the LETKF. Within the LETKF, the balance operator can only propagate information one way, for example, from streamfunction to temperature, but not vice versa as in the hybrid 4DEnVar. Many applications contain variables that are physically unrelated and should not be correlated, but contain nonzero correlations. The second part of this dissertation presents two forms of variable localization in a unified framework: observation space variable localization (VO) and model space variable localization (VM). VO restricts the impact that observations have to certain model variables. VM removes the cross-correlations during the computation of the background error covariance. VM is more computationally expensive, but it has the added advantages of not requiring knowledge of observation types and allowing a single observation to impact multiple model variables whose cross-correlations have been removed.