Nonlinear and Multiresolution Error Covariance Estimation in Ensemble Data Assimilation

Ensemble Kalman Filters perform data assimilation by forming a background covariance matrix from an ensemble forecast. The spread of the ensemble is intended to represent the algorithm's uncertainty about the state of the physical system that produces the data. Usually the ensemble members are evolved with the same model. The first part of my dissertation presents and tests a modified Local Ensemble Transform Kalman Filter (LETKF) that takes its background covariance from a combination of a high resolution ensemble and a low resolution ensemble. The computational time and the accuracy of this mixed-resolution LETKF are explored and compared to the standard LETKF on a high resolution ensemble, using simulated observation experiments with the Lorenz Models II and III (more complex versions of the Lorenz 96 model). The results show that, for the same computation time, mixed resolution ensemble analysis achieves higher accuracy than standard ensemble analysis. The second part of my dissertation demonstrates that it can be fruitful to rescale the ensemble spread prior to the forecast and then reverse this rescaling after the forecast. This technique, denoted ``forecast spread adjustment'' provides a tunable parameter that is complementary to covariance inflation, which cumulatively increases the ensemble spread to compensate for underestimation of uncertainty. As the adjustable parameter approaches zero, the filter approaches the Extended Kalman Filter when the ensemble size is sufficiently large. The improvement provided by forecast spread adjustment depends on ensemble size, observation error, and model error. The results indicate that it is most effective for smaller ensembles, smaller observation errors, and larger model error, though the effectiveness depends significantly on the type of model error