| Summary: | <p>Data assimilation considers the problem of using a variety of data to calibrate model-based estimates of dynamic variables and static parameters. Geoscientific examples include (i) satellite observations and atmospheric models for weather forecasting, and (ii) well-log data and reservoir flow simulators for oil production optimization. Approximate solutions are provided by the set of techniques deriving from the ensemble Kalman filter (EnKF), which combines a Monte Carlo approach with assumptions of linearity and Gaussianity. This thesis proposes some improvements to the accuracy and understanding of such ensemble methods.</p> <p>Firstly, a new scheme is developed to account for model noise in the forecast step of the EnKF. The main aim is to eliminate the sampling errors of additive, simulated noise. The scheme is based on the previously developed "square root" schemes for the analysis step, but requires further consideration due to the limited subspace spanned by the ensemble. The properties of the square root scheme in general are surveyed.</p> <p>Secondly, the "finite size" ensemble Kalman filter (EnKF-N) is reviewed. The EnKF-N explicitly considers the uncertainty in the forecast moments (mean and covariance), thereby not requiring the multiplicative inflation commonly used to compensate for an intrinsic bias of the analysis step of the standard EnKF. Thus, in the perfect model setting, it avoids the process of tuning the inflation factor. This presentation consolidates the earlier literature on the EnKF-N, substantiates the scalar inflation perspective, and rectifies a deficiency.</p> <p>Thirdly, two ensemble "smoothers" expressed by different recursions, used in different applications, and hitherto thought to yield different results, are shown to be equivalent. The theory is revisited under practical considerations, where equivalence is broken due to inflation and localization, but the methods remain equally capable.</p> <p>In each case, the theory is tested and the accuracy performance is benchmarked against standard methods using numerical twin experiments.</p>
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