Abstract
Ensemble methods are widely used in data assimilation for numerical weather prediction. These methods utilize sample covariance matrices that are subject to sampling error, which is commonly addressed by application of a localisation. The form of the localisation is usually ad-hoc. This thesis develops a series of theoretically optimal localisations based on statistics of the state sampling processes. The theoretical localisations are: (a) Optimal localisation for a Single True Covariance (OSTC); (b) Optimal localisation for a Variable True Covariance (OVTC) which includes knowledge of the climatology; and (c) Hybrid Optimal localisation for a Variable True Covariance (HOVTC) which damps the difference from the mean gain/covariance as opposed to the gain/covariance itself. The optimal localisations are computed to optimise the direct localisation of either the background error covariance or the Kalman gain for assimilating a single observation. The performances of the theoretical localisations are compared to a tuned Gaussian localisation in a series of ideal scenarios where the sampling processes that generate the states are known.
The scenarios used to test the localisation include; a Gaussian shaped covariance model and three scenarios of increasing complexity based on 1D geostrophic balance. Localisation can introduce additional imbalance into the analysis state so investigations explore the impact of the localisations on the balance.
Results have shown that the theoretical localisations perform comparably to or better than the Gaussian localisation for single observation assimilation but break down for dense observations. HOVTC localisation is shown to outperform traditional forms of localisation in the single observation cases. HOVTC localisation introduces less imbalance than OVTC localisation. It is shown that neither directly optimising the gain nor the covariance are ideal as it is desirable to optimise the gain whilst applying localisation in the numerator and denominator of the gain. In the Gaussian model experiments, a tuned hybrid localisation is proposed based on the form of the optimal hybrid localisation and this is shown to perform well in all ranges of observation density and assimilation strengths.
The thesis explores the factors that affect the form and performance of the optimal localisation. It shows that theoretically derived localisations can produce improved assimilation performance for a range of observation densities and assimilation strengths. Finally, it shows that studying the optimal localisation can inform the improvement of localisation regimes for more complex models.