Abstract
Ensemble methods are increasingly 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 paper presents results from applying a series of theoretically optimal localisations, derived for assimilating a single observation (sparse density), to a Gaussian model state. The theoretical localisations included are optimal localisation for a single true covariance (OSTC), optimal localisation for a variable true covariance (OVTC), which includes knowledge of the climatology and optimal hybrid localisation for a variable true covariance (HOVTC) which damps the difference from the mean covariance as opposed to the covariance itself. The optimal localisations and Gaussian localisation perform similarly for sparse observations. For dense observations, the theoretical assumptions do not hold, and the optimal localisations break down, but the Gaussian, which is retuned, continues to perform well. HOVTC localisation is shown to outperform traditional forms of localisation in the single observation cases. 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 paper shows that theoretically derived localisations can produce improved assimilation performance for a range of observation densities and assimilation strengths in a Gaussian model scenario. It provides the proof of concept that studying the optimal localisation can inform the improvement of localisation regimes for more complex models.