Towards a theory of optimal localisation

Abstract

All practical ensemble-based atmospheric data assimilation (DA) systems use localisation to reduce the damaging impact of spurious long-range correlations arising from the finite ensemble size. However, the form of the localisation function is generally ad-hoc, and requires expensive tuning to optimise the system. For the case of a single observation and known true background error correlation, we derive an expression for the localisation factor that minimises the expected root-mean-square (RMS) analysis error. Idealised tests show this formulation performs well for multiple observations provided their density is not too high. The width of the optimal localisation function scales with the width of the underlying correlation, but does not have the same shape. The optimal observation-space localisation for a single spatially integrating observation depends on the observation-to-gridpoint background error correlation, making it broader than the optimal localisation for point observations and potentially competitive with model-space localisation. A new form of hybrid DA is proposed in which localisation damps the sample correlations towards their climatological mean rather than zero, reducing the RMS error and potentially improving the dynamic balance of the analysis. The presence of variance errors causes the optimal localisation factor to depend on the ratio of observation to background error variance, and raises the possibility that a small amount of variance damping may be beneficial. For dense observations, a more elaborate theory is required, which will almost certainly depend on the observation network. We present some preliminary analysis of the features of the multi-observation problem, which for instance suggests that the optimal solution may involve different localisation factors in the numerator and denominator of the Kalman filter equation. We note that even optimal localisation gives an expected RMS error which exceeds that of perfect DA, contrary to the assumption made by ‘deterministic’ ensemble filters.