A simpler formulation of forecast sensitivity to observations: application to ensemble Kalman filters

We introduce a new formulation of the ensemble forecast sensitivity developed by Liu and Kalnay with a small correction from Li et al. The new formulation, like the original one, is tested on the simple Lorenz 40-variable model. We find that, except for short-range forecasts, the use of localization in the analysis, necessary in ensemble Kalman filter (EnKF) when the number of ensemble members is much smaller than the model's degrees of freedom, has a negative impact on the accuracy of the sensitivity. This is because the impact of an observation during the analysis (i.e. the analysis increment associated with the observation) is transported by the flow during the integration, and this is ignored when the ensemble sensitivity uses a fixed localization. To address this problem, we introduce two approaches that could be adapted to evolve the localization during the estimation of forecast sensitivity to the observations. The first one estimates the non-linear evolution of the initial localization but is computationally expensive. The second one moves the localization with a constant estimation of the group velocity. Both methods succeed in improving the ensemble estimations for longer forecasts.

Overall, the adjoint and ensemble forecast impact estimations give similarly accurate results for short-range forecasts, except that the new formulation gives an estimation of the fraction of observations that improve the forecast closer to that obtained by data denial (Observing System Experiments). For longer-range forecasts, they both deteriorate for different reasons. The adjoint sensitivity becomes noisy due to the forecast non-linearities not captured in the linear tangent model and the adjoint. The ensemble sensitivity becomes less accurate due to the use of a fixed localization, a problem that could be ameliorated with an evolving adaptive localization. Advantages of the new formulation include it being simpler than the original formulation and computationally more efficient and that it can be applied to other EnKF methods in addition to the local ensemble transform Kalman filter.