Ensemble sensitivity analysis: performance and mitigation over mesoscale complex terrain

Recent expansion in availability of relocatable near-surface atmospheric observing sensors introduces the question of where placement maximizes gain in forecast accuracy. As part of a larger effort to establish the viability of ensemble sensitivities to inform observing strategies for mesoscale flows, the potential for ensemble sensitivity analysis (ESA) is examined for high-resolution (Dx=4 km) predictions in complex terrain. The primary objective is to determine whether a mesoscale ESA applied at these scales is useful for identifying potential observing locations in weak flow. ESA can be inaccurate when the underlying assumptions of linear dynamics (and Gaussian statistics) are violated, or when the sensitivity cannot be robustly sampled. A case study of a fog event at the Salt Lake City airport (KSLC) provides a useful period for examining these issues, with the additional influence of complex terrain. A realistic upper-air observing network is used in perfect-model ensemble data assimilation experiments, providing the statistics for ESA. Results show that water vapor mixing ratios over KSLC are sensitive to potential temperature on the first model layer tens of km away, 6 h prior to verification and prior to the onset of fog. Sensitivity 12 h prior is weaker but leads to qualitatively similar results. Potential temperatures are shown to be a predictor of inversion strength in the Salt Lake basin; the ESA predicts southerly flow and strengthened inversions with warmer potential temperatures in a few locations. Simple linearity tests show that small perturbations do not lead to the expected forecast change, but larger perturbations do, suggesting that noise can dominate a small perturbation. Assimilating a perfect observation at the maximum sensitivity location produces forecasts more closely agreeing with the ESA. Sampling error evaluation show that similar conclusions can be reached with ensembles as small as 48 members, but smaller ensembles do not produce accurate sensitivity estimates. One experience from this study is that unlike in strongly forced, smooth, synoptic-scale weather the sensitivity fields are not straightforward to analyze. A strongly forced mountain wave case in the lee of the Rockies is more straightforward to analyze, but the sensitivities are still weak.

Motivated by the experiences described above, we investigate some mathematical and theoretical aspects of ensemble sensitivity, with the goal of understanding the limits of ensemble sensitivities in these flow scenarios. So far in the literature, covariance localization (tapering) has not been applied when performing ensemble sensitivity analysis. Sampling error in computing the sensitivities via lagged covariances leads to an over-estimation of the impact of a perturbation. Most commonly when computing sensitivities, the analysis covariance is approximated with the corresponding diagonal matrix. Two consequences follow: (1) the multi-variate sensitivity is approximated by a univariate sensitivity, and (2) sampling error in off-diagonal elements are obviated. It is unknown, however, how much information is lost by ignoring the off-diagonal elements in the full covariance. When forecasts depend on many details of the previous analysis, it is reasonable to expect that the diagonal approximation is too severe. The purpose of this presentation is to clarify the effects of the diagonal approximation, and investigate the need for localization when off-diagonal elements are considered. Motivated by examples arising from sensitivities estimated within a cycling mesoscale ensemble data assimilation system, for easier interpretation we turn to the two-scale model first presented by Lorenz in 2005. For most problems, an efficient matrix inversion is possible by finding a minimum-norm solution, and employing appropriate matrix factorization. Factors for localizing/tapering the spatio-temporal covariances can be found by a Bayesian estimation technique following Anderson (2007). Skill in predicting a nonlinear response from the linear sensitivities is superior when localized multivariate sensitivities are used, particularly when fast scales are present, model error is present, and the observing network is sparse. The results provide a set of best practices for using ensemble sensitivities over mesoscale complex terrain, particularly where sparsely observed.