Ensemble sensitivity analysis (ESA) augments ensemble data assimilation by eliminating the need for tangent-linear and adjoint models to identify the sensitivity of forecast errors to initial conditions and hypothetical observations. ESA can be performed off-line with an existing dataset produced from an ensemble filter. Because sensitivity estimates are based on ensemble statistics, it too suffers from sampling error and can likely benefit from some form of localization.
Sampling error in ESA arises in two places. First, when hypothetical observations are introduced to test the sensitivity estimates for linearity. Here the same localization that was used in the filter itself can be simply applied. Second and more critical, localization should be considered within the sensitivity calculations. Sensitivity to hypothetical observation, estimated without re-running the ensemble, includes regression of a sample of a final-time (forecast) metric onto a sample of initial states. Derivation to include localization results in the localization squared applied directly to the regression. Because the forecast metric is usually a sum, and can also include a sum over a spatial region and multiple physical variables, a spatial localization function is difficult to specify. We present results from experiments to empirically estimate localization factors for ESA to test hypothetical observations for mesoscale data assimilation in a mountainous environment. Localization factors are first derived for an ensemble filter following the empirical localization methodology to optimize the assimilation. Sensitivities for a fog event over Salt Lake City are tested for linearity by approximating assimilation of perfect observations at points of maximum sensitivity, both with and without localization. Observation sensitivity is then estimated, with and without localization, and tested for linearity. Results demonstrate the need for ESA localization in complex terrain.