Empirical covariance localization for ensemble sensitivity analysis in complex terrain

Ensemble filter data assimilation suffers from sampling error and potential difficulty in estimating weak correlations and covariances between an observed variable and other state variables. Mountainous environments and mesoscale simulations heighten the challenge. A widely accepted approach for mitigating the effects of sampling error is application of a covariance reduction factor. The factor typically decays with physical distance, such that it is zero at a distance where we heuristically expect covariance to be meaningless. Spurious covariances thought to arise purely from sampling error are eliminated. The ensemble filter community calls this method “covariance localization” because it limits covariances to short length scales. Given the nature of sampling error and the flow dependence intrinsic to ensemble filters, nothing fundamental dictates that the covariance must be localized in space. Spatial localizations are typically isotropic and stationary, which is arguably inappropriate in the lower atmosphere over complex terrain. Other recent work focuses on the sample statistics themselves to determine the factor by which covariance is reduced, ignoring spatial relationships. Covariance between two co-located variables, or a variable and a parameter, can be reduced if the correlation between the two variables has a high probability of being spurious. A third approach bases the covariance factor on short-range forecast performance. Empirical localization is derived by finding coefficients to reduce covariances that minimize forecast errors. Covariance localization has not been thoroughly explored in mountainous environments, where the terrain can determine spatial correlations.

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.