On the Foundation and Different Interpretations of Ensemble Sensitivity

Abstract

Abstract In sensitivity analysis, ensemble sensitivity is defined as the regression coefficients resulting from a simple linear regression of changes of a response function on initial perturbations. One of the interpretations for ensemble sensitivity considers this a simplified version of regression-based adjoint sensitivity called univariate ensemble sensitivity whose derivation involves the so-called diagonal approximation. This approximation, which replaces the analysis error covariance matrix by a diagonal matrix with the same diagonal, helps to avoid inversion of the analysis error covariance, but, at the same time causes confusion in understanding and practical application of ensemble sensitivity. However, some authors have challenged such a controversial interpretation by showing that univariate ensemble sensitivity is multivariate in nature, which raises the necessity for the foundation of ensemble sensitivity. In this study, we have tried to resolve the confusion by establishing a robust foundation for ensemble sensitivity without relying on the controversial diagonality assumption. As employed in some studies, we adopt an impact-based definition for ensemble sensitivity by taking into account probability distributions of analysis perturbations. The mathematical results show that standardized ensemble sensitivity carries in itself three important quantities at the same time: 1) standardized changes of the forecast response with one standard deviation changes of individual state variables, 2) correlations between the forecast response and individual state variables, and 3) the most sensitive analysis perturbation. The theory guarantees validity of ensemble sensitivity, demonstrates its multivariate nature, and explains why ensemble sensitivity is effective in practice.