Optimal distributions of growing‐type initial perturbations for ensemble forecasts: Theory and application in the Lorenz‐96 model

Abstract Ensemble forecasts are frequently utilized to assess the uncertainties of prediction systems. There is a consensus that generating initial perturbations with specific structures is more conducive to characterizing the growth dynamics of analysis errors, demonstrating higher forecast skills. However, the widely used methods, such as linear singular vectors (SVs) and orthogonal conditional nonlinear optimal perturbations (O‐CNOPs) exhibit strong linear‐assumption dependence and the overestimation of growing properties for analysis errors, respectively, severely limiting the ability to capture analysis errors and forecast skills. To tackle these challenges, a theoretical framework is established to solve the optimal distribution of the nonlinear growing‐type initial perturbations by variational inference (VI) incorporated with the concept of CNOPs, marked as VI‐CNOPs. As the distribution is obtained, diverse initial perturbations for ensemble forecasts can be easily sampled in it. To evaluate the reliability of VI‐CNOPs, a series of ensemble forecast experiments are then conducted using the Lorenz‐96 model. We compare the deterministic and probabilistic forecast skills of VI‐CNOPs, O‐CNOPs, and SVs under various optimization durations. The results reveal that, as the optimization durations extend, the forecast skills of VI‐CNOPs progressively improve, consistently outperforming O‐CNOPs and SVs. This trend remains consistent across various forecast lead times. Further analysis reveals that VI‐CNOPs more effectively capture the covariance matrix of analysis errors, aligning with the fundamental concept of perturbation generation methods for ensemble forecasts. Moreover, unlike O‐CNOPs and SVs, VI‐CNOPs do not require the utilization of adjoint and tangent linear models, largely expanding its application. These results indicate the novelty and efficacy of VI‐CNOPs for ensemble forecasts.