Cost‐Effective Convective‐Scale Static Background‐Error Covariances: Methodology and Experiment in RRFS Hybrid EnVar System for Direct Radar Reflectivity Data Assimilation Over the CONUS

Abstract This study proposes a cost‐effective convective‐scale static background‐error covariance matrix (CSS B ) approach and evaluates the approach for direct radar reflectivity data assimilation (DA) using the Finite‐Volume Cubed‐Sphere limited‐area model (FV3‐LAM)‐based DA system. The relative importance of cross‐variable correlations between model variables is first identified. To reduce the variational minimization cost, B _CRITICAL is established by only including the 43 critical cross‐correlations for convective scales. Compared to the baseline CSS B that consists of all 66 cross‐correlations, B _CRITICAL is cost‐effective, saving the minimization cost by ∼28.3% while achieving similar performance. Emulating the Rapid Refresh Forecast System, the impacts of incorporating B _CRITICAL into the Gridpoint Statistical Interpolation (GSI)‐based three‐dimensional variational (3DVar) and hybrid ensemble–variational (EnVar) DA frameworks are explored for a convective storm event. Three experiments using B _CRITICAL (Exp‐3DVar), a pure ensemble‐based B (Exp‐EnVar), and a blended B with a 30% weighting factor for B _CRITICAL (Exp‐Hybrid) are conducted. Detailed comparisons and evaluations show the superiority of B _CRITICAL over ensemble‐based B in adding observed reflectivity and the positive impacts of hybridizing B _CRITICAL and pure ensemble‐based B on analysis and forecast improvements. Applying B _CRITICAL to observed clear‐air areas for the first time facilitates spurious background reflectivity reduction. The analyses in Exp‐Hybrid match reflectivity and low‐level temperature observations better than Exp‐3DVar, followed by Exp‐EnVar. In the subsequent forecasts, Exp‐3DVar better predicts the storm spin‐up than Exp‐EnVar. Exp‐Hybrid better maintains the added storms and suppresses the overestimated storms compared to Exp‐3DVar to achieve the highest forecast skill.