Scale-Dependent Inflation Algorithms for Ensemble Kalman Filters

Abstract Ensemble-based data assimilation methods often suffer sampling errors due to limited ensemble sizes and model errors, which can result in filter divergence. One method to avoid filter divergence is inflation, which inflates ensemble perturbations to increase ensemble spread and account for model errors. The commonly applied inflation methods, including the multiplicative inflation, relaxation to prior spread (RTPS), and additive inflation, often use a constant inflation parameter. To capture different error growths at different scales, a scale-dependent inflation is proposed here, which applies different inflation magnitudes for variables associated with different scales. Results from the two-scale Lorenz05 model III show that for the ensemble square root filter (EnSRF) and integrated hybrid ensemble Kalman filter with ensemble mean updated by hybrid background error covariances (IHEnKF-Mean), scale-dependent inflation is superior to constant inflation. Constant inflation overinflates small-scale variables and results in increased small-scale errors, which then propagate to large-scale variables through the coupling between large- and small-scale variables and lead to increased large-scale errors. Scale-dependent inflation applies larger inflation for large-scale variables and imposes no inflation for small-scale variables, since large-scale errors have larger magnitudes than small-scale ones and small-scale errors grow faster than large-scale ones. But IHEnKF-Ensemble that updates both the ensemble mean and perturbations with hybrid background error covariances is much less sensitive to scale-dependent inflation, compared to EnSRF and IHEnKF-Mean, because updating ensemble perturbations with hybrid background error covariances can play a role similar to the scale-dependent inflation. Significance Statement Ensemble-based data assimilation and ensemble forecasts often have smaller ensemble spread than errors. Strategies used by ensemble-based data assimilation to combat insufficient ensemble spread usually focus on the short-term ensemble forecasts, rather than considering the whole ensemble forecasts over different lead times. Thus, there are obvious gaps between the ensemble-based data assimilation and ensemble forecasts. A scale-dependent inflation that can capture different error growths at different scales is proposed, which obtains improved consistency between ensemble spread and errors at different lead times and effectively links the ensemble-based data assimilation and ensemble forecasts.