Scale-dependent inflation algorithms for ensemble Kalman filters

Abstract

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, relax 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.