Scale-dependent Inflation to Improve Multi-scale Ensemble-based Data Assimilation

The degree of the background ensemble deficiency, often manifested as ensemble underdispersion, can vary at different scales in the ensemble-based data assimilation. This study develops two scale-dependent inflation (SDI) methods based on two scale-unaware inflation approaches, RTPS (Relaxation To Prior Spread, Whitaker & Hamill 2012) and SE (Sampling-Error treatment inflation, Hodyss et al. 2016), respectively. In the scale-dependent RTPS inflation (RTPS-SDI), the posterior ensemble spread is relaxed toward the prior ensemble spread at each scale separately. In the scale-dependent SE inflation (SE-SDI), mathematical equations are derived so that posterior ensemble error variance is individually adjusted toward the Bayesian posterior error variance at each scale. The impact of RTPS-SDI and SE-SDI are examined and evaluated by implementing these two approaches within the Multiscale Local Gain Form Ensemble Transform Kalman Filter (MLGETKF, Wang et al. 2021) and a two-layer surface quasi-geostrophic turbulence model. During the analysis cycling, RTPS-SDI and SE-SDI outperform RTPS and SE, respectively, in the reduction of analysis errors and the enhancement of ensemble spread nearly at all scales and all cycles. In addition, the improvements in RTPS-SDI over RTPS are greater than those of SE-SDI over SE. These improvements in both SDI methods are associated with greater inflation magnitudes at all scales, especially at larger scales, compared to their scale-unaware counterparts. Regarding subsequent forecasts, both SDI methods show statistically significantly better forecast performance than their scale-unaware counterparts.