Sampling error mitigation through spectrum smoothing: first experiments with ensemble transform Kalman filters and Lorenz models

In data assimilation, an ensemble provides a way to propagate the probability density of a system described by a nonlinear prediction model. Although a large ensemble size is required for statistical accuracy, the ensemble size is typically limited to a small number due to the computational cost of running the prediction model, which leads to a sampling error. Several methods, such as localization and inflation, exist to mitigate the sampling error, often requiring problem-dependent fine-tuning and design. This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum. It turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations. The efficacy of the new idea is validated through a suite of stringent test regimes of the Lorenz 96 turbulent model.