  In data assimilation, an ensemble provides a nonintrusive way to evolve a
probability density 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, exist to mitigate the sampling error, often requiring
problem-dependent fine-tuning and design. This work introduces another sampling
error mitigation method using a smoothness constraint in the Fourier space. In
particular, this work smoothes out the spectrum of the system to increase the
stability and accuracy even under a small ensemble size. The efficacy of the
new idea is validated through a suite of stringent test problems, including
Lorenz 96 and Kuramoto-Sivashinsky turbulence models.
