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.

翻译：在数据同化中，集合提供了一种非侵入式方式来演化由非线性预测模型描述的概率密度。尽管统计精度需要较大的集合规模，但由于运行预测模型的计算成本，集合规模通常限制在较小数量，这会导致采样误差。现有多种缓解采样误差的方法（如局地化），但这些方法通常需要根据具体问题进行精细调整和设计。本研究提出一种利用傅里叶空间平滑约束的采样误差缓解新方法，具体通过对系统谱进行平滑处理，在较小集合规模下提升稳定性和精度。该方法的有效性通过洛伦兹96模型和库拉莫托-西瓦辛斯基湍流模型等一系列严格测试问题得到验证。