Sampling error mitigation through spectrum smoothing in ensemble data assimilation

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.