Many topical problems in science require making real-time predictions of nonlinear spatially extended systems with physical instabilities across a wide range of scales based on partial observations and an imperfect knowledge of the true dynamics with many degrees of freedom. In such cases data assimilation is usually necessary for mitigating the model error and constraining the unresolved turbulent fluxes in order to improve the stability and skill of the imperfect predictions. However, various data assimilation/filtering strategies developed recently for these applications are also imperfect and not optimal due to the formidably complex nature of the problem.
The lack of resolution in the noisy observations is particularly severe in turbulent geophysical systems with an enormous range of interacting spatio-temporal scales and rough energy spectra near the mesh scale of the discretized imperfect models.