Ensemble Kalman filters for dynamical systems with unresolved turbulence

Ensemble Kalman filters are developed for turbulent dynamical systems where the forecast model does not resolve all the active scales of motion. These filters are based on the assumption that a coarse-resolution model is intended to predict the large-scale part of the true dynamics; since observations invariably include contributions from both the resolved large scales and the unresolved small scales, careful treatment of a filter’s observation model is required. A general ensemble Kalman filtering framework is developed here centered around novel treatment of the observation model. The framework requires prediction of the mean and covariance of the unresolved small scales; we show that the use of a time-independent ‘background’ climatological mean and covariance for the small scales, analogous to the use of static ‘background’ statistics in optimal interpolation and variational data assimilation, leads to good results in a difficult test problem. These results are significantly improved through the use of superparameterization – a multiscale modeling strategy that reduces large-scale model error and simultaneously provides forecast estimates of the mean and covariance of the unresolved scales. The one-dimensional test problem from dispersive wave turbulence used here is computationally tractable yet is particularly difficult for filtering because of the substantial small scale turbulence: a shallow energy spectrum proportional to k where k is the wavenumber, non-Gaussian extreme event statistics, and with two-thirds of the climatological variance carried by the unresolved small scales. The new methods are far better at capturing the extreme event distribution and have smaller RMS error than standard methods, even with sparse observations. The standard methods are unable to beat the climatological error even with accurate observations of all the coarse model variables, covariance localization and inflation.