Effects of mis-specified time-correlated model error in the (ensemble) Kalman Smoother

Data assimilation is often performed under the perfect model assumption. Although there is an increasing amount of research accounting for model errors in data assimilation, the impact of an incorrect specification of the model errors on the data assimilation results has not been thoroughly assessed. We investigate the effect that an inaccurate time correlation in the model error description can have on data assimilation results, deriving analytical results using a Kalman Smoother for a one-dimensional system. The analytical results are evaluated numerically to generate useful illustrations. For a higher-dimensional system, we use an ensemble Kalman Smoother. Strong dependence on observation density is found. For a single observation at the end of the window, the posterior variance is a concave function of the guessed decorrelation time-scale used in the data assimilation process. This is due to an increasing prior variance with that time-scale, combined with a decreasing tendency from larger observation influence. With an increasing number of observations, the posterior variance decreases with increasing guessed decorrelation time-scale because the prior variance effect becomes less important. On the other hand, the posterior mean-square error has a convex shape as a function of the guessed time-scale with a minimum where the guessed time-scale is equal to the real decorrelation time-scale. With more observations, the impact of the difference between two decorrelation time-scales on the posterior mean-square error reduces. Furthermore, we show that the correct model error decorrelation time-scale can be estimated over several time windows using state augmentation in the ensemble Kalman Smoother. Since model errors are significant and significantly time correlated in real geophysical systems such as the atmosphere, this contribution opens up a next step in improving prediction of these systems.