RESUMO
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.
RESUMO
Data assimilation is often performed in a perfect-model scenario, where only errors in initial conditions and observations are considered. Errors in model equations are increasingly being included, but typically using rather adhoc approximations with limited understanding of how these approximations affect the solution and how these approximations interfere with approximations inherent in finite-size ensembles. We provide the first systematic evaluation of the influence of approximations to model errors within a time window of weak-constraint ensemble smoothers. In particular, we study the effects of prescribing temporal correlations in the model errors incorrectly in a Kalman smoother, and in interaction with finite-ensemble-size effects in an ensemble Kalman smoother. For the Kalman smoother we find that an incorrect correlation time-scale for additive model errors can have substantial negative effects on the solutions, and we find that overestimating of the correlation time-scale leads to worse results than underestimating. In the ensemble Kalman smoother case, the resulting ensemble-based space-time gain can be written as the true gain multiplied by two factors, a linear factor containing the errors due to both time-correlation errors and finite ensemble effects, and a nonlinear factor related to the inverse part of the gain. Assuming that both errors are relatively small, we are able to disentangle the contributions from the different approximations. The analysis mean is affected by the time-correlation errors, but also substantially by finite-ensemble effects, which was unexpected. The analysis covariance is affected by both time-correlation errors and an in-breeding term. This first thorough analysis of the influence of time-correlation errors and finite-ensemble-size errors on weak-constraint ensemble smoothers will aid further development of these methods and help to make them robust for e.g. numerical weather prediction.
