Time-correlated model error in the (ensemble) Kalman smoother

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.