Data assimilation with autocorrelated model error

Data assimilation has often been performed under the perfect model assumption, but in reality, numerical models often contain model errors with spatial and temporal correlations. The objective of this thesis is to thoroughly investigate the impact of an inaccurate time correlation in the model error description on data assimilation results, both analytically and numerically using the ensemble Kalman Smoother (EnKS). Furthermore, we try to develop an efficient way to perform online estimation of certain model error autocorrelation parameters with the data assimilation scheme. With a simple linear model and a single-parameter autocorrelation, we find that the performance of the data assimilation scheme can be impacted by the departures between the actual values of the parameter and the value proposed in the data assimilation process with sparse observations. However, the impact of the incorrect parameter can be diminished with dense observations. Furthermore, we show that the correct model error decorrelation timescale can be estimated after multiple simulation windows using the state augmentation method with the linear system. More complex autocorrelation, in which decaying and oscillatory scales are considered, is later examined on the linear model and, furthermore, a nonlinear logistic map. It seems impossible for the EnKS to track both decaying and oscillatory parameters in the autocorrelation, and the iterative variant of the EnKS (IEnKS) is required. With the nonlinear logistic map, even the IEnKS fails to find the correct values for the two parameters and can get stuck in local minima. Fortunately, we can find the correct values of the parameters with careful tuning of the IEnKS and transformation of the solution space. When the problem confronts a high-dimensional nonlinear system such as the quasigeostrophic model, a large part of the state has to be observed in space, even for the simplest case of the model error autocorrelation. In this case, it shows the limitations of our method for practical weather forecast systems since the observation cannot be as dense as needed for the parameter estimation to work, and result in an affordable scheme