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This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator (BLUE) and its covariance. Some of the Kalman filter’s detailed properties are reviewed here: linear transformations of the state and observations, extending the state vector to include observed variables, and temporal correlation in the model or observation errors. The Kalman filter can be applied to nonlinear and non-Gaussian systems via either the extended Kalman filter or the BLUE, although both approaches are clearly sub-optimal. The ensemble Kalman filter (EnKF) employs sample covariances from an ensemble of forecasts at each update time and allows practical implementation of an approximate Kalman filter. The EnKF is consistent with a Monte- Carlo implementation of the BLUE. Many of the EnKF’s properties, including basic effects of sampling error, can be understood in the context of Kalman-filter theory.

This chapter, compares results from the Canadian global ensemble Kalman filter (EnKF) with observations. This inevitably leads to discrepancies between the observed real atmosphere and its modelled equivalent. These discrepancies originate from system error. In a system simulation experiment, an attempt is made to obtain a coherent picture of the error evolution of a system. Errors can be due to things as different as an inappropriate closure assumption in a forecast model and inaccurate observations of surface pressure. This chapter, first describes Monte- Carlo methods in general to arrive at a definition of “‘system error”.‘. This is followed by an elimination procedure. First, medium-range ensemble forecasts are used to quantify the understanding of weaknesses of the forecast model. Subsequently, consideration turns to the data-assimilation context to see what additional error sources must be present. The chapter ends with some speculation on the types of errors that should be included.

This chapter describes the use of smoothers in data assimilation. The filtering problem in data assimilation consists in estimating the state of a system based on past and present observations. In contrast to filters, amoothers implement Bayesian data assimilation using future observations. Smoothing problems can be posed in different ways. The main formulations in geophysics are fixed-point, fixed-interval, and fixed-lag smoothers. In this chapter, these problems are first introduced in a Bayesian framework, and the most straightforward Bayesian solutions are formulated. Common linear, Gaussian implementations, many of which are based on the classical Kalman filter, are then derived, followed by their ensemble counterparts, based on the usual ensemble Kalman filter techniques. Finally, the pros and cons, as well as the computational complexities, of all the schemes are discussed.

In this chapter, an experimental environment built around the Lorenz III toy model is used to demonstrate some points concerning localization. In an ensemble Kalman filter, localization is almost always necessary because of restrictions on the size of the ensembles. In fact, localization is the key technique that makes the ensemble approximation to the Kalman filter computationally feasible. How localization is best applied depends on aspects of the model dynamics and the observational network. A reasonable choice often leads to a substantial improvement in performance. Fortunately, as shown in this chapter, the statistics from the ensemble itself can provide guidance in the selection of a reasonable localization method.

This chapter compares results from the Canadian global ensemble Kalman filter (EnKF) with observations. This inevitably leads to discrepancies between the observed real atmosphere and its modelled equivalent. These discrepancies originate from system error. In a system simulation experiment, an attempt is made to obtain a coherent picture of the error evolution of a system. Errors can be due to things as different as an inappropriate closure assumption in a forecast model and inaccurate observations of surface pressure. This chapter first describes Monte Carlo methods in general to arrive at a definition of ‘system error’. This is followed by an elimination procedure. First, medium-range ensemble forecasts are used to quantify the understanding of weaknesses of the forecast model. Subsequently, consideration turns to the data-assimilation context to see what additional error sources must be present. The chapter ends with some speculation on the types of errors that should be included.

This chapter deals with short-range error statistics in the context of the ensemble Kalman filter (EnKF). To arrive at an optimal data-assimilation system, a good description of the uncertainty in the background field is needed. Historically, different approaches, with a solid comparison against either a ground truth or observations, have been used to obtain limited descriptions. The first category includes observation system simulation experiments (OSSEs), while the second includes methods based on statistical analysis of innovations. The EnKF is a relatively new method that simulates the effect of known sources of error to arrive at a Monte Carlo estimate of flow-dependent background error statistics. It is necessary to validate the ensemble statistics–in part by comparison with results from established methods–to identify areas of improvement for the EnKF. This chapter first summarizes existing methods and then studies the properties of a research version of the Canadian global EnKF.