Effective data assimilation crucially depends on being able to quantify the extent to which one can trust various aspects of the weather forecast that is used as the "first guess" field in the data assimilation procedure. For example, one might be certain that a cold front was approaching a small island in a large ocean basin and uncertain about the exact location and orientation of the front. In such a situation, one would ideally use meteorological data from the small island to get a better estimate of the position of the front. The unapproximated Kalman filter provides an optimal way of making such an estimate. It does so by using flow dependent error covariance matrices whose leading eigenvectors describe the structure of the forecast errors most likely to occur when a front is approaching an island. The class of sub-optimal Kalman filters known as Ensemble Kalman Filters approximate the error covariance matrices of the Kalman filter with covariance matrices constructed from ensemble perturbations about the mean (or control) of an ensemble forecast. Inaccuracies in ensemble based estimates of error covariance matrices arise from (a) model error (b) poor initialization of the ensemble c) a poor representation of the contribution of model error to prediction error variance. The Ensemble Transform Kalman Filter (ET KF) attempts to ameliorate these problems by using the difference between the first guess field and observations together with maximal likelihood estimation theory to transform the original ensemble perturbations into the most likely set of ensemble perturbations given that the difference between the first guess field and the observations is what it is. As a side benefit, the ensemble transformation also provides an extremely computationally efficient means of obtaining the Kalman gain matrix from the ensemble perturbations that appears to have been previously overlooked. The technique should be of interest to Fluid Dynamicists because it provides a means of using observations to quantify the difference between the predictability of a model and the predictability of the atmosphere. The effectiveness of the ET KF will be demonstrated by comparing its performance as a data assimilation tool against the performance of a range of other sorts of data assimilation schemes.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics