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Time series are often assumed to arise as observations from an underlying dynamical system. The observations though need not be one-to-one mappings of the full state of the underlying dynamical system, which is thus only partially observed. Both for the purpose of analyzing such systems as well as forecasting future observations, it is usually necessary to compute trajectories which are on the one hand consistent with some proposed model of the dynamics, but which on the other hand closely follow (or `shadow') the recent history of observations. This process (referred to as data assimilation in the atmospheric sciences or smoothing in the engineering community) is revisited in this contribution.
As currently employed (for example in weather forecasting), variational methods meet with the fundamental difficulty that the corresponding normal equations are ill-posed. For this reason, only very short observation windows can be taken into account. In this contribution, an approach to data assimilation using concepts from nonlinear control theory will be presented[1]. The model dynamics are augmented by a perturbation, which is chosen so as to make the discrepancy between the trajectory and the actual observations, or "tracking error", small. At the same time, large control actions are penalized as well, in order to create trajectories which are as consistent with the (uncontrolled) model dynamics as possible. In the presence of model error, a small but non-vanishing perturbation will remain necessary to keep the trajectory close to the observations. In addition to being an effective means to regularize the problem, the perturbations hold interesting clues about the original dynamics. We demonstrate that an ex-post analysis of the perturbations provides information about degrees of freedom that have not been taken into account by the model. Therefore, the presented approach is not only useful in the context of prediction, but also as a diagnostic tool. References: [1] JB. Data Assimilation in Nonlinear Systems Through Tracking. submitted to Physica D, 2009. |
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