Improving the prediction of complex nonlinear turbulent dynamical systems using nonlinear filter, smoother and backward
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RESEARCH
Improving the prediction of complex nonlinear turbulent dynamical systems using nonlinear filter, smoother and backward sampling techniques Nan Chen * Correspondence:
[email protected] Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Abstract Predicting complex nonlinear turbulent dynamical systems using partial observations is an important topic. Despite the simplicity of the forecast based on the ensemble mean time series, several critical shortcomings in the ensemble mean forecast and using path-wise measurements to quantify the prediction error are illustrated in this article. Then, a new ensemble method is developed for improving the long-range forecast. This new approach utilizes a mixture of the posterior distributions from data assimilation and is more skillful in predicting non-Gaussian statistics and extreme events than the traditional method by simply running the forecast model forward. Next, a systematic framework of improving forecast models is established, aiming at advancing the predictions at all ranges. The starting model in this new framework belongs to a rich class of nonlinear systems with conditional Gaussian structures. These models allow an efficient nonlinear smoother for state estimation using partial observations, which in turn facilitates a rapid parameter estimation based on an expectation–maximization algorithm. Conditioned on the partially observed time series, the nonlinear smoother further advances an efficient backward sampling of the hidden trajectories, the dynamical and statistical characteristics from which allow a systematic quantification of model error through information theory. The sampled trajectories then serve as the recovered observations of the hidden variables that promote the use of general nonlinear data-driven modeling techniques for a further improvement of the forecast model. A low-order model of the layered topographic equations with regime switching and rare events is used as a test example to illustrate this framework. Keywords: Prediction, Data assimilation, Nonlinear smoother, Backward sampling, Information theory, Extreme events
1 Introduction Predicting complex nonlinear turbulent dynamical systems is a notoriously hard topic [7,42,66,67,73,74]. One of the main reasons is that the perfect knowledge of nature is never known in practice. Due to the lack of a complete understanding of physics and other interconnected processes, model error is inevitable in the development of approx-
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imate forecast models. The model bias and model uncertainty are well-known barriers that greatly affect the prediction of various nonlinear and non-Gaussian features including extreme events [28,31,74]. Another practical difficulty is that nature typically involves multiple temporal and spatial scales as well as complex nonlinear interactions between state variables in these different scales. To charac
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