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# Predictability of Ensemble Forecasting Estimated Using the Kullback-Leibler Divergence in the Lorenz Model

• A new method to quantify the predictability limit of ensemble forecasting is presented using the Kullback-Leibler (KL) divergence (also called the relative entropy), which provides a measure of the difference between the probability distributions of ensemble forecasts and local reference (true) states. The KL divergence is applicable to a non-normal distribution of ensemble forecasts, which is a substantial improvement over the previous method using the ensemble spread. An example from the three-variable Lorenz model illustrates the effectiveness of the KL divergence, which can effectively quantify the predictability limit of ensemble forecasting. On this basis, the KL divergence is used to investigate the dependence of the predictability limit of ensemble forecasting on the initial states and the magnitude of initial errors. The local predictability limit of ensemble forecasting varies considerably with the initial states, as well as with the magnitude of initial errors. Further research is needed to examine the real-world applications of the KL divergence in measuring the predictability of ensemble weather forecasts.
摘要: Kullback–Leibler（KL）散度（相对熵）可以定量表征集合预报和局部参考状态的概率分布之差。本文提出了利用KL散度来定量估计集合预报可预报期限的新方法。KL散度方法不但适用于概率分布呈现正态分布的集合预报，而且适用于概率分布呈现非正态分布的集合预报，比传统方法具有更广的适用性。将KL散度方法应用于Lorenz模型中，研究表明它可以有效的定量确定集合预报的可预报期限。在此基础上，利用KL散度方法研究了集合预报的可预报期限对于初始状态和初始误差大小的依赖性。结果表明，集合预报的局部可预报期限会随着初始状态以及初始误差的大小发生较大的变化。在未来研究中，我们将KL散度方法应用于定量估计真实天气集合预报的可预报性。
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## Manuscript History

Manuscript revised: 18 April 2019
Manuscript accepted: 30 April 2019
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Predictability of Ensemble Forecasting Estimated Using the Kullback-Leibler Divergence in the Lorenz Model

###### Corresponding author: Ruiqiang DING, drq@mail.iap.ac.cn;
• 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
• 2. College of Earth Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
• 3. Institute of Space Weather, School of Math and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
• 4. College of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
• 5. College of Global Change and Earth System Sciences, Beijing Normal University, Beijing 100875, China

Abstract: A new method to quantify the predictability limit of ensemble forecasting is presented using the Kullback-Leibler (KL) divergence (also called the relative entropy), which provides a measure of the difference between the probability distributions of ensemble forecasts and local reference (true) states. The KL divergence is applicable to a non-normal distribution of ensemble forecasts, which is a substantial improvement over the previous method using the ensemble spread. An example from the three-variable Lorenz model illustrates the effectiveness of the KL divergence, which can effectively quantify the predictability limit of ensemble forecasting. On this basis, the KL divergence is used to investigate the dependence of the predictability limit of ensemble forecasting on the initial states and the magnitude of initial errors. The local predictability limit of ensemble forecasting varies considerably with the initial states, as well as with the magnitude of initial errors. Further research is needed to examine the real-world applications of the KL divergence in measuring the predictability of ensemble weather forecasts.

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