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# The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii

• It has been demonstrated that ensemble mean forecasts, in the context of the sample mean, have higher forecasting skill than deterministic (or single) forecasts. However, few studies have focused on quantifying the relationship between their forecast errors, especially in individual prediction cases. Clarification of the characteristics of deterministic and ensemble mean forecasts from the perspective of attractors of dynamical systems has also rarely been involved. In this paper, two attractor statistics——namely, the global and local attractor radii (GAR and LAR, respectively)——are applied to reveal the relationship between deterministic and ensemble mean forecast errors. The practical forecast experiments are implemented in a perfect model scenario with the Lorenz96 model as the numerical results for verification. The sample mean errors of deterministic and ensemble mean forecasts can be expressed by GAR and LAR, respectively, and their ratio is found to approach $\sqrt{2}$ with lead time. Meanwhile, the LAR can provide the expected ratio of the ensemble mean and deterministic forecast errors in individual cases.
摘要: 前人研究表明集合平均预报在大样本平均的情况下比确定性（或单一）预报有更高的预报技巧。然而，很少研究关注它们预报误差之间的定量关系，尤其在一些个例预报中。同时，从动力系统吸引子的角度对确定性和集合平均预报的特征进行的研究也很少。本文利用吸引子的两个统计量即全局和局部吸引子半径来揭示确定性和集合平均预报误差的关系。基于Lorenz96模型的完美模式情景下的实际预报试验结果用来作为理论的检验。确定性预报和集合平均预报的样本平均误差可以分别用全局和局部吸引子半径来表达，它们的比值随着预报时间接近$\sqrt{2}$。同时，局部吸引子半径提供了确定性和集合平均预报误差在不同个例中的期望比值。
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Cambridge University Press, 230 pp.10.1198/tech.2005.s32624ad4992e47c8e246389c1c272eda9b9http%3A%2F%2Famstat.tandfonline.com%2Fdoi%2Fpdf%2F10.1198%2Ftech.2005.s326http://amstat.tandfonline.com/doi/pdf/10.1198/tech.2005.s326This comprehensive text and reference work on numerical weather prediction covers for the first time, not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a...This comprehensive text and reference work on numerical weather prediction covers for the first time, not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. 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## Manuscript History

Manuscript revised: 07 September 2018
Manuscript accepted: 10 October 2018
###### 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

## The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii

###### Corresponding author: Jianping LI, ljp@bnu.edu.cn;
• 1. School of Meteorology, University of Oklahoma, Norman, OK 73072, USA
• 2. College of Global Change and Earth System Science (GCESS), Beijing Normal University, Beijing 100875, China
• 3. Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266237, China
• 4. Cooperative Institute for Research in the Atmosphere, GSD/ESRL/OAR/NOAA, Boulder, CO 80305, USA
• 5. Fujian Meteorological Observatory, Fuzhou 350001, China
• 6. Wuyishan National Park Meteorological Observatory, Wuyishan 354306, China
• 7. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
• 8. Plateau Atmosphere and Environment Key Laboratory of Sichuan Province, Chengdu University of Information Technology, Chengdu 610225, China

Abstract: It has been demonstrated that ensemble mean forecasts, in the context of the sample mean, have higher forecasting skill than deterministic (or single) forecasts. However, few studies have focused on quantifying the relationship between their forecast errors, especially in individual prediction cases. Clarification of the characteristics of deterministic and ensemble mean forecasts from the perspective of attractors of dynamical systems has also rarely been involved. In this paper, two attractor statistics——namely, the global and local attractor radii (GAR and LAR, respectively)——are applied to reveal the relationship between deterministic and ensemble mean forecast errors. The practical forecast experiments are implemented in a perfect model scenario with the Lorenz96 model as the numerical results for verification. The sample mean errors of deterministic and ensemble mean forecasts can be expressed by GAR and LAR, respectively, and their ratio is found to approach $\sqrt{2}$ with lead time. Meanwhile, the LAR can provide the expected ratio of the ensemble mean and deterministic forecast errors in individual cases.

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