﻿ 基于本征正交分解的平流层风场建模与预测<sup>*</sup>
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Modeling and prediction of stratospheric wind field based on proper orthogonal decomposition
LI Kui, DENG Xiaolong, YANG Xixiang, HOU Zhongxi, ZHOU Xin
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
Received: 2017-11-06; Accepted: 2018-04-08; Published online: 2018-04-18 15:40
Foundation item: National Ministries and Commissions Foundation Item of China (GFZX0X0201-1)
Corresponding author. YANG Xixiang.E-mail:nkyangxixiang@163.com
Abstract: The stratospheric wind field has an important influence on the design and trajectory control of the near-space low dynamic aerostat. Focused on the modeling of the stratospheric wind field, this paper proposes a reduced order analysis method of the wind field data based on the proper orthogonal decomposition (POD) method. On this basis, a Fourier model which can predict the stratospheric wind field is proposed in this paper. This paper takes the wind field in Changsha from 2005 to 2009 as an example, uses the proposed POD method and Fourier prediction model to model and predict the wind field, and analyzes the accuracy of Fourier prediction model. The results show that the POD method can be used to model the east-west wind field efficiently and accurately. The Fourier prediction model can be used to predict the east-west wind field accurately and the prediction accuracy is closely related to the regularity of the actual wind field. The more compact the wind field data are, the more obvious the periodicity is, the higher the prediction accuracy is.
Keywords: stratospheric wind field     proper orthogonal decomposition method     reduced order modeling     Fourier prediction model     prediction accuracy

1 基于POD方法的风场建模

POD方法是指从一组庞大的数据中获得一组低维最优基, 而这组低维最优基是一种能够表示原复杂系统的降阶模型。也就是说POD方法是一种降维的方法, 对给定的数据进行最优的低维逼近, 用较小的维数将原物理模型的主要特征表现出来[13]。本文通过POD方法获得风场信息的最优标准正交基, 将风场数据中的每个风场信息投影到标准正交基上, 获得每个风场数据在标准正交基上的投影系数[14]。并通过最优标准正交基与相对应的投影系数计算出一组新的数据, 将原风场数据的主要特征表现出来。

1.1 瞬像矩阵的生成

 (1)

1.2 POD模态的获取

POD模态又称为POD基。POD方法的目的是寻找一组最优的POD基φ1, φ2, …, φr(φi为列向量), 使得数据模型中任意一天特定时刻的风场可以表示为POD基的线性组合[16], 即

 (2)

 (3)

RRL×L, 求取相关矩阵R的非零特征值以及特征向量φi:

 (4)

 (5)

φi(i=1, 2, …, n; nL)是一组标准正交基, 即

 (6)

 (7)

 (8)

1.3 降阶模型误差评估

POD模态表示捕获物理场的主要特征, 截断后的前r阶POD模态较全阶模态所捕获的能量比为

 (9)

 (10)
2 基于Fourier级数的风场预测模型

 (11)

3 实例分析与结果

3.1 风场建模与预测

 图 1 相对模态能量分布 Fig. 1 Relative mode energy distribution

 图 2 采用POD方法重建风场 Fig. 2 Reconstruction of wind field using POD method

 图 3 系数拟合(东西方向) Fig. 3 Coefficient fitting (east-west direction)
 图 4 系数拟合(南北方向) Fig. 4 Coefficient fitting (north-south direction)

 图 5 各种方法下风矢量图的比较 Fig. 5 Comparison of wind vector plots for various methods

 图 6 实际风矢量图与预测风矢量图的对比 Fig. 6 Comparison of actual wind vector with predicted wind vector plots
3.2 Fourier预测精度分析

 图 7 残差分析 Fig. 7 Residual analysis

 图 8 不同高度的风速变化情况 Fig. 8 Change of wind speed at different altitudes
 图 9 不同高度的残差分析 Fig. 9 Residual analysis at different altitudes
4 结论

1) 提出了一种对平流层风场数据进行降阶处理的POD方法, 在POD方法的基础上, 提出了一种可以对平流层风场进行预测的Fourier预测模型。

2) 以长沙地区为例, 选取海拔高度10~30 km的5年风场数据, 采用提出的POD方法与Fourier预测模型对风场进行了建模与预测, 并对Fourier预测精度进行了分析。

3) 研究结果表明:采用POD方法可以对东西方向风场进行高效高精度的降阶建模, 由于南北方向风场变化极其不规则, 不可以采用低阶的POD模型进行建模; 通过Fourier预测模型能对东西方向风场进行准确预测, 预测精度与实际风场随时间变化的规律性有关, 风场数据越紧凑, 周期性越明显, Fourier预测精度越高。

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#### 文章信息

LI Kui, DENG Xiaolong, YANG Xixiang, HOU Zhongxi, ZHOU Xin

Modeling and prediction of stratospheric wind field based on proper orthogonal decomposition

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(9): 2013-2020
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0685