﻿ 纬度因素对季节性时序TEC短期预报模型的影响分析
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 大地测量与地球动力学  2018, Vol. 38 Issue (6): 581-586  DOI: 10.14075/j.jgg.2018.06.007

### 引用本文

CHEN Yutian, LIU Lilong, LIU Zhongliu, et al. Analysis of the Influence of Latitude Factors on Seasonal Time Series TEC Short-Term Forecast Model[J]. Journal of Geodesy and Geodynamics, 2018, 38(6): 581-586.

### Foundation support

Guangxi Bagui Scholar Special Fund of Post and Innovation; National Natural Science Foundation of China, No.41664002, 41704027; Open Fund of Guangxi Key Laboratory of Spatial Information and Geomatics, No.16-380-25-27; Guangxi University (To Enhance the Ability of Youth Education Hall Project) Scientific Research Project, No.KY2016YB189, 2017KY0267.

### 第一作者简介

CHEN Yutian, postgraduate, majors in GNSS technology and application, E-mail:cyt19943@163.com.

### 文章历史

1. 桂林理工大学测绘地理信息学院，桂林市雁山街319号，541006;
2. 广西空间信息与测绘重点实验室，桂林市雁山街319号，541006

1 季节性时序模型 1.1 SARIMA模型简介

ARMA(p, q)模型将d阶差分后的平稳序列进行拟合，再通过d阶差分还原得到预报值。ARMA(p, q)模型的表达式如下：

 $\begin{array}{l} {x_t} = {\varphi _1}{x_{t-1}} + {\varphi _2}{x_{t-2}} + ... + {\varphi _p}{x_{t-p}} + \\ {\varepsilon _t} + {\theta _1}{\varepsilon _{t - 1}} + {\theta _2}{\varepsilon _{t - 2}} + ... + {\theta _q}{\varepsilon _{t - q}} \end{array}$ (1)

 $\varphi (B)\mathit{\Phi} ({B^S}){\nabla ^d}\nabla _S^D{y_t} = \theta (B)\Theta ({B^S}){\varepsilon _t}$ (2)

1.2 Holt-Winters指数平滑法

Holt-Winters是一种指数平滑法，该模型有3种形式：无季节模型、加法模型和乘法模型。其中加法模型和乘法模型可以对样本序列的长期趋势、趋势增量以及季节变动作出估计[12]，对季节性趋势较强的数据具有较好的预报能力[13]

Holt-Winters加法模型：

 $\left\{ \begin{array}{l} {S_{\rm{t}}} = \alpha ({X_t}-{I_{t-L}}) + (1-\alpha )({S_{t - 1}} - {b_{t - 1}})\\ {I_t} = \beta ({X_t} - {S_t}) + (1 - \beta ){I_{t - L}}\\ {b_t} = \gamma ({S_t} - {S_{t - 1}}) + (1 - \gamma ){b_{t - 1}}\\ {F_{t + m}} = {S_t} + m{b_t} + {I_{t - L + m}} \end{array} \right.$ (3)

Holt-Winters乘法模型：

 $\left\{ \begin{array}{l} {S_{\rm{t}}} = \alpha (\frac{{{X_t}}}{{{I_{t-L}}}}) + (1-\alpha )({S_{t-1}} - {b_{t - 1}})\\ {I_t} = \beta (\frac{{{X_t}}}{{{S_t}}}) + (1 - \beta ){I_{t - L}}\\ {b_t} = \gamma ({S_t} - {S_{t - 1}}) + (1 - \gamma ){b_{t - 1}}\\ {F_{t + m}} = ({S_t} + m{b_t}){I_{t - L + m}} \end{array} \right.$ (4)

2 实验分析 2.1 实验数据准备

Kp指数用来描述地磁活动的强度，每天共8个值，一般认为日Kp指数之和大于30则地磁活动强烈。从图 1可以看出，预报期内各个时段的Kp值均小于5且每天的Kp指数之和小于25，地磁活动平稳。Dst指数用于表征全球的环电流强度，时间分辨率为1 h，一般当Dst指数低于-30 nT时可能发生小磁暴，处于-30 ~-50 nT时可能发生中等磁暴，处于-100 nT以下时可能发生大磁暴。图 2给出预测期的Dst指数均不低于-30 nT，不可能发生磁暴。综上所述，地磁活动与磁暴对模型预报的影响可以忽略。

 图 1 预报期Kp指数 Fig. 1 Kp index in forecast period

 图 2 预报期Dst指数 Fig. 2 Dst index in forecast period
2.2 预报精度分析

 $P = \frac{1}{n}\sum\limits_{t = 1}^n {(1-\frac{{|{I_{{\rm{pre}}}}-{I_{{\rm{IGS}}}}|}}{{{I_{{\rm{IGS}}}}}} \times 100\% )}$ (5)
 ${\rm{RMSE}} = \sqrt {\frac{1}{n}{{\sum\limits_{t = 1}^n {({I_{{\rm{pre}}}}-{I_{{\rm{IGS}}}})} }^2}}$ (6)

3种模型8个纬度地区TEC值的预报结果与残差分布如图 3所示。从IGS给出的观测值可以看出，TEC的变化幅度随纬度的降低而增大，且低纬度区域的TEC变化周期性更为明显。在任何纬度，3种模型均能较好地反映电离层TEC的变化，且SARIMA模型的预报效果略高于Holt-Winters模型。通过预报值残差可以看出，TEC预报值与观测值的误差在25°N及以下的低纬地区要大于35°N以上的中、高纬地区。

 图 3 不同纬度地区TEC 5 d预报结果及残差 Fig. 3 5 d forecast results and residual of TEC in different latitudes

 图 4 均方根误差随纬度的变化规律 Fig. 4 Variation of root mean square error with latitude
3 结语

1) 从时间上看，3种模型均能较好地反映电离层TEC值的变化趋势，SARIMA模型的预报精度最高，其次为加法模型，乘法模型最低。

2) 通过分析TEC预报结果的日均相对精度和均方根误差发现，北纬25°和55°地区预报精度最低，均方根误差表现为一大一小两个峰值，而在北纬35°~45°区域和65°以上地区的预报精度较高，并且均方根误差随纬度的降低总体呈现出增长趋势。

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Analysis of the Influence of Latitude Factors on Seasonal Time Series TEC Short-Term Forecast Model
CHEN Yutian1,2     LIU Lilong1,2     LIU Zhongliu1,2     HE Chaoshuang1,2
1. College of Geomatic Engineering and Geoinformatics, Guilin University of Technology, 319 Yanshan Road, Guilin 541006, China;
2. Guangxi Key Laboratory of Spatial Information and Geomatics, 319 Yanshan Road, Guilin 541006, China
Abstract: By testing the length of time series, the influence of geomagnetic index on the exclusion of geomagnetic activity and geomagnetic activity is analyzed. For the periodic variation of the TEC value of the ionosphere and its inhomogeneous distribution with latitude, we use the seasonal time series model, SARIMA and the Holt-Winters exponential smoothing model. 48 regions of different latitudes of the northern hemisphere are predicted using the TEC grid data provided by the IGS Center, and the daily mean relative accuracy and root mean square error are defined to assess forecast accuracy. The results show that all three models can reflect the cyclical changes of ionospheric TEC values, but the root mean square error decreases with the latitude and shows an overall trend of growth; at 25° and 55° north latitude it shows a maximum value, while at 45° N, it shows a minimum value.
Key words: geomagnetic index; ionosphere; TEC; SARIMA; Holt-Winters