﻿ 基于不等式约束算法改善高纬度地区GNSS电离层建模精度
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 大地测量与地球动力学  2019, Vol. 39 Issue (7): 722-727  DOI: 10.14075/j.jgg.2019.07.011

### 引用本文

HUANG Xiaodong, TU Rui, LIU Jinhai, et al. Inequality Constraints Algorithm to Improve the High Latitudes GNSS Ionosphere Modeling Accuracy[J]. Journal of Geodesy and Geodynamics, 2019, 39(7): 722-727.

### Foundation support

National Key Research and Development Program of China, No.2016YFB0501804; Open Fund for State Key Laboratories of Geo-Information Engineering, No.SKLGIE2016-M-2-4; National Natural Science Foundation of China, No.41504006, 41674034;One Hundred Person Project of CAS, No.Y620YC1701;Key Research and Development Program of Frontier Science and Technology, No.QYZDB-SSW-DQC028.

### 第一作者简介

HUANG Xiaodong, postgraduate, majors in ionosphere modeling based on GNSS, E-mail:ntschxd@163.com.

### 文章历史

1. 中国科学院国家授时中心，西安市书院东路3号，710600;
2. 中国科学院大学，北京市玉泉路19号(甲)，100049;
3. 中国科学院精密导航定位与定时技术重点实验室，西安市书院东路3号，710600;
4. 地理信息工程国家重点实验室，西安市雁塔路中段1号，710054

1 基于GPS数据建立全球VTEC模型

 $\begin{array}{*{20}{c}} {\tilde P_{2j}^i - \tilde P_{1j}^i = 40.28 \times \frac{{{\rm{TEC}}}}{{f_2^2}} - }\\ {40.28 \times \frac{{{\rm{TEC}}}}{{f_1^2}} + {\rm{d}}{q_j} + {\rm{d}}{q^i}} \end{array}$ (1)

2 附不等式约束的最小二乘算法电离层建模

 $\begin{array}{*{20}{c}} {\sum\limits_{n = 0}^{{n_{\max }}} {\sum\limits_{m = 0}^n {{{\tilde P}_{nm}}\left( {\sin \varphi } \right)\left( {{{\tilde A}_{nm}}\cos \left( {m\mu } \right) + } \right.} } }\\ {\left. {{{\tilde B}_{nm}}\sin \left( {m\mu } \right)} \right) \ge {W_x}} \end{array}$ (5)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{E}} = \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{N}}^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\\ \mathit{\boldsymbol{c}} = \mathit{\boldsymbol{G}}{{\mathit{\boldsymbol{\hat X}}}_0} - \mathit{\boldsymbol{W}} \end{array} \right.$ (6)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{h}} = \mathit{\boldsymbol{E\mu }} + \mathit{\boldsymbol{c}}\\ {\mathit{\boldsymbol{\mu }}^{\rm{T}}}\mathit{\boldsymbol{h}} = 0,\mu ,h \ge 0 \end{array} \right.$ (7)

 ${\mathit{\boldsymbol{\hat X}}_{{\rm{ICLS}}}} = {\mathit{\boldsymbol{\hat X}}_{\rm{0}}} + {\mathit{\boldsymbol{N}}^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{\hat \mu }}$ (8)

 图 1 不等式约束最小二乘算法电离层建模流程 Fig. 1 Flow chart of ICLS ionospheric modeling
3 实验及结果分析

 图 2 2018-01-05 UT11：30~12：00穿刺点分布 Fig. 2 IPP distribution during UT 11:30-12:00 on Jan.5 2018

 $\sum\limits_{i = 1}^N {{\rm{DC}}{{\rm{B}}^i}} = 0$ (9)

 图 3 2018-01-05 UT12：00 NTSCG(LS)全球VTEC分布 Fig. 3 Global VTEC distribution of NTSCG(LS) at UT12:00 on Jan.5 2018

 图 4 2018-01-05 UT12：00 NTSCG(ICLS)全球VTEC分布 Fig. 4 Global VTEC distribution of NTSCG(ICLS) at 12:00 on Jan.5 2018

 图 5 2018-01-05 UT12:00 NTSCG(LS)与CODG全球VTEC差异分布 Fig. 5 Global VTEC of NTSCG(LS)-CODG residual map at UT12:00 on Jan.5 2018

 图 6 2018-01-05 UT12:00 NTSCG(ICLS)与CODG全球VTEC差异分布 Fig. 6 Global VTEC of NTSCG(ICLS)-CODG residual map at UT12:00 on Jan.5 2018

 $\left\{ \begin{array}{l} {\rm{mean}} = \frac{1}{N}\sum\limits_{i = 1}^N {\left( {{\rm{VTE}}{{\rm{C}}_k} - {\rm{VTE}}{{\rm{C}}_{{\rm{CODE}}}}} \right)} \\ {\rm{RMS}} = \sqrt {\frac{{\sum\limits_{i = 1}^N {{{\left( {{\rm{VTE}}{{\rm{C}}_k} - {\rm{VTE}}{{\rm{C}}_{{\rm{CODE}}}}} \right)}^2}} }}{N}} \end{array} \right.$ (10)

 图 7 NTSCG与CODG之间全球TEC差异RMS按纬度统计结果 Fig. 7 Results of global TEC residual of NTSCG and CODG based on latitude

4 结语

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Inequality Constraints Algorithm to Improve the High Latitudes GNSS Ionosphere Modeling Accuracy
HUANG Xiaodong1,2,3     TU Rui1,2,3,4     LIU Jinhai1,2,3     ZHANG Rui1,3     LU Xiaochun1,3
1. National Time Service Center, CAS, 3 East-Shuyuan Road, Xi'an 710600, China;
2. University of Chinese Academy of Sciences, A19 Yuquan Road, Beijing 100049, China;
3. Key Laboratory of Precision Navigation and Timing Technology, CAS, 3 East-Shuyuan Road, Xi'an 710600, China;
4. State Key Laboratory of Geo-Information Engineering, 1 Mid-Yanta Road, Xi'an 710054, China
Abstract: Due to uneven distribution of the international GNSS service (IGS) organization tracking stations in high latitudes, the observation data used for ionospheric fitting modeling are uneven and incomplete, leading to inadequate modeling accuracy of the total ionospheric content in this region. When the ionospheric fit model is established using observation data from these IGS tracking stations, the grid output values of the ionospheric fitting model contain a large number of negative and zero values, which are contrary to the actual physical meaning of the ionosphere. Aiming at this problem, we adopt the ionospheric grid output negative point, zero point to join inequality constraint conditions, and the inequality constraint least square method, to calculate parameters of optimization. Experimental results show that the algorithm shows a significant improvement in reducing the large number of zeros and negative values in the sparse high-latitude areas, and modeling accuracy of the global ionosphere model is also improved.
Key words: IGS; global ionosphere model; TEC; high-latitude areas; inequality constraint