﻿ 对流层延迟对GNSS单点定位影响的全球评估
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 大地测量与地球动力学  2020, Vol. 40 Issue (11): 1194-1199  DOI: 10.14075/j.jgg.2020.11.017

### 引用本文

WANG Jinfang, YANG Ling. Global Assessment on GNSS Single Point Positioning Accuracy Impacted by Tropospheric Delay[J]. Journal of Geodesy and Geodynamics, 2020, 40(11): 1194-1199.

### Foundation support

National Natural Science Foundation of China, No.41504022;Open Fund of State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, No.19R02.

### Corresponding author

YANG Ling, associate professor, majors in GNSS spatial error processing and integrity theory, E-mail: lingyang@tongji.edu.cn.

### About the first author

WANG Jinfang, postgraduate, majors in GNSS data processing, E-mail: jinfang_w@foxmail.com.

### 文章历史

1. 同济大学测绘与地理信息学院，上海市四平路1239号，200092

1 对流层天顶延迟的全球时空变化规律 1.1 数据来源

1.2 统计特性

 图 1 IGS测站分布及ZPD均值 Fig. 1 Distribution of IGS stations and ZPD mean

 图 2 ZPD极差统计和频数分布 Fig. 2 ZPD range statistics and frequency distribution

 图 3 ZPD均值随纬度变化 Fig. 3 ZPD mean values vary with latitude
2 对流层延迟对GNSS单点定位的影响公式

2.1 GNSS基本观测方程

GNSS伪距和载波相位观测方程通常如下：

 $\left\{ \begin{array}{l} P_{r, f}^s = \rho _{r, f}^s + {t_r} - {t^s} + \alpha _r^s{T_z} + {\beta _f}\boldsymbol{I}_r^s + b_{r, f}^s + {\varepsilon _P}\\ \mathit{\Phi} _{r, f}^s = \rho _{r, f}^s + {t_r} - {t^s} + \alpha _r^s{T_z} - {\beta _f}\boldsymbol{I}_r^s + b_{r, f}^sN_{r, f}^s + {\varepsilon _\mathit{\Phi} } \end{array} \right.$ (1)

2.2 对流层延迟对单点定位模型参数估计的影响

 $\left\{ \begin{array}{l} \boldsymbol{l} = \boldsymbol{Ax} + \boldsymbol{e}\\ {\mathop{\rm var}} \left( \boldsymbol{e} \right) = {\boldsymbol{D_{ee}}} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {\boldsymbol{l}_0} = \left\langle {P_{r, f}^s - \rho _{r, 0}^s} \right\rangle \\ {\boldsymbol{l}_1} = \left\langle {P_{r, f}^s - \rho _{r, 0}^s - T_r^s} \right\rangle \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} {{\boldsymbol{\hat x}}_i} = {\boldsymbol{D}_{{{\boldsymbol{\hat x}}_i}{{\boldsymbol{\hat x}}_i}}}{\boldsymbol{A}^{\rm{T}}}\boldsymbol{D_{ee}}^{ - 1}{\boldsymbol{l}_i}\\ {\mathop{\rm var}} \left( {{{\boldsymbol{\hat x}}_i}} \right) = {\boldsymbol{D}_{{{\boldsymbol{\hat x}}_i}{{\boldsymbol{\hat x}}_i}}} = {\left( {{\boldsymbol{A}^{\rm{T}}}\boldsymbol{D_{ee}}^{ - 1}\boldsymbol{A}} \right)^{ - 1}}, i = 0, 1 \end{array} \right.$ (4)

 $\left\{ \begin{array}{l} {{\boldsymbol{\hat e}}_i} = \boldsymbol{R}{\boldsymbol{l}_i}\\ \boldsymbol{R} = \boldsymbol{l} - \boldsymbol{A}{\boldsymbol{D}_{{{\boldsymbol{\hat x}}_i}{{\boldsymbol{\hat x}}_i}}}{\boldsymbol{A}^{\rm{T}}}\boldsymbol{D_{ee}}^{ - 1}\\ {\mathop{\rm var}} \left( {{{\boldsymbol{\hat e}}_i}} \right) = {\boldsymbol{D}_{{{\boldsymbol{\hat e}}_i}{{\boldsymbol{\hat e}}_i}}} - {\boldsymbol{D_{ee}}} - \boldsymbol{A}{\boldsymbol{D}_{{{\boldsymbol{\hat x}}_i}{{\boldsymbol{\hat x}}_i}}}{\boldsymbol{A}^{\rm{T}}}, i = 0, 1 \end{array} \right.$ (5)

 $\left\{ \begin{array}{l} \Delta {{\boldsymbol{\hat x}}_1} = {{\boldsymbol{\hat x}}_0} - {{\boldsymbol{\hat x}}_1} = {\boldsymbol{D}_{{{\boldsymbol{\hat x}}_i}{{\boldsymbol{\hat x}}_i}}}{\boldsymbol{A}^{\rm{T}}}\boldsymbol{D_{ee}}^{ - 1}\boldsymbol{T}\\ \Delta {{\boldsymbol{\hat e}}_1} = {{\boldsymbol{\hat e}}_0} - {{\boldsymbol{\hat e}}_1} = \boldsymbol{RT} \end{array} \right.$ (6)

3 实验分析

 图 4 春分日各测站N、E、U三个方向的均值及STD值 Fig. 4 Mean values and STD values of each station in N, E, U directions on the spring equinox

 图 5 夏至日各测站N、E、U三个方向的均值及STD值 Fig. 5 Mean values and STD values of each station in N, E, U> directions on the summer solstice

 图 6 秋分日各测站N、E、U三个方向的均值及STD值 Fig. 6 Mean values and STD values of each station in N, E, U directions on the autumnal equinox

 图 7 冬至日各测站N、E、U三个方向的均值均值及STD值 Fig. 7 Mean values and STD values of each station in N, E, U directions on the winter solstice

4 结语

1) 全球ZPD统计结果。在全球范围内ZPD的空间分布特征主要与纬度相关，其大小随纬度增加而减小，浮动范围为2.2~2.7 m，均值约为2.4 m；ZPD空间分布沿赤道不完全对称，在北半球的离散度略大；在北纬地区，夏季和秋季的ZPD大于冬季和春季。

2) 对流层延迟引起的定位偏差的统计结果。对流层延迟引起的单点定位偏差在U方向最大，达7~15 m；E方向最小，在±0.2 m以内；N方向的影响居中，在±0.6 m以内。

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Global Assessment on GNSS Single Point Positioning Accuracy Impacted by Tropospheric Delay
WANG Jinfang1     YANG Ling1
1. College of Surveying and Geo-Informatics, Tongji University, 1239 Siping Road, Shanghai 200092, China
Abstract: In order to study the uncertainty of single point positioning deviation caused by the spatiotemporal difference of tropospheric delays, firstly, we use the IGS ZPD product to analyze the correlation between the maximum, minimum, mean, STD and spatial distribution of the station. The results show that the ZPD average is around 2.4 m. There is an overall tendency to decrease with increasing latitude. However, it is not completely symmetrical along with the equator, and the dispersion is slightly larger in the northern hemisphere. Then, based on single point positioning model, the influence formula of tropospheric delay on the calculation of positioning parameters is derived, and its influence on single point positioning is evaluated. The results show that the positioning deviations caused by tropospheric delay are the largest in the U direction, reaching 7 to 15 m; the most insignificant deviations are shown in the E direction, within ±0.2 m; and the deviations in the N direction are centered within ±0.6 m.
Key words: tropospheric delay; ZPD; single point positioning; accuracy analysis