﻿ 附加先验信息约束的BDS-3精密授时算法
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 大地测量与地球动力学  2024, Vol. 44 Issue (6): 566-571  DOI: 10.14075/j.jgg.2023.09.197

### 引用本文

AI Qingsong, ZHANG Bin, XU Li, et al. BDS-3 Precision Timing Algorithm with Additional Prior Information Constraints[J]. Journal of Geodesy and Geodynamics, 2024, 44(6): 566-571.

### Foundation support

Natural Science Foundation of Hubei Province, No.2023AFB435; Postdoctoral Innovation Research Project of Hubei Province, No.R22R6201; Open Research Fund of State Key Laboratory of Geodesy and Earth's Dynamics, Innovation Academy for Precision Measurement Science and Technology, CAS, No.SKLGED2024-3-3; National Natural Science Foundation of China, No.42304038.

### 第一作者简介

AI Qingsong, PhD, engineer, majors in GNSS satellite clock offset estimation and precise point positioning algorithm, E-mail: aiqingsong@asch.whigg.ac.cn.

### 文章历史

1. 长江空间信息技术工程有限公司(武汉)，武汉市解放大道1863号，430010;
2. 长江设计集团有限公司，武汉市解放大道1863号，430010;
3. 广州市城市规划勘测设计研究院，广州市建设大马路10号，510000

1 基于B1C/B2b组合观测值的PPP授时模型构建

1.1 PPP授时函数模型构建

GNSS原始非组合PPP伪距和载波相位观测方程可表示为[6-7]

 $\left\{\begin{array}{l} E\left[\Delta P_{\mathrm{r}, f}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\mathrm{d} t_{\mathrm{r}}(i)-\mathrm{d} t^{\mathrm{s}}(i)+\mu_{f} I_{\mathrm{r}, 1}^{\mathrm{s}}(i)+b_{\mathrm{r}, f}-b_{f}^{\mathrm{s}} \\ E\left[\Delta L_{\mathrm{r}, f}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\mathrm{d} t_{\mathrm{r}}(i)-\mathrm{d} t^{\mathrm{s}}(i)-\mu_{f} I_{\mathrm{r}, 1}^{\mathrm{s}}(i)+N_{\mathrm{r}, f}^{\mathrm{s}} \end{array}\right.$ (1)

 $\left\{\begin{array}{l} E\left[\Delta P_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+[\tilde{\mathrm{d}}t_{\mathrm{r}}(i)+\underbrace{\alpha_{12} b_{\mathrm{r}, 1}}_{\tilde{\mathrm{d}} t_{\mathrm{r}, i f 12}{(i)}}+\beta_{12} b_{\mathrm{r}, 2}]-[\mathrm{d} t^{\mathrm{s}}(i)+\underbrace{\alpha_{12} b_{1}^{\mathrm{s}}+\beta_{12} b_{\mathrm{s}}^{\mathrm{s}}}_{\tilde{\mathrm{d}}t^{\mathrm{s}}(i)}] \\ E\left[\Delta P_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\underbrace{\left[{\mathrm{d}}t_{\mathrm{r}}(i)+b_{\mathrm{r}, i f 12}\right]}_{\tilde{\mathrm{d}}t_{\mathrm{r}, i f 12}(i)}-\underbrace{\left[\mathrm{d} t^{\mathrm{s}}(i)+b_{if12}^{\mathrm{s}}\right]}_{\tilde{\mathrm{d}}t^{\mathrm{s}}(i)} \end{array}\right.$ (2)
 $\left\{\begin{array}{l} E\left[\Delta L_{\mathrm{r}, f}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\mathrm{d} t_{\mathrm{r}}(i)-\mathrm{d} t^{\mathrm{s}}(i)-\mu_f I_{\mathrm{r}, 1}^{\mathrm{s}}(i)+N_{\mathrm{r}, f}^{\mathrm{s}} \\ E\left[\Delta L_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\mathrm{d} t_{\mathrm{r}}(i)+b_{i f 12}^{\mathrm{s}}-\mathrm{d} t^{\mathrm{s}}(i)-b_{\mathrm{r}, i f 12}+\alpha_{12} N_{\mathrm{r}, 1}^{\mathrm{s}}+\beta_{12} N_{\mathrm{r}, 2}^{\mathrm{s}} \\ E\left[\Delta L_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)+\tilde{\mathrm{d}} t^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\tilde{\mathrm{d}} t_{\mathrm{r}, i f 12}(i)+\underbrace{b_{i f 12}^{\mathrm{s}}-b_{\mathrm{r}, i f 12}+\alpha_{12} N_{\mathrm{r}, 1}^{\mathrm{s}}+\beta_{12} N_{\mathrm{r}, 2}^{\mathrm{s}}}_{\tilde{N}_{\mathrm{r}, i f 12}^{\mathrm{s}}} \end{array}\right.$ (3)

 $\left\{\begin{array}{l} E\left[P_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)+\tilde{\mathrm{d}} t^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\tilde{\mathrm{d}} t_{\mathrm{r}, i f 12}(i) \\ E\left[L_{\mathrm{r}, i f 12}^{\mathrm{s}}(i)+\tilde{\mathrm{d}} t^{\mathrm{s}}(i)\right]=e_{\mathrm{r}}^{\mathrm{s}} x+m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\tilde{\mathrm{d}} t_{\mathrm{r}, i f 12}(i)+\widetilde{N}_{\mathrm{r}, i f 12}^{\mathrm{s}} \end{array}\right.$ (4)

 $\left\{\begin{array}{l} \tilde{\mathrm{d}} t_{\mathrm{r}, i f 12}(i)=\mathrm{d} t_{\mathrm{r}}(i)+b_{\mathrm{r}, i f 12}, \tilde{\mathrm{d}} t^{\mathrm{s}}(i)=\mathrm{d} t^{\mathrm{s}}(i)+b_{i f 12}^{\mathrm{s}} \\ b_{\mathrm{r}, i f 12}=\alpha_{12} b_{\mathrm{r}, 1}+\beta_{12} b_{\mathrm{r}, 2}, b_{i f 12}^{\mathrm{s}}=\alpha_{12} b_{1}^{\mathrm{s}}+\beta_{12} b_{2}^{\mathrm{s}} \\ \widetilde{N}_{\mathrm{r}, i f 12}^{\mathrm{s}}=\alpha_{12} N_{\mathrm{r}, 1}^{\mathrm{s}}+\beta_{12} N_{\mathrm{r}, 2}^{\mathrm{s}}-b_{\mathrm{r}, i f 12}+b_{i f 12}^{\mathrm{s}}, \alpha_{12}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}, \beta_{12}=\frac{-f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}} \end{array}\right.$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {E\left[ {\Delta P_{{\rm{r}}, if34}^{\rm{s}}(i)} \right] = e_{\rm{r}}^{\rm{s}}x + m_{\rm{r}}^{\rm{s}}(i){T_{\rm{r}}}(i) + {\rm{d}}{t_{\rm{r}}}(i) - {\rm{d}}{t^{\rm{s}}}(i) + {\alpha _{34}}\left( {{b_{{\rm{r}}, 3}} - b_3^{\rm{s}}} \right) + {\beta _{34}}\left( {{b_{{\rm{r}}, 4}} - b_4^{\rm{s}}} \right)}\\ {E\left[ {\Delta P_{{\rm{r}}, if34}^{\rm{s}}(i) + \widetilde {\rm{d}}{t^{\rm{s}}}(i)} \right] = e_{\rm{r}}^{\rm{s}}x + m_{\rm{r}}^{\rm{s}}(i) \cdot {T_{\rm{r}}}(i) + }\\ {\underbrace {\left[ {{\rm{d}}{t_{\rm{r}}}(i) + {\alpha _{34}}{b_{{\rm{r}}, 3}} + {\beta _{34}}{b_{{\rm{r}}, 4}}} \right]}_{{\rm{\tilde d}}{t_{{\rm{r}}, if13}}(i)} + \underbrace {\left( {{\alpha _{12}}b_1^{\rm{s}} - {\alpha _{34}}b_3^{\rm{s}}} \right) + \left( {{\beta _{12}}b_2^{\rm{s}} - {\beta _{34}}b_4^{\rm{s}}} \right)}_{\varOmega _{if12, 34}^{\rm{s}}}} \end{array}} \right.$ (6)
 $\left\{ \begin{array}{l} E\left[ {\Delta L_{{\rm{r}}, if34}^{\rm{s}}(i)} \right] = e_{\rm{r}}^{\rm{s}}x + m_{\rm{r}}^{\rm{s}}(i){T_{\rm{r}}}(i) + {\rm{d}}{t_{\rm{r}}}(i) - {\rm{d}}{t^{\rm{s}}}(i) + {\alpha _{34}}N_{{\rm{r}}, 3}^{\rm{s}} + {\beta _{34}}N_{{\rm{r}}, 4}^{\rm{s}}\\ E\left[ {\Delta L_{{\rm{r}}, if34}^{\rm{s}}(i) + \widetilde {\rm{d}}{t^{\rm{s}}}(i)} \right] = e_{\rm{r}}^{\rm{s}}x + m_{\rm{r}}^{\rm{s}}(i){T_{\rm{r}}}(i) + \widetilde {\rm{d}}{t_{{\rm{r}}, if34}}(i) + \underbrace {{\alpha _{34}}N_{{\rm{r}}, 3}^{\rm{s}} + {\beta _{34}}N_{{\rm{r}}, 4}^{\rm{s}} - {b_{{\rm{r}}, if34}} + b_{if12}^{\rm{s}}}_{\tilde N_{{\rm{r}}, if{\rm{ }}34}^{\rm{s}}} \end{array} \right.$ (7)

 $\left\{\begin{array}{l} E\left[\Delta P_{\mathrm{r}, i f 34}^{\mathrm{s}}(i)+\tilde{\mathrm{d}} t^{s}(i)\right]= \\ \quad m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\tilde{\mathrm{d}} t_{\mathrm{r}, i j 34}(i)+\varOmega_{i f 12, 34}^{\mathrm{s}} \\ E\left[\Delta L_{\mathrm{r}, i f 34}^{\mathrm{s}}(i)+\tilde{\mathrm{d}} t^{\mathrm{s}}(i)\right]= \\ \quad m_{\mathrm{r}}^{\mathrm{s}}(i) T_{\mathrm{r}}(i)+\tilde{\mathrm{d}} t_{\mathrm{r}, i f 34}(i)+\widetilde{N}_{\mathrm{r}, i f 34}^{\mathrm{s}} \end{array}\right.$ (8)

 \begin{aligned} & \varOmega_{i f 12, 34}^{\mathrm{s}}=\left(\alpha_{12} b_{1}^{\mathrm{s}}-\alpha_{34} b_{3}^{\mathrm{s}}\right)+\left(\beta_{12} b_{2}^{\mathrm{s}}-\beta_{34} b_{4}^{\mathrm{s}}\right)= \\ & \quad\left(\mathrm{DCB}_{12}^{\mathrm{s}}-\mathrm{DCB}_{23}^{\mathrm{s}}\right)-\beta_{12} \mathrm{DCB}_{12}^{\mathrm{s}}-\beta_{34} \mathrm{DCB}_{34}^{\mathrm{s}} \end{aligned} (9)

 $\left\{\begin{array}{l} \tilde{\mathrm{d}} t_{\mathrm{r}, i f 34}(i)=\mathrm{d} t_{\mathrm{r}}(i)+b_{\mathrm{r}, i f 34} \\ b_{\mathrm{r}, i f 34}=\alpha_{34} b_{\mathrm{r}, 3}+\beta_{34} b_{\mathrm{r}, 4} \\ \widetilde{N}_{\mathrm{r}, i f 34}^{\mathrm{s}}=\alpha_{34} N_{\mathrm{r}, 3}^{\mathrm{s}}+\beta_{34} N_{\mathrm{r}, 4}^{\mathrm{s}}-b_{\mathrm{r}, i f 34}+b_{i f 12}^{\mathrm{s}} \\ \alpha_{34}=\frac{f_{3}^{2}}{f_{3}^{2}-f_{4}^{2}}, \beta_{34}=-\frac{f_{4}^{2}}{f_{3}^{2}-f_{4}^{2}} \\ \varOmega_{i f 12, 34}^{\mathrm{s}}=\left(\mathrm{DCB}_{12}^{\mathrm{s}}-\mathrm{DCB}_{23}^{\mathrm{s}}\right)- \\ \quad \beta_{12} \mathrm{DCB}_{12}^{\mathrm{s}}-\beta_{34} \mathrm{DCB}_{34}^{\mathrm{s}} \end{array}\right.$ (10)
1.2 附加先验信息模型的PPP授时模型构建

 $\delta_{\text {Allan }}^{2}=3 q_{0} \tau^{-2}+q_{1} \tau^{-1}+\frac{q_{2} \tau}{3}+\frac{1}{20} q_{3} \tau^{3}$ (11)
 $\sigma_{w}^{2}=\left(\delta_{\text {Allan }} \tau c\right)^{2}$ (12)

 $\left\{\begin{array}{l} \tilde{\mathrm{d}} t_{k, k-1}=\varPhi_{k, k-1} \tilde{\mathrm{d}} t_{k-1} \\ D_{\tilde{\mathrm{d}} t_{k, k-1}}=\varPhi_{k, k-1} D_{\tilde{\mathrm{d}} t_{k-1}} \varPhi_{k, k-1}^{\mathrm{T}}+D_{w_{k-1}} \end{array}\right.$ (13)

 $D_{\tilde{\mathrm{d}} t_{k, k-1}}=D_{\tilde{\mathrm{d}} t_{k-1}}+\sigma_{w}^{2}$ (14)

2 实验结果与分析

2.1 BDS的B2b观测信息分析

 图 1 接收B2b信号的MGEX站点分布 Fig. 1 Distribution of MGEX stations track B2b signals

 图 2 全球范围内观测到B2b信号的可视卫星数 Fig. 2 The amount of visible satellites track B2b signals on a global scale
2.2 先验信息确定

 图 3 接收B2b信号的氢钟站点对应的站钟稳定性 Fig. 3 Stability of station clocks with hydrogen clock track B2b signals

2.3 BDS-3精密定位和精密授时性能测试

 图 4 BDS-3不同组合观测值精密定位精度统计 Fig. 4 Accuracy of BDS-3 precise positioning with different combination observations

 图 5 基于不同站钟处理模式的单站授时性能 Fig. 5 The timing performance based on different clock processing modes

 图 6 基于spt0站钟处理模式的接收机钟差和历元间差分序列 Fig. 6 Receiver clock bias and epoch differential series based on spt0 station clock processing modes
3 结语

 [1] 张小红, 程世来, 李星星, 等. 单站GPS载波平滑伪距精密授时研究[J]. 武汉大学学报: 信息科学版, 2009, 34(4): 463-465 (Zhang Xiaohong, Cheng Shilai, Li Xingxing, et al. Precise Timing Using Carrier Phase Smoothed Pseudorange from Single Receiver[J]. Geomatics and Information Science of Wuhan University, 2009, 34(4): 463-465) (0) [2] Tu R, Zhang P F, Zhang R, et al. Modeling and Performance Analysis of Precise Time Transfer Based on BDS Triple-Frequency Un-Combined Observations[J]. Journal of Geodesy, 2019, 93(6): 837-847 DOI:10.1007/s00190-018-1206-3 (0) [3] 葛玉龙. 多频多系统精密单点定位时间传递方法研究[D]. 西安: 中国科学院国家授时中心, 2020 (Ge Yulong. Research on Methodology of Multi-Frequency and Multi-GNSS Precise Point Positioning Time Transfer[D]. Xi'an: National Time Service Center, CAS, 2020) (0) [4] 张鹏飞. GNSS载波相位时间传递关键技术与方法研究[D]. 西安: 中国科学院国家授时中心, 2019 (Zhang Pengfei. The Research of Key Technology and Approach for Time and Frequency Transfer Based on GNSS Carrier Phase Observation[D]. Xi'an: National Time Service Center, CAS, 2019) (0) [5] 艾青松. GNSS卫星钟差估计与精密单点定位时间传递方法研究[D]. 武汉: 中国科学院精密测量科学与技术创新研宄院, 2021 (Ai Qingsong. Research on Methodology of GNSS Satellite Clock Offset Estimation and Precise Point Positioning Time Transfer[D]. Wuhan: Innovation Academy for Precision Measurement Science and Technology, CAS, 2021) (0) [6] 章红平, 高周正, 牛小骥, 等. GPS非差非组合精密单点定位算法研究[J]. 武汉大学学报: 信息科学版, 2013, 38(12): 1 396-1 399 (Zhang Hongping, Gao Zhouzheng, Niu Xiaoji, et al. Research on GPS Precise Point Positioning with Un-Differential and Un-Combined Observations[J]. Geomatics and Information Science of Wuhan University, 2013, 38(12): 1 396-1 399) (0) [7] 张小红, 左翔, 李盼. 非组合与组合PPP模型比较及定位性能分析[J]. 武汉大学学报: 信息科学版, 2013, 38(5): 561-565 (Zhang Xiaohong, Zuo Xiang, Li Pan. Mathematic Model and Performance Comparison between Ionosphere-Free Combined and Uncombined Precise Point Positioning[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5): 561-565) (0) [8] Wang K, Rothacher M. Stochastic Modeling of High-Stability Ground Clocks in GPS Analysis[J]. Journal of Geodesy, 2013, 87(5): 427-437 DOI:10.1007/s00190-013-0616-5 (0) [9] Ge Y L, Zhou F, Liu T J, et al. Enhancing Real-Time Precise Point Positioning Time and Frequency Transfer with Receiver Clock Modeling[J]. GPS Solutions, 2019, 23(1) (0) [10] Tan B F, Ai Q S, Yuan Y B. Analysis of Precise Orbit Determination of BDS-3 MEO and IGSO Satellites Based on Several Dual-Frequency Measurement Combinations[J]. Remote Sensing, 2022, 14(23) (0)
BDS-3 Precision Timing Algorithm with Additional Prior Information Constraints
AI Qingsong1,2     ZHANG Bin1,2     XU Li1,2     TANG Zhao1,2     ZHA Jiuping3
1. Changjiang Space Information Technology Engineering Co Ltd, 1863 Jiefang Road, Wuhan 430010, China;
2. Changjiang Institute of Survey, Planning, Design and Research Co Ltd, 1863 Jiefang Road, Wuhan 430010, China;
3. Guangzhou Urban Planning and Design Survey Research Institute, 10 Jianshedama Road, 510000
Abstract: Considering the high stability of high-performance hydrogen clocks and the strong correlation between clock offset series epochs, the receiver clock offset with additional prior information constraint is tested for precise point positioning(PPP) timing. The results based on B1C/B2b combination of BDS-3 show that compared to traditional timing models, the precision timing models with additional prior information have higher timing performance, especially for the short-term stability which has an improvement up to 60%. In addition, the PPP accuracy of B1I/B3I and B1C/B2b combinations of BDS-3 is basically equivalent, which can achieve positioning accuracy better than 0.5 cm and about 2 cm in horizontal and vertical direction, respectively. Owing to the better observation data quality of B1C/B2b, the positioning accuracy is improved by 13.8% compared to the B1I/B3I combination.
Key words: precision timing; prior information; precise point positioning; BDS-3; B1C/B2b