China Satellite Navigation Conference (CSNC 2024) Proceedings: Volume III (Lecture Notes in Electrical Engineering, 1094) 9819969433, 9789819969432

Citation preview

Lecture Notes in Electrical Engineering 1094

Changfeng Yang Jun Xie   Editors

China Satellite Navigation Conference (CSNC 2024) Proceedings Volume III

Lecture Notes in Electrical Engineering

1094

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Napoli, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, München, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, University of Karlsruhe (TH) IAIM, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Dipartimento di Ingegneria dell’Informazione, Sede Scientifica Università degli Studi di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Intelligent Systems Laboratory, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, Department of Mechatronics Engineering, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Intrinsic Innovation, Mountain View, CA, USA Yong Li, College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Subhas Mukhopadhyay, School of Engineering, Macquarie University, NSW, Australia Cun-Zheng Ning, Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Department of Intelligence Science and Technology, Kyoto University, Kyoto, Japan Luca Oneto, Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Genova, Genova, Italy Bijaya Ketan Panigrahi, Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Federica Pascucci, Department di Ingegneria, Università degli Studi Roma Tre, Roma, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany Germano Veiga, FEUP Campus, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Haidian District Beijing, China Walter Zamboni, Department of Computer Engineering, Electrical Engineering and Applied Mathematics, DIEM—Università degli studi di Salerno, Fisciano, Salerno, Italy Junjie James Zhang, Charlotte, NC, USA Kay Chen Tan, Department of Computing, Hong Kong Polytechnic University, Kowloon Tong, Hong Kong

The book series Lecture Notes in Electrical Engineering (LNEE) publishes the latest developments in Electrical Engineering—quickly, informally and in high quality. While original research reported in proceedings and monographs has traditionally formed the core of LNEE, we also encourage authors to submit books devoted to supporting student education and professional training in the various fields and applications areas of electrical engineering. The series cover classical and emerging topics concerning: • • • • • • • • • • • •

Communication Engineering, Information Theory and Networks Electronics Engineering and Microelectronics Signal, Image and Speech Processing Wireless and Mobile Communication Circuits and Systems Energy Systems, Power Electronics and Electrical Machines Electro-optical Engineering Instrumentation Engineering Avionics Engineering Control Systems Internet-of-Things and Cybersecurity Biomedical Devices, MEMS and NEMS

For general information about this book series, comments or suggestions, please contact [email protected]. To submit a proposal or request further information, please contact the Publishing Editor in your country: China Jasmine Dou, Editor ([email protected]) India, Japan, Rest of Asia Swati Meherishi, Editorial Director ([email protected]) Southeast Asia, Australia, New Zealand Ramesh Nath Premnath, Editor ([email protected]) USA, Canada Michael Luby, Senior Editor ([email protected]) All other Countries Leontina Di Cecco, Senior Editor ([email protected]) ** This series is indexed by EI Compendex and Scopus databases. **

Changfeng Yang · Jun Xie Editors

China Satellite Navigation Conference (CSNC 2024) Proceedings Volume III

Editors Changfeng Yang China Satellite Navigation Engineering Centre Beijing, China

Jun Xie China Academy of Space Technology Beijing, China

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-981-99-6943-2 ISBN 978-981-99-6944-9 (eBook) https://doi.org/10.1007/978-981-99-6944-9 © Aerospace Information Research Institute 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Editorial Board

Topic: S01: GNSS Applications Chairman Dangwei Wang, Beijing UniStrong Science and Technology Co., Ltd., Beijing, China Vice-Chairman Shaojun Feng, Qianxun Spatial Intelligence Inc., Shanghai, China Changhui Xu, Chinese Academy of Surveying and Mapping, Beijing, China Caicong Wu, China Agricultural University, Beijing, China Jianping Xing, Shandong University, Jinan, China Jianhua Wei, BeiDou Application and Research Institute Co., Ltd. of Norinco Group, Beijing, China

Topic: S02: GNSS and Their Augmentations Chairman Rui Li, Beihang University, Beijing, China Vice-Chairman Long Yang, Beijing Future Navigation Technology Co., Ltd., Beijing, China Wenxiang Liu, National University of Defense Technology, Hunan, China Xingxing Li, Wuhan University, Hubei, China Yansong Meng, Xi’an Branch of China Academy of Space Technology, Shaanxi, China

Topic: S03: Satellite Orbit Determination and Precise Positioning Chairman Xiaogong Hu, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China Vice-Chairman Jianwen Li, Information Engineering University, Henan, China Jianghui Geng, Wuhan University, Hubei, China Bofeng Li, Tongji University, Shanghai, China Xiaolin Meng, The University of Nottingham, Nottingham, UK

vi

Editorial Board

Topic: S04: Time Frequencies and Precision Timing Chairman Aimin Zhang, National Institute of Metrology, Beijing, China Vice-Chairman Liang Wang, The 203th Research Institute of China Aerospace Science and Industry Corporation, Beijing, China Lijun Du, Xi’an Branch of China Academy of Space Technology, Shaanxi, China Ya Liu, National Time Service Center, Chinese Academy of Sciences, Shaanxi, China

Topic: S05: System Intelligent Operation and Autonomous Navigation Chairman Xingqun Zhan, Shanghai Jiao Tong University, Shanghai, China Vice-Chairman Haihong Wang, Institute of Telecommunication and Navigation Satellites, CAST, Beijing, China Wenbin Gong, Innovation Academy for Microsatellites of Chinese Academy of Sciences, Shanghai, China Yuxin Zhao, Harbin Engineering University, Heilongjiang, China Caibo Hu, Beijing Satellite Navigation Center, Beijing, China

Topic: S06: GNSS Signal Technologies Chairman Xiaochun Lu, National Time Service Center, Chinese Academy of Sciences, Shaanxi, China Vice-Chairman Hongwei Zhou, China Academy of Space Technology, Beijing, China Dun Wang, Space Star Technology Co., LTD. Beijing, China Yang Li, The 29th Research Institute of China Electronics Technology Group Corporation, Sichuan, China Zheng Yao, Tsinghua University, Beijing, China Xiaomei Tang, National University of Defense Technology, Hunan, China

Editorial Board

vii

Topic: S07: GNSS User Terminals Chairman Hong Li, Tsinghua University, Beijing, China Vice-Chairman Wenjun Zhao, Beijing Satellite Navigation Center, Beijing, China Zishen Li, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China Liduan Wang, ComNav Technology Ltd., Shanghai, China Chengjun Guo, University of Electronic Science and Technology of China, Sichuan, China

Topic: S08: PNT Architectures and New Technologies Chairman Zhongliang Deng, Beijing University of Posts and Telecommunications, Beijing, China Vice-Chairman Baoguo Yu, The 54th Research Institute of China Electronics Technology Rong Zhang, Tsinghua University, Beijing, China Jiangning Xu, Naval University of Engineering, Hubei, China Jinsong Ping, The National Astronomical Observatories of the Chinese Academy of Sciences, Beijing, China Dongyan Wei, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China Tianhe Xu, Shandong University, Jinan, China

viii

Editorial Board

Scientific Committee Senior Advisor Qingjun Bu, China National Administration of GNSS and Applications, Beijing, China Liheng Wang, China Aerospace Science and Technology Corporation, Beijing, China Yuzhu Wang, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China Guoxiang Ai, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China Lehao Long, China Aerospace Science and Technology Corporation, Beijing, China Shuhua Ye, Shanghai Astronomical Observatories, Chinese Academy of Sciences, Shanghai, China Jingjun Guo, Tsinghua University, Beijing, China Daren Lv, The Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China Yongcai Liu, China Aerospace Science and Industry Corporation, Beijing, China Jingnan Liu, Wuhan University, Hubei, China Jiadong Sun, China Aerospace Science and Technology Corporation, Beijing, China Zuhong Li, China Academy of Space Technology, Beijing, China Rongjun Shen, China Satellite Navigation System Committee, Beijing, China Chi Zhang, The former Electronic Information Foundation Department of General Equipment Department Xixiang Zhang, The 29th Research Institute of China Electronics Technology Group Corporation, Sichuan, China Lvqian Zhang, China Aerospace Science and Technology Corporation, Beijing, China Junyong Chen, National Administration of Surveying, Mapping and Geo information, Beijing, China Benyao Fan, China Academy of Space Technology, Beijing, China Dongjin Luo, China People’s Liberation Army, Beijing, China Huilin Jiang, Changchun University of Science and Technology, Jilin, China Guohong Xia, China Aerospace Science and Industry Corporation, Beijing, China Peikang Huang, China Aerospace Science and Industry Corporation, Beijing, China Chong Cao, China Research Institute of Radio Wave Propagation, CETC 22), Beijing, China Faren Qi, China Academy of Space Technology, Beijing, China Rongsheng Su, China People’s Liberation Army, Beijing, China Shusen Tan, Beijing Satellite Navigation Center, Beijing, China Ziqing Wei, Xi’an Institute of Surveying and Mapping, Shaanxi, China

Editorial Board

ix

Chairman Changfeng Yang, China Satellite Navigation System Committee, Beijing, China

Vice-Chairman Yuanxi Yang, China National Administration of GNSS and Applications, Beijing, China Shiwei Fan, China Satellite Navigation Engineering Center, Beijing, China

Executive Chairman Jun Xie, China Academy of Space Technology, Beijing, China Lanbo Cai, China Satellite Navigation Office, Beijing, China

Committee Members Qun Ding, The 20th Research Institute of China Electronics Technology Group Corporation, Beijing, China Xiangrong Ding, Legislative Affairs Bureau of the Central Military, Beijing, China Xiancheng Ding, China Electronics Technology Group Corporation, Beijing, China Quan Yu, Peng Cheng Laboratory, Shenzhen, China Zhijian Yu, Taiyuan Satellite Launch Center of China’s Manned Space Project, Shanxi, China Jian Wang, Alibaba Group, Zhejiang, China Wei Wang, China Aerospace Science and Technology Corporation, Beijing, China Feixue Wang, National University of Defense Technology, Hunan, China Zhaoyao Wang, China Satellite Navigation Office, Beijing, China Shafei Wang, Academy of Military Sciences PLA China, Beijing, China Lihong Wang, Legislative Affairs Bureau of the Central Military, Beijing, China Chengqi Ran, China Satellite Navigation Office, Beijing, China Weimin Bao, China Aerospace Science and Technology Corporation, Beijing, China Yueguang Lv, Science and Technology Commission of the CPC Central Military Commission Zhaowen Zhuang, National University of Defense Technology, Hunan, China Chong Sun, Beijing Institute of Tracking and Communication Technology, Beijing, China Yadu Sun, Aerospace Engineering Research Institute of the PLA Strategic Support Force Xianyu Li, Research Institute of the PLA Rocket Force Minling Li, China Society for World Trade Organization Studies, Beijing, China Jun Yang, China Satellite Navigation Office, Beijing, China Hui Yang, China Academy of Space Technology, Beijing, China

x

Editorial Board

Longxu Xiao, Research Institute of the PLA Rocket Force Bin Wu, Beijing Institute of Tracking and Communication Technology, Beijing, China Yirong Wu, The Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China Weiqi Wu, Xichang Satellite Launch Center, Sichuan, China Haitao Wu, Aerospace, Chinese Academy of Sciences, Beijing, China Manqing Wu, China Electronics Technology Group Corporation, Beijing, China Jun Zhang, Beijing Institute of Technology, Beijing, China Zhijie Chen, National Core Laboratory of Airspace Technology Zhonggui Chen, The 5th Research Institute of China Aerospace Science and Technology Corporation, Beijing, China Jinping Chen, Beijing Satellite Navigation Center, Beijing, China Baojun Lin, Innovation Academy for Microsatellites of Chinese Academy of Sciences, Shanghai, China Zhixin Zhou, Space Engineering University, Beijing, China Jianping Zhou, Chief Designer of China’s Manned Space Project Jianhua Zhou, Beijing Satellite Navigation Center, Beijing, China Jiancheng Fang, Beihang University, Beijing, China Wenjun Zhao, Beijing Satellite Navigation Center, Beijing, China Jiang Hu, BeiDou Application and Research Institute Co., Ltd. of Norinco Group, Beijing, China Jie Jiang, China Academy of Launch Vehicle Technology, Beijing, China Weiguang Gao, China Satellite Navigation Engineering Center, Beijing, China Shuren Guo, China Satellite Navigation Engineering Center, Beijing, China Huikang Huang, Ministry of Foreign Affairs of the People’s Republic of China, Beijing, China Xibin Cao, Harbin Institute of Technology, Heilongjiang, China Wenhai Jiao, China Satellite Navigation Engineering Center, Beijing, China Yi Zeng, China Electronics Corporation, Beijing, China Yi Cai, BeiDou ground-based augmentation system Chief Engineer Baoguo Yu, The 54th Research Institute of China Electronics Technology

Executive Members Jun Shen, Beijing Unistrong Science and Technology Co., Ltd. Beijing, China Dangwei Wang, Beijing UniStrong Science and Technology Co., Ltd., Beijing, China Rui Li, Beihang University, Beijing, China Xiaogong Hu, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China Aimin Zhang, National Institute of Metrology, Beijing, China Xingqun Zhan, Shanghai Jiao Tong University, Shanghai, China Xiaochun Lu, National Time Service Center, Chinese Academy of Sciences, Shaanxi, China Hong Li, Tsinghua University, Beijing, China

Editorial Board

xi

Zhongliang Deng, Beijing University of Posts and Telecommunications, Beijing, China Junlin Yang, Beihang University, Beijing, China

Preface

BeiDou Navigation Satellite System (BDS) is China’s global navigation satellite system which has been developed independently. BDS is similar in principle to global positioning system (GPS) and compatible with other global satellite navigation systems (GNSS) worldwide. The BDS will provide highly reliable and precise positioning, navigation and timing (PNT) services as well as short-message communication for all users under all-weather, all-time and worldwide conditions. China Satellite Navigation Conference (CSNC) is an open platform for academic exchanges in the field of satellite navigation. It aims to encourage technological innovation, accelerate GNSS engineering and boost the development of the satellite navigation industry in China and in the world. The 14th China Satellite Navigation Conference (CSNC 2024) is held during 2024, Jinan, China. Including technical seminars, academic exchanges, forums, exhibitions and lectures. The main topics are as followed:

Conference Topics S01 S02 S03 S04 S05 S06 S07 S08

GNSS Applications GNSS and Their Augmentations Satellite Orbit Determination and Precise Positioning Time Frequencies and Precision Timing System Intelligent Operation and Autonomous Navigation GNSS Signal Technologies GNSS User Terminals PNT Architectures and New Technologies

The proceedings (Lecture Notes in Electrical Engineering) have 151 papers in eight topics of the conference, which were selected through a strict peer-review process from 345 papers presented at CSNC2024. In addition, another 170 papers were selected as the electronic proceedings of CSNC2024, which are also indexed by “China Proceedings of Conferences Full-text Database (CPCD)” of CNKI and Wan Fang Data. We thank the contribution of each author and extend our gratitude to 299 referees and 53 session chairmen who are listed as members of editorial board. The assistance of CNSC2024’s organizing committees and the Springer editorial office is highly appreciated. Beijing, China

Changfeng Yang Jun Xie

Contents

Satellite Orbit Determination and Precise Positioning Performance Analysis of BDS-3 Klobuchar Model . . . . . . . . . . . . . . . . . . . . . . . . . Yongxing Zhu, Gang Wan, and Xianggao Yan

3

Analysis of the Impact of Tonga Volcano Eruption on Satellite Navigation Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jingyu Zhang, Bin Wang, Chao Xie, and Junping Chen

16

Prediction of Satellite Solar Radiation Pressure Parameters Based on Recurrent Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianbing Chen, Lei Chen, Shanshi Zhou, and Shuai Huang

30

Real-Time Kinematic Orbit Determination of GEO Satellite with the Onboard GNSS Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xinglong Zhao, Xiaogong Hu, Shaojun Bi, Huichao Zhou, Zheng Song, Qiuli Chen, Meng Wang, Shanshi Zhou, Chengbin Kang, Haihong Wang, and Gong Zhang Analysis of the Contributions of ISL to the Precise Orbit Determination of BDS-3 Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hanbing Peng and Yu Xiang

40

49

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution . . . . . . . . . . . . Lijun Yang, Guofu Pan, Xiang Zuo, Jinsheng Zhang, and Zhihao Yu

59

Effect of Troposphere Parameter Estimation on BDS PPP . . . . . . . . . . . . . . . . . . . Zhimin Liu, Yan Xu, Xing Su, Junli Zhang, Jianhui Cui, Zeyv Ma, Qiang Li, and Baopeng Xu

73

Comprehensive Analysis of the Cycle Slip Detection Threshold in Kinematic PPP During Geomagnetic Storms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiang Li, Xing Su, Chunyan Tao, Junli Zhang, Zhimin Liu, Jianhui Cui, Zeyv Ma, Baopeng Xu, and Yan Xu Analysis of Rapid Re-initialization Performance of Precise Point Positioning for Low-Cost Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fenghua Luo, Lin Zhao, Fuxin Yang, Zhiguo Sun, and Jie Zhang

83

93

xvi

Contents

Prototype of Real-Time Orbit Service for LEO Navigation Satellite System . . . . 103 Guanlong Meng, Haibo Ge, and Bofeng Li BDS3-Based Precise Orbit Determination for LEO Satellites with Single-Receiver Ambiguity Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Houzhe Zhang, Kai Shao, and Xiaojun Duan Assessment About Parameters Selection Strategy of ECOMC Solar Radiation Pressure Model for BDS-3 Satellites During the Earth Eclipsing Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Renlong Wang, Jianhua Cheng, Hui Li, Guojian Sun, and Xitie Lu Assessment of Solar Radiation Pressure Models for BDS-3 MEO Navigation Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Leitong Yuan, Bofeng Li, Weiguang Gao, and Haibo Ge Combined Processing of Outlier and Multipath in GNSS Precise Point Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Haijun Yuan, Xiufeng He, and Zhetao Zhang Polar Motion Prediction Based on the Combination of Weighted Least Squares and Vector Autoregressive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Yu Lei and Danning Zhao Precise Orbit and Clock Offset Determination of LEO Navigation Satellites Based on Multi-constellation and Multi-frequency Spaceborne GNSS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Junjun Yuan, Ertao Liang, Liqian Zhao, Kai Li, Chengpan Tang, Shanshi Zhou, and Xiaogong Hu GNSS Carrier Phase Heading Determination with a Single Array Antenna . . . . . 185 Wenxin Jin, Wenfei Gong, Tianwei Hou, Xin Sun, and Hao Ma Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite Link and Satellite-Ground Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Yinan Meng, Xin Xie, Hongliang Cai, Chao Zhang, Rui Jiang, Xia Guo, and Jun Lu Global Mapping of Ionospheric ROTI Index and Its Preliminary Application in Analysis of Precise Positioning Degradation . . . . . . . . . . . . . . . . . . 211 Haoyang Jia, Zhe Yang, and Bofeng Li Precise Point Positioning Ambiguity Resolution with Multi-frequency Ionosphere-Reduced Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Qing Zhao, Shuguo Pan, Wang Gao, Ji Liu, Yin Lu, and Peng Zhang

Contents

xvii

Simulation Study on Real-Time Orbit Determination Based on GNSS for LEO Satellite Considering the Effect of Electric Propulsion . . . . . . . . . . . . . . 242 Jiapeng Wu, Wanwei Zhang, Fuhong Wang, Meng Wang, and Chengxiang Yin Analysis of the Performance of Broadcast Ionospheric Model for Anti-disturbance Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Xianggao Yan, Xiaolin Jia, Yongxing Zhu, Jialong Liu, and Zhichao Zhang Global Instantaneous Centimeter-Level Multi-constellation and Multi-frequency Precise Point Positioning with Cascading Ambiguity Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Lizhong Qu, Luping Wang, Haoyu Wang, Wei Jiang, and Yiwei Du Time Frequencies and Precision Timing High-Performance Microwave Frequency Standard Systems Based on the Ground-State Hyperfine Splitting of 171 Yb+ and 113 Cd+ Ions . . . . . . . . . . 281 Ying Zheng, Yiting Chen, Nongchao Xin, Shengnan Miao, Haoran Qin, Jianwei Zhang, and Lijun Wang Precision Analysis of Clock Products Data from iGMAS Analysis Centers . . . . . 288 Yifeng Liang, Jiangning Xu, Miao Wu, and Yundong Shang The Research on BeiDou Time Transfer via Precise Point Positioning based on Raw Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Yifan Wu, Qianyi Ren, Xinying Lu, Yongshan Dai, and Yuan Shen An Engineered High Performance Optically Pumped Compact Cesium Atomic Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Qiang Wei, Yu-ao Li, Dong-xu Li, Zhi-bin Wang, and Xing-wen Zhao Ultra-thin Rubidium Atomic Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Fang-fang An, Hu-jun Zhang, Peng Wang, Cheng-yong Liu, Tao Peng, and Xing-wen Zhao A Method for Establishing Elastic Time-Frequency Reference for Navigation Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Richang Dong, Jun Lu, Chengpan Tang, Yinan Meng, and Chengeng Su Research on Time Transfer Method Based on Un-Differenced Combination Model Between Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Dong LV, Genyou Liu, and Run Wang

xviii

Contents

An Improved Method for PPP Time Transfer with Forecast Clock Model and Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Jinyang Han, Jie Zhang, Shiming Zhong, Runmin Lu, and Bibo Peng Research on PPP Time Transfer Method Based on Observable-Specific Signal Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Pan Du, Feng Shen, Dingjie Xu, Juan Yin, Peipei Dai, Qi Li, and Yuqing Zhao Application of Improved GPS Satellite Clock Error Prediction Model in Real-Time Precise Point Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Xiaoming Li and Haojun Li Research on Phase/Frequency Consistent Adjusting Method for Main/Backup Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Guitao Fu, Zhaonan Li, and Baoguo Yu Frequency Performance Evaluation of BeiDou-3 Satellite Clocks . . . . . . . . . . . . . 403 Leyuan Sun, Xiangwei Chang, Wende Huang, Liwen Guo, Bo Liu, and Long Guan Engineering Optical Clocks for GNSS Timing Reference Applications . . . . . . . . 414 Yuan Qian, Huaqing Zhang, Mengyan Zeng, Bin Wang, Yanmei Hao, Lijun Du, Jun Lu, Yuzhuo Wang, Maolei Wang, Yao Huang, Hua Guan, Jun Xie, and Keling Gao A Small High-Temperature Rubidium Frequency Standard . . . . . . . . . . . . . . . . . . 424 Yuxi Li, Shiguang Li, Tongmin Yang, Nina Ma, Zhaohua Liu, Weili Wang, and Liang Wang Accuracy Analysis of BeiDou-3/GPS Real-Time Precise Common View Time Transfer Based on Carrier-Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Hongyuan Zhou, Baoqi Sun, Lingyang Sun, Lihua Zhao, Zhe Zhang, Ge Wang, Kan Wang, and Xuhai Yang System Intelligent Operation and Autonomous Navigation 3D Small Object Detection from Cameras and Point Clouds Using Five-Head Attention in a Fusion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Haogang Mao, Jichao Jiao, Jialun Li, and Yang Fuxing Research on Autonomous Orbit Determination Based on BeiDou Navigation Satellite System Inter-Satellite Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Songhua Hu, Jingshi Tang, and Haihong Wang

Contents

xix

Robust Alignment Base on IKF for SINS/DVL Integrated Navigation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Jingshu Li, Bing Zhu, Ge Tian, Zuohu Li, De Jiang, and Xia Guo Multi-UUVs Cooperative Localization in Asymmetric Large Configuration . . . . 482 Zhenqiang Du, Hongzhou Chai, Minzhi Xiang, Fan Zhang, Jun Hui, and Zhaoying Wang Research on Autonomous Detection and Recovery Technology of Satellite Clock Anomaly During the Autonomous Navigation of BDS-3 Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Shaojun Bi, Guodong Zhang, Haihong Wang, and Zhaofeng Zhong Research on Life Prediction of Navigation Satellite Based on Bayesian Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Hongliang Cai, Ru Feng, Yinan Meng, Bo Zhou, Heng Zheng, and Zongsheng Xie Research on the Constellation Rotation Mechanism and Suppression Method of Distributed Autonomous Orbit Determination . . . . . . . . . . . . . . . . . . . . 523 Haihong Wang, Wei Zhou, Xu Zhang, Songhua Hu, Shanshi Zhou, Qiuli Chen, Jingshi Tang, Hongliang Cai, Jingang Wang, and Fengyu Xia Integrity Monitoring for GNSS/INS Integrated Navigation Based on Improved AIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Qiwei Ye, Yu Gu, Liang Li, Fengze Du, and Ruijie Li Research on Visual Inertia SLAM Technology with Additional Point and Line Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Xiaoze Zheng, Chun Jia, Xiaohan Wang, and Yang Li An Adaptive and Robust Strategy for GPS/IMU/VO Integrated Navigation . . . . . 555 Yeying Dai and Rui Sun RGB-D SLAM Algorithm Based on Clustering and Geometric Residuals in Dynamic Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 Jinjing Chen, Shuguo Pan, Wang Gao, Ji Liu, Yin Lu, and Peng Zhang Research on an Artificial Intelligence Based Diagnosis Algorithm of BDS Telemetry Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Lu Wang, Lei Chen, Jian Wang, Huiyan Zhao, Bohao Cui, and Yueyang Sun

xx

Contents

Evaluation of Satellite Acceleration Determination Methods in GNSS/INS Deep Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Wei Gao, Xingqun Zhan, and Rong Yang Spatiotemporal Alignment and Measurement Accuracy Evaluation of New Point Cloud Devices in Autonomous Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 Jiahui Liu, Xin Zhang, and Xingqun Zhan Calibration Method for LiDAR Cameras in Natural Dynamic Environments . . . . 608 Sheng Hong, Qinghua Zeng, Yineng Li, Xiaorong Sun, and Jizhou Lai Pedestrian Navigation Algorithm Based on EKF Combined with ZUPT + ZARU + Attitude Self-observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 Chen Zhu, Hang Guo, Jian Xiong, and Yujie Wang Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Satellite Orbit Determination and Precise Positioning

Performance Analysis of BDS-3 Klobuchar Model Yongxing Zhu1,2,3 , Gang Wan4(B) , and Xianggao Yan5 1 State Key Laboratory of Geo-Information Engineering, Xi’an 710054, China 2 Xi’an Research Institute of Survey and Mapping, Xi’an 710054, China 3 National Space Science Center, Chinese Academy of Sciences, Beijing 100190, China 4 School of Space Information, Space Engineering University, Beijing 101416, China

[email protected] 5 College of Geological Engineering and Geomatics, Chang’an University, Xi’an 710054, China

Abstract. Since July 2020, the global Beidou Navigation Satellite System (BDS3) satellites provide new B1C and B2a public navigation service signals, as well as the legacy B1I and B3I signals. With the opening of BDS-3 global service, the service area of the broadcast Klobuchar model (BDSKlob) parameters expanded from China and parts of the Asia-Pacific region to the whole world. In this study, the differences between the GPS Klobuchar model (GPSKlob) and the BDSKlob is comprehensively analyzed. The data since the opening of the BDS-3 is collected, and the results show that: (1) the average correction accuracy of the BDSKlob model in global-scale is about 4.61–5.75 TECU; (2) the correction abnormalities of the modified BDSKlob model in high latitude region has been improved significantly; (3) the interchangeability between the BDSKlob and the GPSKlob model algorithm has been verified, is initially verified. Keywords: Beidou Global Navigation Satellite System (BDS-3) · BDS Klobuchar · GPS Klobuchar · Accuracy assessment

1 Introduction Ionosphere is regarded as one of the main error sources of navigation satellite systems due to its effects on radio signals, such as emission, refraction, scattering and absorption (Klobuchar 1987; Yuan et al. 2019; Zhu et al. 2019). For GNSS multi-frequency users, it is common to eliminate the ionospheric delay by using the ionosphere-free observation to improve the accuracy of positioning, velocity and timing. However, for GNSS single-frequency users, broadcast ionospheric models are still the main method to correct the ionospheric delay and improve the accuracy of the real-time service (Klobuchar 1987; Li et al. 2013; Wu et al. 2013; Yang et al. 2014). To serve the real-time correction of the ionospheric delay for global users, GPS broadcasts the Klobuchar model parameters (GPSKlob), Galileo broadcasts the Nequick model parameters (Nequick G), Beidou regional system (BDS-2) broadcasts the improved Klobuchar model parameters (BDSKlob), and Beidou global system (BDS-3) broadcasts the BDGIM model coefficients (BDGIM), respectively. Many experiments have been performed under different © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 3–15, 2024. https://doi.org/10.1007/978-981-99-6944-9_1

4

Y. Zhu et al.

conditions to evaluate the precision and accuracy of broadcast ionospheric models. The results of these experiments showed that the correction ratio of GPSKlob and Nequik G in global-scale is 50–60% and 60–70%, respectively; the correction ratio of BDSKlob in Asia-Pacific region and BDGIM in global-scale is about 60% to 70% and over 75%, respectively (Klobuchar 1987; Wang et al. 2016; Wang 2017; Yuan et al. 2019; Zhu et al. 2019). Among them, the Klobuchar model is widely used by single-frequency users due to its simplified structures, better correction accuracy and wide use of GPS. Also, it is adopted by BDS-2 with good correction effect. The BDS-3 newly broadcasts BDGIM model parameters to serve the global users, and broadcasts BDSKlob parameters at B1I and B3I signals, which is different from GPSKlob (Zhu et al., 2020; Zhu et al. 2020; Mao et al. 2020). From the aspect of the algorithm, the GPSKlob model and the BDSKlob model are different in structures by comparing the GPS and BDS Interface Control Documents. It can be concluded as (Wu et al. 2013; Zhang et al. 2014): (1) different Coordinate frames. GPSKlob adopts the sun-fixed geomagnetic coordinate frame, which better reflects the correlation between ionospheric changes and geomagnetic activities. Whereas, BDSKlob adopts the sun-fixed geographic coordinate frame, which better uniforms the geographic longitude and the time, and coincides with the daily variation of the ionosphere. (2) Different Coefficients Updation. GPSKlob parameters are picked from the existing 370 groups of contents, according to the day of the year (DOY) and the average solar flux values of the previous five days, and updated once every one to ten days. While, he BDSKlob parameters are calculated with the observations of the regional monitoring stations in China and updated every 2 h. (3) Different model constants. The thin-layer height of GPSKlob model is set to 350 km and the radius of the earth is 6378.137 km. However, the thin-layer height of BDSKlob model is 375 km and the radius of the earth is 6378.137 km. From the aspect of the accuracy, the correction accuracy of BDSKlob is better than the GPSKlob in the mid-latitude regions of the Northern Hemisphere (including Asia, Europe, North Africa, etc.). Accordingly, the accuracy of single-frequency single-point positioning is improved by about 7.8–35.3%. In the low-latitude region and the Southern Hemisphere, the correction accuracy of BDSKlob is slightly worse than that of GPSKlob (Wu et al. 2013; Zhang et al. 2014). As is obtained based on the reginal monitoring network in China, BDSKlob show more abnormal corrections in the high-latitude region. The main reasons are that the monitoring stations fail to cover the high-latitude and lowlatitude regions and the correction accuracy of BDSKlob in the Southern Hemisphere was considered symmetrical to that of the Northern Hemisphere in the same latitude. With the opening of BDS-3 global service on July 31, 2020, the B1I and B3I signals services have extended from the Asia-Pacific to the whole world. Accordingly, the parameters of the BDSKlob have also been upgraded to serve the whole world. Nowadays, the system builders and users have paid more attention to the performance of BDSKlob in global-scale, especially in the high-latitude regions and the outside Asia-Pacific region ((Yang et al. 2018). In this paper, the characteristics of Klobuchar (including BDSKlob and GPSKlob) are firstly introduced in Sect. 2. The comparison between the BDSKlob and GPSKlob in algorithm are described and analyzed in Sect. 3. Then, a comprehensive

Performance Analysis of BDS-3 Klobuchar Model

5

accuracy assessment of BDSKlob and GPSKlob based on the International GNSS Service (IGS) ionospheric grid products are carried out in Sect. 4. Finally, the experimental results and conclusions are given.

2 Klobuchar model Klobuchar model was first proposed in 1987, which was simplified based on the Bent model (Klobuchar 1987). Klobuchar is a vertical delay model, assuming that the free electrons are all in a thin-layer with a constant height (350 km). The vertical ionospheric delay at nighttime is set as a constant. The diurnal variation of the vertical ionospheric delay is represented as a half positive cosine wave plus a constant bias. The amplitude and period of the cosine term are described as a third-order polynomial of the geomagnetic latitude, consisting of the eight coefficients. Basically, they can reflect the characteristics of ionospheric variation and ensure the reliability of ionospheric forecasts on a large scale (Zhao and Zhang 2013). As shown in Fig. 1, the Klobuchar model includes 4 parameters: DC for the night-time constant and AMP, PHASE, PER for the amplitude, phase and period of cosine term, respectively.

Fig. 1. Klobuchar model and fitted actual monthly average time-delay data (Klobuchar 1987)

During nighttime, the vertical ionospheric delay is set as a constant value of 5 ns. The phase in Klobuchar model is set as a constant of 14 h LT, when the TEC behavior in many parts of the world has a diurnal maximum. The amplitude and period of the cosine term are described as a third-order polynomial of the geomagnetic latitude φm , consisting of the eight coefficients αn and βn (n = 1, 2, 3, 4). The vertical ionospheric

6

Y. Zhu et al.

time delay computed with Klobuchar model can be expressed as shown in Eq. (1): ⎧   ⎪ ⎨ 5 × 10−9 + AMP · cos 2π(t − PHASE) , |t − PHASE| < PER/4 PER Tion (t) = ⎪ ⎩ 5 × 10−9 , |t − PHASE| ≥ PER/4 ⎧ ⎧ 3 3 ⎪ ⎪   ⎪ ⎪ n ⎨ ⎨ αn φm , AMP > 0 βn φmn , PER > 72000 , PER = n=0 AMP = n=0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 0, AMP ≤ 0 0, PER ≤ 72000 (1)

3 Comparison on model algorithm 3.1 Main function For Eq. (1), the GPSKlob performs Fourier series expansion on the cosine function (cos) and takes the first three terms. ϕm is geomagnetic latitude, ϕg is geographic latitude. The ionospheric correction in the Southern Hemisphere is supposed to be symmetrical with the Northern Hemisphere. The calculation formulas are compared as follows: BDSKlob ⎧   ⎪ ⎨ 5 × 10−9 + AMP · cos 2π(t − 50400) , |t − 50400| < PER/4 PER Tion (t) = ⎪ ⎩ 5 × 10−9 , |t − 50400| ≥ PER/4

AMP =

3

n=0 αn

 n ϕg  , AMP > 0 AMP ≤ 0

0, ⎧ 172800, PER ≥ 172800 ⎪ ⎪ ⎪ 3 ⎨  n βn ϕg  , 172800 > PER ≥ 72000 PER = ⎪ ⎪ ⎪ ⎩ n=0 0, PER < 72000 Tion (t) =

GPSKlob

5 × 10−9 + AMP · 1 −

X2 2

+

X4 24

 , |X | < 1.57

5 × 10−9 , X ≥ 1.57 3 n n=0 αn ϕm , AMP > 0 AMP = 0, AMP ≤ 0 3 n n=0 βn ϕm , PER ≥ 72000 PER = 0, PER < 72000 2π(t − 50400) X = PER

(2)

Performance Analysis of BDS-3 Klobuchar Model

7

Comparing the two formulas above and conclusions are as follows. (1) The GPSKlob replaces the original cosine function with Taylor series expansion and takes the first three terms as shown in Eq. (3): cos(X ) ≈ 1 −

X4 X2 + 2 24

(3)

(2) The latitude of GPSKlob is geomagnetic latitudes and latitude of BDSKlob is geographic latitude, the conversion is given as Eq. 4: sin ϕm = sin ϕg sin ϕp + cos ϕg cos ϕp cos(λg − λp )

(4)

where (λg , ϕg ) represents the geographic longitude and latitude of the ionospheric pierce point (IPP), (λp , ϕp ) is the geomagnetic north pole coordinate (value of 287.33°, 80.33° in 2020), and t is the local time of the station. Actually, the GPSKlob simplifies the formula to ϕm = ϕg + ϕp cos(λg − λp ) for further applications, and the approximate error is less than 1° in the continental United States (mid-to-high latitude area). 3.2 Mapping function As shown in Fig. 2, the arrow indicates the line of sight between the station and the satellite. R0 is the radius of the earth, h is the height of the single ionospheric layer, Z is the observation altitude angle, α is the inclination of the line of sight of the ionospheric pierce point, and ψ is the geocentric angle between the station and ionospheric pierce 1/ cos α. point. The projection function is MF = 

2 0 · cos E , where R0 is 6378.137km For BDSKlob, MF = 1/ cos α = 1/ 1 − R0R+h and h is 375 km. For GPSKlob, the R0 is 6378137 km and the h is 350 km. The MF is simplified to MF ≈ 1.0 + 16.0 × (0.53 − Z)3 . When the altitude angle is more than 5°, the accuracy of this approximate formula is within 2%. 3.3 Other Formulas Other formulas mainly include the geocentric angle ψ between the station and the IPP, the geographic longitude and latitude of the IPP (λg , ϕg ), and local time t. Accurate formulas are used in the BDSKlob model, and simplified formulas are used in the GPSKlob model.

 0 (1) For the GPSKlob, the geocentric angle ψ= π2 −E −arcsin R0R+h · cos E is simplified 0.0137 − 0.022. The simplified formula has an error less than 0.2° when the as ψ ≈ E+0.11 altitude angle is greater than 10°, and is greater than 10° when the altitude angle is 5° and 0°. The error with the precise formula is only 0.4° and 0.3°, respectively.

(2) The model to calculate the geographical longitude and latitude coordinates of the IPP (λg , ϕg ) is: ⎧ ⎪ ⎨ ϕg = arcsin(sin ϕu cos ψ + cos ϕu sin ψ cos A)   (5) sin ψ sin A ⎪ = λ + arctan λ u ⎩ g cos ϕg

8

Y. Zhu et al.

Fig. 2. The spatial relationship diagram of the center of the earth, the station and the satellite

Similarly, the formula is simplified in GPSKlob model which can be given as: ⎧ ⎪ ⎨ ϕg = ϕu + ψ cos A (6) ψ sin A ⎪ ⎩ λg = λu + cos ϕ g where, (λu , ϕu ) are the longitude and latitude of the station, and A is the azimuth of the observed satellite. The latitude range of GPSKlob is 75 °S–75 °N. Users outside of this range directly use the ionospheric correction value at the nearest latitude point. (3) The formula to calculate local time t is t = ts + λg , where ts is GPST or BDT at the time of measurement respectively.

4 Results and analysis In this section, data since the open service of BDS-3 are selected to analyze the accuracy of the BDSKlob and GPSKlob. Among, the broadcast ionospheric coefficients are collected for 20 days from 221 to 240 day of year 2020. Moreover, the IGS final ionospheric grid products are used as the reference to evaluate the accuracy with the statistical index RMS.

Performance Analysis of BDS-3 Klobuchar Model

9

4.1 Average correction accuracy According to the grid interval of 5° × 2.5°(longitude × latitude) from ranges of −180° to 180° and 87.5 °S–87.5 °N, ionospheric TEC calculated by the BDSKlob and GPSKlob are compared with the IGS final ionospheric grid products. The statistical results of daily residuals dvtec and RMS are shown in Table 1 and in Fig. 3. In Fig. 3, the x-axis represents the days of the year, and the y-axis is the ionospheric residuals error which unit is TECU.

error (Unit:Tecu)

6.5 BDSKlob GPSKlob

6

5.5

5

4.5

4 220

222

224

226

228

230

232

234

236

238

240

Day of Year Fig. 3. The global average correction accuracy of BDSKlob and GPSKlob

Table 1. Statistics on the global average correction accuracy of BDSKlob and GPSKlob (Unit: TECU) ME AN

221

222

223

224

225

226

227

228

229

230

BDS Klob 5.10

4.61

5.02

4.75

4.63

5.45

4.99

5.01

5.22

5.05

4.66

GPS Klob 4.63

4.63

4.69

4.6

4.51

4.59

4.69

4.59

4.67

4.64

4.62

231

232

233

234

235

236

237

238

239

240

BDS Klob 4.84

5.51

5.45

5.5

5.24

4.79

5.75

5.29

5.02

5.25

GPS Klob 4.53

4.71

4.82

4.68

4.55

4.57

4.68

4.61

4.51

4.67

From Table 1 and Fig. 3, it can be seen that: (1) the correction accuracy of the GPSKlob is slightly better than that of the BDSKlob in global scale; (2) the correction accuracy of the BDSKlob is within 4.61–5.75 TECU, and the average is 5.10 TECU; (3) the correction accuracy of GPSKlob is 4.51–4.82 TECU, and the average is 4.63 TECU. It is indicated that the accuracy of the BDSKlob still needs to be further improved after the opening of BDS-3.

10

Y. Zhu et al.

4.2 Correction accuracy for different latitudes As mentioned in Sect. 1, the correction accuracy of the BDSKlob in the Asia-Pacific region is about 60–70%, which is better than the GPSKlob excepting the abnormal corrections in high latitude regions. Figure 4 shows the global correction of the BDSKlob in 2017 (Only one representative day was selected.). The abnormal corrections of the BDSKlob in high latitude regions have been observed.

(a)

(b) Fig. 4. The global correction accuracy distribution of the BDSKlob in 2017 (Unit: TECU)

With the opening of BDS-3 global services, the BDSKlob has also been upgraded to improve the correction accuracy in areas outside the high-latitude and the Asia-Pacific

Performance Analysis of BDS-3 Klobuchar Model

11

regions. Figure 5 shows the global corrections of BDSKlob at 14:00 UTC on days of year 221 and 240 in 2021. It can be seen that the model has been improved in correcting the abnormal conditions outside the high latitude and Asia-Pacific regions after the BDSKlob is upgraded.

(a)DOY-221 14:00 UTC

(b) DOY-240 14:00 UTC Fig. 5. The global correction accuracy distribution of the improved BDSKlob model (Unit: TECU)

Referring to Sect. 4.1, the ionospheric TEC correction residuals are firstly calculated for all grid points, and then the ionospheric TEC correction accuracy was counted at different latitudes. The daily RMS of 20 days (day of year 221 to 240 in 2021) was taken for the statistical results and shown in Fig. 6, where the x-axis is the ionospheric

12

Y. Zhu et al.

correction error and the y-axis is the geographic latitude. From Fig. 6 we can see that: (1) the correction accuracy of BDSKlob and GPSKlob are relatively low in the high-latitude region of the Southern Hemisphere, which is probably due to the lack of the observations in the Southern Hemisphere; (2) in the mid-latitude region, the correction accuracy of GPSKlob and BDSKlob are basically comparable; (3) in the low-latitude region, the correction accuracy of BDSKlob is slightly better than that of GPSKlob, which is due to the fact that the coefficients of BDSKlob are obtained from the measured data and updated timely; (4) the correction accuracy of BDSKlob is relatively low in the highlatitude region of Northern Hemisphere and more data are needed for further analysis, though the abnormal corrections of the model has been improved in the high-latitude region. 87.5 75.0

Latitude (Uint:Degree)

62.5

BDSKlob GPSKlob BDSKlob BDSKlob

BDSKlob

50.0 37.5 25.0 12.5 0 -12.5 -25.0 -37.5 -50.0 -62.5 -75.0 -87.5

0

2

4

6

8

10

2

4

6

8

10

error (Unit:Tecu) Fig. 6. The correction accuracy distribution of BDSKlob and GPSKlob models at different latitude

4.3 Interchangeability of BDSKlob and GPSKlob models As analyzed in Sect. 3, the BDSKlob adopts the accurate formula of the Klobuchar model, while the GPSKlob uses a simplified formula. Hence, the formulas of the BDSKlob and GPSKlob are quite different. Actually, the main difference between the BDSKlob and GPSKlob is the coordinate frame, involving geomagnetic and geographic coordinate frames. Considering that the GPSKlob has been widely used in various terminals, the interchangeability of BDSKlob and GPSKlob are proposed in the study. If the BDSKlob coefficients can be applied in the GPSKlob algorithm, it is of great significance for the further promotion of the BDSKlob. Firstly, the BDSKlob coefficients are collected to calculate the ionospheric TEC with the formulas of BDSKlob and GPSKlob (GPSKlob_BDS) at each grid respectively.

Performance Analysis of BDS-3 Klobuchar Model

13

Then, comparing the calculated TEC with the IGS final ionospheric grid products, the statistics on daily RMS of BDSKlob and GPSKlob_BDS based on the grid interval of 5° × 2.5° (longitude × latitude) from ranges of −180° to 180° and 87.5°S–87.5°N are shown in Fig. 7. The statistical results indicate that the correction accuracy of BDSKlob and GPSKlob_BDS are basically the same, which means the broadcasted coefficients of the BDSKlob can be directly applied to the GPSKlob. Therefore, there is a certain interchangeability between them, which is of great significance for the promotion of BDSKlob coefficients. However, limited by the correction accuracy of the current Klobuchar model, the applicability of this interchangeability needs to be further verified and the correction accuracy of the Klobuchar model coefficients need to be improved. 6.5

error (Unit㸸Tecu)

BDSKlob GPSKlob-BDS 6

5.5

5

4.5

4 220

225

230

235

240

Day of Year Fig. 7. The global average correction accuracy of BDSKlob and GPSKlob_BDS models

5 Conclusions Broadcast ionospheric model of GNSS is considered as one of the main methods for GNSS single-frequency users to correct ionospheric delay and improve the accuracy of real-time services. With the opening of the global service of BDS-3, the coverage of B1I and B3I navigation service signals have been expanded from the regional to the world. Accordingly, the BDSKlob coefficients are broadcasted all over the world and service for the global users. In this study, the comparison between BDSKlob and GPSKlob from aspects of accuracy and algorithm are analyzed. The main conclusions are summarized as follows. (1) In terms of the main functions, mapping functions and other algorithm, the BDSKlob adopts the exact formulas of the Klobuchar model, whereas the GPSKlob used the simplified ones. Among, the major difference is the coordinate frame, referring to the geomagnetic and geographic coordinate frames adopted by GPSKlob and BDSKlob, respectively.

14

Y. Zhu et al.

(2) The average correction accuracy of the BDSKlob and GPSKlob are analyzed in global-scale. It is indicated that the correction accuracy of GPSKlob is slightly better than the BDSKlob. BDSKlob is within 4.61–5.75 TECU, and the GPSKlob is about 4.51–4.82 from the statistics in 2020. (3) The correction accuracy of BDSKlob and GPSKlob at different latitudes were analyzed. In general, the correction accuracy of the BDSKlob is slightly better than that of the GPSKlob in the low latitude region, and is basically comparable in midlatitude areas. The correction accuracy of BDSKlob in high latitude areas is a little worse, though the abnormal corrections at high latitude has been improved. (4) The current correction accuracy of Klobuchar model, BDSKlob and GPSKlob are interchangeable to a certain extent, which is of great significance for the promotion of the BDSKlob coefficients. This study comprehensively analyze the performance of the BDSKlob since the service area is expanded to the whole world. Moreover, more data will be collected for further analysis, such as abnormal correction and annual variation of ionosphere, especially for high latitude regions. Acknowledgments. This study was funded by State Key Laboratory of Geo-Information Engineering, No. SKLGIE2020-M-1-1; the National Natural Science Foundation of China(42074006, 41874041, 41904042).

References Klobuchar J (1987) Ionospheric time-delay algorithms for single-frequency GPS users. IEEE Trans Aerosp Electr Syst AES 23(3):325–331 Li W, L M, Hu Z, et al (2013) Comparative analysis of COMPASS and GPS ionospheric model on positioning and navigation precision. J Navig Position Mao Y, ZY, Song X (2020) Accuracy analysis of broadcast ionospheric model of global navigation satellite system. J Geodesy Geodyn 40(9):4 Wu X, Hu XG, Wang G, Zhong H et al (2013) Evaluation of COMPASS ionospheric model in GNSS positioning. Adv Space Res 51(6):959–968. https://doi.org/10.1016/j.asr.2012.09.039 Wang N (2017) Study on GNSS differential code biases and global broadcast ionospheric models of GPS, Galileo and BDS Wang N, Yuan Y, Li Z et al (2016) Improvement of Klobuchar model for GNSS single-frequency ionospheric delay corrections. Adv Space Res 57(7):1555–1569. https://doi.org/10.1016/j.asr. 2016.01.010 Yuan Y, W N, Li Z, et al (2019) The BeiDou global broadcast ionospheric delay correction model (BDGIM) and its preliminary performance evaluation results. Navig J Inst Navig 66(1):55–69. https://doi.org/10.1002/navi.292 Yang Y, X Y, Li J, et al (2018) Progress and performance evaluation of BeiDou global navigation satellite system: data analysis based on BDS-3 demonstration system. Sci China Earth Sci 61:614–624. https://doi.org/10.1007/s11430-017-9186-9 Yang Y, L J L, Wang A, et al (2014) Preliminary assessment of the navigation and positioning performance of BeiDou regional navigation satellite system. Sci China Earth Sci 57:144–152. https://doi.org/10.1007/s11430-013-4769-0

Performance Analysis of BDS-3 Klobuchar Model

15

Zhu Y, T S, Zhang Q, et al (2019) Accuracy evaluation of the latest BDGIM for BDS-3 satellites. Adv Space Res 64(6):1217–1224. https://doi.org/10.1016/j.asr.2019.06.021 Zhang Q, Z Q, Zhang H, et al (2014) Evaluation on the precision of Klobuchar model for BeiDou navigation satellite system. Geomat Inf Sci Wuhan Univ 39(2):5. https://doi.org/10.13203/j. whugis20120716 Zhao W, Zhang C (2013) Practical analysis and improvement of Klobuchar model. Chin J Space Sci 33(6):624–628 Zhu Y, Tan S, Feng L et al (2020) Estimation of the DCB for the BDS-3 new signals based on BDGIM constraints. Adv Space Res 66(6):1405–1414 Zhu Y, TS, Ren X, et al (2020) Accuracy analysis of GNSS global broadcast ionospheric model. Geomat Inf Sci Wuhan Univ 45(5):8

Analysis of the Impact of Tonga Volcano Eruption on Satellite Navigation Services Jingyu Zhang1,2(B) , Bin Wang1 , Chao Xie2 , and Junping Chen1 1 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China

[email protected] 2 Beijing Satellite Navigation Center, Beijing, China

Abstract. On January 15, 2022, the violent eruption of Tonga volcano in the South Pacific Ocean triggered tsunami waves and atmospheric gravity waves around the world, which had a significant impact on the Earth’s magnetic field and atmosphere. Based on GNSS observations, this paper analyzes the impact of Tonga volcanic eruption on satellite navigation services from the aspects of signal propagation delay and user positioning. The results show that the troposphere over China has a short-term fluctuation of 2–3% due to the impact of atmospheric gravity waves triggered by Tonga volcanic eruption. For the middle and low latitudes, the ionosphere shows VTEC abnormal values from southeast to northwest, and the propagation speed of VTEC abnormal is basically equivalent to that of atmospheric gravity waves. The volcanic eruption caused obvious fluctuations in the up direction of positioning results. Dynamic PPP can be used for real-time monitoring of large-scale anomalies of satellite navigation services. Keywords: Tonga volcano eruption · Satellite navigation services · Vertical total electron content · Dynamic precise point positioning

1 Introduction The violent eruption of Tonga volcano on January 15, 2022 is the biggest blast in 21st century. It has attracted the common attention of experts in many fields such as meteorology, seismology and remote sensing. Preliminary observations confirm that the eruption went through multiple processes, involving physics, atmosphere, chemistry and other disciplines, and affected multiple layers of the Earth, including from the outer space of the ionosphere, 100 km away, to the underground. The atmospheric gravity waves generated after the eruption spread globally in the form of Lamb waves, with a propagation speed of about 310 m/s. Multiple monitoring methods have observed the secondary propagation signals of atmospheric gravity waves after a circle around the earth. According to the monitoring data of GNSS station around Tonga [1], Tonga volcanic eruption has caused a certain coseismic displacement to its surroundings, resulting in the northwest horizontal displacement around Tonga and obvious uplift in the elevation direction.

© Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 16–29, 2024. https://doi.org/10.1007/978-981-99-6944-9_2

Analysis of the Impact of Tonga Volcano Eruption

17

2 Analysis of the Tonga volcano eruption on satellite signal propagation The Tongan eruption caused significant atmospheric gravity waves that spread across the globe. Theoretically, the atmospheric ionosphere and troposphere through which satellite navigation signals travel will be significantly affected, which may have an impact on users of satellite navigation and positioning services. Detailed analysis will be made below. 2.1 The Analysis of Troposphere Delay Since the troposphere is a non-dispersive medium, the satellite navigation signals with different frequencies have the same propagation speed when propagating in the troposphere, and the propagation delay is mainly affected by the atmospheric pressure, temperature and humidity, so mathematical models can be used for accurate modeling and correction. Taking the commonly used Saastamoinen model as an example, the dry and wet delays in the zenith direction of the user are respectively expressed as Td = 0.002277(1 + 0.0026cos2φ + 0.00028H )P   1255 + 0.05 e Tw = 0.002277 T

(1)

where, φ is latitude, H is user antenna height, P is atmospheric pressure, T is atmospheric temperature (Kelvin), and e is local pressure caused by water vapor. Experience shows that the dry delay Td is usually less than 3m at low altitude and can be reduced to 1m at high altitude. The wet delay is usually less than several centimeters. The drier the climate, the smaller the delay. The research [2] shows that, the atmospheric gravity wave disturbance of global diffusion has little impact on the water vapor and temperature of various places, but has caused significant fluctuations in the atmospheric pressure. According to the actual monitoring results, as shown in Fig. 1, severe pressure fluctuations have been observed successively by meteorological stations in the central and eastern parts of China. According to Eq. (1), dry delay fluctuation caused by atmospheric pressure fluctuation is Td = 0.002277(1 + 0.0026cos2φ + 0.00028H )P

(2)

It can be seen from the above that the greater the atmospheric pressure fluctuation, the greater the dry delay fluctuation. It can be seen from Fig. 1 that the atmospheric gravity wave disturbance has an impact on the atmospheric pressure in the central and western regions of China of about 2% ~ 3%, so the resulting dry delay change is about 2% ~ 3% of the original dry delay, that is, millimeter scale, less than the accuracy of the tropospheric model correction. To sum up, the atmospheric gravity wave generated by the Tonga volcanic eruption can have a certain impact on the troposphere, but it has little impact on user positioning services. After the correction of the troposphere model, the relevant impact is basically not perceived.

18

J. Zhang et al.

Fig. 1. Change of atmospheric pressure in parts of China on January 15, 2022 (The horizontal axis is the BeiDou Time, and the vertical axis is the local atmospheric pressure)

2.2 The Analysis of Ionosphere Delay As a dispersive medium, the ionosphere is mainly located in the atmospheric region of 70–1000 km. The free electrons generated by the atmospheric ionization caused by solar radiation affect the propagation of satellite navigation signals. The relationship between the ionospheric delay and the free electron concentration in the ionospheric region is diono =

1 40.3 · ·TEC sin φ f 2

(3)

where, TEC represents the content of free electrons in the ionosphere, φ represents the elevation of the ionospheric puncture point, and f represents the frequency of the satellite navigation signal. It can be seen that the latter two parameters are usually unchanged, so the ionospheric delay disturbance caused by ionospheric fluctuation depends on the change of free electron content, namely diono =

1 40.3 · ·TEC sin φ f 2

(4)

Ionospheric fluctuations may be affected by two types of factors: first, solar activity, which shows the influence of daily cycle, seasonal cycle and 11 year cycle, and is related to geographical location, that is, the lower the latitude, the more intense the impact of solar activity on the ionosphere, and the smaller the impact on high latitude areas; The second

Analysis of the Impact of Tonga Volcano Eruption

19

is the ionospheric traveling wave disturbance caused by volcanic eruptions, earthquakes, tsunamis and other natural activities or violent human activities, such as nuclear tests. In order to analyze the ionospheric disturbance caused by the Tonga volcanic eruption, the influence of solar activity should be excluded first. 2.2.1 The Influence of Solar Activity on the Ionosphere The intense solar activity will produce strong electromagnetic radiation and high-energy particle radiation, and the speed of tens of billions of tons of charged particles generated by it can reach hundreds or even thousands of kilometers per hour. These charged particles will cause strong geomagnetic storms after reaching the earth, and then induce ionospheric storms, affecting short wave radio communication, sky wave over the horizon reflection radar and satellite navigation system services. The prediction of the Solar Activity Prediction Center of the National Astronomical Observatory of the Chinese Academy of Sciences shows that from January 12, 2022, the Sun will have obvious coronal mass ejections. On January 14, the X-ray flare will reach level M, and then there will be strong geomagnetic activity in the next three days, which may cause ionospheric anomalies. Table 1 shows that strong geomagnetic activities occurred on January 14 and 15, and the ionospheric anomaly caused by geomagnetic activities can be excluded through comparative analysis. Four domestic GNSS monitoring stations are selected to calculate the zenith direction total electron content (VTEC) between the four monitoring stations and BeiDou GEO-3 satellite and the fitted ionospheric model results, and analyze the ionospheric changes under different space weather conditions over the station area. By comparing the ionosphere in the quiet period with the ionosphere in the case of strong geomagnetic activity, the daily variation characteristics of the ionosphere caused by geomagnetic activity are obtained, so as to effectively separate the influence of geomagnetic activity and the influence of ionospheric traveling wave disturbance caused by atmospheric gravity waves. As shown in Figs. 2, 3 and 4, according to the distance from Tonga, the fitting results of the VTEC and ionospheric models of the four GNSS monitoring stations in Shantou (about 8800 km), Haikou (about 9300 km), Chengdu (about 10300 km) and Lhasa (about 11500 km) regions on January 12 when the geomagnetic field is quiet and January 14 when the geomagnetic field is active are given in turn. B13 (combination of B1 and B3 frequency points) represents the real-time VTEC extracted from Beidou dual frequency observation measurement, and KLB8 represents the Klobuchar 8 parameter model of the day of calculation. Both can reflect the changes of VTEC. Here we take VTEC of B13 as the main analysis object, and the calculation formula is VTEC13 =

f12 · f32 40.28 · (f12

− f32 )

  sat rec − DCB13 − ε13 · cos zipp P13 − DCB13

(5)

where, VTEC 13 represents the VTEC value of B13 combined observation, Z IPP represents the zenith angle of the satellite at the puncture point, f 1 and f 3 represents the carrier frequencies of B1 and B3 respectively, P13 represents the difference of pseudo range observations of B1 and B3, DCB sat 13 and DCB rec 13 represent the difference

20

J. Zhang et al. Table 1. Solar activity and geomagnetic activity prediction

Release date

CME

Geomagnetic activity

X-ray flare

Proton event

10-January-22

None

Calm

None

None

11- January-22

None

Calm

None

None

12- January-22

CME

Calm

None

None

13- January-22

CME

Calm

None

None

14- January-22

CME

Strong

M1.8

None

15- January-22

CME

Strong

C4.2

None

16- January-22

CME

Strong

C2.6

None

17- January-22

CME

Perturbation

C2.8

None

18- January-22

CME

Medium

M1.5

None

19- January-22

CME

Medium

None

None

code deviation of satellite and receiver respectively, and E13 represents the observation error including multipath. For the calculation method of KLOBUCHAR 8 parameter ionospheric model, refer to the relevant method in [3]. For clear comparison, Fig. 3 has drawn a red dotted line representing the maximum VTEC of B13 on January 12. By comparing the ionospheric conditions on January 12 and 14, it can be seen that when the geomagnetic activity was strong on January 14, the VTEC peak of Shantou Station at low latitudes (20 °N, 33 m above sea level), Haikou Station (23 °N, altitude 25 m) at about 7 o’clock at B13 increase by about 30%, and the impact period is mainly reflected in the BeiDou time (BDT) between 2 o’clock and 14 o’clock during the day, resulting in a possible increase of 2.5m in ionospheric delay for satellite navigation users, and relatively constant ionospheric delay at night (after 14 o’clock). In comparison, the VTEC peak of。mid-latitude Chengdu station (31 °N, altitude 492 m) increased by about 10% around 5 o’clock, the VTEC peak of Lhasa station (30 °N, altitude 3674 m) increased by nearly 50% around 6 o’clock. Because Chengdu and Lhasa are similar in latitude but there exist nearly 3000 m in altitude, this indicates that different altitudes lead to different fluctuations in ionospheric delay caused by solar activity. As a summary, solar activity leads to more active geomagnetic activity, which eventually increases the peak of ionospheric VTEC during the day (2–14 BDT) at low latitudes, but has little effect on the morphology of ionospheric daily cycle changes. In contrast, active geomagnetic activity may have a slightly smaller ionospheric impact at mid-latitudes, but active geomagnetic activity can still be significantly detected at high altitudes. 2.2.2 The Influence of Solar Activity on the Ionosphere According to the monitoring of satellites at home and abroad [4], the shock wave generated by the eruption of Tonga volcano has spread to the whole world. The volcanic

Analysis of the Impact of Tonga Volcano Eruption

(a) Shantou

(b)Haikou

(c) Chengdu

(d) Lhasa

21

Fig. 2. Ionosphere of different observation stations on January 12

eruption generates Rayleigh wave (propagation velocity is about 2.3 km/s) and atmospheric gravity wave (propagation velocity is about 0.35 km/s) respectively. Through horizontal and vertical propagation, the ionospheric traveling wave disturbance caused by the volcanic eruption greatly increases the ionospheric wave fluctuation in a short time. The specific process is described as follows: after the volcanic eruption, the vertical acoustic wave excited by the volcanic eruption first reaches the ionosphere height, and then propagates along the horizontal direction with Rayleigh wave speed. The change of ionosphere caused by the Rayleigh wave is the fastest, and it takes more than one hour to spread from Tonga volcano eruption to China (the nearest distance is about 8000 km). Then, the volcanic eruption produced inclined upward atmospheric gravity waves. After continuous propagation, reflection and refraction, the atmospheric waves caused by the volcanic eruption in Tonga spread to Chinese Mainland in about 7 h. Using the data from four GNSS monitoring stations in Shantou, Haikou, Chengdu and Lhasa to analyze the impact of the Tonga volcanic eruption on the ionosphere in China on January 15, as shown in Fig. 4, the VTEC results after being superimposed by the traveling ionospheric disturbances and geomagnetic activities on January 15 are given according to the distance from Tonga. The maximum VTEC value of B13 on January 12 is shown in the red dotted line in the figure, and the red ellipse represents the abnormal changes compared with the VTEC on January 14.

22

J. Zhang et al.

Maximum of VTEC on Jan.12

(a) Shantou

(b)Haikou

Maximum of VTEC on Jan.12

(c) Chengdu

(d) Lhasa

Fig. 3. Ionosphere of different observation stations on January 14.

First, analyze the abnormal changes of VTEC caused by volcanic eruption. On January 15, except for the normal noon peak of VTEC, the four stations all showed significant abnormal fluctuations of VTEC after 10 o’clock BeiDou Time. According to the propagation speed of atmospheric gravity wave 0.35 km/s, combined with the distance from the four stations to the eruption point of Tonga volcano, the propagation time is about 7 h (Shantou), 7.4 h (Haikou), 8.2 h (Chengdu Station), 9.1 h (Lhasa Station), and the impact time of atmospheric gravity wave is about 11 h (Shantou), 11.4 h (Haikou Station), 12.2 h (Chengdu), 13.1 h (Lhasa), It is basically consistent with the time of the first peak in the ellipses of each sub graph in Fig. 4. This shows that from the monitoring stations in China, the impact of the Tonga volcanic eruption on the ionosphere propagates at the speed of atmospheric gravity waves, resulting in significant abnormal fluctuations in the ionosphere in the middle and low latitudes of China. In addition, according to the analysis of the impact of terrain on the propagation of atmospheric gravity waves [4], atmospheric gravity waves are affected by the terrain. When approaching the high-altitude terrain, they are similar to waves approaching the beach. The closer to the beach, the shorter the wavelength, the greater the amplitude. This can explain the second VTEC spike in (a) and (b) of Fig. 4 (noted in the red ellipses). After the ionospheric traveling wave disturbance generated by atmospheric

Analysis of the Impact of Tonga Volcano Eruption

(a) Shantou

(b)Haikou

(c) Chengdu

(d) Lhasa

23

Fig. 4. Ionosphere of different observation stations on January 15

gravity waves reaches Shantou and Haikou, the ionospheric violent fluctuations caused by topography produce the VTEC peak. Secondly, based on the ionospheric changes monitored by domestic GNSS stations, the impact boundary of Tonga volcanic eruption is analyzed. Taking prefecture level cities in major domestic provinces as samples, it can be found that the ionospheric disturbance almost cannot be perceptible in the area north of 34 °N (Zhengzhou), but are still perceptible to the west of 80。E (Ali). This shows that the impact of Tonga volcanic eruption on the ionosphere in the high latitudes (to the north of 34 °N) is limited, while the impact on the middle and low latitudes (to the south of 34 °N) has exceeded western China (Fig. 5). Thirdly, analyze the influence of geomagnetic activity on January 15. Compared with the diurnal ionospheric amplitudes on January 15 and 12, the VTEC peak values of Shantou Station and Haikou Station B13 in the low latitude region increased by about 20%. In the mid-latitudes, the peak VTEC of Chengdu Station B13 increased by about 80%, while the peak VTEC of Lhasa Station B13 increased by about 110%. This suggests that the continued geomagnetic activity on January 15 still had an impact on the ionosphere, with a slight decline in the VTEC peak at lower latitudes, but a higher ionospheric peak affected by geomagnetic activity at 4–7 o’clock in the mid-latitudes, especially a larger increase in VTEC at Lhasa. In summary, the diurnal variation of

24

J. Zhang et al.

(a) Zhengzhou

(b)Ali

Fig. 5. Ionosphere of Zhengzhou and Ali on January 15

VTEC at each station in Fig. 4 is a combination of geomagnetic activity and Tonga volcanic eruption.

3 Impact of Tonga Volcanic Explosion on Satellite Navigation Service and Monitoring In addition to the transmission of satellite navigation signals, the ionospheric traveling wave disturbance caused by the Tonga volcanic eruption may affect the positioning and navigation service performance finally obtained by satellite navigation users. First, analyze and compare the impact of volcanic eruptions on users around Tonga and in China, and then discuss the monitoring of relevant impacts. 3.1 Impact on User Navigation and Positioning Services The GPS dual frequency (L1 and L2, the same below) observation data sampled at Tonga Station (TONG) about 70 km away from the Tonga volcanic eruption were selected, and the dynamic precise single point positioning (PPP) was used to calculate the positioning results on January 15. Since the data sampling frequency is less than 1 s in [1], the dynamic precision single point positioning results are different from each other, similar to the impact of the data sampling frequency in [5]. It can be seen that in the positioning results in Fig. 6, there is a significant displacement in the elevation direction, up to 9 m, and the positioning results in the horizontal direction are also affected by the volcanic eruption, with a maximum of more than 2 m (Fig. 6). According to previous analysis, Tonga volcanic eruption has had a significant impact on the ionosphere, which may affect the navigation and positioning services of ordinary users. Taking Tonga Station as an example, GPS dual frequency positioning results of users are calculated by three methods, namely, no correction of ionosphere, dual frequency de ionosphere combination, and correction of broadcast ionosphere model (Klobuchar). The positioning results are shown in Fig. 7, and the positioning accuracy is shown in Table 2. The above results show that without ionospheric correction, ionospheric disturbances caused by volcanic eruptions have no significant impact on

Analysis of the Impact of Tonga Volcano Eruption

25

Fig. 6. Processing results of dynamic precision single point positioning at Tonga station on January 15 (Horizontal coordinate-axis sampling interval is 30 s)

dual frequency positioning results; The combination of dual frequency de ionosphere amplifies the observation noise, making the ionosphere correction only improve the sky direction by 26%, and the horizontal positioning accuracy is basically equivalent to the result without correction; However, the adoption of the radio ionospheric model correction can significantly improve the sky direction and horizontal direction, especially the sky direction accuracy is increased by about 60%. Further, 9 days from January 11–19 are selected to compare the difference of elevation and horizontal positioning results of Tonga Station with 3 different ionospheric treatment methods, and confirm the impact of ionospheric fluctuations. It can be seen from Fig. 8 that, from a comprehensive comparison, the volcanic eruption on January 15 has little impact on the positioning accuracy in the north and east directions, and only the accuracy in the sky direction decreases significantly compared with other dates, which indicates that the impact of ionospheric fluctuations is mainly reflected in the sky direction. In the positioning results based on the two ionospheric correction methods, the combination of the ionosphere has only slightly improved the north direction and sky direction in general, but deteriorated the east direction positioning results; The broadcast ionosphere model correction can weaken the influence of the ionosphere and effectively improve the positioning accuracy in all directions, especially the sky direction accuracy is increased by about 74% on average. 3.2 Monitoring of Ionospheric Impacts It can be seen from the above that GNSS dynamic PPP can accurately perceive the impact of volcanic eruptions, while ordinary single point positioning cannot be achieved. Further, it can be seen from the positioning results in Fig. 7 that the sky direction is most affected; The results in Fig. 8 show that after the correction of the broadcast ionosphere model, the correction effect of the sky direction is also the most obvious. On this basis, we further calculate the dynamic PPPs of stations around Tonga and in China, mainly in

26

J. Zhang et al.

(a) Uncorrected ionosphere

(b) Dual frequency correction

(c) Broadcast model correction Fig. 7. Dual frequency positioning results of Tonga station with different ionosphere processing modes on January 15 (Horizontal coordinate-axis sampling interval is 30 s)

Table 2. Positioning accuracy RMS of different ionospheric processing methods (m) Uncorrected dual frequency

Dual frequency correction

Broadcast model correction

N

1.5481

1.4791

1.2993

E

1.5307

1.7825

1.2032

U

4.7404

3.7545

2.9630

the sky direction, and observe the transmission of the impact of Tonga volcanic eruption. Five IGS stations, Fiji, Micronesia, Ogasawara, Taipei and Lhasa, are selected from near to far from Tonga, and the dynamic PPP results on January 15 are calculated respectively using the precise ephemeris. As can be seen from Fig. 9, based on the celestial positioning results of dynamic PPP, it can be monitored that the celestial positioning disturbance caused by Tonga volcanic eruption spreads from near to far. This also shows the feasibility of monitoring the ionospheric abnormal fluctuations based on the celestial positioning results of dynamic

Analysis of the Impact of Tonga Volcano Eruption

(a)North Positioning accuracy

27

(b) East Positioning accuracy

（c）Up Positioning accuracy Fig. 8. Comparison of positioning results of different ionosphere processing methods from January 11–19

PPP. It should be noted that the dynamic PPP can monitor the anomaly of the celestial positioning result caused by the ionospheric change, which is also mainly due to the large ionospheric disturbance caused by the volcanic eruption on January 15.

28

J. Zhang et al.

Fig. 9. Impact of Tonga volcanic explosion on dynamic PPP orientation results at each Site (Horizontal coordinate-axis sampling interval is 30 s)

4 Conclusion The Tonga volcanic eruption on January 15, 2022 has an important impact on the earth’s atmosphere. This paper theoretically analyzes and computes the troposphere and ionosphere fluctuations caused by the volcanic eruption from the perspective of satellite navigation services, discusses the impact of volcanic eruption and its correction methods from the perspective of user navigation and positioning, and studies the means of monitoring the impact of volcanic eruption. The conclusions are as follows. (1) The Tonga volcanic eruption caused a wide range of pressure fluctuations, which led to tropospheric fluctuations. According to the pressure monitoring in China, the impact of tropospheric fluctuations caused by 2–3% pressure fluctuations is very small. (2) The volcanic eruption caused a wide range of ionospheric fluctuations. The monitoring results show that the ionosphere in the middle and high latitudes (north of 34 °N) of China is limited, while the ionosphere in the middle and low latitudes (south of 34 °N) is greatly affected. The ionospheric VTEC along the southeast to northwest direction successively shows abnormal wave peaks, which are equivalent to the propagation speed of atmospheric gravity waves generated by the volcanic eruption. (3) From the user navigation and positioning results, the volcanic eruption caused significant fluctuations in the user’s sky orientation positioning results. By using the broadcast ionospheric model correction, the dual frequency single point positioning accuracy can be significantly improved, while the dual frequency combined ionospheric method has limited improvement effect.

Analysis of the Impact of Tonga Volcano Eruption

29

(4) Using dynamic PPP, we can calculate the large scale satellite navigation and positioning service anomaly in middle and low latitudes caused by volcanic eruption. Therefore, based on high-precision dynamic positioning technologies such as dynamic PPP or PPP-RTK, real-time monitoring and perception of anomalies serving satellite navigation can be realized.

References 1. Yufeng H, Zhenhong L et al (2022) Comprehensive remote sensing fast interpretation analysis of Tonga volcanic eruption in 2022. J Wuhan Univ Inf Sci Ed 47(2):242–251 2. Misra P, Enge P (2006) Global positioning system: signals, measurements, and performance. Ganga-Jamuna Press, Lincoln 3. Hongping Z (2006) Research on China’s regional ionosphere monitoring and delay correction based on ground based GPS. Graduate School of Chinese Academy of Sciences (Shanghai Astronomical Observatory) 4. Xingbao W (1996) The influence of topography on the propagation and development of gravitational inertial waves. Meteorol Sci 16(3):1–10 5. Xiaohong Z, Fei G, Xingxing L (2010) Impact of IGS satellite clock error product sampling interval on PPP accuracy. J Wuhan Univ Inf Sci Ed 35(2):152–155 6. Baocheng Z (2012) Theoretical methods and application research of GNSS nondifference noncombination precision single point positioning. Graduate School of Chinese Academy of Sciences 7. Rongxin F, Chuang S, Weiwei S et al (2013) Real time GNSS seismometer system implementation and accuracy analysis. J Geophys 56(2):450–458 8. Guanbin W, Chen Junping W, Xiaomeng, et al (2020) Atmospheric correction model based on non-difference and non-combination PPP-RTK and its performance verification. J Survey Mapping 49(11):1407–1418

Prediction of Satellite Solar Radiation Pressure Parameters Based on Recurrent Neural Network Jianbing Chen1

, Lei Chen2(B) , Shanshi Zhou3,4 , and Shuai Huang1

1 China Top Communication Co., Ltd., Beijing 100088, China 2 Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China

[email protected]

3 Shanghai Astronomical Observatory, CAS, Shanghai 200030, China 4 Shanghai Key Laboratory of Space Navigation and Positioning Techniques, Shanghai

Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China

Abstract. The BeiDou Navigation Satellite System (BDS) has been put into operation. With the system’s continuous operation, the data accumulated by the BDS has a growing trend. At the same time, the service is diversified and refined. Due to the huge system, operation and maintenance are increasingly difficult. To adapt to the high-quality and refined operation of the BDS, it is proposed to introduce artificial intelligence-related technologies to assist the BDS in achieving highquality intelligent operation and maintenance. Based on the three network models of recurrent neural networks (RNN, LSTM, and GRU), this paper models and analyzes the BDS satellite’s solar radiation pressure parameters. The optimal model and hyperparameters are obtained through data mining and analysis and model training, and the prediction model is used to verify the measured data. It is found that the prediction accuracy of the three recurrent neural network models is equivalent, the average accuracy is more than 90%, and the prediction accuracy from high to low is GRU, RNN, and LSTM. The prediction method based on the cyclic neural network adopted in this paper can be applied to the state prediction of time series data of the BDS and has certain reference significance for the construction of intelligent operation and maintenance of the BDS. Keywords: Beidou operation and maintenance · Service status · Artificial intelligence · Neural network · Deep learning · LSTM · RNN · GRU · Big data · Prediction verification

1 Introduction With the continuous application and development of artificial intelligence technology and big data technology, the development needs of intelligent operation and maintenance business [1]. Isolated forest algorithm, Apriori algorithm, support vector machine [2] and other machine learning algorithms as well as deep learning network models have been applied to intelligent operation and maintenance research. Literature [3] uses the LSTM network to predict faults, which verifies that the prediction accuracy of the cyclic neural network model is better than the traditional machine learning algorithm. © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 30–39, 2024. https://doi.org/10.1007/978-981-99-6944-9_3

Prediction of Satellite Solar Radiation Pressure Parameters

31

In GNSS data processing, RNN and LSTM (cyclic neural network) are used to analyze and predict sequence data [4] and are also applied in network firewall [5] and traffic monitoring [6]. However, research shows that RNN will forget the previous state information with time, so LSTM (short and long-term cyclic neural network) is introduced. In addition to being applied to image analysis, document summarization, speech recognition, handwriting recognition, and other fields, the LSTM network also performs well in time series data prediction. It is mainly used to describe the relationship between the current data and the previous input data, use its memory ability to save the state information before the input network [7], and use the previous state information to influence the exact value and development trend of the subsequent data, and use LSTM to achieve good results in the analysis and prediction of satellite data based on long-term multi data sources [8]. GRU is an improved network model of LSTM, with only two gates and higher operational efficiency. It is also applied in message middleware and log anomaly detection [9], solar radiation prediction [10], traffic flow prediction [11], and GNSS landslide displacement prediction [12] and has achieved good results. By introducing the deep learning neural network [13], we can realize the early perception of the state of the BDS, prepare the response plan in advance on the system side, deeply tap the system potential, and optimize the system operation and maintenance accuracy. On the user side, by sensing the satellite positioning accuracy in advance, the constellation configuration is optimized during positioning to achieve better positioning accuracy. In addition, the prediction of solar radiation pressure parameters of satellites can provide important data support for the long-term autonomous operation of satellites in orbit.

2 Research Review Domestic scholars such as Shi Chuang and Xiao Yun have studied the research progress and development trend of satellite’s solar radiation pressure modeling [14], and some research refined on these models [15]. According to the amount of satellite information used, the modeling strategy can be divided into There are three categories: pure empirical methods do not use the physical information of solar radiation pressure; semi-empirical methods consider part of the satellite’s geometric structure and surface material information; and pure analytical methods use all available satellite structures, surface materials, attitudes, space environments, etc. information. Although there is no clear boundary between these three types of methods, the classification method helps to explain the mainstream modeling methods, but these methods are all modeling and calculating the solar radiation pressure parameter to obtain the correct radiation pressure value. Through these solar radiation pressure models On the premise of not needing to know the satellite structure and material information, the light pressure perturbation force experienced by the satellite during orbital motion can be simulated more accurately by only using some solar radiation pressure parameters. At present, the ECOM radiation pressure model is one of the most widely used empirical solar radiation pressure models, but how to obtain these parameters, in the past, solar radiation pressure can only be calculated with the measured parameter values, and there is no relevant research to use some methods to forecast these parameters.

32

J. Chen et al.

In this situation, based on the principles and methods of recurrent neural networks, this paper uses recurrent neural networks such as RNN, LSTM, and GRU to model and evaluate the accuracy and usability of the model by comparing the predicted parameter values with the actual parameter values. Has reference significance in the occasions where satellite solar radiation pressure prediction is required.

3 Recurrent Neural Network Model RNN, LSTM, and GRU are typically three types of models in recurrent neural net-works [16–18], which are introduced as follows: RNN is a special neural network structure. According to the view, “human cognition is based on past experience and memory”. The reason why RNN is called a cyclic neural network is that the current output of a sequence is also related to the previous output. The specific manifestation is that the network will remember the previous information and apply it to the calculation of the current output; that is, the nodes between the hidden layers are not connected but connected, and the input of the hidden layer includes not only the output of the input layer but also the output of the hidden layer at the previous time. The long-term and short-term memory network (LSTM), a cyclic neural network, is suitable for processing and predicting important events with relatively long intervals and delays in time series [11]. The main purpose is to solve the gradient disappearance and gradient explosion problems in the long sequence training process. In short, LSTM can perform better in longer sequences than ordinary RNN, LSTM has been applied in many fields of science and technology [19]. The output and state of the LSTM model are calculated by formula (1)–(6). it = σ (Wxixt + Wimmt − 1 + Wicct − 1 + bi)

(1)

ft = σ (Wxfxt + Wfmmt − 1 + Wfcct − 1 + bf )

(2)

ct = ft  ct − 1 + it  tanh(Wcxxt + Wcmmt − 1 + bc)

(3)

ot = σ (Wxoxt + Wommt − 1 + Wocct − 1 + bo)

(4)

mt = ot  tanh(ct)

(5)

yt = Wymmt + by

(6)

In the formula: it, ot and ft indicate that the output value at t is the input gate, the forgetting gate and the output gate respectively; mt Indicates the activation state of the memory cell at t; ct Indicates the neuron at t; W Indicates the weight coefficient matrix of the LSTM model; b Indicates the bias; tanh and σ Represents the hyperbolic tangent activation function and the activation function.

Prediction of Satellite Solar Radiation Pressure Parameters

33

LSTM is widely used in many sequence tasks (including natural gas load forecasting [20], stock market forecasting, language modeling [21], and machine translation [22]), and performs better than other sequence models (such as RNN), especially when there is a large amount of data. LSTM is carefully designed to avoid the gradient disappearance of RNN. The main limitation of the vanishing gradient is that the model cannot learn longterm dependence. However, by avoiding the gradient disappearance problem, LSTM can store more memory (hundreds of time steps) than conventional RNN. Compared with the RNN, which only maintains a single hidden state, the LSTM has more parameters and can better control which memories are saved and which are discarded at a specific time step. For example, the hidden state must be updated in each training step, and the RNN cannot determine whether the memory is saved and the memory is discarded. GRU (recurrent gate unit) is a kind of recurrent neural network (RNN). Like LSTM, it is also proposed to solve the gradient problem in long-term memory and backpropagation. Three gate functions are introduced into LSTM: input gate, forgetting gate, and output gate to control input value, memory value, and output value. In the GRU model, there are only two doors: an updated door and a reset door. The structure of GRU input and output is similar to that of ordinary RNN, and its internal idea is similar to that of LSTM.

4 Prediction Method of Satellite Solar Radiation Pressure Parameters Based on Recurrent Neural Network The BDS uses the ECOM model to model the solar radiation pressure, The ECOM model was described by five parameters: x-bias, y-bias, z-bias, x-cos, and x-sin. After obtaining the satellite solar radiation pressure parameters, first, preprocess the obtained satellite solar radiation pressure parameter data, remove the invalid value and zero value that are obviously out of range, then normalize the data, and divide the data into training data and verification data by 80% and 20%. Then, the RNN, LSTM, and GRU neural network models are designed and established. The model is trained with the training data. After multiple iterations, the optimal model is obtained. The model is used to predict the validation data. Then the predicted value is compared with the actual value to score the model and evaluate its accuracy of the model. After several times training and optimization, a ten-layer neural network model design is finally obtained. The layers of three different types of neural networks are consistent, and the neural network units and dropout values are consistent. The corresponding network layer types are replaced to control the experimental variables. The neural network model comprises an input layer, simple RNN layer or LSTM layer or GRU layer, a dropout layer, and a full link layer (dense). There are ten layers of network structure, the only distinction among the three neural network models is the layer type of layer 1, 3, 5, 7, 8 (signed with different colors),and all the parameters are the same. The network design structure and network parameters are shown in Fig. 1.

34

J. Chen et al.

Fig. 1. Design diagram of RNN/LSTM/GRU neural network model

5 Test Results and Analysis After the model training is completed, the model’s accuracy is compared and evaluated with the measured data. The optimized prediction results of the three kinds of recurrent neural network solar radiation pressure parameter models are shown in Tables 1, 2, and 3. Statistics are made on the total data volume and accuracy (1-RMSE). As show in the Table 1, the average accuracy of LSTM is 94.13%, the highest accuracy is 98.60%, and the lowest accuracy is 86.87%. As show in the Table 2,the average accuracy of RNN was 94.52%, the highest accuracy was 99.25%, and the lowest accuracy was 88.28%; As show in the Table 3,the average accuracy of GRU was 94.56%, the highest accuracy was 99.27%, and the lowest accuracy was 88.33%. Through this experiment, the following conclusions are found: 1. The prediction accuracy of most data is above 90%, and the average accuracy is 94.40%, the residual error of about 3d-11 m/s2 , and the effect on the orbit integration for one day is about dm level. 2. The accuracy of a few satellite data is over 95%, which is characterized by a strong periodic trend. 3. For the data of this level dimension and order of magnitude, the difference between LSTM and RNN and the prediction accuracy of GRU is not particularly large. GRU has a slight leading advantage, RNN is second, and LSTM is slightly poor. however, the accuracy difference between the three network models is very small.

Prediction of Satellite Solar Radiation Pressure Parameters

35

Table 1. Summary of solar radiation pressure LSTM training results. Sat no.

Amount

X-bias (%)

Y-bias (%)

Z-bias (%)

X-cos (%)

19

614

96.40

93.13

96.89

97.09

20

465

96.00

91.37

96.69

93.38

21

454

96.88

86.87

96.38

92.01

25

810

94.13

93.90

96.29

96.51

26

810

94.13

94.68

96.28

96.20

27

810

94.70

93.47

97.08

96.82

28

810

95.64

90.97

97.26

96.55

29

810

92.25

90.46

93.78

93.37

30

810

94.35

93.74

95.74

95.10

36

810

94.15

92.73

96.50

95.61

37

810

94.20

93.59

96.12

95.14

38

810

95.36

94.30

95.79

91.97

39

810

94.05

94.35

96.45

92.75

40

810

93.27

94.50

97.88

95.67

41

810

90.78

93.67

94.84

93.57

42

810

93.06

93.58

94.87

94.48

43

810

94.43

95.56

94.62

93.73

44

810

94.74

92.35

94.33

95.84

45

810

92.65

91.07

96.68

93.45

46

810

92.71

91.23

96.23

95.42

47

810

92.28

93.48

94.25

93.25

48

429

93.65

89.35

94.00

93.43

49

426

94.45

89.43

92.70

91.64

50

440

91.22

92.06

93.16

92.36

51

440

87.79

93.14

89.60

89.76

52

463

95.83

98.60

95.91

94.98

53

463

96.09

98.28

95.83

94.91

Average

695

93.90

92.96

95.41

94.26

Average

94.13%

6 Conclusions Based on the three network models of deep learning recurrent neural networks (RNN, LSTM, and GRU), this paper model the solar radiation pressure parameters and positioning accuracy of the BDS satellites and uses the measured data of the BDS from 2019

36

J. Chen et al. Table 2. Summary of solar radiation pressure RNN training data results.

Sat no.

Amount

X-bias (%)

Y-bias (%)

Z-bias (%)

X-cos (%)

19

614

97.02

94.18

97.57

97.42

20

465

96.37

93.40

97.14

92.94

21

454

96.82

93.53

97.00

93.31

25

810

94.14

93.80

96.55

96.83

26

810

94.20

94.65

96.39

96.76

27

810

94.54

93.70

97.61

96.74

28

810

95.61

90.65

97.60

96.59

29

810

92.47

90.48

94.68

93.43

30

810

94.44

92.91

95.85

94.93

36

810

94.16

92.84

96.55

95.85

37

810

94.18

93.82

96.59

95.26

38

810

95.61

94.15

95.07

92.48

39

810

94.00

94.48

96.78

93.02

40

810

93.83

94.45

97.96

95.73

41

810

90.76

93.50

95.72

92.40

42

810

93.22

93.69

95.12

93.28

43

810

94.45

95.48

95.10

94.41

44

810

94.73

92.34

94.21

95.35

45

810

92.72

90.77

96.67

93.57

46

810

92.73

91.04

96.93

95.35

47

810

92.37

94.08

95.55

93.55

48

429

93.77

91.88

94.30

93.82

49

426

95.75

91.98

93.18

91.66

50

440

91.22

92.19

95.02

93.50

51

440

88.28

93.73

91.40

90.65

52

463

96.00

99.25

96.02

97.48

53

463

96.15

99.15

96.12

97.41

Average

695

94.06

93.56

95.88

94.58

Average

94.52%

to 2021 for data mining analysis and model training to obtain the best model and model parameters, and uses the prediction model to predict and verify the measured data, The prediction accuracy of the three cyclic neural network models is equivalent, with an average accuracy of more than 90%. The prediction accuracy ranges from high to low, including GRU, RNN, and LSTM. Moreover, the calculation speed of GRU is faster, and

Prediction of Satellite Solar Radiation Pressure Parameters

37

Table 3. Summary of solar radiation pressure GRU training data summary. Sat no.

Amount

X-bias (%)

Y-bias (%)

Z-bias (%)

X-cos (%)

19

614

97.06

94.08

97.89

97.33

20

465

96.53

93.48

97.74

93.25

21

454

96.90

91.11

96.66

93.47

25

810

94.16

94.07

96.62

96.80

26

810

94.21

94.79

96.42

96.80

27

810

94.79

93.66

97.39

96.90

28

810

94.81

91.04

97.62

96.59

29

810

91.91

90.56

94.76

93.42

30

810

94.58

93.63

95.82

95.15

36

810

94.16

92.80

97.03

95.63

37

810

94.09

93.65

96.80

95.37

38

810

95.53

94.23

96.12

92.32

39

810

94.03

94.44

97.03

93.00

40

810

93.80

94.56

97.80

95.59

41

810

90.78

93.60

96.24

93.39

42

810

93.32

93.65

95.21

93.96

43

810

94.41

95.63

95.22

94.23

44

810

94.80

92.36

94.29

95.84

45

810

92.83

91.31

96.65

93.56

46

810

92.80

91.21

97.18

95.40

47

810

92.37

94.44

95.45

93.56

48

429

93.77

90.87

94.24

93.72

49

426

95.81

90.92

93.18

92.04

50

440

91.24

92.06

95.06

93.37

51

440

88.33

93.52

91.62

90.93

52

463

95.94

99.27

95.93

97.80

53

463

96.19

99.26

95.93

97.36

Average

695

94.04

93.49

96.00

94.70

Average

94.56%

the training loss and verification loss can be minimized as quickly as possible. Considering the vast data volume of the BDS, It is the best choice to use the GRU cyclic neural network for state prediction, which has specific reference significance for the evolution and promotion of intelligent operation and maintenance of the BDS.

38

J. Chen et al.

Acknowledgments. This study was supported by the National Natural Science Foundation of China (Grant Nos. 12173072).

References 1. Vichare NM, Pecht MG (2006) Prognostics and health management of electronics. IEEE Trans Compon Packag Technol 29(1):222–229 2. Sapankevych NI, Sankar R (2009) Time series prediction using support vector machines: a survey. IEEE Comput Intell Mag 4(2):24–38 3. Wang X, Wu J, Liu C, et al (2016) Application of singular spectrum analysis for failure time series. J Beijing Univ Aeronaut Astronaut 42(11):2321–2331 4. Gao W, Gao J, Yang L, Wang M, Yao W (2021) A novel modeling strategy of weighted mean temperature in China using RNN and LSTM. Rem Sens 13:3004. https://doi.org/10.3390/rs1 3153004 5. Zhu S, Du R, Chen J, He K. Scheme for hardening WAF based on the RNN model. Comput Eng. https://doi.org/10.19678/j.issn.1000-3428.0063518 6. Xinfeng X, Wei X. Research on network abnormal traffic detection based on improved RNN. Wireless Internet Technology. Jiangsu Golden Shield Detection Technology Co., Ltd., Nanjing 210042, China 7. Chen K, Zhou Y, Dai F (2015) A LSTM-based method for stock returns prediction: a case study of China stock market. In: IEEE international conference on big data, pp 2823–2824 8. Ye W, Zhang F, Du Z (2022) Machine learning in extreme value analysis, an approach to detecting harmful algal blooms with long-term multisource satellite data. Rem Sens 14:3918. https://doi.org/10.3390/rs14163918 9. Jia X, Fang W, Zhang W. Log anomoly detection of distributed system based on message middle-ware and GRU. Comput Digital Eng 1. School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing 210044; 2. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081; 3. Nanjing Xinda Institute of Meteorological Science and Technology Co., Ltd., Nanjing 210044 10. Manguo Z, Yanguo H, Jinfeng D. Short term prediction of soral irradiance based on GRURF model. Acta Energiae Solaris Sinica. School of Electrical Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou 341000, China 11. Lisheng Y, Yangyang W. Traffic flow combination prediction model based on improved VMDGAT-GRU [J/OL. J Electron Measur Instrum. https://kns.cnki.net/kcms/detail/11.2488.TN. 20220705.0938.006.html 12. Jiang Y et al (2022) A graph convolutional incorporating GRU network for landslide displacement forecasting based on spatiotemporal analysis of GNSS observations. Rem Sens 14:1016. https://doi.org/10.3390/rs14041016 13. Shi C, Xiao Y, Fan L, et al (2022) Research progress of radiation pressure modelling for navigation satellites. Acta Aeronautica et Astronautica Sinica 43(10):527389(in Chinese). https://doi.org/10.7527/S10006893.2022.27389 14. Li J (2022) Solar radiation pressure modeling for Beidou-3 satellites. School of Geodesy and Geomatics Wuhan University, Wuhan 15. Tang Y, Jiang J, Liu J, Yan P, Tao Y, Liu J (2022) A GRU and AKF-based hybrid algorithm for improving INS/GNSS navigation accuracy during GNSS outage. Rem Sens 14:752. https:// doi.org/10.3390/rs14030752 16. Xia Yu-lu (2019) A review of the development of recurrent neural network. Comput Knowl Technol 1009–3044. Central China Normal University, Wuhan 430079, China

Prediction of Satellite Solar Radiation Pressure Parameters

39

17. Jian-wei L, Zhi-yan S. Overview of recurrent neural networks. Control Decis. https://doi.org/ 10.13195/j.kzyjc.2021.1241 18. Yang L, Wu Y, Wang J, Liu Y. Research on recurrent neural network. 1. College of Electronics and Information Engineering, Tongji University, Shanghai 201804, China; 2. College of Economics and Management, Tongji University, Shanghai 201804, China 19. Wang J, Jiang W, Li Z, Lu Y (2021) A new multi-scale sliding window LSTM framework (MSSW-LSTM): a case study for GNSS time-series prediction. Rem Sens 13:3328. https:// doi.org/10.3390/rs13163328 20. Huang J-Q, Qin L-x (2021) Study on association prediction of temperature and precipitation using ALSTM. J Guangxi Univ (Natural Science Edition) 46(4):1024–1035 21. Yi W, Juan X, Ying C. Deep neural networks language model based on CNN and LSTM hybrid architecture. School of Information Management, Nanjing University, Nanjing 210023 22. Tian H.-N, Guo X, Yuan W. Research on interactive neural machine translation method based on LSTM. [A] 1. Hebei University of Technology, College of Artificial Intelligence and Data Science, Tianjin 300130, China; 2.Qinhuangdao Research Institute, National Rehabilitation Auxiliary Research Center, Qinhuangdao Hebei 066000, China

Real-Time Kinematic Orbit Determination of GEO Satellite with the Onboard GNSS Receiver Xinglong Zhao1 , Xiaogong Hu2,3(B) , Shaojun Bi1 , Huichao Zhou1 , Zheng Song1 , Qiuli Chen1 , Meng Wang4 , Shanshi Zhou2,3 , Chengbin Kang1 , Haihong Wang1 , and Gong Zhang1 1 Institute of Telecommunication and Navigation Satellites, China Academy of Space

Technology, Beijing 100094, China 2 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China

[email protected]

3 Shanghai Key Laboratory for Space Positioning and Navigation, Shanghai 200030, China 4 Space Star Technology Co., Ltd., Beijing 100086, China

Abstract. The spacecraft has widely used Global Navigation Satellite System (GNSS) in precise navigation, attitude determination, formation flight and time synchronization. For high earth orbit spacecraft, the space environment is very challenging for GNSS receivers to work. Taking the No. 2 Telecommunication Technology Test satellite (TTS-2) as an example, under both preprocessing strategies of the carrier phase smoothing pseudorange algorithm and single receiver fault detection algorithm based on residual, this paper analyzes the real-time kinematic orbit determination capability of Geostationary Earth Orbit (GEO) satellite by standard point positioning (SPP) and kinematic precise positioning (KPP). The experimental results show that the 3D error of the SPP kinematic orbit is about 100 m, and that of the KPP kinematic orbit is about 20 m. Carrier phase smoothing pseudorange and single receiver fault detection algorithm based on residual are used to preprocess the onboard GPS data, they reduce the 3D error of the SPP kinematic orbit to 49.4 m and radial error to 46.4 m, which decreased by 45%. And they reduce the 3D error of the KPP kinematic orbit to 16 m, a reduction of 23%, and the radial error to 10 m, a reduction of 34%. The kinematic orbit is convenient to implement the hardware and software of the onboard GNSS receiver, and the real-time and high-precision position provides favorable conditions for the accurate orbit control of the spacecraft. This paper’s research work and conclusion have important reference value and significance for the application of high-orbit GNSS, and will promote the development of space-borne GNSS application. Keywords: GEO satellite · Kinematic orbit · Standard point positioning · Kinematic precise positioning

© Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 40–48, 2024. https://doi.org/10.1007/978-981-99-6944-9_4

Real-Time Kinematic Orbit Determination of GEO Satellite

41

1 Introduction The rapid development of modern space technology has promoted the application of Global Navigation Satellite System (GNSS) in the navigation and timing service of spacecraft. The most typical example is to provide the real-time precise Orbit Determination (OD) service for Low Earth Orbit (LEO) satellites through the onboard GNSS receiver. Similarly, GNSS can provide the same service to high earth orbit spacecraft. In the 1980s, the United States proposed to apply GPS to navigation and timing services of the spacecraft in low orbit, high orbit and even deep space. Thereafter, the United States conducted the first application experiment to test the navigation performance of receiver in processing sidelobe signals for high-orbit satellite [1, 8]. Subsequently, the United States and Europe continued to develop high-orbit GNSS receivers and on-orbit tests [3, 4]. Researchers have found in the processing of GNSS measurement data by high-orbit spacecraft that combining GNSS measurement and simple dynamics model with the adaptive orbital filter could achieve higher position accuracy [5]. In 2014, China’s Chang’E-5T spacecraft was equipped with a GNSS receiver with high sensitivity for navigation. The test results show that the position accuracy is less than 42 m, and the velocity accuracy is less than 0.1 m/s [6, 7]. Dr. Fan Min from Shanghai Astronomical Observatory of Chinese Academy of Sciences conducted detailed processing of onboard GNSS data of Chang’E-5T spacecraft, and verified the effectiveness of GNSS supporting spacecraft OD during lunar exploration and return [8]. The No. 2 Telecommunication Technology Test Satellite (TTS-2), launched in 2017, has carried a GPS receiver and solved the autonomous navigation in orbit. According to detailed analysis of the data quality of TTS-2, the consistency of orbit overlapping in the 30 h arc is 2.14 m, which significantly improves the navigation accuracy compared with the traditional ground-based OD technology of Geostationary Earth Orbit (GEO) satellite [9, 10]. Meanwhile, by using GEO satellite as occultation, ionospheric inversion can be achieved with its onboard GNSS data [11]. This paper collects the TTS-2 onboard GPS data for 8 days in 2018 to study the real-time kinematic OD performance of GEO satellite. Firstly, TTS-2 and its OD results are introduced. Then, the carrier phase smoothing pseudorange algorithm and the single receiver fault detection algorithm based on residual are introduced. Then, by designing test scenarios, different algorithms were used to preprocess the onboard GPS data, and the real-time kinematic orbit accuracy of GEO satellite realized by standard point positioning (SPP) and kinematic precise positioning (KPP) was analyzed. The research work of this paper will have important reference value and significance for the application fields of GEO satellite real-time OD based on the GNSS receiver.

2 TTS-2 TTS-2 is a GEO satellite launched in 2017, equipped with a GPS receiver whose antenna points at the center of the earth to real-time collect GPS measurements. Because the main lobe navigation signal beam angle of GPS satellite is about 42.6°, and the orbit altitude of TTS-2 is higher than that of GPS satellites, the onboard receiver receives the main lobe and side lobe signals from the GPS satellite of the opposite side of the earth, as shown in

42

X. Zhao et al.

Fig. 1. In this paper, the single frequency C1/L1 navigation signal were collected, with a sampling interval of 2 s and a starting and ending time of day of year (DOY) 100 16:00 to DOY 108 16:00 in 2018.

Fig. 1. The main lobe and side lobe GPS signals from the opposite side of the earth of TTS-2

2.1 Onboard GPS Data In the onboard data processing, measurement noise is directly related to the receiver position results. The noises of the GPS pseudorange and carrier phase of TTS-2 are analyzed. The onboard GPS C1 pseudorange noise and L1 carrier phase noise of TTS-2 are calculated by the third-difference algorithm of epochs, and the results are shown in Fig. 2. It could be found that the noise variation of C1 pseudorange after removing the gross error is −40 to 40 m, and the noise variation of L1 carrier phase after removing the gross error is −6 to 6 cm. According to statistics, the Root Mean Square (RMS) values is 7.3 m for C1 noise and 1.3 cm for L1 noise. 2.2 Post-Process OD Results The high-precision dynamic orbit of TTS-2 could be obtained by the onboard GPS data, with an accuracy of up to meter-level [9, 10]. After employing the OD method with the prior dynamic parameter error constraint [12], the consistency of the orbit overlapping arcs of TTS-2 is about 1 m, and the optimal consistency can reach submeter level, as shown in Fig. 3. So, the post-process orbit for GEO satellites is sufficed as the reference to evaluate the real-time kinematic orbit accuracy.

Real-Time Kinematic Orbit Determination of GEO Satellite

43

Fig. 2. The time series of C1 noise and L1 noise. (The different colors represent different GPS satellites’ data)

Fig. 3. The consistency of the orbit overlapping arcs. The orbits were solved by the orbit determination method with the prior dynamic parameter error constraint.

3 Test Strategy 3.1 Data Preprocessing In this paper, carrier phase smoothing pseudorange algorithm and single receiver fault detection algorithm based on residual are used for data preprocessing. Carrier Phase Smoothing Pseudorange Algorithm. In the absence of the carrier phase cycle jump, the noise level of carrier phase is two orders of magnitude lower than that

44

X. Zhao et al.

of pseudorange, and the noise level of pseudorange can be reduced by sliding the time window of carrier phase. Single Receiver Fault Detection Algorithm Based on Residual. This algorithm is firstly proposed in the application of the receiver autonomous integrity monitoring (RAIM). When the number of visible satellites of the receiver is greater than 4, which is signed as N, N visible satellites are arranged and combined. There are C n 4 + C n 5 + … + C n n combination cases for the point positioning solutions. The pseudorange residual statistics for each combination are constructed to detect the fault satellite. In this paper, the single receiver fault detection algorithm based on the residual of the RAIM is adopt. When the number of visible satellites is greater than 4, the positioning result of the visible satellite combination with the minimum pseudorange residual statistic is selected as the final positioning result of the epoch. 3.2 OD Strategy In the test of this paper, the GPS products of orbit and clock correction are obtained from the GPS broadcast messages. By setting the cutoff elevating angle, a small amount of the onboard measurements containing tropospheric delay and ionospheric delay could be eliminated. And the Time Group Delay (TGD) is corrected. The receiver clock correction and position result are solved simultaneously by least squares algorithm. The detailed data processing of OD strategy is shown in the following Table 1. Table 1. The data processing of OD strategy Item

Data or method

Orbit of GPS satellite

GPS broadcast messages

Clock correction of GPS satellite

GPS broadcast messages

Tropospheric delay

The cutoff elevating angle of −78°

Ionospheric delay

The cutoff elevating angle of −78°

TGD

GPS interface control document

Receiver clock correction

Estimation

The antenna phase center offset of GPS receiver

Nominal value

The satellite attitude of TTS-2

The stable zero-yaw attitude mode

3.3 Test Design In this paper, there are two positioning algorithms employed to determine the real-time kinematic orbit of TTS-2 based on the onboard GPS data. One is the Standard Point Positioning (SPP) based on pseudorange solved with least-squares method per epoch, the other is the Kinematic Precise Positioning (KPP) based on pseudorange and carrier

Real-Time Kinematic Orbit Determination of GEO Satellite

45

phase with Kalman filter method per epoch. The different data preprocessing algorithms are adopted to carry out two scenario tests. Scenario 1: OD with SPP. There are four tests were designed: a. Kinematic orbit is solved by SPP based on the onboard original C1 pseudorange, which is marked as SPP. b. Kinematic orbit is solved by SPP based on the C1 pseudorange with the single receiver fault detection algorithm based on residual, which is marked RAIM. c. Kinematic orbit is solved by SPP based on the C1 pseudorange with the carrier phase smoothing pseudorange algorithm, which is marked as Smooth-SPP. d. Kinematic orbit is solved by SPP based on the C1 pseudorange with the both preprocessing algorithms of the single receiver fault detection algorithm based on residual and the carrier phase smoothing pseudorange algorithm, which is marked as Smooth-RAIM. Scenario 2: OD with KPP. The weight of C1/L1 is set to 1:50, and the cycle jump of L1 carrier phase is detected by the third-difference algorithm of epochs. there are two tests: a. OD with KPP is carried out directly with the onboard original C1/L1 measurements, which is marked as KPP. b. OD with KPP is carried out with the both preprocessing algorithms of the single receiver fault detection algorithm based on residual and the carrier phase smoothing pseudorange algorithm, which is marked as SR-KPP.

4 Analysis and Results 4.1 Scenario 1 Figure 4 shows the error sequence of the kinematic orbit results of each test in scenario 1. It is directly seen that the along-track and cross-track errors of the OD with SPP of scenario 1 are close to each other and are obviously smaller than the radial errors. The RMS statistical values of the orbit error are counted in scenario 1, as shown in Table 2. The three-dimensional (3D) RMS value and radial RMS value of the SPP kinematic orbit are 92 m and 88.2 m, respectively. The 3D RMS values of the RAIM and Smooth-SPP are about 70 m, and the radial RMS values are about 67 m. Compared with SPP, the 3D RMS values and radial RMS values of the RAIM and Smooth-SPP are reduced by more than 20%. For Smooth-RAIM, the 3D RMS value is 49 m, and the radial RMS value is 46.4 m, which is more than 45% lower than that of SPP. The above analysis shows that both preprocessing algorithms of the single receiver fault detection algorithm based on residual and the carrier phase smoothing pseudorange algorithm can promote the orbit position accuracy, and would be better when using both algorithms. 4.2 Scenario 2 Figure 5 shows the error sequence of the kinematic precise position results of each test in scenario 2. It is obvious from the Fig. 5 that the kinematic orbit error of SR-KPP is

46

X. Zhao et al.

Fig. 4. The error sequence of the kinematic orbit results in scenario 1

Table 2. The RMS statistical values of the orbit error in scenario 1 (Unit: cm). Test

SPP

RAIM

Smooth-SPP

Smooth-RAIM

Reduced radio in 3D (%)

/

23

24

46

Reduced radio in radial (%)

/

22

24

45

3D

92

70.8

69.7

49.4

Radial

88.2

67.7

66.1

46.6

Along-track

17.4

15

14.5

11.7

Cross-track

19.5

14.6

16.5

11.5

significantly smaller than that of KPP, when the onboard original GPS C1/L1 measurements is preprocessed by the both algorithms of the carrier phase smoothing pseudorange algorithm and the single receiver fault detection algorithm based on residual. The RMS statistical values of the orbit error are counted in scenario 2, as shown in Table 3. The 3D RMS values of KPP kinematic orbit error is 20.9 m, and the RMS values of KPP kinematic orbit error are about 12 m in radial, along-track and cross-track. The

Real-Time Kinematic Orbit Determination of GEO Satellite

47

Fig. 5. The error sequence of the kinematic orbit results in scenario 2.

3D RMS values of SR-KPP kinematic orbit error is 16.1 m, and the radial, along-track and cross-track RMS values are respectively 7.8 m, 10.1 m and 9.8 m. Compared with the KPP orbit, the radial, along-track and cross-track errors are reduced by 34%, 17%, and 19%, respectively. The 3D orbit accuracy is promoted by 23%. Table 3. The RMS statistical values of the orbit error in scenario 2 (Unit: cm). Test

PPP

SR-PPP

Reduced radio of SR-PPP (%)

3D

20.9

16.1

23

Radial

11.8

7.8

34

Along-track

12.2

10.1

17

Cross-track

12.2

9.8

19

5 Summarize In this paper, the both algorithms of the carrier phase smoothing pseudorange algorithm and the single receiver fault detection algorithm based on residual are applied to the preprocessing of TTS-2 onboard GPS data, and the real-time kinematic OD tests of SPP and KPP are carried out respectively. The tests results show that both data preprocessing algorithms could improve the kinematic OD accuracy. When both are preprocessed at the same time, the higher-precision orbit could be obtained. Compared with the original data, the kinematic orbit accuracy of Smooth-RAIM is improved by 45%. The 3D orbit accuracy of SR-KPP is better than 20 m, improved by 23%, and the radial orbit accuracy is better than 8 m, improved by 34%.

48

X. Zhao et al.

The data preprocessing algorithm adopted in this paper can be directly applied to the GNSS receiver of the GEO satellite. The research work and conclusion will provide important reference value and significance for the GNSS application of high earth orbit spacecraft. When the Multi-GNSS receiver is employed on the high earth orbit spacecraft, the orbit accuracy would be improved, which is our next work. Funding. This research was funded by the Qian Xuesen Youth Innovation Fund of China Aerospace Science and Technology Corporation In 2021.

References 1. Balbach O et al (1998) Tracking GPS above GPS satellite altitude: first results of the GPS experiment on the HEO mission Equator-S. IEEE 1998 Position Location and Navigation Symposium, pp 243–249. Palm Springs, CA, USA (1998) 2. Mehlen C et al (2001) Real-time GEO orbit determination using TOPSTAR 3000 GPS receiver. Navigation 48(3):169–179 3. Broquet R, Perrimon N, Polle B, et al (2010) HiNAV inertial/GNSS hybrid navigation system for launchers and re-entry vehicles. In: 2010 5th ESA workshop on satellite navigation technologies and European workshop on GNSS signals and signal processing (NAVITEC), pp 1–6. Noordwijk (2010) 4. Capuano V et al (2016) Standalone GPS L1 C/A receiver for lunar missions. Sensors 16:347 5. Capuano V et al (2015) GNSS-based orbital filter for earth moon transfer orbits. J Navig 69:745–764 6. Su X et al (2017) Chang’E-5T orbit determination using onboard GPS observations. Sensors 17:1260 7. Wang M et al (2018) Performance of GPS and GPS/SINS navigation in the CE-5T1 skip re-entry mission. GPS Solut 22(56) 8. Fan M et al (2015) Orbit improvement for Chang’E-5T lunar returning probe with GNSS technique. Adv Space Res 56(11):2473–2482 9. Jiang K et al (2018) TJS-2 geostationary satellite orbit determination using onboard GPS measurements. GPS Solut 22(87) 10. Zhu J et al (2020) High accuracy navigation for geostationary satellite TTS-II via space-borne GPS. In: 2020 39th Chinese Control Conference (CCC), Shenyang, China, pp 3362–3367 11. Li W et al (2019) Extraction of electron density profiles with geostationary satellite-based GPS side lobe occultation signals. GPS Solut 27(18) 12. Zhao X et al (2022) GEO precise orbit determination based on GPS Single frequency C1/L1 leakage signal. Progr Astron 40(001):117–129

Analysis of the Contributions of ISL to the Precise Orbit Determination of BDS-3 Satellites Hanbing Peng1,2(B) and Yu Xiang2 1 Technische Universität Berlin, Berlin, Germany

[email protected] 2 Beijing Satellite Navigation Center, Beijing, China

Abstract. The 3rd generation of the Chinese Beidou Navigation Satellite System (BDS-3) is the first GNSS deployed with Inter-Satellite Link (ISL) payloads within the whole constellation. A series of Precise Orbit Determination (POD) experiments have been conducted to investigate the contributions of ISL to the orbits of BDS-3 satellites in cases of different ground tracking coverage. Results show that the improvement in orbit precisions is most significant when only a regional ground network is available. When integrating with ISL observations, the single-day solution from a regional network is good enough to meet the performance requirement. For a globally distributed sparse tracking network, additional ISL measurements also better the orbit precision by 51% from the point of view of orbit Day Boundary Discontinuity (DBD) 1D RMS, the average of which decreases from 8.22 cm to 4.05 cm. Surprisingly, benefits of integrating ISL with ground tracking are still visible even with a denser global network. In this case, an average reduction of ~28% in orbit DBD 1D RMS can be observed after integrating ISL measurements, decreasing from 5.12 cm to 3.59 cm. In addition to range measurements, the profit of ISL clock measurements is also explored. It shows that combining ISL clock measurements with ground tracking reduces the orbit 1D RMS by 19%. And the orbit radial component exhibits the most benefits from additional ISL clock observations, with a DBD RMS reduction from 4.02 cm to 2.44 cm. Results from this research indicate the great potential of the ISL of BDS-3, which can not only guarantee the performance of standard services under unfavorable conditions but also improve the high-precision service capability of BDS-3. Keywords: BDS-3 · Precise Orbit Determination · Inter-Satellite Link

1 Introduction Starting from the experimental satellites, Inter-Satellite Link (ISL) has become a standard payload of every BDS-3 space vehicle (Tang et al. 2017; Ren et al. 2017; Wang et al. 2017a, b). Utilizing Ka-band (26.5–40 GHz) radio signals, cross-links established between mutually visible satellites provide communication channels within the whole © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 49–58, 2024. https://doi.org/10.1007/978-981-99-6944-9_5

50

H. Peng and Y. Xiang

constellation. Outward control instructions and inward telemetry messages can be transmitted to and from all satellites without considering much about the coverage of ground facilities of the Operation Control System (OCS). Additionally, ISL can also serve as an innovative technique for precisely measuring ranges between satellites. Orbit determinations conducted in the OCS can benefit a lot from those new observations. When the ground network is partially unavailable or cannot support whole-arc tracking, ISL observations can be exploited to fulfill the operational mission of generating broadcast ephemerides. Under extreme circumstances, without any contact with the OCS, relying only on ISL can maintain the service ability of a GNSS for a certain period. Some previous studies have demonstrated the profits brought by ISL for BDS-3 orbit determinations. Song et al. (2017) and Yang et al. (2017) compared the orbit determination results from L-band observations and that from L- and Ka-band observations for three BDS-3 experimental satellites. Yang et al. (2019) presented the orbit improvements brought by ISL observations for the BDS-3 basic system compared to results from a regional network. Wang et al. (2019) studied the benefits of additional ISL observations for BDS-3 orbit determinations under conditions of two different ground tracking networks. In this study, we want to comprehensively demonstrate the contribution of ISL to the precise orbit determination of BDS-3. Along with ISL derived range observations, clock observations which until now, have only been used for the time synchronization procedure are also exploited. Furthermore, the combination of ISL range and clock observations has been firstly investigated in terms of orbit determination.

2 Inter-Satellite Links 2.1 Observation Model Assumed that satellite i transmits an outward ranging signal at its apparent clock time t1i j which is received by satellite j at its own apparent clock time t2 . Within a short period i with respect to t1 , satellite i receives an inward ranging signal at its apparent clock time j t4i which is, in turn, sent by satellite j at the apparent clock time t3 of satellite j. The two ranging events between satellites i and j then constitute a complete dual one-way ranging, which can be modeled by Eq. (1).   j ρ i,j = X j (t2 ) − X i (t1 ) + c δt j (t2 ) − δt i (t1 ) + dsi + dr + εi,j j

ρ j,i = X i (t4 ) − X j (t3 ) + c[δt i (t4 ) − δt j (t3 )] + ds + dri + εj,i

(1)

In Eq. (1), ρ i,j and ρ j,i are the one-way pseudorange observations from satellite i to satellite j and from satellite j to satellite i; X i (∗) and X j (∗) are the position vectors of j satellite i and j; t∗i and t∗ are event epochs given in the apparent clocks of satellite i and j, respectively; t∗ denotes event epochs given in the imagined system clock; δt i (∗) and δt j (∗) are clock offsets of satellite i and j with respect to the system time; dsi and dri are j j the transmission and receipt delays of the ISL device onboard satellite i while ds and dr are the transmission and receipt delays of the ISL device onboard satellite j. Hardware delays of ISLs are normally assumed to be constant over a typical processing session

Analysis of the Contributions of ISL to the Precise Orbit

51

of 24–72 h and therefore are not related to specific epochs in the above observation equations. εi,j and εj,i denote other errors that affect the dual one-way ranging of ISL, including Ka-band antenna phase center corrections, relativistic effects, and random noises. For inter-satellite links, which barely go through the atmosphere, tropospheric and ionospheric errors are not considered. According to a procedure described by Tang et al. (2018), original dual one-way measurements can be transformed into two instantaneous ranges between the linked satellites, as denoted by Eq. (2). In Eq. (2), t means the derivation epoch which is j usually selected to be the closest epoch to t2 and t4i with an integer multiple of three seconds.   i,j j i,j ρ∗ = X j (t) − X i (t) + c δt j (t) − δt i (t) + dsi + dr + ε∗   j,i j j,i ρ∗ = X i (t) − X j (t) + c δt i (t) − δt j (t) + ds + dri + ε∗ (2) To decouple geometric and clock information, those two instantaneous ranges are summed and differenced to obtain the derived range and clock observations, respectively, as depicted by Eq. (3). i,j

j,i

j

j

i,j

j,i

ds + dr ε∗ + ε∗ ρ∗ + ρ∗ d i + dri = X j − X i  + s + + 2 2 2 2 i,j j,i j j i,j j,i   i i ds − dr ε∗ − ε∗ d − dr ρ∗ − ρ∗ ij = c δt j − δt i + s − + ρ− = 2 2 2 2 ij

ρ+ =

(3)

2.2 Data Processing Derived range and clock observations of ISL of the period of 12.01–12.31, 2019 are used in this study. To comprehensively evaluate the contributions of ISL measurements, three ground tracking networks selected from the IGS Multi-GNSS Experiment/Pilot Project (MGEX) are used. They differ in the aspects of the number and denseness of stations as well as their geographical coverage (see Fig. 1). The regional network consisting of 14 stations within the Asia-Pacific area (Reg) is used to simulate as realistically as possible the current OCS monitoring facilities of BDS-3. Network GloS, which is comprised of 45 sparsely but evenly distributed global stations, can provide moderately optimal ground tracking. The third network (GloA), incorporating all 143 stations within the MGEX which can track BDS-3 satellites during the study period, is expected to provide probably the best ground tracking. As given in Sect. 2.1, ISL original dual one-way ranging observations are transformed into “independent” range and clock measurements after the derivation procedure. Normally, the derived clock measurements are only utilized in a separate clock estimation process in the BDS-3 OCS, which does not involve the legacy L-band GNSS observations but measurements from other two-way time synchronization techniques (Pan et al. 2018, 2021). As the coupling of clock and radial orbit errors has always been one of the pain points that hinder precise orbit determination, those additional clock measurements may also benefit the orbit determination by decreasing this correlation. Therefore, in addition to ISL range measurements, the derived clock observations are also investigated for

52

H. Peng and Y. Xiang

Fig. 1. Geographical distributions of network Reg, GloS, and GloA used in this study

their contributions to the POD. In principle, the combination of derived range and clock measurements should contain both advantages in the two types of measurements and perform similarly as the original dual one-way observations. For this reason, the contribution of combined ISL derived range and clock observations is also examined. To sum it up, depending on the used ground tracking network and involved ISL measurements, eight different processing schemes have been examined in this study. Schemes named Reg, Reg+ISLR, GloS, GloS+ISLR, GloA, and GloA+ISLR are designed to evaluate contributions of ISL Range (ISLR) measurements in cases of different ground networks. Solutions without ISL only rely on L-band GNSS measurements from the corresponding networks. Schemes GloA+ISLC and GloA+ISLD are devoted to exploring the benefits of ISL Clock (ISLC) observations and the combination of the two types of ISL Derived (ISLD) measurements, respectively. Except for the ground network and ISL observation types, all other data processing strategies and adopted models are kept the same among different solutions. One-day processing arc with a 5 min sampling interval for the ground observations is adopted. Because of the derivation procedure, ISL derived observations are spaced with intervals of integer multiples of three seconds.

3 Results and Analysis In this section, orbit precision resulting from different processing schemes are compared in terms of Day Boundary Discontinuity (DBD). Radial orbital accuracy of four BDS-3 MEOs, which are routinely tracked by the ground facilities of the International Laser Ranging Service (ILRS), are also evaluated by Satellite Laser Ranging (SLR) observations. 3.1 Incorporate Only ISL Range Observations In Figs. 2, 3 and 4, comparisons of orbit DBD 1D RMS for processing schemes with or without additional ISL rang measurements under the support of different ground networks are presented. In the case of using a regional network, i.e. Figure 2, because of

Analysis of the Contributions of ISL to the Precise Orbit

53

large values, orbit DBD 1D RMS for scheme Reg are showed by the left y-axis in the unit of meters while that of scheme Reg+ISLR by the right y-axis in the unit of centimeters. Due to the very weak observability, orbits of BDS-3 MEOs can only be determined with a precision of several to tens of meters by daily observing sessions from a regional network. Especially bad quality of the orbital along-track component can be observed. Adopting longer processing arcs, e.g. two- or three-day arcs would improve the situation significantly (Peng et al. 2018). After adding ISL range observations, the average orbit DBD 1D RMS decreases from ~16.3 m to ~7.5 cm, and that of the along-track, crosstrack, and radial components are reduced to 8.2, 9.0, 4.1 cm from 27.6, 1.8, and 5.0 m, respectively. With the network GloS, incorporating ISL observations reduces the 1D RMS by ~51% on average, from 8.22 cm to 4.05 cm. Even using the network GloA, which consists of all capable tracking stations from MGEX at that moment, apparent improvement in the orbit precision can be observed for almost all satellites. The average orbit DBD 1D RMS of all BDS-3 MEOs declines from 5.12 cm to 3.59 cm, i.e., around 30% on the whole.

Fig. 2. Orbit DBD 1D RMS of solutions Reg and Reg+ISLR. Note that, results from solution Reg are shown using the left y-axis in the unit of meters while that from solution Reg+ISL are shown by the right y-axis in the unit of centimeters.

Fig. 3. Orbit DBD 1D RMS of solutions GloS and GloS+ISLR

In Table 2, statistics of SLR residuals of orbits obtained from different solutions are given. In the case of the regional network (Reg), radial orbit accuracy gets greatly improved as orbit precision showed by DBDs after introducing ISL observations. But in cases of global networks GloS and GloA, reductions of SLR residuals are not very large

54

H. Peng and Y. Xiang

Fig. 4. Orbit DBD 1D RMS of solutions GloA and GloA+ISLR

for solutions with additional ISL observations. This may be related to the hardware delays that existed in the ISL observations. In this study, ISL hardware delays are modeled as constants and estimated together with all other parameters for each session. Some initial results show apparent variations of those bias estimates, which suggests that refined modeling for ISL hardware delays may be needed. Table 1. Mean RMS of orbit DBD from different solutions [cm] Schemes

Along-track

Cross-track

Radial

1D

Reg

2762.26

175.69

503.63

1627.63

8.17

9.01

4.10

7.48

Reg+ISLR GloS

11.32

5.28

6.54

8.22

GloS+ISLR

4.47

4.62

2.63

4.05

GloA

6.82

3.81

4.02

5.12

GloA+ISLR

4.11

3.73

2.70

3.59

GloA+ISLC

5.49

3.72

2.44

4.14

GloA+ISLD

4.19

3.97

2.34

3.65

3.2 Incorporate Only ISL Clock Observations Along with relative ranges between linked satellites, the deriving of original ISL dual one-way observations also generates measurements of relative clock offsets of satellite pairs. Uncertainties of estimated clock offsets from an Orbit Determination and Time Synchronization (ODTS) procedure usually correlate with orbit errors, especially when using a less optimal ground network. In this case, if the coupling of clock and orbit errors could be reduced, orbit quality would probably be improved. In Figs. 5 and 6, standard derivations of satellite clock offset differences when compared with IGS MGEX gbm and wum products for solutions GloA and GloA+ISLC are

Analysis of the Contributions of ISL to the Precise Orbit

55

Table 2. Statistics of orbit SLR residuals from different solutions [cm] PRN C20 Mean STD RMS C21 Mean STD RMS C29 Mean STD RMS C30 Mean STD RMS

Reg Reg+ISLR GloS GloS+ISLR GloA GloA+ISLR GloA+ISLC GloA+ISLD 6.19 6.49 6.37 6.30 6.33 6.13 6.18 6.16 29.18 2.35 2.28 1.75 1.86 1.70 1.81 1.88 29.83 6.90 6.77 6.53 6.60 6.37 6.44 6.44 0.97 5.84 6.34 6.05 5.98 5.86 5.95 5.84 33.02 3.31 2.53 1.82 2.02 1.85 1.93 1.94 33.04 6.71 6.82 6.32 6.31 6.14 6.25 6.15 13.52 0.18 −2.14 −2.06 −2.19 −2.13 −2.27 −2.69 28.31 4.63 3.02 3.04 2.70 2.66 2.64 2.69 31.37 4.64 3.70 3.67 3.48 3.41 3.48 3.80 10.39 −4.87 −3.12 −3.40 −3.15 −2.96 −2.41 −4.97 24.76 5.17 3.34 3.45 3.13 3.10 2.92 5.06 26.85 7.11 4.57 4.85 4.44 4.29 3.79 7.09

presented. Although several satellites show different STDs when compared with different external products, both comparisons indicate clear STD reductions for most satellites after incorporating ISL clock observations.

Fig. 5. Standard deviations of satellite clock offset differences wrt IGS MGEX gbm products for solutions GloA and GloA+ISLC

Fig. 6. Standard deviations of satellite clock offset differences wrt IGS MGEX wum products for solutions GloA and GloA+ISLC

56

H. Peng and Y. Xiang

In Fig. 7, orbit DBD 1D RMS of solutions GloA and GloA+ISLC are compared. As expected, apparent improvements in the orbit precisions can be observed when using additional ISL clock observations. Similar to range observations, ISL clock observations also help to homogenize orbit precisions among satellites. Satellites with inferior orbit quality when using only ground tracking measurements get improved more after incorporating ISL clock observations, e.g. C27, C34, and C37. The average RMS of radial and 1D DBD are 2.44 and 4.14 cm, respectively, amounting to reductions of 39% and 19% with respect to that using only ground observations.

Fig. 7. Comparison of orbit DBD 1D RMS of solutions GloA and GloA+ISLC

3.3 Incorporate Both ISL Range and Clock Observations Theoretically, the combination of ISL derived range and clock observations should be equivalent to ISL original dual one-way ranging measurements. Incorporating both ISL derived range and clock observations into the processing with ground tracking measurements should be the ultimate choice if the goal is to exploit as thoroughly as possible the advantage of this new technique. In Fig. 8, orbit DBD 1D RMS of solutions wherein ISL observations have been incorporated in varying degrees are compared. As revealed before, solutions with either ISL range or clock observations introduced are superior to that relying only on ground tracking facilities. For most satellites, ISL range observations show a greater advantage than clock observations in improving orbit precisions. Not surprisingly, integrating range and clock observations simultaneously with the ground tracking reduces orbit DBDs obviously. However, except for satellites C19-C22 and C26, solution GloA+ISLD shows similar results as solution GloA+ISLR. In other words, the additional clock measurements in solution GloA+ISLD do not present many extra benefits for improving orbit precisions. From the results of SLR validation (See Table 2), solution GloA+ISLD shows even worse overall performance compared with solution GloA+ISLR or GloA+ISLC, especially for satellite C30. This may indicate inconsistency existed between those two types of ISL derived observations. It would be very interesting to compare with results from the processing of original dual one-way observations.

Analysis of the Contributions of ISL to the Precise Orbit

57

Fig. 8. Comparison of orbit DBD 1D RMS of solutions GloA and GloA+ISLD

4 Conclusion In this study, the contributions of ISL observations to the orbit determination of BDS-3 satellites have been investigated comprehensively. Not only the derived range, as studied in previous research, but also the derived clock measurements have been firstly used in the orbit determination. For a regional network, integrating ISL observations greatly improves the orbit precision, reducing orbit DBD 1D RMS from tens of meters to 7–8 cm. This means the current 3-day processing arc adopted in the OCS of BDS-3 could be shortened to the 1-day arc if ISL observations are operationally used. Adopting a shorter processing arc for broadcast ephemerides generation not only improves the system’s robustness but also decreases its complexity. For global networks, the benefits of additional ISL observations are also apparent. This indicates that, instead of being used only for supporting the space segment of BDS-3, the potential of ISL in high-precision applications like precise PNT services and geodetic sciences is promising. The derived clock observations have been firstly used in orbit determination and proved to be beneficial to improving orbit precisions. However, when combining the range and clock observations together with the ground tracking, the result has not been further improved as might be expected. And unlike the precision showed by DBDs, the orbit accuracy (mainly the radial component) does not present many profits from incorporating ISL observations. Systematic inconsistency, either between the two types of ISL observations or between the ground tracking and ISLs is suspected to be the cause of those unsolved issues. In the following research, taking the processing of ISL original dual one-way observations as the comparison would help to understand these questions.

References Pan J et al (2018) Time synchronization of new-generation BDS satellites using inter-satellite link measurements. Adv Space Res 61(1):145–153 Peng H, Yang Y, Wang G, Li Y (2018) Effect analysis of GPS/BDS combined orbit determination on Beidou satellite orbits. J Geodesy Geodyn 38(12):6 Pan J et al (2021) Full-ISL clock offset estimation and prediction algorithm for BDS3. GPS Solut 25(4):140

58

H. Peng and Y. Xiang

Ren X, Yang Y, Zhu J, Xu T (2017) Orbit determination of the next-generation Beidou satellites with intersatellite link measurements and a priori orbit constraints. Adv Space Res 60(10):2155–2165 Song X, Mao Y, Feng L, Jia X, Ji J (2017) The preliminary result and analysis for bd orbit determination with inter-satellite link data. Acta Geodaetica et Cartographica Sinica 46(5):547– 553 Tang C et al (2017) Centralized autonomous orbit determination of Beidou navigation satellites with inter-satellite link measurements: preliminary results. Scientia Sinica Physica, Mechanica & Astronomica 47(2):1–11 Tang C et al (2018) Initial results of centralized autonomous orbit determination of the newgeneration BDS satellites with inter-satellite link measurements. J Geodesy 92(10):1155–1169 Wang C, Zhao Q, Guo J, Liu J, Chen G (2019) The contribution of intersatellite links to BDS-3 orbit determination: model refinement and comparisons. Navigation 66(1):71–82 Wang H, Chen Q, Jia W, Tang C (2017a) Research on autonomous orbit determination test based on BDS inter-satellite-link on-orbit data. In: Sun J, Liu J, Yang Y, Fan S, Yu W (eds) China Satellite Navigation Conference (CSNC) 2017 Proceedings: Volume III, Singapore, 2017// 2017. Springer Singapore, pp 89–99 Wang H, Xie J, Zhuang J, Wang Z (2017b) Performance analysis and progress of inter-satellitelink of Beidou system. In: ION GNSS+ 2017, Portland, Oregon, September 25–29 2017, pp 1178–1185 Xie X et al (2019) Precise orbit determination for BDS-3 satellites using satellite-ground and inter-satellite link observations. GPS Solut 23(2):40 Yang D, Yang J, Li G, Zhou Y, Tang C (2017) Globalization highlight: orbit determination using BeiDou inter-satellite ranging measurements. GPS Solut 21(3):1395–1404 Yang Y et al (2019) Inter-satellite link enhanced orbit determination for BeiDou-3. J Navig 1–16 Yang Y, Yang Y, Hu X, Tang C, Zhao L, Xu J (2019) Comparison and analysis of two orbit determination methods for BDS-3 satellites. Acta Geodaetica et Cartographica Sinica 48(7):831–839 Yang Y et al (2021) BeiDou-3 broadcast clock estimation by integration of observations of regional tracking stations and inter-satellite links. GPS Solut 25(2):57

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution Lijun Yang(B)

, Guofu Pan, Xiang Zuo, Jinsheng Zhang, and Zhihao Yu

Guangzhou Hi-Target Navigation Tech Co. Ltd., Guangzhou 510000, China [email protected]

Abstract. The four available civil frequencies of BDS-3 provide strong support for multi-frequency multi-system PPP-AR. It has been shown that multi-frequency improves the PPP-AR positioning performance of BDS-3/GNSS significantly, but most of the IGS station 30 s sampling rate observations are used to evaluate the positioning performance, while the high sampling rate is closer to the actual application scenario. To fill this gap, the multi-frequency multi-system BIAS product of CNES is used, combined with the 1 s sampling rate observations of the European EUREF station, to verify the multi-frequency BDS-3/GNSS is used to verify the possibility of achieving 1 s ambiguity resolution. The multi-frequency and multi-system single-epoch PPP-WAR is performed for smart driving applications with sub-metre positioning requirements, and the necessary conditions to meet the accuracy are analysed. Therefore, this paper analyses the stability differences between CNES BIAS phase fractional deviation product systems, between satellites and between frequencies; achieves fast convergence of the BDS-3/GNSS full-frequency PPP-AR by optimising the partial ambiguity fixation algorithm, and analyses the contribution of the number of wide lane fixes in the single-epoch PPP-WAR. The results show that the inter-system standard deviation of BIAS products: GPS < Galileo < BDS-3; the BDS-3 four-frequency PPP-AR imitates a dynamic mean convergence time of 15.32 min, with horizen and vertical accuracy of 4.3 cm and 5.3 cm; in the BDS-3/GNSS multi-frequency PPP-AR experiments, fast convergence within 3 min can be achieved with 90% confidence. In the BDS3/GNSS multi-frequency PPP-AR experiment, the horizen and vertical accuracy can be 2.2 cm and 8.9 cm with 90% confidence, and some samples can be converged with 1 s fixation; in the single-epoch PPP-WAR, the horizen accuracy of 20 cm and vertical accuracy of 30 cm can be achieved when the number of WL fixation reaches 13. Keywords: BDS-3 · Multi-frequency · PPP-AR · Single-epoch PPP-WAR

1 Introduction Precision point positioning (PPP) can obtain high accuracy position information globally with only a single station, but the ambiguity is float resolution making its convergence time above 10 min, which limits its widespread use, while PPP ambiguity resolution © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 59–72, 2024. https://doi.org/10.1007/978-981-99-6944-9_6

60

L. Yang et al.

(AR) can accelerate convergence and improve positioning accuracy; multi-system compared to single-system PPP-AR can increase the number of ambiguity subsets, improve redundancy and enhance overall AR performance, and partial ambiguity resolution can significantly shorten time to first fix; compared to dual-frequency, multi-frequency can constitute extra-wide-lane(EWL) ambiguity, and first fixation and convergence time can be accelerated by level-by-level AR. Cao et al. [1] conducted multi-frequency GNSS PPP-AR based on the OSB product of CNES, verifying that GPS IFCB had been absorbed by the OSB product, and the single BDS-3 quad-frequency PPP-AR had the imitation dynamic and static convergence times of 39 min and 25.1 min, respectively, and the accuracy reached cm level after AR, and the multi-frequency GNSS PPP-AR had the horizontal direction accuracy of mm level after AR, and the imitation dynamic and static convergence times were 17 min and 15.7 min, respectively. Zhao et al. [2] also conducted GPS L1/L2/L5 and Galileo E1/E5a/E5b/E6 PPP-AR based on CNES OSB products, and transformed the Uncombined ambiguity to be estimated in the function model into each EWL and widelane(WL) ambiguity to be estimated, and use Best Integer Equivariant Estimation (BIE) for AR, the average convergence time is 29.2 min, double the medium error, single-epoch PPP-WAR reaches 0.44 m in horizontal and 1.27 m in vertical relative to single-epoch PPP; Jiang Guo et al. [3] used Galileo E1/E5a/E6 and BeiDou3 B1C/B2a/B3I signals for multi-frequency PPP-AR, and estimated the third frequency clock difference to eliminate the third frequency IFCB, satellite PCO correction problem relative to the conventional triple-frequency, and the above combination can achieve 62% sample 1 min convergence at 10 fixed satellites. The vehicle static and dynamic convergence times are 14 min and 33 min respectively; Zhao et al. [4] used continuous cut-off elevation angle to select a subset of optimal ambiguity in multi-frequency GNSS PPP step-by-step ambiguity fixing for the first fixing of about 2 min. Tao et al. [5] used triple-frequency BDS/Galileo/GPS/QZSS observations with a sampling rate of 1 s in the Chinese region data to generate UPD and perform PPP-AR at the terminal, the full-system model uses 34.7 satellites and achieves a fast convergence of 1 s horizontal and 2 s vertical at 95% confidence level. Single-epoch PPP-WAR can provide instantaneous sub-metre resolutions for autonomous driving. Jianghui Geng et al. [6] estimated EWL and WL UPD based on triple-frequency GPS/BDS/Galileo/QZSS observations, and achieved singleepoch PPP-WAR positioning accuracy of 22.8 cm in mean horizontal and 41 cm in vertical at the terminal, taking into account that triple-frequency data are highly susceptible to loss in urban complex environment is highly susceptible to loss, and considering the larger number of Galileo/BDS-3 frequency, it also analysed in 2020 [7] the conversion of UPD products into OSB products for terminal Galileo/BDS-3 single-epoch full-frequency PPP-WAR, with real-time planimetric accuracy of 27 cm and vertical of 52 cm under 8–12 visible satellites in on-board experiments. The above literature has not fully considered the potential of observations with a sampling rate of 1 s for multi-frequency BDS-3/GNSS PPP-AR in practical scenarios. For this reason, this paper uses the CNES BIAS product to target fast convergence of GNSS multi-frequency PPP-AR based on 1 s observations from the EUREF site in Europe; the multi-frequency GNSS single-epoch PPP-WAR instantaneous positioning performance is analysed.

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

61

2 Multi-frequency and Multi-system PPP-AR Principle This section first introduces the multi-frequency GNSS PPP function model, focusing on the PPP-AR step-by-step ambiguity resolution strategy, including the partial ambiguity resolution subset preference strategy. 2.1 Multi-frequency Undifferenced and Uncombined PPP Observation Equations The original pseudorange and the carrier observations form the basic function model [8] Q,s

Q,s

Pr,f = ρr Q,s

Q,s

Lr,f = ρr

Q

Q,s

+ c(dtr − dt Q,s ) + Tr

Q

Q,s

+ c(dtr − dt Q,s ) + Tr

Q,s

Q,s

Q,s

Q,s

+ Ir,f + c(br,f + bf ) + εr,f

Q,s

Q,s

Q,s

Q,s

Q,s

(1) Q,s

− Ir,f + λf (Nr,f + Br,f + Bf ) + ξr,f

(2)

where Q, s, r, f denotes the PRN number, receiver ID, and signal frequency of the satellite systems G (GPS), C (BDS), and E (Galileo) corresponding to the satellite, respecQ,s Q,s tively; Pr,f , Lr,f denotes the pseudorange and carrier phase observations in meters, Q,s

respectively; λf

Q,s

is the carrier phase wavelength f for the frequency; ρr

is the geo-

Q dtr ,

dt Q,s denotes the receiver metric distance between the station and the satellite; Q,s Q,s and satellite clock errors; Tr , Ir,f the tropospheric and ionospheric oblique delays, Q,s

Q,s

Q,s

respectively; Nr,f the carrier phase full-period ambiguity; br,f , bf

the frequencyQ,s

Q,s

dependent receiver and satellite pseudorange hardware delays, respectively; Br,f , Bf

Q,s

the frequency-dependent receiver and satellite phase hardware delays, respectively; εr,f , Q,s

ξr,f the sum of pseudorange and carrier observation noise, multipath effects and other un The sum of the modelled errors. Of these, the carrier hardware delay can be divided into a constant component (estimated) and a time-varying component (temperature-dependent, absorbed). Q,s

Q,s

Q,s

Q,s

Br,f = Br,f + δBr,f , Bf Q,s

Q,s

where Br,f , Bf

Q,s

= Bf

Q,s

+ δBf

(3)

are the constant parts of the receiver and satellite carrier hardware Q,s

Q,s

delays, respectively, and δBr,f , δBf correspond to the time-varying parts indicated. Parameters in the model (e.g. receiver clock difference and pseudorange hardware delay, ionospheric slope delay and pseudorange hardware delay, hardware delay between frequencies, phase ambiguity and its hardware delay) are linearly correlated, leading to a model rank deficit. The parameters causing the rank deficit are reorganised using parameter reorganisation to eliminate the rank defect. The pseudorange hardware delay and phase hardware delay parameters are reorganised and absorbed by the corresponding satellite clock difference, receiver clock difference and ionospheric delay. In addition to the multipath errors and observation noise indicated above, there are some modelled errors in the observation equations and are derived from the relevant theorised models and subsumed into the residuals.

62

L. Yang et al.

The pseudorange and carrier observation equations were linearized and reparameterized to obtain. Q,s

Q,s

pr,f = gr Q,s

Q,s

lr,f = gr

Q,s

· x + cd t r Q,s

· x + cd t r

Q,s

Q,s

Q,s

Q,s

Q,s

+ mr,w · Tr,w + μf · I r,1 + r,f + δd r,f + εr,f Q,s

Q,s

Q,s

+ mr,w · Tr,w − μf · I r,1 + λf

Q,s

Q,s

Q,s

· N r,f + r,f + ξr,f

(4) (5)

The above equation has ⎧ Q,s Q Q,s Q,s cd ¯tr = c(dtr + br,IF12 ) + δBr,IF12 ⎪ ⎪ ⎪ ⎪ ⎪ Q,s Q,s Q,s Q,s Q,s Q,s ⎪ ⎪ I¯r,1 = Ir,1 + cβ12 (DCB12 + DCBr,12 ) − β12 (δDPB12 + δDPBr,12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q,s Q,s Q,s Q,s Q,s Q,s Q,s Q,s ⎪ ⎨ r,f = β12 DCB12 − DCB13 − λf cβ12 (DCB12 + DCBr,12 ) + br,3 − br,IF12 (f = 3) Q,s Q,s Q,s Q,s Q,s Q,s Q,s ⎪ ⎪ r,f = δBf − δBIF12 − μf β12 (δDPB12 + δDPBr,12 ) + δBr,f − δBr,IF12 (f = 3) ⎪ ⎪ ⎪ ⎪ ⎪ Q,s Q,s Q,s Q,s Q,s ⎪ ⎪ δ d¯ r,f = μf β12 (δDPB12 + δDPBr,12 ) − (δBIF12 + δBr,IF12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯ Q,s Q,s Q,s Q,s Q,s Q,s Q,s Q,s Nr,f = Nr,f + B¯ r,f + B¯ f − f · bIF12 − f · br,IF12 + f · μf β12 (DCB12 + DCBr,12 ) (f = 1, 2, 3)

(6) where 3 represents multi-frequency other than frequency 1, 2. In summary, the vector of parameters to be estimated for the Uncombined PPP model is   Q,s Q,s Q,s Q,s T Q,s Q,s Q,s X = x, cd t r , Tr,w , I r,1 , r,f , r,f , N r,1 , N r,2 , N r,3

(7)

The parameters to be estimated consist of five categories: the amount of receiver position coordinate variation, receiver clock aberration correction, zenith tropospheric wet delay, L1 ionospheric oblique delay (including the time-varying portion of the receiver, satellite side DCB and both phase hardware delays), and carrier phase ambiguity over L1 and L2. Note: The satellite clock difference parameters have been corrected by the precision clock difference for the inter-frequency deviation IFB for the remaining freQ,s quency relative to 1 and 2, r,f for the inter-frequency clock deviation IFCB for the remaining frequency relative to 1 and 2, and for the time-varying part of the pseudorange hardware delay, which is not estimated here because its error is small and absorbed by the pseudorange residuals. The inter-system bias (ISB) between different systems is considered and estimated with a white noise process, while the BDS-3:GPS:Galileo weighting is 1:3:3. 2.2 Multi-frequency PPP-AR Based on BIAS Products The IGS (International GNSS Service) proposes the BIAS-SINEX standard format of phase fractional deviation product, which can flexibly realize any frequency fixation requirement. In this paper, the multi-frequency ambiguity resolution is realized based on the BIAS product of CNES [11], and only the ambiguity resolution of CDMA system is considered, and the FDMA system is only involved in the float resolution. The difference between multi-frequency PPP-AR and dual-frequency PPP-AR is shown in Fig. 1, which

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

63

Fig. 1. Dual-frequency (left)/multi-frequency (right) Uncombined PPP step-by-step AR process

mainly adds the acquisition of EWL ambiguity as well as AR and normal equation constraint update. (1) BIAS product composition The uncombined pseudorange deviations in second and uncombined phase deviations in weeks in the BIAS product are  ⎧ Q,s Q,s Q,s Q,s ˜ ⎪ b b = β12 · DCB12 = β − b 12 1 ⎪ 1 2 ⎪ ⎨  ˜ Q,s = −α12 bQ,s − bQ,s = −α12 · DCBQ,s (8) b 2 1 2 12 ⎪ ⎪ ⎪ ⎩ Q,s Q,s Q,s b˜ 3 = DCB13 − β12 · DCB12 ⎧ Q,s Q,s Q,s ⎪ B˜ = B¯ 1 − f1 · bIF12 ⎪ ⎨ 1 Q,s Q,s Q,s (9) B˜ 2 = B¯ 2 − f2 · bIF12 ⎪ ⎪ ⎩ ˜ Q,s Q,s Q,s Q,s Q,s Q,s B3 = B¯ 3 − f3 · bIF12 + δB3 − δBIF12 − μ3 β12 δDPB12 In the equation, the remaining frequency absorb more IFCB errors on the satellite side compared to frequency 1 and 2, and the study shows that only GPS IIF needs IFCB correction, while BDS-3, Galileo and GPS III have negligible errors. The above GNSS pseudorange deviation and phase deviation are directly corrected to the pseudorange and phase observation values of the observation equation, then the satellite side pseudorange and phase hardware delay error terms are eliminated and the parameters to be estimated

64

L. Yang et al.

become ⎧ Q,s Q Q,s Q,s ⎪ cd ¯t = c(dtr + br,IF12 ) + δBr,IF12 ⎪ ⎪ r ⎪ ⎪ Q,s Q,s Q,s Q,s ⎪ ⎪ I¯r,1 = Ir,1 + cβ12 DCBr,12 − β12 δDPBr,12 ⎪ ⎪ ⎨ Q,s Q,s Q,s Q,s Q,s r,f = −λf cβ12 DCBr,12 + br,3 − br,IF12 (f = 3) ⎪ ⎪ ⎪ Q,s Q,s Q,s Q,s ⎪ ⎪ r,f = −μf β12 δDPBr,12 + δBr,f − δBr,IF12 (f = 3) ⎪ ⎪ ⎪ ⎪ ⎩ N¯ Q,s = N Q,s + B¯ Q,s − f · bQ,s + f · μ β DCBQ,s (f = 1, 2, 3) f 12 r,12 r,IF12 r,f r,f r,f

(10)

(2) EWL/WL ambiguity resolution The PPP float resolution yields the float ambiguity at each frequency, and EWL/WL float ambiguity is formed by the following equation. ⎞ ⎛

Q,s 

 N Q,s r,1 N r,ewl 0 1 −1 ⎜ Q,s ⎟ (11) = ⎝ N r,2 ⎠ Q,s 1 −1 0 N r,wl Q,s N r,3 Due to the number of GPS, Galileo and BDS-3 frequency and the composition of the EWL/WL noise factor, options are given here for the EWL/WL combination of each system.

Fig. 2. BDS-3/GPS/Galileo EWL/WL combinations for each frequency

EWL/WL ambiguity parameter contains hardware delay at the satellite side and hardware delay at the receiver side, and the receiver hardware delay is eliminated by using inter-satellite single difference. C,ij

C,j

C,ij

C,ij

C,ij

C,ij

C C LMW = LC,i MW − LMW = λWL N r,WL = λWL Nr,WL + BMW + ξr,MW Q,ij

(12)

The hardware delay on the satellite side BMW has been eliminated by the BIAS product, and WL ambiguity is fixed to an integer using multi-epoch smoothing in combination with the Rounding method when no weekly jumps occur. More accurate EWL/WL ambiguity is obtained by backgeneration constrained filtering according to the observation equation. In the literature [4], the EWL/WL AR are determined using LAMBDA search, and this paper combines the above two methods to determine the EWL/WL

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

65

ambiguity. It should be noted that multiple EWL ambiguity fixed need to be constrained simultaneously with the method equation. (3) Narrow-lane (NL) ambiguity resolution The receiver-side hardware delay in the NL ambiguity parameter is also eliminated by Q,s Q,s the inter-star single difference, while the satellite side hardware delay B1 − f1 · bIF12 is directly eliminated by the BIAS product. The inter-star single difference NL ambiguity is fixed by the LAMBDA algorithm, and the PPP fixed resolution is obtained by constrained filtering of the floating NL ambiguity with the fixed NL ambiguity. 2.3 Rapid Ambiguity Resolution Strategy Figure 3 gives this paper single system part of the ambiguity resolution flow chart, BIAS product correction, WL/NL ambiguity fractional part of the satellite ambiguity over the threshold value to eliminate, EWL/WL/NL ambiguity fractional part of the threshold value is generally set to: 0.25 and 0.15 weeks, while WL ambiguity need to pass Bootstrapping test, the threshold value is set to 0.999, greater than the threshold value is If the ambiguity fails to be fixed, the worst ambiguity will be removed according to the height angle lift of 5°, the number of continuous tracking epoch, the signal-tonoise ratio and the carrier-to-noise ratio, and the variance of the float ambiguity, and LAMBDA will be fixed and tested again, and the cycle will be repeated until the optimal ambiguity subset is obtained. If the number of remaining ambiguity is less than 4, the ambiguity fixing fails and the float resolution is output.

Fig. 3. PPP partial ambiguity resolution strategy

3 Experiment Results To verify the contribution of the BIAS product to the terminal multi-frequency and multi-system PPP-AR, five stations were arbitrarily selected from EUREF sites for a total of 4 days in 2022 DOY 151/153/156/157 with a sampling rate of 1 s observations

66

L. Yang et al.

and set to restart every 2 h, and the distribution of stations is shown in Fig. 3. Firstly, based on its post-processing BIAS product, the standard deviation of each satellite of the BDS-3/GPS/Galileo system is given to evaluate its stability; secondly, multi-frequency PPP-AR is performed in combination with the available frequency of BDS-3 to evaluate the positioning performance; finally, PPP-AR is implemented considering all available frequency of BDS-3/GNSS to give its fast convergence confidence level within 3 min.

Fig. 4. Site map

3.1 CNES BIAS Product Performance To analyse the stability of the multi-frequency and multi-system BIAS product. The phase fractional deviations of all available frequencies for DOY 151 days in 2022 were selected to plot the standard deviation STD of BDS-3/GPS/Galileo by satellite system and different satellites, respectively, as shown in Figs. 5, 6 and 7; Fig. 5 shows that except for C23, C29, C33, C43, C44 and C46, the STD of all satellites is within 0.2 ns, and from the different frequency of individual satellites From the different frequency of individual satellites, the STDs of B1I and B1C are comparable, followed by B3I, and B2a is the most unstable; Fig. 6 shows that for the L1/L2 dual frequency, the STDs of all satellites are within 0.05 ns, except G12 and G19, and L1 is slightly better than L2, and for the L5 frequency, compared to the 5 satellites of BLOCK III type that provide PCO/PCV correction for the third frequency, the 12 satellites of BLOCK The 12 satellites of type IIF do not provide correlation for the time being, resulting in their L5 frequency being unstable; the STD of the Galileo system in Fig. 7 is comparable and better than 0.1 ns for all frequency except E02, E08, E18, E24, E25 and E31. From the mean STD statistics of different frequency in Table 1, GPS is the best and BDS-3 is lower than Galileo.

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

67

Fig. 5. BDS-3 satellite phase fractional deviation STD

Fig. 6. GPS satellite phase fractional deviation STD

Fig. 7. Galileo satellite phase fractional deviation STD

3.2 BDS-3 Multi-frequency PPP-AR Four civil frequency provided by BDS-3 were used for multi-frequency PPP-AR [12], and this paper sets 10 consecutive epoch less than 5 cm in horizontal and less than 10cm in vertical as convergence. Figure 8 gives the positioning timing diagram of station

68

L. Yang et al. Table 1. Average BDS-3/GPS/Galileo phase fractional deviation STD for DOY 2022,151

System

STD/ns B1I/L1/E1

B3I/L2/E5a

B2a/L5/E5b

B1C/E6

BDS-3

0.132

0.162

0.244

0.131

GPS

0.048

0.062

0.08

–

Galileo

0.075

0.101

0.097

0.093

OLK2 for 2022 DOY 156 days from 2:00:00 to 12:00:00 UTC. It can be seen that the four frequencies are better than the triple-frequencies in the first reliable fixing time in the imitation dynamic mode, which speeds up the convergence time. From the full time, the probability of the triple-frequencies fixing error is greater than that of the four frequencies, and the triple-frequencies have an error fixing near the 25200s.

Fig. 8. OLK2 station triple/four-frequency PPP-AR positioning

Table 2 gives multi-station, multi-day and multi-frequency PPP-AR positioning statistics, with the four-frequency convergence time being slightly better than the triplefrequency, which is a result of more EWL being provided in the four-frequency, while both have comparable accuracy. Table 2. Multi-station, multi-day BDS-3 quad-frequency PPP-AR average accuracy statistics Number of Frequency

Horizon RMS/m

Vertical RMS/m

Convergence time/min

Triple

0.043

0.053

15.32

Four

0.043

0.057

17.60

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

69

3.3 GNSS Multi-frequency PPP-AR Compared with single BDS-3 multi-frequency PPP-AR, GNSS multi-frequency PPPAR brings more satellites, optimizes the spatial geometry, forms more ambiguity subsets [9], and the optimal subset found by partial ambiguity resolution is also more reliable, reducing the problem of re-convergence due to insufficient number of satellites. To evaluate the contribution of multi-frequency PPP-AR in accelerating the ambiguity resolution and convergence, Fig. 9 gives the positioning timing diagram of the four systems of HELG station dual-frequency and multi-frequency PPP-AR in imitation dynamic mode from 12:00:00 to 14:00:00 UTC on day 153 of DOY 2022. In the figure, the dual-frequency PPP-AR needs 10 min of filtering before convergence, and then the blurring starts to be fixed, while the multi-frequency PPP-AR is fixed from the beginning, and the positioning accuracy is comparable at the end of the filtering.

Fig. 9. HELG station dual/triple/full frequency PPP-AR positioning

The right view shows the first 19 s zoom-in of the left view of Fig. 9. The dualfrequency PPP-AR does not show significant convergence in the E/N/U direction within 19 s of the second, while the multi-frequency starts to fix and converge directly in the first 1 s. This indicates that the fixation constraint in the multi-frequency PPP-AR accelerates the fixation of WL/NL, thus increasing the speed and overall convergence rate. This indicates that the EWL fixation constraint in multi-frequency PPP-AR accelerates both wide and NL fixation, thus enhancing the convergence rate and overall fixation rate. Table 3 presents the multi-station, multi-day and multi-frequency PPP-AR statistics for confidence, horizontal and vertical accuracy and their first fix times for the 3min convergence requirements, with the percentages of triple/full frequency meeting the requirements being 70% and 90% respectively. Compared to the triple-frequency, the full frequency PPP-AR has better positioning accuracy and meets more of the requirements. It is worth noting that the first fix time is longer for the full frequency than the triple frequency, which is due to the stability of the BIAS product and the partial error fixing strategy. 3.4 Single-Epoch PPP-WAR For single-epoch multi-frequency PPP-WAR, all parameters to be estimated are reinitialised for each epoch without considering the cycle slip, so no convergence is

70

L. Yang et al.

Table 3. Multi-site, multi-day, multi-frequency PPP-AR reaching 3min convergence confidence and its accuracy statistics Number of Frequency

Confidence

Horizon RMS/m

Vertical RMS/m

TTFF/min

Triple

70%

0.024

0.095

0.67

Full

90%

0.022

0.089

1.08

required and the PPP float resolution is performed directly, first EWL ambiguity is fixed constrained WL, WL ambiguity is fixed and the WL fixed resolution is output [10]. The single-epoch PPP float resolution without EWL and WL fixing and constraint. Figure 10 shows the positioning accuracy statistics for the TORN station GREC system triple/ full frequency PPP-WAR and triple-frequency PPP float resolution timing diagrams from 1900 to 2400 UTC on day 156 of DOY 2022.

Fig. 10. Single-epoch PPP-WAR positioning at TORN stations

As can be seen from the figure, the single-epoch PPP float resolution has the largest positioning error. After fixing EWL and WL, the positioning accuracy and its stability will be further improved as the number of correct fixes in the WL increases. As can be seen from Table 4, the single-epoch element PPP-WAR fix rate is close to 100%, and all of them can achieve decimetre-level positioning accuracy. Table 5 gives the mean values of RMS in the E, N and U directions corresponding to different number of fixes in the WL for multi-station multi-day statistics, reaching 0.18m in horizontal and 0.23m in vertical when the number of fixes is greater than 15. Table 6 gives the multi-day and multi-station average RMS accuracy statistics, with both single-epoch full-frequency PPP-WAR relative to single-epoch PPP float resolution horizontal and vertical positioning accuracy improvements of over 40%. Compared to the time period in Table 4, the overall accuracy has all decreased, which is a result of the final number of wide alley fixes not being stable above 7.

BDS-3/GNSS Multi-frequency PPP Rapid Ambiguity Resolution

71

Table 4. Positioning accuracy statistics for different single-epoch PPP-WAR Model

Horizon RMS/m

Vertical RMS/m

Fix rate/%

PPP

0.61

0.96

–

Triple PPP-WAR

0.27

0.4

Full PPP-WAR 0.22

0.3

Horizon Improve rate/%

Vertical Improve rate/%

–

–

99.9

55.7

58.3

100

63.9

68.8

Table 5. Multi-station multi-day RMS averages of fixed numbers for different WL ambiguities Single-epoch WL AR number

E/m

N/m

U/m

4–6

0.42

0.3

0.73

7–9

0.26

0.18

0.55

10–12

0.2

0.14

0.39

13–15

0.17

0.12

0.28

>15

0.14

0.11

0.23

Table 6. Positioning accuracy statistics for different single-epoch PPP-WAR Model

Horizon RMS/m

Vertical RMS/m

PPP

0.65

1.04

Triple PPP-WAR

0.42

0.68

Full PPP-WAR 0.38

0.61

Fix rate/%

–

Horizon Improve rate/%

Vertical Improve rate/%

–

–

99.23

34.4

34.6

99.26

41.5

41.3

4 Conclusion In this paper, based on the real-time European EUREF station 1 s observation data, the BIAS product of CNES is used to implement multi-frequency and multi-system PPP-WAR. The stability of the BIAS phase fractional deviation product is firstly evaluated, and the BDS-3/GNSS multi-frequency PPP-WAR is carried out based on the optimization of the partial ambiguity fixation strategy, with statistical positioning performance. Finally, the multi-frequency and multi-system single-epoch PPP-WAR positioning performance and the necessary conditions to achieve stable decimetre accuracy are analysed.

72

L. Yang et al.

In terms of BIAS phase fractional deviation stability, the B2a frequency in BDS-3 is slightly worse than other frequency in the system, the L5 frequency of some satellites lacking PCO/PCV correction in GPS is slightly worse, and there is no significant difference between Galileo frequency; in terms of BDS-3/GNSS multi-frequency PPP-AR, the single BDS-3 triple-frequency and four-frequency PPP-AR in imitation dynamic mode in GNSS multi-frequency PPP-AR, some samples were fixed at 1 s, and converged quickly within 3 min at 90% confidence level; in single-epoch PPP-WAR, the fixed rate of WL was above 99%, but only when the number of fixed WL reached 13 could a stable resolution of 20 cm in horizontal and 40 cm in vertical be output. From the multi-station and multi-day statistical results, the GNSS multi-frequency single-epoch PPP-WAR has an accuracy of 38 cm in horizontal and 61 cm in vertical. The results of this study demonstrate that fast convergence of BDS-3/GNSS multifrequency PPP-AR can be achieved without relying on atmospheric products, which provides a basis for its application to in-vehicle high accuracy positioning.

References 1. Cao X, Yu X, Ge Y et al (2022) BDS-3/GNSS multi-frequency precise point positioning ambiguity resolution using observable-specific signal bias. Measurement 195:111134 2. Zhao L, Blunt P, Yang L (2022) Performance analysis of zero-difference GPS L1/L2/L5 and Galileo E1/E5a/E5b/E6 point positioning using CNES uncombined bias products. Rem Sens 14(3):650 3. Guo J, Zhang Q, Li G et al (2021) Assessment of multi-frequency PPP ambiguity resolution using Galileo and BeiDou-3 signals. Rem Sens 13(23):4746 4. Zhao Q, Pan S, Gao W, et al (2022) Multi-GNSS fast precise point positioning with multifrequency uncombined model and cascading ambiguity resolution. Mathematical Problems in Engineering 5. Tao J, Chen G, Guo J et al (2022) Toward BDS/Galileo/GPS/QZSS triple-frequency PPP instantaneous integer ambiguity resolutions without atmosphere corrections. GPS Solut 26(4):1–14 6. Geng J, Guo J, Chang H, et al (2019) Instantaneous decimeter-level positioning using triplefrequency GPS/BeiDou/Galileo/QZSS data over wide areas. In: Proceedings of the ION 2019 Pacific PNT meeting, pp 1013–1022 7. Geng J, Guo J (2020) Beyond three frequencies: an extendable model for single-epoch decimeter-level point positioning by exploiting Galileo and BeiDou-3 signals. J Geodesy 94(1):1–15 8. Zhou F, Dong D, Li W et al (2018) GAMP: an open-source software of multi-GNSS precise point positioning using undifferenced and uncombined observations. GPS Solut 22(2):1–10 9. Li X, Liu G, Li X, et al (2020) Galileo PPP rapid ambiguity resolution with five-frequency observations. GPS Solut 24(1):1–13 10. Qu L, Wang L, Wang H, et al (2022) GPS/Galileo/BDS-2/BDS-3 full-frequency uncombined precise point positioning with fast ambiguity resolution and single-epoch ambiguity resolution on a global scale 11. Liu T, Chen Q, Geng T et al (2022) The benefit of Galileo E6 signals and their application in the real-time instantaneous decimeter-level precise point positioning with ambiguity resolution. Adv Space Res 69(9):3319–3332 12. Li P, Jiang X, Zhang X et al (2020) GPS + Galileo + BeiDou precise point positioning with triple-frequency ambiguity resolution. GPS Solut 24(3):1–13

Effect of Troposphere Parameter Estimation on BDS PPP Zhimin Liu1 , Yan Xu1 , Xing Su1(B) , Junli Zhang2 , Jianhui Cui1 , Zeyv Ma1 , Qiang Li1 , and Baopeng Xu1 1 Shandong University of Science and Technology, Qingdao 266590, China

[email protected] 2 32039 Troops, Beijing 100094, China

Abstract. To investigate the effect of the tropospheric wet delay parameter estimation on BDS precision point positioning (PPP), observations from 13 global MGEX stations are selected over 24 days for static PPP using four parameter estimation methods: Random walk (RW), Piecewise Constant estimation (PWC: 120/180/360, estimated every 120/180/360 min). The results of the tropospheric zenith delay (ZTD) and PPP positioning are evaluated using the ZTD product and the weekly solution product provided by IGS. The ZTD time series and ZTD accuracy of the four parameter estimation methods are analyzed, and the applicability of the four parameter estimation methods is further validated by the PPP positioning accuracy. The experiments show that RW and PWC:120 are suitable for the case of drastic overall ZTD variation, whereas PWC:180 and PWC:360 are suitable for the case of gentle overall ZTD variation; furthermore, it is observed that selecting a suitable parameter estimation method leads to an improvement in both ZTD accuracy and PPP positioning accuracy. According to those results, we recommend the following order of preference for the four parameter estimation methods: RW, PWC:120, PWC:180, and PWC:360, based on their respective strengths. Keywords: ZTD · PPP · RW · PWC · Parameter Estimation

1 Introduction The BeiDou Navigation Satellite System (BDS) is an independent Global Navigation Satellite System (GNSS) developed by China. It has completed full construction and opened to provide global positioning, navigation, and timing services [1–3]. When GNSS is used for high-precision positioning, electromagnetic wave signals emitted by satellites are refracted and delayed as they pass through the ionosphere and troposphere [4–8]. The tropospheric delay is one of the important errors affecting precise point positioning (PPP) [9–16]. Then mapping functions such as GMF (Global Mapping Function, GMF) and NMF (Neil Mapping Function, NMF) convert zenith tropospheric delay (ZTD) into delay of other inclined paths. The ZTD is made up of two components: Zenith Hydrostatic Delay (ZHD) and Zenith Wet Delay (ZWD). The zenith dry delay accounts for approximately 90% of the weight of the zenith tropospheric delay and is primarily © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 73–82, 2024. https://doi.org/10.1007/978-981-99-6944-9_7

74

Z. Liu et al.

influenced by factors such as air pressure and temperature, which can be accurately estimated using models such as the Saastamoinen, Black, and Hopfield [17–19]. There are several factors that may affect the wet delay in the troposphere, including temperature, humidity, atmospheric pressure, cloud cover and precipitation, and parametric estimation methods are usually used. The most commonly used parameter estimation methods today are the Piecewise Constant (PWC) method, the Random Walk (RW) method, and White Noise (WN). The PWC method divides the entire observation time, according to a given constant, into multiple time periods. And introduces a wet delay parameter for each time period, with the same amount of wet delay variation estimated within the same time period. The RW method estimates the zenith tropospheric wet delay by the correlation between epochs. The WN method is more commonly used for receiver clock difference and intersystem bias (ISB). A large number of scholars have conducted research on the estimation methods of tropospheric wet delay parameters, and the overall scheme includes the comparison of piecewise constant and piecewise linear methods considering the horizontal gradient [20]. It has also been analyzed for the performance of the piecewise linear method in terms of data processing, where 2 h estimates a tropospheric wet delay parameter optimally [21]. There are also comparative analyses of PPP performance using Kalman filter and least squares to solve the tropospheric residuals [22]. While random wandering is more often applied in system deviation [23], there are relatively few random walk and piecewise constants to compare and analyze PPP performance, and this paper addresses this problem by comparing and analyzing random walk with different piecewise constants and using PPP performance to test the feasibility.

2 Mathematical Models and Methods 2.1 Precision Point Positioning Model Precision point positioning is a single-point positioning method that employs precision ephemeris and precision clock difference products, along with model correction or parameter estimation for a variety of errors [24–29]. Following are the pseudo-range and carrier phase observation models for precision point positioning. s s s s − Ni λi + dorb − Ii,r − Trs + λi (bi,R − bsi ) + εi,r Lsi,r = ρrs − cVTR + cVTS s s s s s Pi,r = ρrs − cVTR + cVTS + dorb − Ii,r − Trs + c(bi,R − bsi ) + εi,r

(1)

s are the carrier phase observations and pseudo-range observations at where, Lsi,r and Pi,r the frequency i from the satellite s to the receiver r, respectively. λi is the wavelengths. Ni is the ambiguities of the carrier phase. c is the velocities of light in vacuum. ρrs represents s are the receiver the geometric distances from the satellite to the receiver. VTR and VTS s clock differences and satellite clock differences, respectively. dorb is the orbital errors; s and T s are the ionospheric and tropospheric delays, respectively. b s Ii,r i,R and bi are the r s hardware delays of the receiver and the satellite, respectively. εi,r is the observation noises.

Effect of Troposphere Parameter Estimation on BDS PPP

75

2.2 Tropospheric Delay Estimation Methods First, the initial values of the tropospheric dry delay and of the tropospheric wet delay are calculated using the Saastamoinen model, where the meteorological parameters required for the model are used in the GPT3 (Global Pressure and Temperature, GPT) meteorological model, and then the wet delay parameters are estimated using the least squares method. A Markov process is a memoryless stochastic process in which the probability of the current state distribution is related to the previous moment only [30]. The wet delay parameter estimation is well suited to the theory of first-order Markov processes, and the random walk specific expression is as follows [31]. ZWDrs (t + t) = ZWDrs (t) + ωe ωe ∼N (0, σ 2 )

(2)

where, ZWDrs (t + t) and ZWDrs (t) are the wet delays of satellite s and receiver r at moments t + t and t, respectively. ωe zero-mean Gaussian white noise √ with a deviation from the mean of σ and a random wander noise constraint of 2.5mm/ h. The t in the random walk (RW) method represents one epoch, and the t in the piecewise constant method (e.g., PWC:120) is 120 min.

3 Data Sources and Experimental Protocols 3.1 Data Sources Observations from 13 MGEX (Muti-GNSS Experiment, MGEX) stations around the world on days 60–83 of 2022 are used for the experiments in this paper, and the distribution of stations is shown in Fig. 1. Wuhan University’s iGMAS (International GNSS Monitoring and Assessment System) Center provides precision ephemeris and precision clock difference products.

Fig. 1. Distribution of 13 MGEX stations

The IGS weekly solution is used as the coordinate reference value for positioning results, and RMS is used as the accuracy indicator. The tropospheric delay product (ftp:/igs.gnsswhu.cn/pub/gps/products/troposphere) of IGS, which includes tropospheric delay solutions from a large number of tracking stations around the world, was

76

Z. Liu et al.

used to generate the ZTD results. The tropospheric zenith delay every two hours is used as a parameter to be determined during the leveling process according to the delay parameter estimation method, and the tropospheric delay estimation results are then calculated by weighted averaging, and the IGS product has a high accuracy [32, 33]. 3.2 Experimental Protocols To investigate the effects of tropospheric wet delay parameter estimation methods on ZTD and positioning accuracy, four wet delay parameter estimation methods, RW, PWC:120, PWC:180 and PWC:360, were used for static accuracy point positioning, and the detailed solution strategies are shown in Table 1. The ZTD is calculated by adding the ZHD calculated with the tropospheric model and the ZWD estimated with the parameter estimation method, and the ZTD accuracy is calculated using the IGS ZTD product. To further validate and analyze the effect of the above four schemes on the performance of the PPP, the positioning results of the PPP are compared with the weekly solutions of the IGS to obtain the positioning accuracy of the PPP in the E, N, and U directions. Table 1. PPP processing strategy. Treatment term

Strategy

Observation type

Ionosphere-Free combined observation value

Sampling interval

30 s

Stochastic model

Elevation angle

Orbital products

Precision ephemeris and precision clock difference

Cut-off altitude angle

7°

Tropospheric delay model

ZTD: SASS + GPT3 ZWD: RW/PWC:120/PWC:180/PWC:360 for parameter estimation Mapping function: GMF

Receiver clock difference

White noise estimation

Ambiguity resolution

Float

Phase winding Model and Solid tide Model correction

The software platform used for the experiments in this paper is PRIDE PPP-AR [34]. PRIDE PPP-AR is an open-source software package developed by the PRIDE Laboratory, GNSS Research Center of Wuhan University with multi-system precision point positioning ambiguity fixation. When using the RW parameter estimation method, the wet delay parameter is estimated for each epoch; while when using PWC:120/180/360, only one wet delay parameter is introduced in every 120/180/360 min.

Effect of Troposphere Parameter Estimation on BDS PPP

77

4 Analysis of Experimental Results 4.1 ZTD Analysis Figure 2 shows the time series of ZTD, ZWD, and ZTD bias. ZWD and ZTD have the same trend and direction, and the accuracy of ZTD can also reflect the accuracy of ZWD parameter estimation. In the AMC4 station and VILL station, the ZTD value solved by the four parameter estimation methods has good consistency with the ZTD reference value of IGS, and the trend is basically the same. The ZTD time series change is flatter at the AMC4 station, while the ZTD time series change is more severe at the VILL station.

Fig. 2. ZTD, ZWD, and ZTD bias for four methods at AMC4 (left three) and VILL (right three) stations

Further comparisons show that the ZTD accuracies of RW and PWC:120 at station AMC4 are basically within 15 mm, and those of PWC180 and PWC:360 are both within 10 mm. The ZTD accuracy of RW at station VILL is within 15 mm, except for a few points, and is the best of the four schemes. The ZTD accuracy of PWC:120 is almost always within 20 mm, and The ZTD accuracy of PWC:180 is mostly within 20 mm, with a few epochs reaching 30 mm, and the ZTD accuracy of PWC:360 is relatively poor, with the worst reaching 45 mm. The ZTD of AMC4 station fluctuates around 7 cm between days 80 and 81, while the estimated ZTD accuracy of PWC:360 is close to 30 mm. At the VILL station, the ZTD varied about 9 cm between days 71 and 73, while PWC:360 calculated the ZTD with an accuracy of up to 45 mm. This is due to the fact that PWC:360 uses the same wet delay solution settings during the longer period of large ZTD fluctuations, resulting in some significant changes in some epochs. Figure 3 depicts the numerical distribution between the ZTD values estimated by the four methods and the IGS ZTD reference value; the diagonal line allows for a clearer distribution of the numerical relationship, while the RMS is a measure of the ZTD accuracy. Since the ZTD estimates of the same constant segment of the piecewise constant method are the same, in order to better reflect the numerical distribution relationship

78

Z. Liu et al.

Fig. 3. Comparison of the IGS ZTD and the ZTD estimated using the four methods at AMC4 (top) and VILL (bottom)

between the ZTD estimates and the IGS reference values, the ZTD values in the same constant segment of IGS ZTD are averaged. As can be seen from Fig. 3, the divergence between the ZTD estimates of RW and the reference ZTD values of IGS is obvious among the four parameter estimation methods of AMC4 at the measurement station. Compared to RW and PWC:120, the ZTD values estimated for PWC:180 and PWC:360 are evenly distributed and significantly concentrated on both sides of the line. For the RW, PWC:120, PWC:180, and PWC:360 estimates, the effective values of ZTD accuracy are 5.7 mm, 5.23 mm, 4.90 mm, and 5.00 mm, respectively. At the VILL station, the ZTD estimates for RW and PWC:120 were more uniformly distributed than those for PWC:180 and PWC:360. The deviation from the straight path was greatest for PWC:180 and PWC:360. The effective values of ZTD accuracy for RW, PWC:120, PWC:180, and PWC:360 were 6.85 mm, 6.93 mm, 7.00 mm, and 5.00 mm. 6.93 mm, 7.00 mm and 8.43 mm are the dimensions. 4.2 Analysis Based on PPP and ZTD Accuracy The applicability of these four parameter estimation methods was further verified using the positioning accuracy of PPP. Figure 4(a) shows PPP positioning accuracy and ZTD accuracy for 13 sites. In general, the difference between E and N directions of the four parameter estimation methods is obviously very small, except for station WUH2, all stations could be within 4 mm. U direction positioning accuracy, except for AMC4, GCGO and WUH2 stations, PPP positioning accuracy and ZTD accuracy are RW, PWC: 120, PWC: 180 and PWC: 360. Figure 4(b) shows the magnitude of the IGS ZTD change at each of the 13 sites. The red dot represents a single-day standard deviation of 24 days. It can be seen from Fig. 4(b) that the standard deviation of IGS ZTD of AMC4 station and GCGO station is in the range of 8 mm and 5 mm, respectively, and the ZTD change is relatively mild, and the PPP positioning accuracy of PWC:120, PWC:180 and PWC:360 is higher than that of RW. The standard deviation of IGS ZTD of the WUH2 station exceeds 20 mm, and the ZTD change is relatively drastic, while the ZTD accuracy of RW is the highest, indicating that RW is more suitable for the situation of sharp changes in ZTD.

Effect of Troposphere Parameter Estimation on BDS PPP

79

Fig. 4. Accuracy of Positioning and ZTD (a), and Standard deviation of IGS ZTD (b)

The single-day positioning accuracy of the four parameter estimation methods was compared with the single-day distribution of the IGS ZTD using the AMC4 and VIL stations as examples. As shown in Fig. 5, the boxplot is used to reflect the degree of variation of ZTD values over a single day. In the boxplot, the upper and lower whisker lines represent the maximum and minimum values, the upper and lower box lines represent the upper and lower quartiles, and the red dot in the middle represents the mean value. From Fig. 5, it can be seen that the positioning accuracy of the four parameter estimation methods for AMC4 and VIL stations is the same in the E and N directions, because these four parameter estimation methods mainly affect the tropospheric zenith delay, and the effect of the U direction is very significant compared with the E and N directions. The ZTD variation of AMC4 station is flat, while the ZTD variation of VILL station is drastic. For station AMC4, the trend of U-directional positioning accuracy is generally consistent for the four parameter estimation methods. PWC:120, PWC:180, and PWC:360 are more consistent in the U-direction. At the VILL station, PWC:120 and PWC:180 differ less from each other except for days 70 to 73, where RW is the best. It can be seen that the accuracy of PWC:360 at VILL becomes significantly worse in the U-direction from days 70 to 73. This is because the ZTD changed drastically during this period, and PWC:360 used the same wet delay parameters for a long time, which made the wet delay residuals larger and the positioning accuracy worse.

80

Z. Liu et al.

Fig. 5. Accuracy of E, N, U time series and ZTD time series of four parameter estimation methods for AMC4 (left) and VILL (right)

5 Conclusion In this paper, observations from 13 global MGEX stations are chosen for static precision point positioning using four tropospheric wet delay parameter estimation methods: RW, PWC:120, PWC:180, and PWC:360, respectively. The results of the ZTD and PPP solutions are evaluated using IGS’s ZTD and weekly solution products, and the adaptability of the four parameter estimation methods is examined. The conclusions are as follows: (1) When the ZTD fluctuates variably, the effect on PWC:360’s ZTD accuracy is very significant. This is because PWC:360 uses the same wet delay solution parameters over a longer time interval, resulting in significant variability in some epochs. (2) The four parameter estimation methods have insignificant effects on PPP in the E and N directions but significant effects on positioning accuracy in the U direction. This is because the influence of tropospheric delay on PPP positioning results is primarily reflected in the U direction. (3) Based on these results, we suggest the following order of preference for the four parameter estimation methods: RW, PWC:120, PWC:180, and PWC:360. Short estimation time frames (RW and PWC:120) are appropriate for severe ZTD variation, whereas long estimation (PWC:180 and PWC:360) are appropriate for flat ZTD variation. This is because when ZTD changes are gentle, longer estimation periods reduce the number of least-squares parameters to be estimated while increasing constraints on other parameters, resulting in improved localization accuracy. Whereas when ZTD changes are severe, shorter estimation periods reduce the wet delay residuals, resulting in improved localization accuracy. Choosing the best-wet delay parameter estimation method has theoretical and practical implications for improving ZTD and localization accuracy.

Effect of Troposphere Parameter Estimation on BDS PPP

81

Acknowledgments. The work was supported by the Shandong Provincial Natural Science Foundation, China (Grant number ZR2023MD054); National Natural Science Foundation of China (No. 42304036); Key Laboratory of Geomatics and Digital Technology of Shandong Province; and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant number 2017RCJJ072).

References 1. Yang Y et al (2021) Featured services and performance of BDS-3. Sci Bull 66(20):2135–2143 2. Hu Z, Chen G, Zhang Q, Guo J, Su X, Li X, Zhao Q, Liu J (2013) An initial evaluation about BDS navigation message accuracy. In: China satellite navigation conference (CSNC) 2013 proceedings. Springer, Berlin Heidelberg, pp 479–491 3. Su X, Chen H, Zhang J, Geng T, Liu Z, Xie X, Li Q (2021) Navigation performance analysis of LEO augmented BDS-3 navigation constellation. In: China satellite navigation conference (CSNC) proceedings. Springer, Singapore, pp 521–530 4. Elsobeiey M (2020) Characteristic differences between IGS final and ray-traced tropospheric delays and their impact on precise point positioning and tropospheric delay estimates. GPS Solut 24(4):97 5. Li Q, Su X, Xie X, Tao C, Cui J, Chen H, Liu Z (2013) Accuracy analysis of error compensation in the ionospheric model of BDS broadcasting based on ABC-BP neural network. In: China satellite navigation conference (CSNC) 2013 proceedings. Springer Nature, Singapore, pp 54–63 6. Wang G, Yin Z, Hu Z, Chen G, Li W, Bo Y (2021) Analysis of the BDGIM performance in BDS single point positioning. Rem Sens 13(19):3888 7. Ma H, Psychas D, Xing X, Zhao Q, Verhagen S, Liu X (2021) Influence of the inhomogeneous troposphere on GNSS positioning and integer ambiguity resolution. Adv Space Res 67(6):1914–1928 8. Li Q et al (2022) Performance analysis of GPS/BDS broadcast ionospheric models in standard point positioning during 2021 strong geomagnetic storms. Rem Sens 14(17):4424 9. Kouba J, Héroux P (2001) Precise point positioning using IGS orbit and clock products. GPS Solut 5:12–28 10. Li M, Qu L, Zhao Q, Guo J, Su X, Li X (2014) Precise point positioning with the BeiDou navigation satellite system. Sensors 14(1):927–943 11. Sun S, Wang M, Liu C, Meng X, Ji R (2023) Long-term performance analysis of BDS-3 precise point positioning (PPP-B2b) service. GPS Solut 27(2):69 12. Li M et al (2015) Assessment of precipitable water vapor derived from ground-based BeiDou observations with Precise Point Positioning approach. Adv Space Res 55(01):150–162 13. Xu Y, Yang Y, Li J (2021) Performance evaluation of BDS-3 PPP-B2b precise point positioning service. GPS Solut 25(4):142 14. Geng T, Xie X, Fang R, Su X, Zhao Q, Liu G, Li H, Shi C, Liu J (2016) Real-time capture of seismic waves using high-rate multi-GNSS observations: application to the 2015 Mw 7.8 Nepal earthquake. Geophys Res Lett 43(1):161–167 15. Ma H, Verhagen S (2020) Precise point positioning on the reliable detection of tropospheric model errors. Sensors 20(6):1634 16. Qu L, Zhao Q, Li M, Guo J, Su X, Liu J (2013) Precise point positioning using combined Beidou and GPS observations. In: China satellite navigation conference (CSNC) 2013 proceedings. Springer, Berlin, Heidelberg, pp 241–252 17. Saastamoinen: Atmospheric correction for the troposphere and stratosphere in radio ranging of Satellites. Use Artif Satell Geodesy 15(6):247–251

82

Z. Liu et al.

18. Black H (1978) An easily implemented algorithm for the tropospheric range correction. J Geophys Res 83(B4):1825–1828 19. Hopfield H (1969) Two-quartic tropospheric refractivity profile for correcting satellite data. J Geophys Res 74(18):4487–4499 20. Li P, Li X, Wang L (2010) Wet troposphere delay estimation method for the influence of the PPP data processing. J Wuhan Univ 35(7):850–853 21. Huang Z, Mei L (2013) Research on the estimation of tropospheric parameters in high precision GPS data processing. Geomat Geoinf 38(05):12–18 22. Li K, Ouyang Y, Lu X, Wu T (2011) Estimation of tropospheric delay in static precision single point positioning. Oceanograph Survey Mapping 31(06):12–15 23. Chen H (2022) Different ISB handling strategy research on the influence of the PPP. J Henan Sci Technol 9(9):28–31 24. Wang B, Lou Y, Liu J, Zhao Q, Su X (2015) Analysis of BDS satellite clocks in orbit. GPS Solut 20(4):783–794 25. Chen H, Niu F, Su X, Geng T, Liu Z, Li Q (2021) Initial results of modeling and improvement of BDS-2/GPS broadcast ephemeris satellite orbit based on BP and PSO-BP neural networks. Rem Sens 13:4801 26. Ma H, Zhao Q, Verhagen S, Psychas D, Liu X (2020) Assessing the performance of multiGNSS PPP-RTK in the local area. Rem Sens 12(20):3343 27. Su M, Su X, Zhao Q, Liu J (2019) BeiDou augmented navigation from low earth orbit satellites. Sensors 19(01):198 28. Miko´s M, Kazmierski K, So´snica K (2023) Characteristics of the IGS receiver clock performance from multi-GNSS PPP solutions. GPS Solut 27(1):55 29. Su X, Geng T, Zhao Q, Qu L, Li X (2012) Research on integrated orbit determination combined satellite-ground and inter-satellite observation based on helmert method of variance components estimate. In: China satellite navigation conference (CSNC) 2012 proceedings. Springer-Verlag, Berlin Heidelberg, pp 349–359 30. Syski R (1963) Elements of the theory of Markov processes and their applications (A T Bharucha-Reid). SIAM Rev 5(2):173–174 31. Hadas T, Teferle FN, Kazmierski K, Hordyniec P, Bosy J (2016) Optimum stochastic modeling for GNSS tropospheric delay estimation in real-time. GPS Solut 21(3):1069–1081 32. Shi X (2009) Calculating wet component of tropospheric propagation delay using IGS products. Beijing Survey Mapping 17(3):6–9 33. Yao Y, Xu X, Hu Y (2017) Analysis of GGOS tropospheric delay product accuracy and its application in PPP. Acta Geodaetica et Cartographica Sinica 46(03):278–287 34. Geng J et al (2019) PRIDE PPP-AR: an open-source software for GPS PPP ambiguity resolution. GPS Solut 23(4):91

Comprehensive Analysis of the Cycle Slip Detection Threshold in Kinematic PPP During Geomagnetic Storms Qiang Li1 , Xing Su1(B) , Chunyan Tao2 , Junli Zhang3 , Zhimin Liu1 , Jianhui Cui1 , Zeyv Ma1 , Baopeng Xu1 , and Yan Xu1 1 Shandong University of Science and Technology, Qingdao 266590, China

[email protected]

2 Beijing Satellite Navigation Center, Beijing 100094, China 3 32039 Troops, Beijing 100094, China

Abstract. Space weather events may affect the magnetosphere and ionosphere, and thus reduce the positioning capacity of global navigation satellite systems. The paper analyzes the BDS/GPS kinematic Precise Point Positioning (PPP) accuracy during the November 2021 geomagnetic storm to investigate the impact of the cycle slip (CS) detection on positioning accuracy under an active ionosphere. The results show that when using the conventional constant threshold of geometryfree (GF) CS detection, several stations in high latitudes present obvious accuracy anomalies and the maximum positioning error reaches 6 m. It is also found that the CS incidence increases significantly during the period of accuracy anomalies. When using the adaptive GF threshold, the kinematic PPP positioning accuracy and CS incidence return to the normal level, which proves that the underprivilege of conventional GF threshold that the geomagnetic storm could generate CS misjudgments and lead to an abnormal positioning accuracy. The average 3D positioning accuracies of 40 stations worldwide located at different latitudes show that the adaptive GF threshold could improve the average positioning accuracy of stations in high latitudes by 42.5%. Keywords: Kinematic Precise Point Positioning · Geomagnetic Storm · Cycle Slip Detection · Adaptive Threshold

1 Introduction The development of global navigation satellite systems plays an increasingly important role in various fields [1–5]. Autonomous navigation of constellations is a new mode of operation and control, which is a supplement and improvement to the existing mode of operation and control mainly based on ground-based control stations [6–10]. In autonomous navigation and intelligent operation, the influence of the space environment is a factor that should be considered. Space weather events with short time scale variations such as flares and coronal mass ejections caused by solar activity can affect and harm the Earth’s magnetosphere, © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 83–92, 2024. https://doi.org/10.1007/978-981-99-6944-9_8

84

Q. Li et al.

ionosphere, and middle and upper atmosphere. The ionosphere, an important component of near-Earth space, contains a large number of free electrons and positively charged ions that have a large impact on global navigation satellite systems [11–13]. During geomagnetic storms, huge amounts of energy are injected into the upper atmosphere in the form of enhanced electric fields, and energetic particles. This is accompanied by a complex response of the ionosphere and thermosphere to geomagnetic storms through the propagation of energy-momentum and thermosphere-ionosphere coupling [14]. Due to the prevalent thermosphere-ionosphere coupling process, perturbations in the thermosphere may affect the behavior of the ionosphere (including positive and negative ionospheric responses) during geomagnetic storms through wind field transport and compositional changes [15, 16]. The disturbance of the ionosphere by geomagnetic storms can lead to a significant impact on positioning performance [17, 18]. Alcay et al. [19] and Poniatowski et al. [20] studied several geomagnetic storms and found that the error of the kinematic precise point positioning (PPP) increases significantly during geomagnetic storms, and the degree of accuracy loss is related to the intensity of Total Electron Content (TEC) fluctuations. In PPP, cycle slip (CS) detection is an important task [21–23]. Many methods have been proposed for CS detection, including Melbourne-Wübbena (MW) observations, geometry-free (GF) phase combination method, polynomial fitting method, Kalman filtering method, etc. [24]. Different methods are suitable for different situations, for instance, MW observations cannot detect the same CSs at dual frequency observations, the polynomial fitting method is suitable for detecting large CSs, and the geometryfree phase combination method suffers from ionospheric system bias, etc. [25]. The ionospheric perturbation is large during geomagnetic storms, and the geometry-free phase combination method may suffer from CS misclassification under low sampling rate conditions [26, 27]. The global kinematic positioning accuracy during the November 2021 geomagnetic storm is analyzed to investigate the influence of the GF CS detection threshold on positioning accuracy. The geomagnetic storm is a strong geomagnetic storm (the minimum Dst is less than −100 nT), and the BDS/GPS combined system is used in the experiment to improve the positioning robustness. The paper is organized as follows: Sect. 2 introduces the commonly used CS detection methods and the existing GF threshold model; Sect. 3 analyzes the space weather indicators during the geomagnetic storm; Sect. 4 details data processing strategies; In Sect. 5, we analyzes the experimental results; Finally, the conclusion is given in Sect. 6.

2 Theory and Methods 2.1 TurboEdit Cycle Slip Detection Method The current widely used method for CS detection in GNSS data processing is the TurboEdit method [28], which has the advantage of single station detection, a high success rate, and is suitable for CS detection of non-differential data. The TurboEdit method uses MW combination observable and GF combination observable for CS detection.

Comprehensive Analysis of the Cycle Slip Detection Threshold

85

The MW combined observable equation is as follows [29, 30]. ϕ λ  −

f1 P1 + f2 P2 + N λ = 0 f1 + f2

(1)

where ϕ , λ represent the wide lane observable and their wavelengths, f 1 , f 2 represent the frequencies of the two signals, P1 , P2 represent the pseudo-range observable of the two signals, and N represent the ambiguity of the wide aisle observable. The ambiguity of the wide lane observable is used as the CS detection: N = ϕ −

f1 P1 − f2 P2 λ (f1 + f2 )

(2)

MW observable eliminates ionospheric errors, satellite and receiver clock differences, and satellite-station geometric distances. It is only affected by measurement noise and multipath errors. However, it cannot detect equaled CSs on two signals, therefore the phase GF combined observable is used as the CS detection quantity to continue the detection, and the GF phase observable is   LGF = λ1 ϕ1 − λ2 ϕ2 = I f12 − I f22 + λ1 N1 − λ2 N2 + εGF (3)   where, I f12 , I f22 represent the ionospheric delay at two frequencies, and εGF represent the combined observation noise. The GF observable eliminates the effects of receiver clock difference, satellite clock difference, and tropospheric delay. It contains only ionospheric errors and frequency-dependent measurement noise, therefore it is also sensitive to equaled CSs on two signals. 2.2 Adaptive GF Threshold Model The difference between adjacent epochs is generally used for real-time CS detection. In practice, there is no uniform threshold for all cases, and the detection thresholds for MW and GF are usually set to 1–2 cycles and 5–15 cm to detect smaller CSs [31]. For instance, in the open-source package RTKLIB, the GF threshold is set to 5 cm by default [32]. The GF CS detection is highly accurate, and could detect small CSs when set to 5 cm, but for large sampling intervals or when the ionosphere is active, the threshold is slightly more stringent, and could easily lead to misclassification [33]. To reduce the misjudgment of CS caused by improper GF threshold setting, adaptive thresholds can be established using certain methods. The GF adaptive threshold models applicable to GPS and BDS satellites are introduced, respectively. Both models are developed by analyzing the characteristics of the difference of GF observations between adjacent epochs under different ionospheric conditions and considering the effects of data sampling interval and satellite elevation. For GPS satellites, the GF threshold model is as follows [33]: T = l × T0

(4)

86

Q. Li et al.

where l is the weighting factor associated with the satellite elevation and is calculated as follows.  1, e ≥ E (5) l=  sin(E)/ sin(e), e < E where e denotes the satellite elevation of the current epoch; E denotes the critical satellite elevation, where the weighting factor takes effect when e is less than E, which is usually taken as 30°. Where T 0 is the empirical threshold value related to the data sampling interval R, calculated as follows. ⎧ ⎪ ⎨ 0.05, 0 < R ≤ 5 s (6) T0 = 0.03 + 0.004 × R, 5 s k1 where

α=

β=

⎧ ⎪ ⎨ ⎪ ⎩

k0 |˜vi |

·

k0 |˜vi |

·

 

1 1 Si2

k1 −|˜vi | k1 −k0 k1 −|˜vi | k1 −k0

|Si | ≤ k2 |Si | > k2 2 2

|Si | ≤ k2 ·

1 Si2

|Si | > k2

(9)

(10)



where Si = CN0i,obs −CN0i,ref , k2 = μ · STDCN0i,ref . μ is an adjustment factor and usually set to 2.0 (approximately 95% confidence level in normal distribution). Specifically, it can be divided into two steps. In the first step, the IGG scheme is used to conduct the processing. When |˜vi | > k1 , the significant outlier is considered to remain in the observation, which must be excluded (Case I, γ˜i = 0). When k0 < |˜vi | ≤ k1 , the observation is considered to be affected by outlier, and the weight should be reduced (Case 2  vi | ). When |˜vi | ≤ k0 , the observation is not considered to be affected II, γ˜i = |˜kv0i | · kk11−|˜ −k0 by outlier, and the weight should be fixed (Case III, γ˜i = 1). Then, the idea of multipath

160

H. Yuan et al.

detection is used to conduct the second processing for the segment of k0 < |˜vi | ≤ k1 or |˜vi | ≤ k0 . Specifically, only when |Si | > k2 , the significant multipath is considered to remain in the observation. In this case, the weight of the observation should be addi2  vi | · S12 , tionally reduced by considering the multipath effect (Case IV, γ˜i = |˜kv0i | · kk11−|˜ −k0 i

or Case V, γ˜i = S12 ). The flow chart of the proposed outlier and multipath processing i method is shown in Fig. 1.

Fig. 1. Flow chart of the combined processing method

3 Experiment Results and Analysis To ensure the effectiveness of the proposed method, one static monitoring receiver of High-Gain brand is placed on the roof of one high building in Hohai University, Nanjing, China. The low-multipath environment is shown in Fig. 2. Then, the GPS and BDS CN0 template functions of different orbit types are established by using the 24-h CN0 observations on DOY 306, 2021. It is noted that only the GPS L1 and L2 observations and BDS B1 and B2 observations are included in the dataset. In addition, the BDS GEO satellites are excluded to establish the CN0 template function due to their invariable elevation. In conclusion, the CN0 template functions for L1/L2 frequency of GPS MEO satellite, B1/B2 frequency of BDS MEO satellite and B1/B2 frequency of BDS IGSO

Combined Processing of Outlier and Multipath in GNSS Precise Point Positioning

161

satellite are established. Due to space considerations, only the CN0 template functions of B1 frequency of BDS MEO and IGSO satellite are depicted in Fig. 3. It can be found that one certain relationship between the CN0 observation and satellite elevation is existed under the low-multipath environment, i.e., the CN0 template function.

Fig. 2. Static dataset under the low-multipath environment

Fig. 3. CN0 observations and template functions of B1 frequency of BDS MEO and IGSO satellites

With the established CN0 template functions, the GNSS observation dataset from the same receiver type under the real complex environments is used to verify the effectiveness of the proposed method. Specifically, one 24-h GNSS monitoring dataset of the same receiver type is used on DOY 288, 2021. The GNSS dataset is processed by using the HHU-PPP positioning software developed by the author team. The BDS GEO satellites are excluded because the CN0 template function is not established for them. The kinematic PPP processing strategy is used. k0 = 2.0, k1 = 5.0, and μ = 2.0 for the proposed combined processing method. The reference coordinates in each epoch are obtained by using the post-processing differential precise positioning. The other specific processing strategies [13] are listed in Table 1.

162

H. Yuan et al. Table 1. Processing strategies of the kinematic PPP experiment

Item

Strategy

Observation

Raw code and phase observation

Observation model

Undifferenced and uncombined model

Frequency

GPS L1/L2; BDS B1/ B2

Sampling interval

10 s

Cut-off elevation

10°

Weighted model

Elevation-dependent model

Satellite orbit and clock

GFZ precise product

DCB correction

CAS DCB product

Cycle slip

MW and GF combination

Ambiguity parameter

Constant estimation; float value

Ionospheric delay

White noise estimation

Tropospheric delay

Dry: saastamoinen model Wet: Random walk estimation

Receiver coordinates

White noise estimation

Receiver clock

White noise estimation

Inter-system bias

Constant estimation

Parameter estimation

Kalman filter

The numbers of visible satellites and the CN0 values of each frequency under the complex environment are depicted in Figs. 4 and 5, respectively. It can be found that the number of visible satellites of GPS and BDS are mostly around 5, and the numbers of visible satellites are low. Moreover, the numbers of visible satellites are also fluctuated due to the influence of the complex surrounding environment. The observed CN0 values of each frequency of GPS and BDS are depicted in Fig. 5. It can be found that the observed CN0 values between 10 degrees and 30 degrees are relatively low and fluctuated. Hence, the GNSS observations of this experiment dataset are easily to be contaminated by outlier and multipath. The SPP is one pre-processing procedure for the kinematic PPP, and thus the SPP positioning performance is carefully compared and analyzed under three different processing schemes. Specifically, the outlier and multipath are both not processed in Scheme I. Only the outlier is processed by using the IGG method in Scheme II. Both the outlier and multipath are simultaneously processed by using the proposed combined processing method in Scheme III. The SPP positioning results in E, N, and U directions under the three processing schemes are depicted in Fig. 6. It can be found that the positioning results of Scheme I are relatively worse, and there are obvious positioning outliers. The positioning results of Scheme II are improved compared with Scheme I. By using the combined processing method, the positioning results of Scheme III are the best. Furthermore, the positioning accuracy of different schemes in E, N, U, and three-dimensional

Combined Processing of Outlier and Multipath in GNSS Precise Point Positioning

163

Fig. 4. Numbers of the visible satellites of GPS and BDS

Fig. 5. CN0 observations of the GPS and BDS satellites

(3D) direction is listed in Table 2. Compared with Scheme I and II, the improvements of Scheme III are also listed in Table 2, where the bold values are the improvements compared with Scheme II. It can be found that the positioning accuracy of Scheme III is the best compared with Scheme I and II. Compared with Scheme II, the positioning accuracy is improved from 1.128, 1.340, and 3.224 m to 0.877, 1.150, and 2.849 m in E, N, and U directions, and the improvements are 22.3%, 14.2% and 11.6%, respectively.

Fig. 6. SPP positioning results of the three different processing schemes

164

H. Yuan et al.

Table 2. SPP positioning accuracy (unit: m), and the improvements (%) of Scheme III compared with Scheme I and II Item

E

N

U

3D

Scheme I

1.375

1.502

3.930

4.427

Scheme II

1.128

1.340

3.224

3.669

Scheme III

0.877

1.150

2.849

3.195

Improvement

36.2/22.3

23.4/14.2

27.5/11.6

27.8/12.9

Besides, the same processing schemes are used to conduct the kinematic PPP experiment. The PPP positioning performance of the three processing schemes is carefully compared and analyzed. The PPP positioning results in E, N, and U directions for the different processing schemes are depicted in Fig. 7. It can be found that the positioning results of Scheme I are relatively the worse due to the influence of outlier and multipath. Compared with Scheme I, the positioning results of Scheme II are improved by using the outlier processing, whereas there still exist some worse results. However, the positioning results of Scheme III are the best, especially in the convergence stage compared with Scheme I and II.

Fig. 7. PPP positioning results of the three different processing schemes

The PPP positioning accuracy of the three processing schemes in E, N, U, and 3D directions are listed in Table 3. Compared with Scheme I and II the improvements of Scheme III are also listed in Table 3, where the bold values are the improvements compared with Scheme II. It can be found that the PPP positioning accuracy of Scheme III is the best. Compared with Scheme I, the improvements in E, N, U, and 3D directions are 23.4%, 54.7%, 18.8% and 39.2%, respectively. Compared with Scheme II, the improvements are 20.3%, 49.2%, 13.6% and 33.5%, respectively. Hence, the proposed outlier

Combined Processing of Outlier and Multipath in GNSS Precise Point Positioning

165

and multipath processing method can effectively improve the positioning performance of kinematic PPP, especially under complex environments. Table 3. PPP positioning accuracy (unit: m), and the improvements (%) of Scheme III compared with Scheme I and II. Item

E

N

U

3D

Scheme I

0.154

0.362

0.234

0.457

Scheme II

0.148

0.323

0.220

0.418

Scheme III

0.118

0.164

0.190

0.278

Improvement

23.4/20.3

54.7/49.2

18.8/13.6

39.2/33.5

4 Conclusion Aiming at the effect of both outlier and multipath in GNSS PPP technology, a combined processing method is proposed in this paper. The combined processing method is based on the idea of the robust estimation and multipath detection. The GNSS observations under real complex environments are used to conduct the SPP and PPP experiments with three different processing schemes. For SPP, the positioning results of the proposed method are the best. Compared with Scheme II, the positioning accuracy of Scheme III in E, N, and U directions is improved from 1.128, 1.340, and 3.224 m to 0.877, 1.150, and 2.849 m, respectively. For PPP, the positioning performance of the proposed method is significantly improved. Compared with scheme I, the improvements of positioning accuracy in E, N, U, and 3D directions are 23.4%, 54.7%, 18.8% and 39.2%, respectively. Compared with Scheme II, the improvements are 20.3%, 49.2%, 13.6% and 33.5%, respectively. In conclusion, the proposed processing method of both outlier and multipath can effectively improve the PPP positioning performance, especially in complex environments. Acknowledgments. This work is supported by The National Natural Science Foundation of China (41830110, 42004014), Natural Science Foundation of Jiangsu Province (BK20200530). The authors would like to acknowledge the reviewers for their constructive comments.

References 1. Yang Y, Mao Y, Sun B (2020) Basic performance and future developments of BeiDou global navigation satellite system. Satell Navig 1(1):1 2. Zhang X, Hu J, Ren X (2020) New progress of PPP/PPP-RTK and positioning performance comparison of BDS/GNSS PPP. Acta Geodaetica et Cartographica Sinica 49(9):1084–1100 3. Geng J, Chang H, Guo J et al (2020) Three multi-frequency and multi-system GNSS highprecision point positioning methods and their performance in complex urban environment. Acta Geodaetica et Carto-graphica Sinica 49(1):1–13

166

H. Yuan et al.

4. Wang Y, Feng Y, Zheng F (2016) Geometry-free stochastic analysis of BDS triple frequency signals. In: Proceedings of the 2016 international technical meeting of the institute of navigation, pp 956–969 5. Baarda W (1968) A testing procedure for use in geodetic networks. Netherlands Geodetic Commission Publication on Geodesy, Delft 6. Teunissen PJG (2017) Distributional theory for the DIA method. J Geodesy 92(1):59–80. https://doi.org/10.1007/s00190-017-1045-7 7. Zaminpardaz S, Teunissen P (2019) DIA-datasnooping and identifiability. J Geodesy 93(1):85–101 8. Yang L, Shen Y (2020) Robust M estimation for 3D correlated vector observations based on modified bifactor weight reduction model. J Geodesy 94(3):1–17 9. Wu J, Yang Y (2001) Robust estimation for correlated GPS baseline vetro network. Acta Geodaetica et Cartographica Sinica 03:247–251 10. Yang Y (1991) Robust bayesian estimation. Bulletin géodésique 65(3):145–150 11. Strode P, Groves P (2016) GNSS multipath detection using three-frequency signal-to-noise measurements. GPS Solutions 20:399–412 12. Zhang Z, Li B, Gao Y et al (2019) Real-time carrier phase multipath detection based on dual-frequency C/N0 data. GPS Solutions 23(1):1–13 13. Li B, Ge H, Shen Y (2015) Comparison of ionosphere-free, Uofc and uncombined PPP observation models. Acta Geodaetica et Cartographica Sinica 44(7):734–740

Polar Motion Prediction Based on the Combination of Weighted Least Squares and Vector Autoregressive Models Yu Lei1(B) and Danning Zhao2 1 School of Computer Science and Technology, Xi’an University of Posts and

Telecommunications, Xi’an 710121, China [email protected] 2 School of Electronic and Electrical Engineering, Baoji University of Arts and Sciences, Baoji 721016, China

Abstract. This paper presents the application of weighted least squares (WLS) extrapolation and vector autoregressive (VAR) modelling in polar motion prediction. The simultaneous predictions of pole coordinates xp , yp are generated by the combination of (1) WLS extrapolation of harmonic models for the linear trend, Chandler and annual wobbles, and (2) VAR stochastic prediction of the WLS residuals (WLS+VAR). For WLS fit the weights are computed by a power function to accurately extract the Chandler and annual wobbles. Moreover, the VAR technology is used to model and predict the WLS residuals of pole coordinates xp , yp together considering the correlation between xp and yp . . The results show that the accuracy of the ultra short-term predictions up to 10 days into the future obtained by WLS+VAR is equal to or slightly better than that by LS+AR, whilst the predictions out 10 days are substantially more accurate than those by LS+AR. Furthermore, the medium-term predictions beyond 90 days can be improved by WLS+VAR in comparison with the prediction values provided by the IERS Bulletin A. The improvement in the prediction accuracy can reach up to 20%. Keywords: Polar motion · Prediction · Weighted least squares · Weighting function · Vector autoregressive model

1 Introduction Movement of the Earth’s rotation axis with respect to the Earth surface is denoted as polar motion (PM), which forms together with variations of the Earth’s rotation rate as well as precession and nutation the whole set of Earth orientation parameters (EOP). The study of EOP provides some information about irregular motion of fluids on a global scale, therefore serves to the requirement for research on geophysical, hydrology, oceanography, meteorology and other disciplines. Daily EOP values are routinely released by the International Earth Rotation and Reference Systems Service (IERS). However, nearreal time EOP estimates are currently unavailable for users owing to the delay caused © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 167–176, 2024. https://doi.org/10.1007/978-981-99-6944-9_15

168

Y. Lei and D. Zhao

by complex data preprocessing and heavy computation procedures. Accurate short-term predictions up to 10 days into the future of the EOP are therefore essential for several real-time applications such as navigation and tracking of interplanetary spacecrafts and precise orbit determination of Earth satellites [1]. Long-term predicted EOP at least 60 days in advance are also required for climate forecasting and for long-term satellite orbit prediction for the purpose of autonomous satellite navigation, which is of practical importance [3]. Many prediction methods and techniques have been used during the past years to decrease errors of PM predictions. Some of these methods and techniques utilize only information within PM time-series, among which least-squares (LS) extrapolation of the harmonic model in combination with a stochastic method (e.g., autoregressive model [3, 4], neural networks [5, 6]) is the most commonly used approach for PM prediction, and some of them use the effective angular momentum from atmosphere, ocean, and hydrology as the explanatory variable to forecast PM [7]. In October 2005, the 1st Earth Orientation Parameters Prediction Comparison Campaign (1st EOP PCC) was launched to examine fundamental properties of different approaches for EOP prediction. The results of the campaign indicate that the combination of LS extrapolation of the harmonic model and AR filtering is one of the most accurate methods participating in the 1st EOP PCC for PM prediction. No individual prediction method or technique, however, performs the best for all prediction ranges and all components of EOP [8]. The requirement for accurate EOP predictions prompted the IERS for more activity on this subject. Starting in 2021, the 2nd EOP PCC is being performed under the auspice of the IERS to evaluate the state-of-art of the existing prediction methods and techniques such as accuracies and reliabilities (http://eoppcc.cbk.waw.pl/). At present, PM predictions are regularly generated in the IERS Bulletin A at daily intervals based on a statistical extrapolation of the most recent PM observations by the combination of LS extrapolation of deterministic models consisting of the linear, Chandler annual and semi-annual terms, and AR stochastic predictions of LS residuals, referred to as LS+AR. The classic LS+AR method has two drawbacks: (1) the LS algorithm does not take into account the influence of observation geometry on fit results, which results in decreased prediction accuracies with increasing prediction horizons; (2) the AR technology models the pole coordinates xp , yp separately and therefore ignores a priori correlation between xp and yp , which might limit the improvement of prediction accuracies. In order to improve the accuracy of PM predictions, this paper develop a new PM prediction approach based on the combination of weighted least squares extrapolation and vector autoregressive modelling, abbreviated as WLS+VAR. On one hand, this approach considers the effect of observation geometry on results of LS fit by assigning the larger weight to recent observations for enhancing extrapolation of the time-variable Chandler and annual wobbles. On the other hand, this approach uses the VAR technique model and predict the residuals for LS misfit of the pole coordinates xp , yp together rather individually to take into account the correlation of two components. The results of predictions have demonstrated that PM time-series can be predicted 365 days in advance with an enhanced accuracy using the developed WLS+VAR approach.

Polar Motion Prediction Based on the Combination

169

2 Prediction Methodology For polar motion prediction a procedure is applied as described in the flowchart of Fig. 1. The observed polar motion can be split up into a first part, which is rather well known because a function model exists, and second, known stochastic part. The known components, further called the deterministic model, consists of periodic effects such as the liner trend, Chandler and annual wobbles. After reduction of the observed timeseries by subtracting the deterministic model, the stochastic residuals of WLS misfit for the pole coordinates xp , yp (the difference between the deterministic model and data themselves) are formed out of a vector for VAR modelling. The subsequently predicted residuals are then added to the deterministic model to obtain the predicted values of polar motion.

Fig. 1 Flowchart of the WLS+VAR approach for polar motion

2.1 Weighted Least Squares Extrapolation The pole coordinates xp , yp contain a few well-known periodicities such as Chandler and annual wobbles. A priori model hence consists of a purely linear part (offset and drift parameters a, b, c, d ) and two periodical elliptical motions representing the Chandler wobble (parameters f1 , A1,x , B1,x , A1,y , B1,y ) and the annual wobble (parameters f2 , A2,x , B2,x , A2,y , B2,y ), as written in the following. 

xp (t) = a + bt + A1,x cos(2π f1 t) + B1,x sin(2π f1 t) + A2,x cos(2π f2 t) + B2,x sin(2π f2 t) yp (t) = c + dt + A1,y cos(2π f1 t) + B1,y sin(2π f1 t) + A2,y cos(2π f2 t) + B2,y sin(2π f2 t)

(1) The semi-major and semi-minor axes of an elliptical motion correspond to the amplitude of circular motion. For completeness, bias and drift of the linear part and parameters of the CW and AW in this a priori model are estimated simultaneously from xp and yp observations by a WLS fit as β = (CT WC)−1 CT WL

(2)

where β represents the vector of estimated parameter (Eq. 3), L represents the matrix of xp and yp observations (Eq. 4), in which n is the length of observed time-series, C

170

Y. Lei and D. Zhao

represents the coefficient matrix (Eq. 5), and W represents a diagonal matrix of the weights, i.e., W = diag[w1 , w2 , · · · , wn ].   a b · · · A1,x B1,x A2,x B2,x β= (3) c d · · · A1,y B1,y A2,y B2,y  T xp (t1 ) xp (t2 ) · · · xp (tn ) (4) L= yp (t1 ) yp (t2 ) · · · yp (tn ) ⎡ ⎤ 1 t1 cos(2π f1 t1 ) sin(2π f1 t1 ) cos(2π f2 t1 ) sin(2π f2 t1 ) ⎢ 1 t2 cos(2π f1 t2 ) sin(2π f1 t2 ) cos(2π f2 t2 ) sin(2π f2 t2 )⎥ ⎢ ⎥ ⎥ (5) C=⎢ ⎢ .. ⎥ ⎣. ⎦ 1 tn cos(2π f1 tn ) sin(2π f1 tn ) cos(2π f2 tn ) sin(2π f2 tn ) The deterministic model as shown in Eq. 1 aims to extrapolate the present trend, Chandler and annual wobbles within xp and yp data. This can be carried out β = (CT WC)−1 CT WL by the WLS extrapolation. This deterministic models are subsequently utilized for two purposes: (1) to extrapolate the deterministic components in time-series and (2) to get stochastic residuals for WLS misfit (the difference between the deterministic models and data themselves). 2.2 Choice of Weighting Function A key issue in WLS fit is the appropriate choice of a weighting matrix W, i.e., the determination of a proper weight wi assigned to the observation at time ti . When observations are closer to the epochs of predictions, the greater the effect on predictions are. Considering the effect of the geometry on results of LS extrapolation, the weights wi are selected according to the following power weighting function in current work.  wi =

1 tn+1 − ti

0.5 (6)

The function is a very simple method to determine the weights for WLS fit, but takes the geometry into account. It generates the larger weights for recent observations, whilst assigns the smaller weights for measurement far from the prediction points. This power function for WLS fit can therefore take into consideration the influence of geometry on fit results. Moreover, WLS with the weighting function can fit the time-variable Chandler and annual wobbles accurately. 2.3 Vector Autoregressive Modelling The VAR model developed by Love and Zicchino (2006) allows one to account for unobserved individual heterogeneity for the entire time-series via the introduction of fixed effects that enhance the consistency and coherence of measurements [9]. Moreover, this model has some practical advantages that make it a very suitable method to model pole

Polar Motion Prediction Based on the Combination

171

coordinates xp and yp together. First, unlike multivariate autoregressive (MAR) process, the VAR model makes no distinction between exogenous and endogenous variables, in accordance with the interdependence reality. Instead, all variables are treated as endogenous mutually. Each variable entered a VAR model not only depends on its historical realization but also correlates with other variables, conforming with real simultaneities and a priori correlation between xp and yp . Second, VAR is very uncomplicated for efficient and coherent estimations for a small number of variables, especially for two variables in case of xp and yp . Let us denote the bivariate xp and yp residual data as a vector V = (υx , υy ) , where υx and υy are xp and yp residual time-series, respectively. Modelling the residual bivariate time-series V(t) may be performed utilizing the VAR technique. Given a vector of residuals V(t) = (υx (t), υy (t)) at time t, the basic VAR model of a p-order without deterministic components has the form V(t) = R1 V(t − 1) + R2 V(t − 2) + · · · + Rp V(t − p) + U(t)

(7)

where Ri is the 2 × 2 matrix of VAR coefficients at lags i (i = 1, 2, . . . , p), and U(t) = (ux (t), uy (t)) is an unobservable zero mean white noise vector with timeinvariant positive definite covariance matrix, in which ux (t) and uy (t) are independent or serially uncorrelated. The work estimates values of the coefficient matrix Ri resulting from fitting the VAR model to the two-dimensional residuals using maximum likelihood. For choice of the order, this work applies the popular Akaike information criterion (AIC) to choose an appropriate order [10]. The fitted VAR model to V(t) = (υx (t), υy (t)) , t = 1, 2, · · · , n is used to predict these residuals using the linear prediction operator. The predictions for several steps in the future are calculated recursively.

3 Result Analysis 3.1 Data Source The IERS regularly releases daily values of the EOP C04 series with a 30-day delay after the combination of solutions of space geodetic technologies including Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) and Global Navigation Satellite System (GNSS). The combined EOP C04 series is routinely published at one-day intervals at the IRES official website (https://www.iers.org/IERS/EN/DataProducts/data.html). The series consists of observations of the pole coordinates xp , yp .from 1962 until 30 days from now with an accuracy of 0.01 ms. The IERS directing board adopted the EOP 14 C04 as the IERS reference series in February 1, 2017. In this work, the EOP 14 C04 series is taken as data source of the pole coordinates xp , yp . 3.2 Correlation Between the Pole Coordinates Figure 2 shows the observed time-series of the pole coordinates xp , yp , their fitted series and their LS fit residuals between Jan 1, 2007 and Dec 31, 2018. It can be seen that the

172

Y. Lei and D. Zhao

fitted curves of pole coordinates by the harmonic model given in Eq. 1 are very close to the observed pole coordinates, indicating the harmonic model can accurately fit the trend, Chandler and annual wobbles in pole coordinates. Furthermore, the residuals for LS misfit show the irregular quasi-periodic variations. The variations might be modeled and then predicted using a time-series analysis method such as AR and VAR. In both the original time-series of the pole coordinates xp , yp , and their residuals, the correlations between xp and yp often exceed 0.9. The correlations are independence of data span. Moreover, the Hurst exponents of the xp and yp residuals by calculated from an R/S analysis are 0.87 and 0.86, respectively, which means that the xp and yp have same predictability and might hence be predicted together.

Fig. 2 Observed polar motion, WLS fitting and residual series

3.3 Comparison with the LS+AR Predictions of the pole coordinates xp , yp are computed by means of two prediction techniques: LS+AR and WLS+VAR. The range of all predictions is equal to 365 days. Approximating 150 predictions are generated every 7 days at different starting prediction days from January 1, 2019 to June 22, 2021 to calculate the mean absolute error. The parameters of the linear part, CW and AW in LS and WLS fit are solved by a sliding window analysis for every 365-day prediction. The window size for LS and WLS fit is set to 12 years, approximately twice the beating period of the Chandler and annual wobbles. For WLS fit the presented power weighting function is applied to compute the weights assigned to different observation points. The predicted values of pole coordinates are calculated as sum of the LS/WLS extrapolations and univariate or vector predictions of

Polar Motion Prediction Based on the Combination

173

xp and yp . The mean absolute error (MAE), used as the official 1st EOP PCC statistics, is selected for evaluating the prediction accuracy. The MAE is expressed for the ith day into the future as follows. MAE(i) =

N  1  (i) (i)  Pj − Oj  N

(8)

j=1

(i)

(i)

where N is the number of predictions, Oj and Pj are the observed pole coordinates and their ith point of jth prediction, respectively. Figure 3 gives the MAE of the predictions out 30 days into the future as well as up to 365 days computed by LS+AR and WLS+VAR. The results show that the accuracy of the ultra short-term predictions up to 10 days obtained by WLS+VAR is comparable with or slightly better than that by LS+AR. However, the short- and medium-term predictions are more accurate than those generated by LS+AR. In particular, the enhancement in accuracies is mostly seen in the case of the medium-term predictions for horizons of 60 to 365 days based on WLS+VAR. One key reason is that VAR works better than AR in modelling and prediction of the residuals for WLS misfit. Furthermore, it can be seen that the choice of appropriate weights is crucial. The weights calculated by the power functions presented seems to be reasonable for WLS fit and might improve the extrapolations of the harmonic model for the trend, time-variable Chandler and annual wobbles. That is, if an appropriate weights are assigned to PM measurement, WLS fit will work better. The figure of the MAE not only illustrate that PM predictions would be biased if the correlation between the xp and yp was ignored, but also show that WLS+MAR is a very accurate method for PM prediction. The essential results regarding the accuracy of two techniques come from an analysis of the MAE for different prediction horizons (Table 1). What can be said with the information from Table 1 is that the improvement in the prediction accuracy is less than 10% in contrast to LS+AR. For the predictions up to 60 days, the improvement is no more than 21%. The improvement is mostly found in the case of the medium-term predictions for intervals of 60 to 365 days, which is 21% at least and can reach 34% at most. 3.4 Comparison with the IERS Bulletin A In order to further evaluate the performance of WLS+VAR, the 365-d predictions in advance are compared with the predicted values of the pole coordinates for the period of 1–52 weeks spanning from January 3, 2020 to December 31, 2021 published in the IERS Bulletin A. Figure 3 shows the MAE of the predictions results obtained by WLS+VAR and the IERS Bulletin A. Table 2 compares the performance of PM predictions based on WLS+VAR with the predictions from the IERS Bulletin A for different horizons. It is seen that the accuracy of the predictions up to 90 days by WLS+VAR is equal to that by the IERS Bulletin A, whilst the predictions out 90 days into the future is more accurate than those provided by the IERS Bulletin A. The enhancement in the prediction accuracy can reach up to 21%. It is also found that the degree of improvement of WLS+VAR for different prediction horizons is somewhat different. This is owing to not only the prediction method, but also results from the prediction staring day (Fig. 4).

174

Y. Lei and D. Zhao

Table 1 Mean absolute error (in mas) statistics of the polar motion predictions by the WLS+VAR and LS+AR models for different prediction horizons Prediction day

LS+AR

WLS+VAR

Improvement percentage (%)

MAExp

MAEyp

MAExp

MAEyp

1

0.21

0.17

0.20

0.16

5

10

3.01

1.88

2.85

1.66

8

20

5.59

3.32

5.13

2.65

13

30

7.37

4.55

6.68

3.32

16

60

10.59

7.42

9.20

4.94

21

90

12.91

10.90

10.27

7.03

27

180

15.62

16.34

11.43

12.59

25

270

20.08

19.68

16.13

14.97

22

365

24.42

22.57

15.57

15.41

34

Fig. 3 Comparison of the mean absolute error (in mas) of the polar motion predictions by the WLS+VAR and LS+AR models

4 Conclusion PM time-series can be predicted 365 days in advance into the future with enhance accuracies by means of the presented WLS+VAR approach. This approach solves simultaneously all parameters of the trend, Chandler and annual wobbles in the pole coordinates

Polar Motion Prediction Based on the Combination

175

Fig. 4 Comparison of the mean absolute error of the polar motion predictions by the WLS+VAR model and IERS Bulletin A Table 2 Mean absolute error (in mas) statistics of the polar motion predictions by the WLS+VAR and IERS Bulletin A for different prediction horizons Prediction day

IERS Bulletin A

WLS+VAR

MAExp

MAEyp

MAExp

MAEyp

Improvement percentage (%)

1

0.25

0.20

0.21

0.21

7

10

2.91

1.73

2.94

1.62

2

20

5.00

2.44

5.22

2.43

−3

30

6.40

3.35

6.46

3.40

−1

60

9.11

4.53

9.35

4.76

−4

90

13.71

6.03

12.86

6.61

2

180

18.84

17.63

16.01

13.06

20

270

20.31

23.54

18.96

19.66

12

365

22.50

20.35

18.38

18.31

14

xp , yp observations by a WLS algorithm for fitting the variability of the periodicities more accurately. For the WLS fit the power weighting function is proposed to take into the effect of the observations geometry on the results of the WLS fit by assigning larger weights to recent observations. Moreover, considering the correlation between xp and yp

176

Y. Lei and D. Zhao

the VAR technology is used to model and predict the WLS residuals of xp and yp together. A set of predictions indicate that the extrapolation of the harmonic model for the trend, Chandler and annual wobbles supported by the WLS algorithm can improve the PM predictions. The results also show that the VAR technology can be applied successfully to together model and predict the residuals for WLS misfit of pole coordinates xp , yp . In the cases analyzed, the VAR predictions are more accurate than the univariate ones. The comparison with the predictions released by the IERS Bulletin A further illustrates that presented WLS+VAR is an accurate approach for PM prediction. This approach is very simple to use and can therefore take place of the commonly used LS+AR method to generate PM predictions.

References 1. Dill R, Dobslaw H (2010) Short-term polar motion forecasts from earth system modelling data. J Geodesy 84(9):529–536 2. Su XQ, Liu LT, Houtse H et al (2014) Long-term polar motion prediction using normal time-frequency transform. J Geodesy 88(2):145–155 3. Guo JY, Li YB, Dai CL et al (2013) A technique to improve the accuracy of Earth orientation prediction algorithms based on least squares extrapolation. J Geodyn 70:36–48 4. Xu XQ, Zhou YH (2019) EOP prediction using least square fitting and autoregressive filter over optimized data intervals. Adv Space Res 56(10):2248–2253 5. Schuh H, Ulrich M, Egger D et al (2002) Prediction of Earth orientation parameters by artificial neural networks. J Geodesy 76(5):247–258 6. Jin X, Guo JY, Shen Y et al (2021) Application of singular spectrum analysis and multilayer perceptron in the mid-long-term polar motion prediction. Adv Space Res 68(9):3562–3573 7. Dill R, Dobslaw H, Thomas M (2019) Improved 90-day Earth orientation predictions from angular momentum forecasts of atmosphere, ocean, and terrestrial hydrosphere. J Geodesy 93(3):287–295 8. Kalarus M, Schuh H, Kosek W et al (2010) Achievements of the Earth orientation parameters prediction comparison campaign. J Geodesy 84(10):587–596 9. Love I, Zicchiino L (2006) Financial development and dynamic investment behavior: evidence from panel VAR. Q Rev Econ Financ 46(2):190–210 10. Akaike H (1998) Autoregressive model fitting for control. In: Selected papers of hirotugu akaike. Springer, New York, pp: 153–170

Precise Orbit and Clock Offset Determination of LEO Navigation Satellites Based on Multi-constellation and Multi-frequency Spaceborne GNSS Data Junjun Yuan1,2 , Ertao Liang1(B) , Liqian Zhao3 , Kai Li2,4 , Chengpan Tang2,4 , Shanshi Zhou2,4 , and Xiaogong Hu2,4 1 Insight Position Digital Intelligence Technology Service Co., Ltd, Shanghai 201109, China

[email protected], [email protected]

2 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China 3 Space Star Technology Co., Ltd, Beijing 100094, China 4 Shanghai Key Laboratory of Space Navigation and Positioning Techniques, Shanghai

Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China

Abstract. With massive low earth orbit (LEO) satellites, LEO navigation enhancement system is important to improve the service performance of GNSS system. The high precision orbit and clock offset calculation of LEO satellites is one key to establish the stable and reliable space-time reference. Based on GPS/BDS/GALILEO data measured by LEO navigation satellites, this paper determines the precise LEO orbit and clock offset, and evaluates the precision of LEO orbit and clock offset solution. The orbit determination accuracy using GPS (L1/L2), GPS (L1/L5), BDS (B1C/B2a), and GALILEO (E1/E5a) combination is 2.42, 2.93, 2.67, and 2.87 cm, respectively. The clock offset solution accuracy using the above four combination is 0.167, 0.194, 0.186, and 0.180 ns, respectively. These results can provide high-quality product for navigation enhancement system services. Keywords: LEO navigation enhancement · Multi-constellation · Orbit · Clock offset

1 Introduction Global Navigation Satellite System (GNSS) can provide all-day, high-precision positioning, navigation and time (PNT) service, and has become an important infrastructure to safeguard national information security and promote economic development [1]. The GNSS systems represented by GPS/BDS/GALILEO not only keep the stable maintenance, but also improve the service performance through satellite-based, ground-based and other enhancement means. Because of its unique advantages in shortening the precision positioning convergence time and improving the positioning accuracy, LEO navigation enhancement system has become the potential choice to further enhance the GNSS system’s service capability [2]. © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 177–184, 2024. https://doi.org/10.1007/978-981-99-6944-9_16

178

J. Yuan et al.

The precise orbit and clock offset products of LEO satellites provide important space and time benchmarks for LEO navigation enhancement, and their solution accuracy is an prerequisite to ensure their tasks. Many LEO satellites, such as GRACE [3], CHAMP [4], SWARM [5], Fengyun [6] and Haiyang [7], have been launched into orbit continuously. Many achievements have been made in the aspects of perturbation refinement and measurement error modelling related to satellite orbit determination. With the continuous maturity of GPS, BDS, GALILEO and other satellite systems, it is possible to use multi-system data for LEO satellite precise orbit determination (POD). Li et al. [6] analyzed the data quality of onboard GPS and BDS data of FY3C satellite, and carried out POD based on the data of two navigation systems. After disabling BDS GEO data, the orbit accuracy was further improved. Based on the BDS data of Luojia-1 satellite, the final solution accuracy can reach 2 cm by adjusting the weight of GEO satellite data [8]. The POD accuracy using GPS, BDS and GPS/BDS satellites is discussed by using the measured data of Tianping satellite, and the orbit determination accuracy is evaluated by means of overlapping arcs and satellite laser range (SLR) check [9]. Using GPS/GALILEO data of Sentinel-6A satellite, the orbit calculation of single system and multi-systems is realized [10]. Yang [11] discussed the accuracy of LEO satellite clock difference solution based on simulation data, which can be better than 1 ns. Zhang [12] made an in-depth analysis of LEO satellite real-time calculation, prediction and other aspects, and studied the performance of different types of clocks. Based on the measured GNSS data of LEO navigation satellite, this paper analyzes the influence of antenna phase center deviation (PCO), different code bias (DCB) correction on the orbit accuracy, and discusses the orbit determination accuracy using single system, multi-system processing strategies. LEO satellite clock offset is also an important product of LEO navigation enhancement system. This paper also discusses the accuracy and characteristics of LEO satellite clock offset calculation. Finally, the relevant conclusions are carefully given.

2 Data Source and Calculation Principle The test data in this paper are from Tianshu-1 (abbreviated as TS01) of Insight Position Digital Intelligence Technology Service Co., Ltd. and one test satellite of a Chinese LEO enhancement system (abbreviated as SH01). The TS01 satellite was launched in October 2021, with an orbital height of about 520 km and an orbital inclination of 97.5°. It can receive L1 and L5 data from GPS satellites and transmit navigation signals to the ground. The SH01 satellite was launched in August 2021, with an orbital height of about 1100 km and an orbital inclination of 86.4°. It mainly verifies the LEO navigation enhancement and communication functions. The SH01 satellite can receive GPS (L1/L2/L5), BDS (B1C/B2a), GALILEO (E1/E5a) data, and broadcast navigation signals. The LEO satellite POD software used in this paper is SHAOOD (Shanghai Astronomical Observatory Orbit Determination) developed by Shanghai Astronomical Observatory, Chinese Academy of Sciences, and its accuracy can reach 2 ~ 3 cm [13]. The date of TS01 satellite test data in this paper is December 2021, and SH01 is February 2022. The GNSS product is the precise product of Wuhan University (WUM). More detailed processing strategies are given in Table 1.

Precise Orbit and Clock Offset Determination of LEO Navigation Satellites

179

Table 1. Precision orbit and clock offset determination strategy of LEO satellite Model

Description

Measurement model Observation

Non-differentiated ionosphere-free linear combination

Arc length and interval

12 h, 10 s

Weighting strategy

PC (a priori sigma of 1 m), LC (a priori sigma of 1 cm)

GPS products

WUM final products

Elevation cut-off angle

5°

GNSS PCO

igs14.atx

LEO PCO

Discussed

Dynamic model Earth gravity

EIGEN_GL04C, 120 × 120

N-body

JPL DE405

Relativity

IERS 2010

Solid earth tide and pole tide IERS 2010 Ocean tides

FES 2004 (30 × 30)

Solar radiation pressure

Cannonball model

Atmospheric drag

NRLMSISE-00

Empirical forces

Piecewise periodical estimation of the sin and cos coefficients in the track and normal directions

Estimated parameters LEO initial state

Position and velocity at the initial state

Receiver clock

Epoch-wise estimated

Ambiguities

Floated solution

Solar coefficients

One per 3 h

Drag coefficients

One per 3 h

Empirical coefficients

One per 3 h

3 Precision Orbit Determination Analysis 3.1 PCO Estimation Accurate phase center of LEO satellite antenna is an important prerequisite for POD. Although a group of PCO values need to be calibrated before the satellite is launched, the real PCO deviates from the ground calibration values due to mass changes, environmental anomalies, and other reasons. Therefore, the PCO estimation is one of the important links of LEO satellite POD. Since it is necessary to apply empirical force in the T and N directions to compensate the force that has not been modeled, the related PCO components in the X and Y directions are often estimated incorrectly [13], so

180

J. Yuan et al.

this paper only estimates the PCO in the Z direction. Here, both ground calibration and on-orbit estimation values are used for POD to analyze the influence of PCO on orbit determination accuracy.

Fig. 1. Estimation results of SH01 satellite PCO-Z (GPS: L1/L2)

Figure 1 shows the estimation of PCO-Z component of SH01 satellite. Compared with the ground calibration value, PCO-Z has changed about 2 cm after the satellite entered orbit. Similarly, the PCO-Z of TS01 satellite changed from −0.2936 m (the nominal PCO) to −0.2742 m (the estimation PCO). Without correction, this PCO error will be obviously reflected in the orbit determination results, especially the phase residuals. The root mean square (RMS) of SH01 satellite POD phase residuals using the ground nominal values is 11.03 mm, while the RMS using the estimated PCO values is 10.63 mm. The RMS of TS01 satellite POD phase residuals using nominal and estimated PCO values is 10.33 mm and 8.47 mm, respectively. Solar storms, fuel changes, orbital maneuvers and other special circumstances may lead to abnormal jumps of PCO [7]. Therefore, it is of great significance to strengthen the on-orbit monitoring of PCO to ensure the satellite orbit accuracy. 3.2 DCB Correction For the purpose of improving system service performance and meeting various user needs, GPS/BDS/GALILEO and other navigation systems gradually broadcast navigation signals of various frequencies. The POD of LEO satellite using navigation data of multiple frequency combinations has become a new processing method. However, the GNSS clock offset products provided by GNSS analysis centers often use fixed frequency pseudo range data as the benchmark, such as L1/L2 combination for GPS, B1I/B3I for BDS, and E1/E5a combination for GALILEO. Therefore, it is necessary to consider the pseudo range reference deviation between navigation data and satellite clock difference. The DCB correction product used in this paper is from the Aerospace Information Research Institute, Chinese Academy of Sciences [14]. Figure 2 shows the impact of DCB on SH01 satellite orbit determination residuals. The RMS of SH01 satellite pseudo range residuals without and with DCB correction are

Precise Orbit and Clock Offset Determination of LEO Navigation Satellites

181

2.83 m and 1.00 m respectively, while the RMS of TS01 satellite pseudo range residuals without and with DCB correction are 3.54 m and 0.66 m, respectively. Obviously, ignoring the DCB correction will have a significant impact on POD.

Fig. 2. Effect of DCB correction on SH01 orbit determination accuracy

3.3 Orbit Accuracy This paper mainly analyzes the orbit determination accuracy using multi-system and multi-frequency navigation data, and TS01 satellite can only receive GPS L1/L5 data. As a result, SH01 satellite is selected as the analysis object in this section and Sect. 4. Due to the lack of external precision orbit and SLR data, overlap arc comparison (10 h) are used to evaluate the orbit quality. Figure 3 shows the overlapping orbit accuracy of SH01 satellite using GPS (L1/L2), GPS (L1/L5), BDS (B1C/B2a), GAL (E1/E5a) data, and Table 2 shows the POD accuracy of single- and multi-systems. First, the orbit determination accuracy of both single- and multi-system solutions is better than 3 cm, which shows the processing strategy and orbit determination results in this paper are reliable. Second, compared with the single system results, the multi-system orbit determination strategy has more observation data, which increases the reliability of parameter estimation. Therefore, the accuracy of multi-system orbit determination results is generally higher. The orbit determination accuracy of GPS (L1/L2)/BDS, GPS (L1/L2)/GAL, BDS/GAL, GPS/BDS/GAL is 2.28 cm, 2.30 cm, 2.36 cm, and 2.21 cm, respectively. However, it must be noted that the adoption of multi-system data also has more observation data and parameters to be estimated, which will increase the calculation pressure in POD.

4 Precision Clock Determination Analysis When using GPS/BDS/GALILEO data for POD, inter system bias (ISB) of different navigation systems must be considered [15]. Generally, we have two methods to handle ISB parameter. First, one system can be selected as the reference system, and an ISB parameter to be estimated can be added when other system data is used. The other way is that the LEO clock offset calculated by different navigation systems can be estimated separately. The second method is adopted in this paper. Figure 4a shows the SH01 clock

182

J. Yuan et al.

(a) GPS(L1/L2)

(b) GPS(L1/L5)

(c) BDS(B1C/B2a)

(d) GAL(E1/E5a)

Fig. 3. Overlapping orbit comparison results of SH01 Satellite

Table 2 POD Results of SH01 Satellite using single and multi-system Data

R/cm

T/cm

N/cm

3D/cm

GPS(L1/L2)

0.80

1.86

1.27

2.42

GPS(L1/L5)

1.00

2.18

1.62

2.93

BDS(B1C/B2a)

0.88

2.08

1.37

2.67

GAL(E1/E5a)

1.02

2.18

1.51

2.87

GPS(L1/L2)/BDS

0.73

1.78

1.18

2.28

GPS(L1/L2)/GAL

0.75

1.75

1.23

2.30

BDS/GAL

0.78

1.83

1.22

2.36

GPS/BDS/GAL

0.71

1.71

1.18

2.21

offset calculated in one day using GPS (L1/L2), GPS (L1/L5), BDS (B1C/B2a), GAL (E1/E5a) combinations. The LEO clock offset calculated by each combination has a relatively stable consistency. However, it can be seen from Fig. 4b that the difference between SH01 satellite clock offset calculated by different combination data is not very stable, even up to 1–2 ns, and shows a certain period. This may be due to the simultaneous

Precise Orbit and Clock Offset Determination of LEO Navigation Satellites

183

calculation of satellite orbit and satellite clock offset, resulting in the difference between SH01 satellite clock offset related to the orbital period. This phenomenon is also found in Sentinel-6A [10].

(a) SH01 clock offset

(b) The difference of SH01 clock offset

Fig. 4. Results of SH01 satellite clock offset of SH01 satellite calculated by different navigation systems data

Due to the lack of external stable and accurate LEO satellite clock offset, overlapping clock offset comparison strategy is adopted when evaluating SH01 satellite clock offset accuracy. Figure 5 shows the clock offset overlapping accuracy of LEO satellite calculated with GPS (L1/L2), GPS (L1/L5), BDS (B1C/B2a), GALILEO (E1/E5a) data. The RMS are 0.167, 0.194, 0.186 and 0.180 ns, respectively. The calculation accuracy of each day is relatively stable, which can provide precise clock offset products for LEO navigation enhancement system.

Fig. 5. Calculation accuracy of SH01 clock offset

184

J. Yuan et al.

5 Conclusion In this paper, the accuracy of orbit determination using single and multi-system data is further improved by estimating the PCO value and applying DCB correction. The clock offset calculated based on different GNSS data has obvious inter system deviation, and presents a certain periodic term. Using multi frequency and multi-system data for precise orbit and clock offset calculation not only expands the understanding of fusion LEO satellite data processing theories and methods, but also ensures that the LEO augmentation system can provide stable and reliable PNT services. Acknowledgements. This project is funded by the National Natural Science Foundation of China (Grant Nos. 12103077 and 12173072) and the Young Elite Scientists Sponsorship Program by CAST(QNRC001).

References 1. Yang Y, Mao Y, Sun B (2020) Basic performance and future developments of BeiDou global navigation satellite system. Satell Navig 1(1):1–8 2. Hein GW (2020) Status, perspectives and trends of satellite navigation. Satell Navig 1(1):1–12. https://doi.org/10.1186/s43020-020-00023-x 3. Kang Z, Nagel P, Pastor R (2003) Precise orbit determination for GRACE. Adv Space Res 31(8):1875–1881 4. Montenbruck O, Kroes R (2003) In-flight performance analysis of the CHAMP BlackJack GPS receiver. GPS Solut 7(2):74–86 5. Oliver M, Stefan H, Jose V et al (2018) Reduced dynamic and kinematic precise orbit determination for the Swarm mission from 4 years of GPS tracking. GPS Solut 22(3):79 6. Li W, Li M, Zhao Q et al (2018) FY3C satellite onboard BDS and GPS data quality evaluation and precise orbit determination. Acta Geod Cartogr Sin 47(S0):9–17 7. Guo J, Zhao Q, Xiang G et al (2015) Quality assessment of onboard GPS receiver and its combination with DORIS and SLR for Haiyang 2A precise orbit determination. Sci China Earth Sci 58(1):138–150 8. Wang L, Xu B, Fu W et al (2020) Centimeter-level precise orbit determination for the Luojia1A satellite using BeiDou observations. Remote Sens 12(12):2063 9. Zhao X, Zhou S, Ci Y et al (2020) High-precision orbit determination for a LEO nanosatellite using BDS-3. GPS Solut 24(4):102 10. Montenbruck O, Hackel S, Wermuth M et al (2021) Sentinel-6A precise orbit determination using a combined GPS/Galileo receiver. J Geodesy 95(9):109 11. Yang Z, Liu H, Qian C et al (2020) Real-time estimation of Low Earth orbit (LEO) satellite clock based on ground tracking stations. Remote Sens 12(12):2050 12. Zhang HC (2021) Method and application of real-time clock bias estimation for LEO-based navigation augmentation GNSS receiver (in Chinese). Wuhan University 13. Yuan J, Zhou S, Hu X et al (2021) Impact of attitude model, phase wind-up and phase center variation on precise orbit and clock offset determination of GRACE-FO and CentiSpace-1. Remote Sens 13:2636 14. Wang N., Yuan Y, Li Z, Montenbruck O, Tan B (2015) Determination of differential code biases with multi-GNSS observations. J Geodesy 90(3): 209–228 15. Mowafy A, Deo M, Rizos C (2016) On biases in precise point positioning with multiconstellation and multi-frequency GNSS data. Meas Sci Technol 27(3):035102

GNSS Carrier Phase Heading Determination with a Single Array Antenna Wenxin Jin , Wenfei Gong, Tianwei Hou(B) , Xin Sun, and Hao Ma Beijing Jiaotong University, Beijing, China {17111022,wfgong,twhou,xsun,16111020}@bjtu.edu.cn

Abstract. In the field of global navigation satellite system (GNSS), low-cost antijamming array antennas are seldom adopted for precise heading determination. In this contribution, we propose a method for carrier phase heading determination with a single array antenna. Using a centimeter-level very-short baseline composed of two antenna elements in the array, the GNSS observations of a single epoch are iteratively solved to remove the impact of antenna-induced phase biases on heading solutions. To improve the accuracy and availability of heading solutions, the baseline constrained LAMBDA method (C-LAMBDA) is adopted to perform the ambiguity resolution (AR). By simulating a set of GNSS observations for very-short baselines of 10 cm, the proposed method is simulated under singlefrequency, single-epoch conditions. Simulation results indicate that the proposed method can achieve heading solutions with root mean square errors (RMSE) of about 2°. The impact of the number of iterations and C-LAMBDA on heading solutions are also analyzed. Keywords: GNSS · Carrier phase heading determination · Array antenna · Phase pattern · Very-short baseline

1 Introduction In global navigation satellite system (GNSS) applications, the carrier phase heading determination technique performs relative positioning between two nearby antennas and then derives a baseline heading solution from the solved baseline vector. However, array antennas, which are widely used in the field of anti-jamming, are difficult to be adopted. The primary challenge of heading determination with array antennas is the phase measurement biases introduced by antennas. Unlike carefully calibrated geodetic antennas, array antennas may introduce centimeter-level phase biases [1, 2]. In our previous research, we proposed a search-based heading determination method applicable to array antennas, which searches for a heading estimate over spatial and time dimensions by performing relative positioning between a single output signal of an array antenna and a fixed base station within 2 km [3]. This method can effectively remove the impact of phase biases, but requires a base station to provide GNSS observations, and the precision of obtained heading solutions is restricted by the heading search step. In this work, we © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 185–198, 2024. https://doi.org/10.1007/978-981-99-6944-9_17

186

W. Jin et al.

intend to perform relative positioning by exploiting independent output signals from two elements of a single array antenna and remove the dependence on additional base stations. However, the anti-jamming array antennas are usually small in size, and the distance between elements is smaller than the carrier wavelength; and the weak GNSS model of very-short baselines, in turn, would raise a challenge to the performance of the integer ambiguity resolution (AR) method. AR is crucial for carrier phase heading determination. The Least-square AMBiguity Decorrelation Adjustment (LAMBDA) is one of the most widely used AR methods [4], but the AR success rate may decrease under weak GNSS models. To address the problem of heading determination under weak GNSS models of single-frequency, single-epoch, Teunissen developed the Constrained LAMBDA (C-LAMBDA) method by incorporating baseline length information as a constraint into the objective function of AR [5]. The C-LAMBDA method can effectively improve the AR success rate and at the same time guarantee a global optimal solution. In order to improve both precision and success rate of our proposed method, we incorporate the C-LAMBDA method into our heading determination method that is designed for very-short baselines between elements on an array antenna. In this contribution, we concentrate on the problem of carrier phase heading determination with two elements in an array antenna. To address the phase biases introduced by antennas, we propose an iteration-based heading determination method; to deal with the weak GNSS model of very-short baselines, we adopt the C-LAMBDA method to perform AR; a set of GNSS data with phase biases at very-short baselines is simulated to verify the performance of the proposed method under single-frequency, single-epoch conditions, and the impacts of the number of iterations and the C-LAMBDA method on heading solutions are analyzed.

2 Model of Carrier Phase Heading Determination With an Array Antenna In our research, two antenna elements in a small array antenna are utilized for carrier phase heading determination. As shown in Fig. 1, two elements on a horizontal array form a very-short baseline of several centimeters (cm). The model of carrier phase heading determination can be expressed as:     a (1) E(y) = A B , D(y) = Qyy , a ∈ Zn , b ∈ R3 b where E(•) and D(•) denote expectation and variance operations, respectively; y comprises double-differenced measurements (pseudorange and carrier phase); a denotes the integer ambiguity vector; b denotes the three-dimensional baseline vector; A and B are design matrices. After solving the float solutions aˆ and bˆ by neglecting the integer constraint on a, we can convert Eq. (1) into a minimization problem based on the least-square and orthogonal decomposition criterion: min

a∈Zn ,b∈R3

2  2  2   ˆ y − Aa − Bb2Qyy = eˆ Q + minn aˆ − aQ + min b(a) − b yy

a∈Z

aˆ aˆ

b∈R3

Qb(a) ˆ b(a) ˆ

(2)

GNSS Carrier Phase Heading Determination with a Single

187

Fig. 1. Model of heading determination with an array antenna

ˆ b(a) ˆ where •2Q = (•)T Q−1 (•); eˆ = y − Aˆa − Bb; denotes the baseline float solutions conditioned on a. The desired baseline and ambiguity fixed solutions are the minimizers of Eq. (2): ⎧ 2  ⎨ a = arg min aˆ − aQ n aˆ aˆ a∈Z (3)   ⎩ b=b a The resolution process of Eq. (2) is referred to as AR. Subsequently, the heading solution can be derived from the baseline solutions: ⎧  ⎪ ⎨ arctan b E , if AR succeeds  bN , h ∈ [0, 2π ) (4) h= ⎪ ⎩ arctan bˆ E , if AR fails ˆ bN

where the subscripts “E” and “N” denote the east and north components of the baseline vector, respectively. For the antenna elements in the anti-jamming array antennas, there is usually a cm-level offset between antenna phase center and geometric center. This offset is characterized in the GNSS observations as antenna-induced phase bias (APB), whose size is dependent on the signal incoming direction. The relationship between APB and signal incoming direction can be calibrated in advance and described by an antenna phase pattern [6]. The carrier phase observations taking into account APB can be expressed as: φ = r + θ + λn + (azi)

(5)

where r denotes the station-satellite geometric distance; θ denotes the spatially dependent errors such as atmospheric delay, which can be eliminated by differencing operation at short baselines; λ and n denote carrier wavelength and integer ambiguity, respectively; (azi) denotes the APB included in the carrier phase observation of the satellite with azimuth of azi (the elevation is dropped to improve conciseness).

188

W. Jin et al.

Since the relative positions of elements on the array are fixed, we can calibrate the phase patterns of elements 1 and 2 in advance, and by differencing them, we can obtain a between-elements single-differenced phase pattern. Then, the APB contained in the between-elements single-differenced carrier phase (SDCP) observations, denoted as 12 (azi − h), can be determined by the difference between the satellite azimuth and the baseline heading. Thus, the SDCP observation can be expressed as: φ12 = r12 + λn12 + 12 (azi − h)

(6)

where the subscript “12” indicates single-difference between elements 1 and 2. Continuing differencing Eq. (6) between satellites and then linearizing the obtained doubledifferenced equation, we can obtain a measurement equation for the model in Eq. (1). Therefore, SDCP can be considered a fundamental measurement in the carrier phase heading determination model.

3 Iteration-Based Heading Determination Method To counter the phase deviation introduced by the array antenna, which can be large, we need to eliminate this deviation with a phase pattern before performing relative positioning. According to Eq. (6), the baseline heading h must be known, whenever the value of the phase bias contained in the SDCP observations is accurately estimated. However, it is incompatible with the fact that h is also the desired value pursued by heading determination method. To solve the contradiction, an iteration-based heading determination method is presented in this contribution. When the measurements in Eq. (1) contain APB, the size of APB will affect the accuracy of heading solutions. Because the distance between elements on anti-jamming array antennas is usually smaller than the carrier wavelength, the size of the corresponding SDCP measurements is also in cm-level. So, the APB in cm-level will significantly affect the accuracy of the SDCP measurements. Consequently, it may be impossible to derive fixed solutions or only fixed solutions with poor accuracy can be obtained. However, since the APB fluctuates continuously in space, we can approximate the APB using a rough estimate of the baseline heading, h(1) , and the corrected SDCP measurement can be expressed as follows:

 (7) φ12,corr = φ12 − 12 azi − h(1)

 where φ12,corr denotes roughly corrected SDCP measurement; 12 azi − h(1) denotes the APB estimate obtained by substituting h(1) . Obviously, the more accurate h(1) is, the more accurate φ12,corr is, and the more reliable the relative positioning and heading solutions are. When the deviation of h(1) is small, the corrected GNSS measurements can be utilized for heading determination (Eqs. 1–4) to obtain a heading solution h(2) , which is more accurate than h(1) ; if h(2) is again substituted into Eq. (7) for better correcting APB, an even more accurate heading solution h(3) will be achieved. After iterating i times in this way until the heading solutions converge, the last heading solution h(i) can be taken

GNSS Carrier Phase Heading Determination with a Single

189

as the final output. In this work, h(1) is set as the heading solution solved by directly solving the original GNSS measurements without considering APB. In the remainder of this section, the process of the proposed iteration-based heading determination method is presented. We adopt the Kalman Filter (KF) method to solve float solutions. However, it should be noted that since this contribution is dealing with the case of single-frequency, singleepoch, the KF here does not exploit the information from previous epochs, but the solutions from the previous iteration. The state vector is set as:  T x= ab

(8)

The parameters in the state vector are same as the unknowns in Eq. (1). In the first iteration, the baseline vector b is initialized as the difference between the single point positioning solutions of two antenna elements; the double-differenced ambiguity vector a is initialized as the double-differenced vectors of the differences between pseudorange and carrier phase observations of visible satellites. Subsequently, we directly adopt the uncorrected GNSS observations for relative positioning and calculate a baseline heading solution according to Eq. (4). The detailed process of solving float solutions with KF can be referred to [7]. KF comprises a time update process and a measurement update process. However, since this contribution is carried out under single-epoch conditions, the original GNSS observations, the satellites that participate in the solution process, and the parameters in the state vector are invariant over iterations. Therefore, our method does not involve the time update process, but resets the state vectors before each iteration. In the ith(i > 1) iteration, we first substitute the heading solution h(i−1) from the previous iteration into Eq. (7) to correct APB; then set the state vector with the solution from the previous iteration: ⎧ T  ⎪ ⎨  a i−1 b i−1 , if AR succeeds (9) xi =  T ⎪ ⎩ aˆ i−1 bˆ i−1 , if AR fails After that, the corrected measurements are utilized to perform relative positioning and baseline heading calculation again. By exploiting observations in the single epoch, this process is iterated for several times until the heading solutions converge:    (i)  (10) h − h(i−1)  < hthreshold where, hthreshold denotes the threshold value to identify the convergence. Since we concentrate on very-short baselines of several centimeters, the threshold can be set to 0.5°. A fluctuation of less than 0.5° on the baseline heading is approximately equivalent to a variation less than 1 mm in the horizontal component of baseline vector. The detailed process of the proposed method is depicted in Fig. 2. After several iterations, the heading solution of the last iteration, h(i) , will be output.

190

W. Jin et al.

Fig. 2. Flowchart of the proposed iteration-based heading determination method

4 Baseline Constrained AR Method In the heading determination process illustrated in Fig. 2, if the heading solution of first iteration can be achieved with small deviation, the subsequent heading solutions will converge rapidly, effectively reducing the number of iterations and improving the efficiency of the algorithm. This can be achieved by improving the success rate of AR and trying to ensure that a fixed solution can be obtained even under the impact of APB. At the same time, in our work, the performance of AR is also challenged by the weak GNSS models of very-short baseline (cm-level) and single-frequency, single-epoch observations. To address the above problems, we attempt to employ the C-LAMBDA method in the very-short baseline heading determination cases to accomplish the work of solving fixed solutions in Fig. 2. We will further compare the performance of C-LAMBDA method with standard LAMBDA method at very-short baselines in the simulation section. In the following, the flows of C-LAMBDA method are briefly sketched. When an a priori baseline length l is incorporated into the AR process, the GNSS model transforms from Eq. (1) to: E(y) = Aa + Bb, D(y) = Qyy , a ∈ Zn E(l) = b, D(l) = σl2 , b ∈ R3

(11)

GNSS Carrier Phase Heading Determination with a Single

191

Consequently, the objective function of AR (minimization problem) also transforms from Eq. (2) to:   y − Aa − Bb2Qyy + l − b2σ 2 min l a∈Zn ,b∈R3   2  2 = eˆ Q + minn aˆ − aQ + H (a, b) (12) yy



aˆ aˆ

a∈Z

2   ˆ − b H (a, b) = min b(a) b∈R3

Qb(a) ˆ b(a) ˆ

 + σl−2 (l

− b)

2

(13)

where the float solution is still solved from Eq. (1). Different from Eq. (2), the ambiguity search space is no longer a hyper-ellipsoid due to the additional baseline length constraint, and the part associated with the baseline vector in Eq. (12), H (a, b), , will not vanish. Therefore, in the ambiguity search process, we need to solve the minimization problem (13) for every integer candidate vector, which can be achieved with the iterative leastsquares method. Moreover, all integer vectors in the ambiguity search space need to be evaluated one by one to guarantee a global optimal solution. To address the problem that frequent solving of Eq. (13) may lead to a decrease in the efficiency of AR, a smaller ambiguity search space can be determined by exploiting the upper and lower bounding functions of Eq. (12). In the end, the fixed solution can be expressed as:   ⎧ 2 ⎪ minn aˆ − aQ + H (a, b) ⎨ a = arg a∈Z aˆ aˆ    (14) ⎪ ⎩ b = b a = arg min {H (a, b)} 

a= a ,b∈R3

The theoretical derivation and verification analysis of the C-LAMBDA method can be referred to [5, 8, 9].

5 Simulation Validation and Analysis 5.1 Simulation Setup The proposed method determines baseline heading by exploiting two elements on an array antenna, compared to a typical carrier phase heading determination scenario, having two characteristics: (1) shorter baseline length (cm-level), and (2) larger phase biases introduced by the antenna. To simulate the above conditions for evaluating the performance of our method, we simulate a set of virtual GNSS observations for very-short baselines using real data, and artificially add simulated phase biases. We employ open data from the GNSS Data Center of Curtin University to simulate the data of 10 cm baseline in this section. Figure 3 depicts the position relationships of four stations (CUTB0, CUT00, CUTA0, and CUTC0) and their correspondence to the three simulated virtual antennas (CUT00_v, CUTA0_v, and CUTC0_v). The suffix “_v” denotes “virtual”. The four GNSS stations are mounted on the same horizontal plane in Curtin University with a distance of several meters from each other. Together

192

W. Jin et al.

with CUTB0, the three virtual antennas form three very-short virtual baselines (BL1, BL2, and BL3) of 10 cm. The GNSS observations of the virtual antennas can be derived according to the geometric relationships between GNSS stations:  φCUTXX_ v = φCUTXX +  /λ , XX ∈ {00,A0,C0} (15) pCUTXX_ v = pCUTXX +  vCUTB0 - sat · vCUTB0 - CUTXX (16) = × (vCUTB0 - CUTXX  − 0.1) vCUTB0 - sat  × vCUTB0 - CUTXX  where φ denotes carrier phase observation; p denotes pseudorange observation; denotes the difference between the station-satellite geometric distance of a virtual antenna and the corresponding real GNSS station; vCUTB0−sat denotes the vector from CUTB0 to the satellite; vCUTB0−CUTXX denotes the vector from CUTB0 to CUTXX.

Fig. 3. Three very-short baselines (10 cm) simulated with real data from Curtin University. Each baseline is composed of CUTB0 and a virtual antenna CUTXX_v

To simulate the APB in carrier phase observations, we constructed a phase pattern (Fig. 4) by referring to the shape of the phase pattern measured in [2]. We calculate the value of APB according to this pattern and artificially add it to the SDCP over virtual very-short baselines to simulate the cm-level APBs. We adopt the data of November 3, 2022 (DOY 307) at 1 Hz to simulate the virtual observations. It should be noted that, since there is an issue of data missing in the observation file of CUTB0, the simulation time is from UTC 00:00:00 to 23:15:44 (83745 epochs). In the following, five set of simulation cases are designed, and the detailed information of which is shown in Table 1. 5.2 Heading Solution Analysis The accuracy of heading solutions is determined by that of relative positioning solutions. The relative positioning solutions of three very-short baselines are shown in Fig. 5. The

GNSS Carrier Phase Heading Determination with a Single

193

Fig. 4. A phase pattern constructed to simulate APB biases

Table 1. Detailed information of five simulation cases Baseline

Length

Heading

AR method

Data

Case (a)

BL 1

10 cm

86.32°

C-LAMBDA

Single-frequency, single-epoch

Case (b)

BL 2

10 cm

150.78°

C-LAMBDA

Single-frequency, single-epoch

Case (c)

BL 3

10 cm

185.72°

C-LAMBDA

Single-frequency, single-epoch

Case (d)

BL 1

10 cm

86.32°

LAMBDA

Single-frequency, single-epoch

Case (e)

BL 1

10 cm

86.32°

LAMBDA

Dual-frequency, single-epoch

root mean square error (RMSE) of horizontal components are 3.60 cm, 3.64 cm, and 3.88 cm, respectively; the RMSE of vertical components are 8.55 cm, 9.44 cm, and 11.02 cm, respectively. In addition, there are only two outliers in Case (a) and (c). We will ignore the two outliers in the following analysis. Figure 6 illustrates the distribution of horizontal deviations in three cases. Figure 7 depicts the deviations of heading solutions in three cases. The RMSE of heading solutions are 2.21°, 2.01°, and 2.16°, respectively, and the 95-percentiles of absolute deviations are 3.96°, 3.94°, and 4.05°, respectively. We can see that the proposed method is able to achieve heading solutions with RMSE of about 2° and 95-percentiles of about 4° for a case similar to the array antenna (very-short baseline, cm-level APB, and single-frequency single-epoch). 5.3 Impact of the Number of Iterations on Heading Solutions In this contribution, we attempt to eliminate APB by iteratively performing relative positioning between antenna elements. The number of iterations determines the efficiency of our method. Figure 8 shows the number of iterations used to converge the heading solution in three cases, and we can see that most epochs iterate less than 5 times, which can be considered efficient. Statistics show that average number of iterations in three cases are 2.9, 3.3, and 2.8, respectively.

194

W. Jin et al.

Fig. 5. Distribution of relative positioning solutions

Fig. 6. Horizontal deviation of relative positioning solutions

Fig. 7. Deviation of heading solutions

The impact of the number of iterations on the relative positioning and heading solutions is also studied. Figure 9 compares the relative positioning solutions solved by iterating once and three times, where the case of iterating once is equivalent to neglecting the effect of APB and performing relative positioning directly. As shown in Fig. 9, multiple iterations can effectively improve the accuracy of relative positioning solutions. Figure 10 indicates the trend of the RMSE of heading solutions and that of the residual errors after correcting APB with the number of iterations. Obviously, as the number of iterations increases, the heading solutions get more accurate, and the correction error gradually decreases to millimeter-level. 5.4 Impact of C-LAMBDA on Heading Solutions As we have remarked, in the algorithm illustrated in Fig. 2, the speed of convergence of heading solutions depends on the accuracy of the solution in first iteration, which is why we adopt C-LAMBDA to perform AR to improve the solution in first iteration.

GNSS Carrier Phase Heading Determination with a Single

195

Fig. 8. Number of iterations

Fig. 9. Distributions of relative positioning solutions solved by iterating once and three times

Fig. 10. Impact of number of iterations on the RMSE of heading solutions and correction errors

In this section, we compare the performance of C-LAMBDA and LAMBDA methods by simulations under baseline BL 1 and with only one iteration; three simulation conditions are adopted: (1) the elevation mask is set to 10° and the simulated APB is turned off; (2) the elevation mask is set to 10° and the simulated APBs are added; and (3) the elevation mask is set to 20° and the simulated APBs are added. Under the three conditions, we test Case (a), (d), and (e) (see Table 1 for case settings), and Tables 2, 3 and 4 show the statistics of the simulation results. Note that in order to compare the performance of the two AR methods accurately, when the accuracy of the positioning and heading solutions are evaluated hereafter, we have excluded the epochs where AR fails or where the ambiguity is fixed incorrectly. Table 2 presents the statistics when simulated APB is turned off, indicating that the AR success rate, horizontal and vertical accuracy of positioning results of C-LAMBDA under single-frequency is slightly better than that of LAMBDA under dual-frequency;

196

W. Jin et al.

Table 2. Comparison between the performance of C-LAMBDA and LAMBDA (elevation mask is set to 10°, simulated APB is turned off) AR success rate (%)

RMSE of relative positioning (cm) Horizontal

Vertical

ba

bp

RMSE of heading solutions (°)

Case (a)

100

0.28

0.71

0.05

0.27

1.60

Case (d)

62.53

0.37

2.25

0.25

0.27

1.54

Case (e)

99.26

0.36

0.74

0.26

0.25

1.43

Table 3. Comparison between the performance of C-LAMBDA and LAMBDA (elevation mask is set to 10°, simulated APBs are added) AR success rate (%)

RMSE of relative positioning (cm) Horizontal

Vertical

ba

bp

RMSE of heading solutions (°)

Case (a)

99.99

0.75

2.43

0.43

0.61

3.66

Case (d)

9.06

1.17

1.58

1.00

0.60

3.88

Case (e)

98.86

0.98

1.84

0.80

0.56

3.54

Table 4. Comparison between the performance of C-LAMBDA and LAMBDA (elevation mask is set to 20°, simulated APBs are added) AR success rate (%)

RMSE of relative positioning (cm)

Case (a)

98.59

1.30

Case (d)

1.88

1.87

Case (e)

95.66

1.40

2.60

Horizontal

ba

bp

RMSE of heading solutions (°)

3.33

0.90

0.94

6.12

35.78

1.31

1.33

8.68

1.08

0.88

5.76

Vertical

while the AR success rate of LAMBDA under single-frequency is only 62%, which is significantly affected by short baseline length. Compared with Tables 2, 3 and 4 present statistics of the cases with simulated APB artificially added. The simulations with respect to Tables 3 and 4 employ different elevation masks to test the performance of our method under different number of satellites; when the elevation masks are raised from 10° to 20°, the average number of visible satellites over 24 h decreases from 8.75 to 6.72. We can see that the AR success rate of C-LAMBDA under single-frequency is still high; the horizontal and vertical positioning accuracy decreases compared to Table 2, but is still at the same level as the positioning accuracy of LAMBDA under dual-frequency. In contrast, the performance, especially

GNSS Carrier Phase Heading Determination with a Single

197

the AR success rate of LAMBDA under single-frequency degrades severely after the adding of APB, indicating that C-LAMBDA is a better choice for our method to be performed under the condition of single-frequency, single-epoch. The C-LAMBDA method can achieve a higher AR success rate with the help of additional baseline length constraint, which at the same time improves the accuracy of relative positioning solutions along the baseline direction, as can be seen from Fig. 5. To investigate this, we derive two horizontal positioning components expressed as follows: ba = bE sin(h) + bN cos(h) bp = bE cos(h) − bN sin(h)

(17)

where ba denotes the horizontal component along the baseline heading; bp denotes the horizontal component perpendicular to the baseline heading; h denotes the true value of the baseline heading. Tables 2, 3 and 4 show a consistent conclusion that, compared with the components of the reference solutions solved by LAMBDA under dual-frequency, the ba component is more accurate, while the accuracy of bp component is not as accurate. In addition, Fig. 11 compares the accuracy of the two horizontal components for three cases in Fig. 5, validating that the ba component is significantly more accurate than the bp component.

Fig. 11. Horizontal deviations of relative positioning solutions

Since the bp component can indicate the deviation from the true direction of baseline heading to baseline solution, the accuracy of the bp component plays a bigger role in the accuracy of heading solutions. As can be seen from Tables 2–4, due to the poor accuracy of the bp component, the accuracy of the heading solutions solved with C-LAMBDA is slightly worse, although still at the same level as Case (d) and (e). However, if considered together with the AR success rate, C-LAMBDA can achieve a higher availability of heading solutions by sacrificing heading accuracy slightly. Therefore, compared with LAMBDA, C-LAMBDA is more suitable for our method.

6 Conclusions In this contribution, a carrier phase heading determination method for array antennas was proposed. Two antenna elements in the array were used for heading determination, and the GNSS observations of a single epoch were iteratively solved by multiple times, attempting to eliminate APB biases. C-LAMBDA was adopted as the AR method to

198

W. Jin et al.

improve the accuracy of the solutions in the first iteration. The performance of the proposed method under the conditions of single-frequency, single-epoch was investigated by simulating a very-short baseline of 10 cm and APB biases. The simulation results indicated that the RMSE of the horizontal component of relative positioning solutions is less than 4 cm, and the RMSE of heading solutions is about 2°. Simulations verified that multiple iterations can effectively eliminate APB, and the average number of iterations required for the convergence of heading solutions is about 3. By comparing with LAMBDA, it was demonstrated that C-LAMBDA is more suitable for our method. Furthermore, under the condition of single-frequency, single-epoch, even the number of visible satellites is decreased, the C-LAMBDA under single-frequency can still guarantee a high success rate for AR and achieve the heading accuracy of same level as that of the LAMBDA under dual-frequency. Acknowledgment. This work was supported by the Young Elite Scientists Sponsorship Program by CAST (Grant No.2022QNRC001) and National Natural Science Foundation for Young Scientists of China under Grant 62201028.

References 1. Willi D, Meindl M, Xu H, Rothacher M (2018) GNSS antenna phase center variation calibration for attitude determination on short baselines. Navigation 65(4):643–654 2. Kim US (2007) Mitigation of signal biases introduced by controlled reception pattern antennas in a high integrity carrier phase differential GPS system. Stanford University, Ann Arbor 3. Jin W, Gong W, Sun X, Hou T (2022) Orientation determination with an array antenna by exploiting its phase pattern characteristics. GPS Solut 27(1):14 4. Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70(1–2):65–82 5. Teunissen PJG (2010) Integer least-squares theory for the GNSS compass. J Geodesy 84(7):433–47 6. Leick A, Rapoport L, Tatarnikov D (2015) GPS satellite surveying (Fourth edition), 4th edn. Wiley, Hoboken, New Jersey 7. Zhao S, Cui X, Guan F, Lu M (2014) A Kalman filter-based short baseline RTK algorithm for single-frequency combination of GPS and BDS. Sensors-Basel 14(8):15415–15433 8. Teunissen PJG, Giorgi G, Buist PJ (2011) Testing of a new single-frequency GNSS carrier phase attitude determination method: land, ship and aircraft experiments. GPS Solut 15(1):15–28 9. Ma L, Lu L, Zhu F, Liu W, Lou Y (2021) Baseline length constraint approaches for enhancing GNSS ambiguity resolution: comparative study. GPS Solut 25(2):40

Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite Link and Satellite-Ground Observations Yinan Meng1 , Xin Xie2(B) , Hongliang Cai1 , Chao Zhang2 , Rui Jiang2 , Xia Guo1 , and Jun Lu1 1 Beijing Institute of Tracking and Telecommunication Technology, BeiJing 100094, China 2 GNSS Research Center, Wuhan University, Hubei, Wuhan 430079, China

[email protected]

Abstract. The onboard atomic clock is responsible for the generation and maintenance of the time and frequency reference signal on the satellite, and its performance directly affects the service performance of the navigation system. Accurate and reliable satellite clock offsets are the basic data source for the performance evaluation of onboard clocks. Based on the satellite clock offsets measured by the Ka-band inter-satellite links (ISL) and the L-band satellite-ground precise orbit determination and time synchronization (ODTS), the performance of BDS-3 onboard rubidium clocks and hydrogen clocks, including the frequency accuracy, the frequency drift rate and the frequency stability are compared and analyzed in this paper. The research results based on the experimental data show that the frequency accuracy and frequency drift rate of the BDS-3 satellite clocks evaluated by the Ka-band ISLs and the L-band ODTS are basically consistent, and the frequency accuracy and drift rate of the hydrogen clock are better than rubidium clock. For the evaluation of frequency stability, the stabilities of BDS-3 satellite clocks for the averaging time of 1000 s and one day reach 2.5 × 10–14 and 2.5 × 10–15 , respectively. The results of hydrogen clocks are also better than those of rubidium clocks. In addition, the satellite clock stability evaluation using the L-band ODTS and Ka-band ISL have their respective advantages. The evaluation results based on the ODTS clock offsets show better short-term stability. When the averaging time of the Allen variance is greater than 5000 s, the ISL clock offsets show better medium and long-term stability. In addition, compared with the ODTS clock offsets, the ISL clock offsets are not affected by the orbit errors, and the frequency stability of IGSO/GEO satellites obtained based on the ISL is significantly better than the ODTS clock. Therefore, the ISL is more suitable for evaluating the frequency stability of the BDS-3 IGSO/GEO satellites. Keywords: BDS-3 · Satellite atomic clock · Inter-satellite link · Precise orbit determination and time synchronization · Frequency stability

© Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 199–210, 2024. https://doi.org/10.1007/978-981-99-6944-9_18

200

Y. Meng et al.

1 Introduction The satellite time in Global Navigation Satellite System (GNSS) is established and maintained by the onboard satellite clocks, and the performance of the satellite clocks determines the accuracy of navigation positioning and timing directly. Therefore, the performance evaluation for the onboard satellite atomic clocks is of great significance to the system construction and user service of satellite navigation system [1]. The Beidou-2 satellite navigation system (BDS-2) was officially announced to provide services for the Asia-Pacific region in 2012 [2, 3], and all the satellites are equipped with rubidium atomic clocks. Some scholars have carried out detailed performance evaluation for BDS-2 onboard atomic clocks. The frequency accuracy average for the onboard atomic clocks is about 2.3 × 10−10 , and the frequency drift rate is stable at (2.5 ∼ 6) × 10−12 s/s2 [4]. In terms of frequency stability, an analysis of precision clock offsets provided by the IGS Analysis Center of Wuhan University shows that the difference of the performance for rubidium atomic clock of three orbital types of BDS2 is non-significant, with the stabilities for the averaging time of 1000 s and 10000 s reach 1 × 10−13 and 2.37 × 10−13 , respectively [5]. The stability for the averaging time of one day can be stabilized at (2 ∼ 4) × 10−14 , which comparable to the accuracy of the GPS Block IIR satellite rubidium clocks [6]. The Beidou-3 satellite navigation system (BDS-3) was fully completed in 2020. The BDS-3 satellites are equipped with new generation rubidium atomic clocks and hydrogen atomic clocks, which have overall superior performance to BDS-2 satellite atomic clocks [7–10]. The data used in the above evaluation work are based on the clock offset products calculated by the Orbit Determination and Time Synchronization (ODTS) method. This type of clock offset products, which provided by each IGS analysis center and freely downloadable on the network, is currently the primary data source for evaluating onorbit atomic clocks. All BDS-3 satellites carry inter-satellite link (ISL) loads to achieve inter-satellite ranging and communication [7]. The Ka-band observations can be used to solve the relative clock offset between satellites, which providing a different time synchronization method. Some scholars analyzed the ISL clock offset and ODTS clock offset products for the 18 MEO satellites. The fit residues for ISL clock offset was significantly smaller than ODTS, and the stability was better in the medium and long term [11]. The Allan variance calculated by using the ISL clock offsets can reach a stability of about (6 ∼ 9) × 10−15 [12]. Using some of the more stable clocks as datum clocks, the constellation quasi-stable atoms can be established. The MEO satellite hydrogen atom clock stability Hadamard Variance is 3.5 × 10−15 , and 2.8 × 10−15 and 8.2 × 10−15 for IGSO satellite hydrogen atomic clock and rubidium atomic clock, respectively [13]. Based on the above, this paper will analyze the consistency and difference of the results in frequency accuracy, frequency drift rate and frequency stability obtained by different time synchronization methods based on the characteristics of the BDS-3 system. On this basis, a comprehensive performance analysis and evaluation of the atomic clock of BDS-3 satellite in orbit is carried out.

Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite

201

2 Method of Satellite Clock Performance Evaluation 2.1 Data Preprocessing Generally, clock offset data can be directly applied to some timing applications, such as clock offset prediction, etc., but clock offset is not sensitive to outliers. The frequency data is the observation quantity describing the internal parameters of the satellite clock, which can be more conveniently applied to the detection of outliers. The frequency data are computed by the following equation: fi =

Li+1 − Li t

(1)

where fi denotes the frequency data at the epoch i; Li+1 and Li denote the clock offset data at the epoch i and at i + 1; and δt denotes the time between two epochs. The outliers often occur during the long-term operation of onboard clocks. In order to better reflect the performance of onboard clocks, the outliers must be detected and eliminated. In this paper, the median absolute deviation (MAD) is applied to detect the outliers. A deviation limit in terms of the MAD is described as follows: MAD = Median{

|fi − m| } 0.675

(2)

where m is the median value of the clock frequency time series. The MAD estimator is approximate to the standard deviation for normally distributed data. If the clock frequency value is outside the range of [m − n × MAD, m + n × MAD], it will be marked as an outlier. The integer n is set as 6 in this paper. It should be noted that frequency data is generally used for outlier detection. 2.2 Frequency Accuracy Accuracy refers to the degree to which the measured value conforms to its definition, and it characterizes the relationship with the nominal value. For example, time deviation refers to the difference between the pulse to be measured and the pulse that fully meets the standard time, while frequency deviation refers to the difference between the frequency to be measured and the nominal frequency. Frequency accuracy can be defined by the following equation: A=

f − f0 f0

(3)

where, A denotes the frequency accuracy, f is the actual output frequency of the onboard clock, f0 is the nominal frequency of the onboard clock. Affected by the internal factors of the onboard clocks and external environmental factors, the actual output frequency in the atom is not a fixed value, but will change within a certain range with time. Therefore, in order to be as close as possible to the actual output frequency in the onboard clock, it is generally necessary to use a longer average time for measurement in a constant environment to weaken the influence of random errors.

202

Y. Meng et al.

In fact, we cannot directly measure the nominal frequency of the onboard clock, so the actual output frequency of the reference atomic clock is used as the nominal frequency to measure the frequency accuracy of the onboard clock. Generally, the frequency accuracy of the reference atomic clock should be more than an order of magnitude higher than the frequency accuracy of the onboard clock. 2.3 Frequency Drift Rate Frequency drift rate, also known as frequency aging rate, is a parameter that characterizes the frequency change characteristics of an onboard clock. During the operation of the onboard clock, affected by various factors such as component aging and external environment changes, the actual output frequency of the onboard clock will decrease or increase over time. Generally, this change shows a linear trend, so the rate at which the output frequency of the onboard clock changes with time is called the frequency drift rate. The calculation formula of frequency drift rate is as follows: D=

fi+1 − fi τ0

(4)

where fi and fi+1 denotes the frequency data at the epoch i and i + 1; τ0 denotes the time between epoch i and i + 1. D is the frequency drift rate. In this paper, we use the least squares adjustment to obtain the average frequency drift rate within a day. The least squares solution of the frequency drift rate is as follows: N 

D=

[yi (τ ) − y(τ )](ti − t)

i=1 N  2  ti − t

(5)

i=1

where D is the frequency drift rate; yi (τ ) denotes the frequency of the onboard clocks at the epochi; τ is the sampling time; ti is the epochi; y(τ ) and t are the average frequency and the average sampling time in the time series, respectively. 2.4 Frequency Stability Clock frequency stability reflects the noise level of the frequency provided by an atomic clock. Allan deviation is a widely used indicator to describe frequency stability and random characteristics. Compared to the classic Allan deviation, the improved overlapping Allan deviation (OADEV) can utilize all possible combinations to calculate variance. The OADEV of a clock at time interval τ can be expressed as: σ 2 (τ ) =

N −2m  2 1 xi+2m − 2xi+m + xi 2 2(N − 2m)τ

(6)

i=1

where M is the number of the time series; m is the smoothing factor; and τ = mτ0 is the smoothing time.

Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite

203

3 Results of Satellite Clock Performance Evaluation In this paper, the precise clock products calculated by different methods are selected to analyze the performance of BDS-3 satellite clocks: (1) the precise clock products are adopted by the international GNSS service (IGS) and the German Geosciences Center (GFZ), the sampling interval is 300s; (2) the precise clock products are calculated by the BDS-3 intersatellite link (ISL) observation data, the sampling interval is 300s. The two data periods above are both during DOY 150 in 2020 to 210 in 2020, a total of 61 days of clock offsets data. 3.1 Frequency Accuracy We calculated the frequency accuracy of the two precise clock products and classified them by satellite types and on-board clock types. The results are shown in Table 1. From Table 1, it is obvious to see that the frequency accuracy statistics of the two methods of calculation have good consistency, except for very few satellites (C21, C34, C41, C42), basically all satellites have frequency accuracy within the range of (−20 ∼ 20) × 10−12 . In this paper, the BDS-3 satellite clocks are classified according to the type of satellite orbit and the type of satellite clock, and it is found that the frequency accuracy of hydrogen clock adopted by BDS-3 is better than that of rubidium clock for the first time. 3.2 Frequency Drift Rate Figure 1 shows the frequency drift rate of the BDS-3 satellite clock offsets calculated by the two clock products. The frequency drift rate of the two clock products shows good consistency on MEO satellites. The frequency drift rate of hydrogen clock is basically stable within (−0.5 ∼ 1) × 10−13 s/s2 , while the frequency drift rate of rubidium clock is slightly worse, stable within (−1.5 ∼ 1) × 10−13 s/s2 . Meanwhile, it was also found that the frequency drift rate of hydrogen clock was positive systematically, while that of rubidium clock was negative systematically. In addition, the products of IGSO satellites and GEO satellites are quite different, which may be due to the fact that ODTS absorbs more satellite radial residuals when calculating satellite clock products, resulting in excessive frequency drift rate, the satellite clock products calculated by ISL are more stable, and the frequency drift rate is smaller, and it presents a systematic positive value like MEO satellites. This shows that the satellite clock products calculated by the observations of the inter-satellite link is better than the satellite clock products calculated by the ODTS. 3.3 Frequency Stability Using the BDS-3 satellite clock offsets calculated from two different data of L-band and Ka-band, we calculate the Allan’s variance of the BDS-3 on-board atomic clock, respectively, and the variation of frequency stability with smoothing time is shown in Figs. 2 and 3. The two auxiliary lines, the black solid line and the black dashed line, indicate the phase white noise characteristics and the random wandering noise characteristics, respectively.

5.15 −38.86

5.54 −8.81 −7.04 −29.94 −38.04

5.27 −38.30 5.39 3.94 10.08 1.32 5.64 −8.73 −6.90 −31.53

C20

C21

C22

C23

C24

C32

C33

C36

C37

C41

2.08 −0.05 −0.07

2.11 0.64 3.51 2.09 −0.03 −0.08

AVERAGE of MEO Rb

C38

C39

C40

AVERAGE of IGSO PHM

C59

C60

3.50

0.63

2.10

13.81

−30.16 13.27

C42

1.18

10.02

3.74

5.30

12.08

12.03

C19

ISL

ODTS

PRN

Table 1. Frequency accuracy computed by ODTS and ISL clock offsets (1 × 10−12 )

(continued)

204 Y. Meng et al.

−0.06 −13.14

17.95 −16.83 −10.76 9.21 −18.63

−0.06 −13.14 14.08 6.50 6.48 5.21 5.57 -24.37 17.95 −16.83 −4.96 9.20 −18.64 11.91

AVERAGE of GEO PHM

C25

C26

C27

C28

C29

C30

C34

C35

C43

C44

C45

C46

AVERAGE of MEO PHM

12.42

-24.68

5.57

5.21

6.44

6.50

14.08

ISL

ODTS

PRN

Table 1. (continued) Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite 205

206

Y. Meng et al.

Fig. 1. Frequency drift rate computed by ODTS and ISL clock offsets(10−13 s/s2 ).

In the comparison of the two figures, it can be found that in the short range of smoothing time (within about 5000s), the Allan variance calculated by ODTS method basically obeys the form of τ −0.5 function, which presents the noise characteristic of random wandering, while the Allan variance calculated by ISL basically obeys the form of τ −1 function, which presents the characteristic of phase white noise, indicating that it is affected by larger white noise. This is due to the large white noise of the interplanetary link observations relative to the carrier phase observations, which makes the Allan variance of the satellite clock difference calculated using ISL slightly worse than that calculated using ODTS in a shorter period of time.

Fig. 2. Frequency stability of BDS-3 satellite clocks computed by ODTS clock offsets

Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite

207

Fig. 3. Frequency stability of BDS-3 satellite clocks computed by ISL clock offsets

Although the ISL observations are affected by large observation noise, the satellite clock offsets calculated using the ISL method is not coupled with the orbit error and will not be affected by the residuals of the orbit determination. It can be found that the satellite clock offsets calculated using the ODTS method show a significant bump around the smoothing time of about 2 × 104 and 6 × 104 , which is caused by the periodic orbital errors. The same situation occurs for satellite clocks of other satellite systems such as GPS, Galileo, while this situation is significantly attenuated for satellite clock offsets calculated using ISL [15]. It can also be found that the orbit type leads to a significant difference in frequency stability. The frequency stability of MEO satellite clocks is better for ODTS clock offsets products due to the residuals of the orbit determination, while IGSO and GEO satellite clocks are less stable. And because the ISL clock offsets products can weaken the effect of the residuals of the orbit determination, the stability of IGSO and GEO satellite clock offsets products is not much different from that of MEO satellite clock offsets products, and much better than that of IGSO and GEO satellite clock offsets products calculated by ODTS. For more detailed comparison, the frequency stability calculated from the ISL clock offsets and ODTS clock offsets data for four representative satellites are given in Fig. 4, where C32 and C43 are MEO satellites with on-board clocks of rubidium and hydrogen, respectively; C38 and C60 are IGSO and GEO satellites, both equipped with hydrogen clocks. For the MEO satellites (C32 and C43), it can be seen that the stability of their ODTS clock offsets turn out to be better than the ISL clock offsets when the smoothing time is relatively short, with the increase of the smoothing time, the ISL clock offsets outperform the ODTS clock offsets at around 10,000 s until they are in agreement at about day stability, While for the IGSO (C38) and GEO satellites (C60), the ISL clock offsets stability results are better than the ODTS clock offsets almost all the time. This is primarily due to the poor orbital accuracy of IGSO and GEO satellites, resulting in poor stability of the on-board clocks solved using ODTS clock differences.

208

Y. Meng et al.

Fig. 4. Comparison of the frequency stability for four BDS-3 satellites calculated by ISL and ODTS clock offsets.

Figure 5 shows the results of the 1000s/10000s/day stabilities of all on-board atomic clocks calculated using the ODTS and ISL clock offsets, respectively. It can be seen that for MEO satellites, the 1000s stability of the ISL clock offsets product is slightly worse, which is due to the influence of larger observation noise. In terms of 10000s stability and day stability, there is no significant difference between the two clock offsets products. The IGSO and GEO satellites, in contrast, are affected by larger residuals of the orbit determination, and the stability of the clock offsets products obtained using ISL is better. This indicates that the ISL observations can provide a better complement to the GEO and IGSO satellites clock offsets.

Fig. 5. 1000s/10000 s/day stability of BDS-3 in-orbit atomic clocks

Performance Evaluation of BDS-3 Satellite Clocks Based on Inter-Satellite

209

4 Conclusions In this paper, ODTS and ISL satellite clock products are used to analyze the performance of BDS-3 onboard satellite clocks, including the frequency accuracy, the frequency drift rate and the frequency stability. The used clock data are from the GFZ ODTS clock products and the satellite clock offsets calculated based on the BDS-3 inter-satellite link data. The sampling interval of clock offsets is 300s. Based on the evaluation results in this paper, the following conclusions can be obtained: 1. The frequency accuracy results of BDS-3 spaceborne clocks evaluated by the two kinds of clock products have good consistency. The frequency accuracy of all satellites is within the range of (−20 ∼ 20) × 10−12 . 2. The frequency drift rate results of BDS-3 MEO satellite clocks evaluated by the two clock products have the same statistical laws. The frequency drift rate of the rubidium clock is slight worse than that of hydrogen clock. 3. In terms of the frequency stability, the ODTS clock products have an obvious bump near the 10,000 s stability, which is caused by periodic orbital errors, while the ISLbased clocks are not affected by orbital errors. In addition, because the IGSO and GEO satellite clocks acquired by ODTS are affected by greater orbital errors, the ISL-based IGSO and GEO satellite clock frequency stability results are significantly better than that of the ODTS clocks. With the completion of the BDS-3 system and the realization of the inter-satellite link technology, the inter-satellite link not only plays an important role in the BDS-3 satellite precise orbit determination and time synchronization, but also provides a more effective way to evaluate the atomic clocks performance.

References 1. Hesselbarth A, Wanninger L (2008) Short-term stability of GNSS satellite clocks and its effects on precise point positioning. In: Proceedings ION GNSS 2008, Savannah, GA, September 2008; pp 1855–1863 2. Yang Y, Tang J, Montenbruck O (2017) Chinese navigation satellite systems. In: Teunissen PJG, Montenbruck O (eds) Springer handbook of global navigation satellite systems. Springer International Publishing: Cham 3. Dou B (2018) CSNO, navigation satellite system signal in space interface control document open service signal B3I (Version 1.0). China satellite navigation office 4. Li T, Zhang W, Wang J, Li X (2022) Characteristic analysis of Beidou satellite-borne atomic clock based on different evaluation. J Heilongjiang Inst Technol 36(4) 5. Huang G, Yu. H, Guo H et al (2017) Analysis of the mid-long term characterization for BDS on-orbit satellite clocks[J]. Geomatics Inf Sci Wuhan Univ 42(7): 982–988 6. Liu S, Jia X., Sun D (2017) Performance evaluation of GNSS on-board atomic clock. Geomatics Inf Sci Wuhan Univ 42(2): 277–284 7. Yang Y, Mao Y, Sun B (2020) Basic performance and future developments of BeiDou global navigation satellite system. Satell Navig 1(1) 8. Qin W, Ge Y, Wei P, Dai P, Yang X (2020) Assessment of the BDS-3 on-board clocks and their impact on the PPP time transfer performance. Measurement 153

210

Y. Meng et al.

9. Wu Z, Zhou S, Hu X, Liu L, Shuai T, Xie Y, Tang C, Pan J, Zhu L, Chang Z (2018) Performance of the BDS3 experimental satellite passive hydrogen maser. GPS Solut 22(2) 10. Cao Y, Huang G, Xie W, Xie S, Wang H (2021) Assessment and comparison of satellite clock offset between BeiDou-3 and other GNSSs. Acta Geod Geoph 56(2):303–319 11. Xie X, Geng T, Zhao Q, Lv Y, Cai H, Liu J (2020) Orbit and clock analysis of BDS-3 satellites using inter-satellite link observations. J Geod 94(7) 12. Pan J, Hu X, Zhou S, Tang C, Wang D, Yang Y, Dong W (2021) Full-ISL clock offset estimation and prediction algorithm for BDS3. GPS Solut 2(4) 13. Yufei Y, Yuanxi Y, Jinping C, Chengpan T et al (2021) Pseudo-stable constellation bias error of BDS-3 and its high-precision prediction. Acta Geod Cartogr Sin 50(12):1728–1737 14. Riley WJ, Howe DA (2008) Handbook of frequency stability analysis. national institute of standards and technology: Gaithersburg, MD, USA 15. Senior KL, Ray JR, Beard RL (2008) Characterization of periodic variations in the GPS satellite clocks. GPS Solut 12(3):211–225

Global Mapping of Ionospheric ROTI Index and Its Preliminary Application in Analysis of Precise Positioning Degradation Haoyang Jia, Zhe Yang(B) , and Bofeng Li College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China [email protected]

Abstract. The Rate Of change of total electron content (TEC) Index (ROTI), derived from the GNSS dual-frequency carrier phase combination, has been widely used as a key indicator of ionospheric disturbances. In this paper, we propose an ordinary kriging approach to generating ionospheric ROTI maps with the aim of characterizing ionospheric irregularities and assessing their impacts on precise point positioning (PPP) at a global scale. In the kriging-based mapping of the ROTI index, we exploit three semi-variogram models (i.e., Gaussian, Spherical and Exponential) and evaluate their performance during the September 6–9, 2017 geomagnetic storm by taking advantage of more than 500 GNSS stations. Results suggest that (1) The overall RMSE of kriging cross-validation for all three models is generally less than 0.4 TECu/min, and the spherical semi-variogram performs best by analyzing the semi-variogram fitting and interpolation accuracy. (2) The ROTI maps can effectively represent the spatial-temporal evolution and intensity of global ionospheric irregularities during the geomagnetic storm. (3) There is a strong positive correlation between the PPP errors and ROTI values. The maximum, average and RMS of 3D positioning errors are 0.33 m, 0.14 m and 0.40 m, respectively, with the level of ROTI values less than 0.5 TECu/min, while they can reach 7.06 m, 2.80 m and 3.63 m, respectively, when the ROTI increases above 2.5 TECu/min. The ground positioning risk derived from the ROTI maps can effectively indicate the degradation of positioning performance. Keywords: GNSS · Ionospheric irregularity · ROTI · Kriging interpolation · Precise point positioning

1 Introduction Ionospheric irregularities can cause effects such as scattering and diffraction of radio signals when they pass through the ionosphere, usually referred to as ionospheric scintillation [1]. For GNSS users, ionospheric scintillation can destroy the stability of the GNSS signals, thus resulting in the degradation of measurements and signal loss in the GNSS receiver [2, 3]. The world-widely distributed GNSS stations has facilitated the study of ionospheric disturbances over large regions. Based on the total electron content (TEC), derived from © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 211–226, 2024. https://doi.org/10.1007/978-981-99-6944-9_19

212

H. Jia et al.

GNSS multi-frequency carrier phase observations, various TEC-based indices have been proposed and utilized to study ionospheric disturbances, such as ROTI (Rate Of change of TEC Index), AATR (Along Arc TEC Rate) and DIX (Disturbance Ionosphere indeX) [4–6]. While studies have shown that ROTI, a widely used index, can serve as a proxy for inverting the evolution of ionospheric irregularities [7–9]. Mapping of ionospheric irregularities using GNSS observations can help investigate ionospheric disturbance characteristics. The International GNSS Service (IGS) has released daily ROTI-map product to characterize the ionospheric irregularity occurrence and intensity, which is created by computing the daily average of the ROTI values within each grid cell of the single-layer ionosphere [10, 11]. In addition, the ionospheric disturbance characteristics can be extended to non-sampling locations through interpolation. Ordinary kriging, one of the widely used spatial interpolation techniques, has been applied to regional mapping of scintillation indices and ROTI, achieving good mapping results [12–14]. However, few past studies have addressed the interpolation methods and the mapping results on a global scale. Many studies on GNSS positioning degradation under disturbed ionospheric activity and geomagnetic conditions have been carried out in recent years. Precise point positioning (PPP) degradation under ionospheric disturbance conditions is primarily attributed to frequent cycle slips, rapid fluctuations in the carrier phase, and loss of lock [15–17]. Studies have shown that there is a strong positive correlation between the PPP errors and ROTI during geomagnetic storms [18, 19], suggesting that the ROTI index can be potentially used to analyze the degradation of GNSS positioning under ionospheric disturbed condition and assess the potential positioning risk. In this paper, we generate the global ionospheric ROTI maps based on the ordinary kriging method, and the interpolation performances of three semi-variogram models are analyzed and discussed. To demonstrate the effectiveness of the ROTI maps, we analyze the characteristics of global ionospheric irregularities during the geomagnetic storm of September 6–9, 2017, and investigate the associated PPP degradation. Moreover, the ROTI maps are used to initially estimate and validate the ground positioning risk. The kriging-based ROTI maps proposed in this paper have significant potential for applications in global ionospheric irregularities monitoring for space weather as well as risk assessment and warning for GNSS navigation and positioning.

2 Data and Methodology 2.1 Data We use GPS observations collected by over 500 GNSS stations evenly distributed worldwide, with data sampled at 30s intervals. Figure 1 presents the geographic distribution of the stations. The observation period is from September 6 to 9, 2017, during which an X9.3 solar flare event occurred, causing strong geomagnetic storms and ionospheric responses observed from various regions of the globe [20, 21]. Figure 2 presents the geomagnetic conditions on September 6–9, 2017. The first main phase commenced at about 23:30 UT on September 7, with a significantly drop of the storm intensity Dst index. The Dst index reached a minimum value of -122 nT until 01:00 UT the next day, and remained negative for several hours. Afterward, the Dst

Global Mapping of Ionospheric ROTI Index and Its Preliminary

213

geomagnetic latitude lines (0°,±20°) 80°N

40°N 20°N

0°

0°

20°S

40°S

80°S 180°

120°W

60°W

60°E

0°

120°E

180°

Fig. 1. Geographical distribution of 500 + GNSS stations used in this study. ap

Kp

(a) 8

300 200

4 100

2

AE/AU/AL (nT)

0

ap

Kp

6

0 (b)

AE

(c)

SYM

AL

AU

2000 1000 0 -1000 -2000 -H

Dst

SYM - H / Dst (nT)

100 50 0 -50 -100 -150 09/06

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

UT

Fig. 2. Variations of geomagnetic activity indices on September 6–9, 2017. (a) Kp and ap index; (b) AE, AU and AL index; (c) SYM-H and Dst index. Two shaded areas indicate two main phases of the geomagnetic storm.

index began to drop again around 11:00 UT on September 8 and continued for about 6 h, reaching a minimum value of -109 nT at 17:00 UT. During the two main phases, the

214

H. Jia et al.

SYM-H index showed the same variation trend; while the Kp and ap index reached their two peaks, respectively; and the auroral electrojet activity indices (i.e., AE, AU, and AL) all had significant fluctuations. 2.2 ROTI Index In this study, we use the ROTI to characterize the evolution of ionospheric plasma irregularities, which is estimated as the standard deviation of ROT (Rate of TEC) using a 5-min sliding window [4]. The TEC time series can be calculated using a geometry-free combination of GNSS dual-frequency carrier phases. To avoid the distortion on ROTI values, we use a modified TurboEdit method to detect cycle slips to obtain continuously available arcs; the moving average detrending method is used to remove non-ionospheric factors such as satellite motion, tropospheric delay and receiver noise; and a 30-degree elevation mask angle is applied to avoid the multipath effect. The ROTI is mapped onto ionospheric pierce points (IPPs) at an altitude of 350 km. 2.3 Mapping Methodology for ROTI Index The spatial covariance structure of the sampled points is measured by a semi-variogram γ (h) in ordinary kriging [22]. For any two IPPs x and x + h with distance h, the semivariogram between their ROTI measurements can be denoted as: γ (h) =

1 E(Z(x + h) − Z(x))2 2

(1)

where Z(x) is the ROTI value at IPP x, and the γ (h) is only related to the distance between the two IPPs. ˆ 0 ) at the target point x 0 can be estimated as a For the ordinary kriging, the value Z(x weighted linear combination of the surrounding known measurements, denoted as: ˆ 0) = Z(x

N 

λi Z(xi )

(2)

i=1

where λi is the weight. To ensure that the estimate is unbiased and optimal, the weights are made to sum to 1, and a Lagrange multiplier ψ(x 0 ) is introduced to achieve minimization: ⎧ N    ⎪ ⎪ λi γ xi , xj + ψ(x0 ) = γ xj , x0 ⎨ i=1 (3) j = 1, 2, ...N N  ⎪ ⎪ ⎩ λi = 1 i=1

where the quantity γ (x i , x j ) is the theoretical semi-variogram between two IPPs. The ˆ 0) solution of the above equations provides the kriging weights and the ROTI value Z(x at the target point x 0 can be then estimated. Choosing an appropriate theoretical semi-variogram model to fit the discrete experimental semi-variograms is the key to generating effective global ionospheric ROTI maps

Global Mapping of Ionospheric ROTI Index and Its Preliminary

215

in our work. There are three theoretical semi-variogram models considered in this study for ROTI mapping, as summarized in Table 1, where d is the distance between the IPP points, p is the partial sill, n is the nugget, and range r represents the farthest distance where correlation exists between ROTIs. The nugget coefficient (denoted as n/(n + p)) can be used as a measure of spatial correlation. When the nugget coefficient is ≤25%, 25% ~75%, and >75%, it indicates strong, moderate, and weak spatial autocorrelation of the variables, respectively [23]. Table 1. Typical theoretical semi-variogram models. Model Gaussian Spherical

Exponential

Function



2  +n p · 1 − exp − (7d )2 (4r) ⎧ ⎨ p · ( 3d − d 3 ) + n d ≤ r 2r 2r 3 ⎩ p+n d >r 



+n p · 1 − exp − 3d r

In our study, the residual sum of squares (RSS) and the coefficient of determination (R2 ) are used to indicate the goodness-of-fit of the theoretical semi-variogram. As the RSS turns to be smaller, the R2 is closer to 1, indicating the better fitting result. In addition, mean error (ME), root mean squared error (RMSE) and correlation coefficient derived from cross-validation are used to evaluate the interpolation performance for each semi-variogram model. 2.4 Processing Strategy for Precise Point Positioning To investigate the effect of ionospheric disturbances on the PPP performances during the geomagnetic storm commenced on September 7–8, 2017, the GPS data are processed for kinematic PPP solutions using RTKLIB (version 2.4.3), which is an open-source program for GNSS positioning [24]. The detailed data processing strategies for GPS kinematic PPP are summarized in Table 2.

3 Results and Discussions 3.1 Comparisons of Semi-Variogram Models for ROTI Mapping A sample of ROTI measurements taken on September 8, 2017, at 14:00 UT is used as an example, with a total number of 2288 recorded at that time. Figure 3 depicts the ionospheric response represented by ROTI on a global scale. Figure 4 and Table 3 present the experimental semi-variogram and fitted theoretical semi-variogram models, respectively. The nugget coefficients of all three models are less than 25%, indicating all three models are suitable for the kriging interpolation.

216

H. Jia et al. Table 2. Strategies for kinematic PPP.

Parameters

Strategies

Observations

GPS observations from L1/L2 signal

Observation weighting

Elevation-dependent weight

Sampling interval

30s

Elevation mask angle

15°

Satellite orbits and clock

Precise final provided by IGS

Ionospheric delay

First-order effect eliminated by IF combination

Tropospheric delay

Saastamoinen model for hydrostatic component Estimated as a random walk for zenith total tropospheric delay NMF is used as tropospheric mapping function

Phase center offset

Corrected with igs14.atx

Ambiguity

Float

Satellite DCB

Corrected with the monthly DCB products from CODE

09/08 14:00 UT

80°N

Latitude

40°N

0°

40°S

80°S

180°

120°W

60°W

0°

60°E

120°E

180°

Longitude

Fig. 3. Geographical distribution of ROTI at 14:00 UT on September 8, 2017.

Comparing RSS and R2 , the gaussian model has the smallest fitting residuals and the highest R2 , followed by the spherical and exponential models. The interpolation performances of the three models are evaluated using the crossvalidation method, and the results are shown in Fig. 5 and Table 4. Figure 5 shows that the distribution of the interpolation errors of the three models is consistent, and the interpolation results are more prone to significant deviations at high latitudes and in the Asian regions where intense ionospheric disturbances occurred. According to the statistics in Table 4, it is found that the model with the best goodnessof-fit is not always the model with the best performance of cross-validation. The correlation coefficients of all three models are greater than 0.8, and the exponential model has the best overall performance, with the lowest ME and RMSE.

Global Mapping of Ionospheric ROTI Index and Its Preliminary

217

Semi-Variogram (log 2 (TECu/min))

1.5

1

Experimental 0.5

Gaussian Spherical Exponential

0 0

20

40

60

80

100

120

Distance (deg.)

Fig. 4. Experimental semi-variogram and corresponding fitted models.

Table 3. Parameters of each semi-variogram model. Parameters

Unit

Model Gaussian

Spherical

Exponential

n/(n + p)

–

0.1425

0

0

RSS

log4 (TECu/min)

0.2327

0.2374

0.2864

R2

–

0.8063

0.8025

0.7617

To determine the optimal semi-variogram model, we apply the methods to the entire time period from September 6 to 9 to further compare and analyze the interpolation performance. Figure 6 depicts the time series of R2 for semi-variogram fitting and RMSE for cross-validation. The R2 of spherical and gaussian models is better overall for both two main phases, while the spherical and exponential models have better crossvalidation results. The cross-validation performance of all the three models decreased in the two main phases but remained better than 0.4 TECu/min overall. From the analysis of both semi-variogram fitting and interpolations, we choose the spherical model to fit the discrete experimental semi-variograms for ROTI mapping. 3.2 Global ROTI Mapping for 2017 September 7–8 Storm Based on the analysis in Sect. 3.1, the spherical model is applied in generating global ROTI maps with a spatial resolution of 1° by 1° and a time interval of 30s. The ionospheric response during the two main phases of the 2017 September 7–8 storm is then analyzed by using the generated ROTI maps, as shown in Fig. 7. Starting at 21:00 UT on September 7, the geomagnetic storm entered the first main phase. At that time, the ionospheric irregularities were primarily observed at high latitudes of the northern and southern hemispheres. As the irregularities reached over South America at 00:00 UT on September 8, the coverage of irregularities over North America

218

H. Jia et al. (a) Gaussian

(b) Spherical

80°N

80°N

40°N

40°N

Latitude

0° 40°S

40°S

80°S

80°S

180°

120°W

60°W

0°

60°E

120°E

180°

180°

120°W

60°W

0°

60°E

120°E

180°

Longitude

(c) Exponential 80°N

Latitude

40°N 0° 40°S ROTI Error (TECu/min)

80°S 180°

120°W

60°W 0° Longitude

60°E

120°E

180°

Fig. 5. Errors distribution of cross-validation. (a) Gaussian; (b) Spherical; (c) Exponential.

Table 4. Statistics of cross-validation for the three semi-variogram models. Statistics

Unit

Model Gaussian

Spherical

Exponential

ME

TECu/min

−0.0085

−0.0057

−0.0046

RMSE

TECu/min

0.1132

0.1035

0.1029

correlation coefficient

–

0.8009

0.8320

0.8340

expanded to the middle and low latitudes, and the large-scale ionospheric irregularities over South America gradually dissipated at 01:00 UT. Except for the high latitudes, irregularities appeared at middle and low latitudes of the Asian region during the second main phase. At about 13:00 UT, irregularities first appeared at low latitudes near 110°E, then spread northwestward along the magnetic field lines with strip-like features at 14:00 UT. Due to the long duration of the second main phase, the enhancement of the ionospheric disturbances was still observed at 15:00 UT. It is clearly seen from the ROTI maps that there are persistent and strong ionospheric disturbances at high latitudes during the geomagnetic storm, whereas the occurrence of ionospheric irregularities at low latitudes is mainly related to the local time and enhanced after sunset. As referring to previous studies [25, 26], the ROTI maps generated from this study also well indicate the evolution and intensity of global ionospheric irregularities during the geomagnetic storm.

Global Mapping of Ionospheric ROTI Index and Its Preliminary Gaussian

Spherical

Exponential

1 (a)

R

2

0.75 0.5 0.25 0

RMSE (TECu/min)

0.4 (b)

0.3 0.2 0.1 0

12:00

09/06

12:00

09/07

12:00

09/08

12:00

09/09

09/10

UT

Fig. 6. Time variations of the R2 and RMSE metrics during September 6–9, 2017

Latitude

(main phase I) 09/07 23:00 UT

(main phase II) 09/08 13:00 UT

80°N

80°N

40°N

40°N

0°

0°

40°S

40°S

80°S

80°S 180°

120°W

60°W

0°

60°E

120°E

180°

180°

Latitude

(main phase I) 09/08 00:00 UT

60°W

0°

60°E

120°E

180°

(main phase II) 09/08 14:00 UT

80°N

80°N

40°N

40°N

0°

0°

40°S

40°S

80°S

80°S 180°

120°W

60°W

0°

60°E

120°E

180°

180°

(main phase I) 09/08 01:00 UT

Latitude

120°W

120°W

60°W

0°

60°E

120°E

180°

(main phase II) 09/08 15:00 UT

80°N

80°N

40°N

40°N

0°

0°

40°S

40°S

80°S

80°S

180°

120°W

60°W

0°

60°E

120°E

180°

Longitude

180°

120°W

60°W

0°

60°E

120°E

180°

Longitude ROTI (TECu/min)

0

0.4

0.8

1.2

1.6

2.0

Fig. 7. Global ionospheric ROTI maps for two main phases of the geomagnetic storm.

219

220

H. Jia et al.

3.3 Kinematic PPP Degradation During the 2017 September Storm We divide the latitude bands by every 10° interval and compute the mean values of ROTI and 3D positioning errors within each band. Figure 8 illustrates the global ROTI values and three-dimensional (3D) positioning errors as a function of geographical latitude and time, which are obtained using observations from over 500 GNSS stations. There is a temporal and spatial consistency between the increase in PPP positioning errors and the development of ionospheric irregularities as shown in Fig. 8. The high latitudes are subjected to more severe ionospheric disturbances, resulting in frequent increases in positioning errors. At approximately 00:00 UT and 15:00 UT on September 8, ionospheric disturbances increased at high and low latitudes, and expanded towards mid-latitudes. An increase in the positioning errors was observed across the high, midand low-latitude regions during both periods. 1

60

0.8 0.6

0 0.4

-30 0.2

-60

0

-90 09/06

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

(b) 3-D Error 90

2

60

1.6

Latitude (deg.)

30

1.2

0 0.8

-30 0.4

-60

0

-90 09/06

Meter

Latitude (deg.)

30

TECu/min

(a) ROTI 90

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

UT

Fig. 8. a ROTI and b 3D positioning errors as a function of geographical latitude and UT

To further investigate the degradation of the PPP under different levels of the ionospheric disturbances, we compute the average value of ROTI over all satellites-in-view every 30 s to match the PPP solution, and then divide the ROTI into different groups and compute the maximum, average and RMS of the horizontal, vertical (absolute value) and 3D PPP errors in each group, respectively. As ROTI >0.5 TECu/min is usually used to indicate the presence of ionospheric irregularities on a scale of a few kilometers [27], the datasets are divided into six groups with an interval of 0.5 TECu/min. We set the maximum error to 95% quantile of its cumulative distribution function to avoid the influence of extreme values in the PPP errors. Figure 9 depicts the kinematic PPP errors under different levels of ionospheric disturbances. As shown in Fig. 9, the positioning errors and RMS show a rising trend as the disturbance level increases. The maximum, average and RMS of the 3D positioning

Global Mapping of Ionospheric ROTI Index and Its Preliminary Horizontal

Maximum (m)

6

Vertical

221

3D

(a)

4 2 0 3 (b)

Average (m)

2 1 0 4

RMS

(m)

3

(c)

2 1 0 1

2

4

3

5

6

Disturbance level

Fig. 9. Kinematic PPP errors under different levels of ionospheric disturbances indicated by ROTI, a Maximum; b Average; c RMS

errors are 0.33 m, 0.14 m and 0.40 m, respectively, with the ROTI values less than 0.5 TECu/min, while they can reach 7.06 m, 2.80 m and 3.63 m, respectively, when the ROTI increases above 2.5 TECu/min. S2 S1

S4

S3

350 NP 527, 0 DS

40°ƒ1

30°ƒ1 90°ƒ(

90°ƒ( Receiver

20°ƒ1 100°ƒ(

110°ƒ(

10°ƒ1

Fig. 10. Schematic showing the projection of the ROTI maps onto the ground.

Both Figs. 8 and 9 indicate that the positioning degradation is strongly related to the development of ionospheric irregularities during the geomagnetic storm. In this regard, we investigate using the ROTI maps to assess potential positioning risks of the ground receivers under disturbed ionospheric conditions. Figure 10 presents the

222

H. Jia et al.

geometry of line-of-sight between satellites and the receiver utilized to project the ROTI map information to ground. The positioning risk for the receiver at any location on the ground can be estimated as follows: Risk =

n 

ωi ROTIi

(4)

i=1

where ROTI i is the ROTI value of IPPi indexed from the ROTI map; ωi is the weight and the sum of weights is 1. An elevation-dependent weighting method is applied and the weight is calculated as follows: 1 sin2 (elei ) (5) ωi = n  2 1 sin (elei ) i=1

where elei is the elevation angle of each satellite. To compare directly with PPP results, we use GPS-only constellation with an elevation mask angle of 15°. We validate the positioning risk in terms of spatial distribution and time variations. For spatial distribution, the notional receivers are designed to be deployed equidistantly on the ground with a resolution of 1° × 1°, and the geodetic height of the receivers is taken as the Earth’s mean radius (6371.393 km). The ground positioning risk map can be generated by computing the positioning risk of all notional receivers. 3D Error (m)

(main phase I) 09/08 00:30 UT

Risk (TECu/min)

(main phase I) 09/08 00:30 UT 2.0

2.0

80°N

80°N 1.6

1.6

40°N

40°N

Latitude

1.2

1.2

0°

0°

0.8

0.8

40°S

40°S 0.4

0.4

80°S

80°S 0

0

180°

120°W

60°W

60°E

(main phase II) 09/08 14:30 UT

120°E

180°

180°

3D Error (m)

120°W

60°W

0°

60°E

120°E

180° Risk (TECu/min)

(main phase II) 09/08 14:30 UT 2.0

2.0

80°N

80°N 1.6

1.6

40°N

40°N

Latitude

1.2

0°

1.2

0° 0.8

0.8

40°S

40°S 0.4

0.4

80°S

80°S 0

Longitude

0

180°

120°W

60°W

0°

60°E

120°E

180°

Longitude

Fig. 11. Spatial validation of ground positioning risks by using ROTI maps.

Figure 11 depicts the 3D positioning errors and geographical distributions of the ground positioning risks at two selected epochs, i.e.,00:30 UT and 14:30 UT, on September 8, 2017. In both time periods, there were severe ionospheric disturbances observed at

Global Mapping of Ionospheric ROTI Index and Its Preliminary

223

high latitudes, and the corresponding ground positioning risks and errors are consistent. When the ionospheric disturbances appeared near the geomagnetic equator, there is an increase in both the ground positioning risks and 3D positioning errors. (a) yel3 (62.32°N,114.48°W)

Risk (TECu/min)

2 1

5 0 09/06

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

3D Error (m)

10

0

UT (b) mgue (35.60°S,69.40°W)

0.5 2

0

1 0 09/06

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

3D Error (m)

Risk (TECu/min)

1

UT (c) cusv (13.65°N,100.53°E) 0.4 0.2 3

0

2 1 0 09/06

12:00

09/07

12:00

09/08

12:00

09/09

12:00

09/10

3D Error (m)

Risk (TECu/min)

0.6

UT

Fig. 12. Time variations of the ground positioning risks observed at three GNSS stations during the time period of September 6–9, 2017.

We perform the continuous positioning risk estimation for the three representative stations located in different areas and compare it to 3D positioning errors. As shown in Fig. 12, the positioning risks at all three stations have different levels of fluctuations during the ionospheric disturbance period, and the 3D positioning errors appear to respond consistently in terms of time and magnitude. The results in Figs. 11 and 12 indicate that the positioning risk, estimated using the ROTI maps generated in this study, could be potentially used as an indicator to assess the positioning risk at the locations where no receivers are actually deployed.

224

H. Jia et al.

4 Conclusions and Future Work In this paper, we conduct the global mapping of ionospheric ROTI index based on the ordinary kriging and assess the kinematic PPP degradation during a severe geomagnetic storm. Major findings are summarized as follows: 1. Three semi-variogram models are applied to the global ionospheric ROTI mapping. The spherical model is chosen for mapping and analysis, considering the goodnessof-fit and performance in the cross-validation by using 500 + GNSS stations during the 2017 September 7–8 storm. 2. The ROTI maps can effectively characterize the spatial-temporal evolution of ionospheric irregularities during the main phases of the geomagnetic storm. 3. The kinematic PPP degradation is consistent with the variation of ROTI during the geomagnetic storm. The maximum, average and RMS of the 3D positioning errors reached 7.06 m, 2.80 m and 3.63 m, respectively, when the ROTI exceeds 2.5 TECu/min. 4. The generated ROTI maps are proposed for the ground-based positioning risk assessment. A preliminary validation through our analysis indicates the ROTI map projected onto the ground is sufficient to indicate the risk of PPP degradation. The kriging-based global ionospheric ROTI maps proposed in this study has the potential for space weather monitoring and GNSS navigation and positioning applications. In future work, we will investigate the use of the expanded kriging method to improve the validity of the ROTI maps in the application of different ionospheric disturbances, such as SID (Sudden Ionospheric Disturbance), TID (Traveling Ionospheric Disturbance), etc., and build a more accurate multi-GNSS ground positioning risk model, with the goal of producing a quantitative assessment of positioning risk and making it available as one of the conventional products in the international GNSS monitoring and assessment system (iGMAS) analysis center. Acknowledgments. This work is supported by the National Natural Science Foundation of China (No.42274027), the Scientific and Technological Innovation Plan from Shanghai Science and Technology Committee (No. 21511103902), and the Fundamental Research Funds for the Central Universities. We would like to thank the GNSS center of Wuhan university (ftp://igs.gnsswhu.cn) for providing the raw GNSS data. The geomagnetic data are obtained from the Data Analysis Center for Geomagnetism and Space Magnetism, Kyoto University, Japan (https://wdc.kugi.kyotou.ac.jp/) and Research Centre for Geosciences, GFZ, Germany (https://www-app3.gfz-potsdam. de/kp_index/Kp_ap_Ap_SN_F107_since_1932.txt). The RTKLIB (version 2.4.3) can be accessed at this link (https://rtklib.com/).

References 1. Béniguel Y (2002) Global Ionospheric Propagation Model (GIM): a propagation model for scintillations of transmitted signals. Radio Sci 37(3):1–14 2. Aquino M, Moore T, Dodson A et al (2005) Implications of ionospheric scintillation for GNSS users in Northern Europe. J Navig 58(2):241–256

Global Mapping of Ionospheric ROTI Index and Its Preliminary

225

3. Dubey S, Wahi R, Gwal AK (2006) Ionospheric effects on GPS positioning. Adv Space Res 38(11):2478–2484. https://doi.org/10.1016/j.asr.2005.07.030 4. Pi X, Mannucci AJ, Lindqwister UJ et al (1997) Monitoring of global ionospheric irregularities using the worldwide GPS network. Geophys Res Lett 24(18):2283–2286 5. Juan JM, Sanz J, Rovira-Garcia A et al (2018) AATR an ionospheric activity indicator specifically based on GNSS measurements. J Space Weather Space Clim 8:A14 6. Jakowski N, Borries C, Wilken V (2012) Introducing a disturbance ionosphere index. Radio Sci 47(04):1–9 7. Basu S, Groves KM, Quinn JM et al (1999) A comparison of TEC fluctuations and scintillations at Ascension Island. J Atmos Sol-Terr Phys 61(16):1219–1226 8. Yang Z, Liu Z (2016) Correlation between ROTI and Ionospheric Scintillation Indices using Hong Kong low-latitude GPS data. Gps Solut 20(4):815–824 9. Carrano CS, Groves KM, Rino CL (2019) On the relationship between the Rate of Change of Total Electron Content Index (ROTI), irregularity strength (CkL), and the scintillation index (S-4). J Geophys Res-Space Phys 124(3):2099–2112 10. Cherniak I, Krankowski A, Zakharenkova I (2014) Observation of the ionospheric irregularities over the Northern Hemisphere: methodology and service. Radio Sci 49(8):653–662. https://doi.org/10.1002/2014rs005433 11. Cherniak I, Krankowski A, Zakharenkova I (2018) ROTI Maps: a new IGS ionospheric product characterizing the ionospheric irregularities occurrence. Gps Solut 22(3):1–12 12. Geng W, Huang W, Liu G et al (2020) Generation of ionospheric scintillation maps over Southern China based on Kriging method. Adv Space Res 65(12):2808–2820 13. Hamel P, Sambou DC, Darces M et al (2014) Kriging method to perform scintillation maps based on measurement and GISM model. Radio Sci 49(9):746–752 14. Beeck SS, Jensen ABO (2021) ROTI maps of Greenland using kriging. J Geod Sci 11(1):83–94 15. Pi X, Iijima BA, Lu W (2017) Effects of ionospheric scintillation on GNSS-based positioning. Navig J Inst Navig 64(1):3–22 16. Juan JM, Sanz J, González-Casado G et al (2018) Feasibility of precise navigation in high and low latitude regions under scintillation conditions. J Space Weather Space Clim 8:A05 17. Chen W, Gao S, Hu C et al (2008) Effects of ionospheric disturbances on GPS observation in low latitude area. Gps Solut 12(1):33–41 18. Jacobsen KS, Andalsvik YL (2016) Overview of the 2015 St. Patrick’s day storm and its consequences for RTK and PPP positioning in Norway. Jx Space Weather Space Clim 6. https://doi.org/10.1051/swsc/2016004 19. Yang Z, Morton YJ, Zakharenkova I et al (2020) Global view of ionospheric disturbance impacts on kinematic GPS positioning solutions during the 2015 St. Patrick’s Day storm. J Geophys Res Space Phys 125(7):e2019JA027681 20. Atıcı R, Sa˘gır S (2020) Global investigation of the ionospheric irregularities during the severe geomagnetic storm on September 7–8, 2017. Geod Geodyn 11(3):211–221 21. Imtiaz N, Younas W, Khan M (2017) Response of the low-to mid-latitude ionosphere to the geomagnetic storm of September 2017. Ann Geophys 38(2):359–372 22. Oliver MA, Webster R (2015) Basic steps in geostatistics: the variogram and kriging. Springer 23. Cambardella CA, Moorman TB, Novak JM et al (1994) Field-scale variability of soil properties in central Iowa soils. Soil Sci Soc Am J 58(5):1501–1511 24. Takasu T (2011) RTKLIB: an open source program package for GNSS positioning 25. Berdermann J, Kriegel M, Bany´s D et al (2018) Ionospheric response to the X9. 3 flare on 6 September 2017 and its implication for navigation services over Eur Space Weather 16(10):1604–1615

226

H. Jia et al.

26. Obana Y, Maruyama N, Shinbori A et al (2019) Response of the ionosphere-plasmasphere coupling to the September 2017 storm: what erodes the plasmasphere so severely? Space Weather 17(6):861–876 27. Ma G, Maruyama T (2006) A super bubble detected by dense GPS network at east Asian longitudes. Geophys Res Lett 33(21)

Precise Point Positioning Ambiguity Resolution with Multi-frequency Ionosphere-Reduced Combination Qing Zhao1,2 , Shuguo Pan1,2(B) , Wang Gao1,2 , Ji Liu2 , Yin Lu2 , and Peng Zhang2 1 School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China

[email protected] 2 Laboratory of Science and Technology On Marine Navigation and Control, China State

Shipbuilding Corporation, Tianjin 300131, China

Abstract. Precise point positioning (PPP) with ambiguity resolution can improve positioning accuracy and reliability. In conventional dual-frequency ionospherefree (IF) PPP (IFPPP), the IF ambiguity loses its integer property, so the fixing process needs to be decomposed into a two-step way of fixing wide-lane (WL) and narrow-lane (NL) ambiguity separately. Only when both are fixed at the same time, can a reliable positioning solution be obtained. In order to take full advantages of multi-frequency signals and simplify the ambiguity resolution process, by proper selecting the combination coefficients, a PPP ambiguity resolution method based on ionosphere-reduced (IR) combination is proposed in this paper. The proposed IRPPP model hardly needs to consider the effect of ionosphere, while has the equivalent wavelength and observation noise compared with conventional IFPPP. In addition, the combined IR ambiguity maintains its integer solvability, thus only one-step of directly fixing the IR ambiguity is need to achieve positioning performance comparable to conventional IFPPP. The model is further evaluated globally, and the results show that the IRPPP is superior to the conventional IFPPP in terms of the number of available NL ambiguity and the time to first fix (TTFF), and the average TTFF can be shortened by 22.6%. As for positioning accuracy, both models are basically the same with centimeter-level accuracy. Keywords: Precise point positioning (PPP) · Ionosphere-reduced (IR) · Multi-frequency · Ambiguity resolution (AR)

1 Introduction The precise point positioning (PPP) can provide high-precision positioning, navigation and timing services globally, and has been widely used in various fields [1, 2]. The conventional PPP does not perform ambiguity resolution, so the convergence time is long and the positioning reliability is poor. Through fixing the un-differenced ambiguity, the convergence time can be shortened, and the positioning accuracy can also be improved, especially in east direction [3]. At present, the dual-frequency ionosphere-free (IF) PPP (IFPPP) is the most widely used due to its simplicity and high computational efficiency. © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 227–241, 2024. https://doi.org/10.1007/978-981-99-6944-9_20

228

Q. Zhao et al.

However, in order to eliminate the influence of ionosphere, the combined ambiguity of IFPPP does not have integer characteristics. Therefore, to obtain ambiguity-fixed solution, it is usually decomposed into a two-step way of fixing wide-lane (WL) and narrow-lane (NL) ambiguity separately [4]. The typical fixing methods including integer phase clock method, decoupled clock method and fractional cycle bias (FCB) method [5–7], and the different methods are equivalent in theory [8, 9]. For the case of conventional dual-frequency, the observation combinations are very limited. With the modernization of global navigation satellite system (GNSS) in recent years, the new satellites can broadcast signals at three or more frequencies, which greatly enriches observation combinations, thus providing new opportunities for reliable ambiguity resolution [10]. Based with simulated observation data, Geng et al. study the triple-frequency PPP ambiguity resolution method with extra-wide-lane (EWL), widelane (WL), narrow-lane (NL) ambiguity fixed sequentially. By using the ambiguity-fixed ionosphere-free (AFIF), composed of the fixed EWL and WL observations, instead of raw pseudorange to assist NL ambiguity fixing, thanks to its lower noise level and multipath effect, the initialization time can be shortened [11]. Li et al. also used a similar idea to study the performance of BDS-2/Galileo multi-frequency PPP ambiguity resolution, and the results show that the results of multi-frequency case are always better than that of dual-frequency case regardless of the time to first fix (TTFF) or positioning accuracy [12]. Also based on the so-called AFIF observation, many scholars have also carried out research on single-epoch wide-lane ambiguity resolution (WAR) positioning. Li et al. studied the performance of different triple-/quad-/five-frequency AFIF combinations, and found that the EWL and WL ambiguity can be fixed instantaneously, and then decimeter-level accuracy can be achieved [13]. For Galileo, Guo and Xin found that compared with other AFIF combinations, the E1/E5a/E6 combination has lower noise level, which is expected to further improve the WAR positioning accuracy to about 10cm [14]. In addition to the aforementioned AIIF-related studies, as the multi-frequency uncombined PPP (UCPPP) model has gradually become the standard model for multifrequency data processing, some scholars have carried out research on ambiguity resolution based on UCPPP. Both Gu et al. and Li et al. conducted BDS-2 triple-frequency UCPPP ambiguity resolution, the former adopts the strategy of EWL/WL/NL ambiguity fixed sequentially, and the results show that the EWL/WL ambiguity can be reliably fixed within 2min, and achieve an accuracy better than 0.5 m [15], while the latter perform ambiguity fixing directly using the raw ambiguity on each frequency, and the results show that the accuracy of triple-frequency ambiguity resolution is higher during the initialization phase [16]. The above-mentioned cascading ambiguity resolution strategy can also be used to assist NL ambiguity resolution [17], with the constraint of EWL/WL ambiguity resolution, the TTFF of NL ambiguity of multi-frequency UCPPP can be shortened to about 2 min [18]. Further based on the multi-frequency observation data of BDS-3 and Galileo, Geng et al. conducted research on the UCPPP-based WAR positioning, and an accuracy of about 20cm can basically be achieved globally [19, 20]. In addition, both Elsobeiey and Duong et al. studied the selection of optimal linear combination coefficients for multi-frequency PPP, and the results show that the convergence speed of PPP can be accelerated to a certain extent. However, in order to eliminate the

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

229

influence of ionosphere, the coefficients are usually real numbers, which means that the combined ambiguity lose its integer solvability [21, 22]. According to the ionosphere parameter processing strategy, the above studies can be roughly classified into two categories, one is to use the IF combination to eliminate its influence, and the other is to estimate it as unknown parameters. Both these two strategies properly deal with the ionosphere parameter, thus ensuring the unbiasedness of the corresponding PPP model. Different from the previous studies mentioned above, this paper is based on the multi-frequency ionosphere-reduced (IR) combination, and by selecting the appropriate IR combination coefficients, the ionosphere is directly ignored in the proposed IRPPP model while achieve positioning performance comparable to conventional IFPPP, especially in the aspect of ambiguity resolution, the two-step of fixing WL and NL ambiguity separately in IFPPP is simplified to only one-step of directly fixing the combined IR ambiguity in IRPPP. The specific model and experiment analysis will be introduced in the following.

2 Conventional IFPPP Ambiguity Resolution For a certain satellite s and receiver r, the un-differenced pseudorange and carrier observation equations on the j-th frequency can be expressed as, s s s Pr,j = ρrs +Trs + c · tr − c · t s +γj · Ir,1 + dr,j − djs + er,j

(1)

s s s Lsr,j = ρrs +Trs + c · tr − c · t s − γj · Ir,1 + λj · (Nr,j + br,j − bsj ) + εr,j

(2)

s and Ls are pseudorange and carrier phase observations; ρ s is geometric where Pr,j r r,j distance; c is the speed of light; tr and t s are receiver and satellite clock bias, respectively; s is the ionosphere delay at the first frequency, and Trs is slant troposphere delay; Ir,1  s is γj = f12 fj2 is the scaling coefficients on the j-th frequency; λj is wavelength; Nr,j integer ambiguity; br,j and bsj are phase bias at receiver and satellite end, respectively; s and ε s are dr,j and djs are code bias at receiver and satellite end, respectively; er,j r,j observation noises of pseudorange and carrier phase observations, respectively. Due to the strong correlation between the estimated parameters, rank deficiency exists in the PPP model. In order to obtain a full-rank observation model, the reparameterization strategy is usually used in data processing. For dual-frequency IFPPP, the receiver and clock bias is usually re-parameterized as follows,  c · ˜t s = c · t s + (α · d1s + β · d2s ) (3) c · ˜tr = c · tr + (α · dr,1 + β · dr,2 )

where ˜t s and ˜tr denote the re-parameterized clock bias; α and β are the coefficients of IF combination, with specific forms as, ⎧  ⎪ ⎨ α = f12 f12 − f22  (4) ⎪ ⎩ β = −f22 f12 − f22

230

Q. Zhao et al.

After applying the above re-parameterization and performing appropriate simplification, the full-rank dual-frequency IFPPP observation model can be obtained as follows,  s s Pr,IF = ρrs +Trs + c · ˜tr − c · ˜t s + er,IF (5) s s Lsr,IF = ρrs +Trs + c · ˜tr − c · ˜t s + λNL · N r,IF + εr,IF s and Lsr,IF denote the IF pseudorange and carrier phase observations, with where Pr,IF

s s , respectively; λ observation noises of er,IF and εr,IF NL = c (f1 + f2 ) is the NL s wavelength; N r,IF is the float IF ambiguity. Since the IF ambiguity does not have integer characteristics, it is usually decomposed into WL and NL ambiguity first, and then fixed separately. The conversion relationship between them is as follows, s

λNL · N r,IF =

c · f2 s s · Nr,WL + λNL · N r,NL − f22

f12

(6)

s

s is the fixed WL ambiguity; N r,NLIF is the float NL ambiguity absorbing where Nr,WL pseudorange and phase hardware bias. For WL ambiguity, it is first estimated using Melbourne-Wübbena (MW) combination with multiple epoch smoothing process [23], and then in view of its long wavelength, it can generally be fixed directly by integer rounding with threshold of 0.2 ~ 0.3 cycle. Once the WL ambiguity is fixed, the float NL ambiguity and the corresponding variance-covariance matrix can be calculated according to Eq. (6) and the error propagation law. On this basis, the FCB is corrected first to restore its integer nature, and then the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) algorithm is adopted to fix it [24]. When both WL and NL ambiguity are fixed simultaneously, the fixed IF ambiguity can be reconstructed to constrain the non-ambiguity parameters to obtain ambiguity-fixed solution.

3 Multi-frequency IRPPP Ambiguity Resolution In theory, the multi-frequency signals can provide more combinations with better properties, such as the IR combination based on integer coefficients. Compared with IFPPP, the IR combination not only almost eliminates the effect of ionosphere, but also maintains integer solvability of the combined IR ambiguity, so that there is no need to adopt the two-step ambiguity fixing process in IFPPP to achieve ambiguity-fixed solution, which provides new opportunities for ambiguity resolution. Currently, the GNSS cable of broadcasting five or more frequency signals include BDS-3 (B1c/B2I/B2a/B6I/B2b/B2ab) and Galileo (E1/E5a/E6/E5b/E5ab) [25]. Considering that the number of stations capable of tracking BDS-3 B2b/B2ab signals around the world is few and their distribution is uneven, therefore, only the Galileo five-frequency signals are used in subsequent chapters for the selection of IR combination coefficients, and further the performance of IRPPP is evaluated globally. It should be noted that although the Galileo five-frequency is used as an example in the following text, this method does not lose generality, that is, it is still applicable to BDS-3 and other GNSS with multiple signals when the combination coefficients are selected appropriately.

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

231

3.1 Multi-frequency IR Combination Selection For Galileo five-frequency case, the multi-frequency combined observations can be expressed as follows [26], Lsr,(i1 ,i2 ,i3 ,i4 ,i5 ) =

i1 · f1 · Lsr,1 + i2 · f2 · Lsr,2 + · · · + i5 · f5 · Lsr,5 i1 · f1 + i2 · f2 + · · · + i5 · f5

(7)

where ik (k = 1, 2, · · · , 5) are integer combination coefficients, and the observation equation corresponding to Eq. (7) can be expressed as, s Lsr,(i1 ,i2 ,i3 ,i4 ,i5 ) = ρrs + c · tr − c · t s +Trs − γ(i1 ,i2 ,i3 ,i4 ,i5 ) · Ir,1  s +λ(i1 ,i2 ,i3 ,i4 ,i5 ) · Nr,(i + br,(i1 ,i2 ,i3 ,i4 ,i5 ) − bs(i1 ,i2 ,i3 ,i4 ,i5 ) 1 ,i2 ,i3 ,i4 ,i5 )

(8)

s +μ(i1 ,i2 ,i3 ,i4 ,i5 ) · εr,1

where γ(i1 ,i2 ,i3 ,i4 ,i5 ) , λ(i1 ,i2 ,i3 ,i4 ,i5 ) , N(i1 ,i2 ,i3 ,i4 ,i5 ) , br,(i1 ,i2 ,i3 ,i4 ,i5 ) , bs(i1 ,i2 ,i3 ,i4 ,i5 ) and μ(i1 ,i2 ,i3 ,i4 ,i5 ) are ionosphere scalar factor, wavelength, integer ambiguity, receiver phase bias, satellite phase bias and noise amplification factor of the combined signal, respectively. Their specific forms are as follows,



⎧ f12 · (i1 f1 + i2 f2 + · · · + i5 f5 ) ⎪ ⎪ γ(i1 ,i2 ,i3 ,i4 ,i5 ) = ⎪ ⎪ ⎪ f(i1 ,i2 ,i3 ,i4 ,i5 ) ⎪ ⎪ ⎪ c ⎪ ⎪ ⎪ λ(i1 ,i2 ,i3 ,i4 ,i5 ) = ⎪ ⎪ f(i1 ,i2 ,i3 ,i4 ,i5 ) ⎪ ⎪ ⎨ s s s s Nr,(i1 ,i2 ,i3 ,i4 ,i5 ) = i1 · Nr,1 + i2 · Nr,2 + · · · + i5 · Nr,5 (9) ⎪ ⎪ ⎪ br,(i1 ,i2 ,i3 ,i4 ,i5 ) = i1 · br,1 + i2 · br,2 + · · · + i5 · br,5 ⎪ ⎪ ⎪ ⎪ ⎪ bs(i1 ,i2 ,i3 ,i4 ,i5 ) = i1 · bs1 + i2 · bs2 + · · · + i5 · bs5 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ (i1 · f1 )2 + (i2 · f2 )2 + · · · + (i5 · f5 )2 ⎪ ⎩ μ(i1 ,i2 ,i3 ,i4 ,i5 ) = f(i1 ,i2 ,i3 ,i4 ,i5 ) where f(i1 ,i2 ,i3 ,i4 ,i5 ) is the frequency of the combined signal, and can be expressed as, f(i1 ,i2 ,i3 ,i4 ,i5 ) = i1 · f1 + i2 · f2 + · · · + i5 · f5

(10)

One thing to note is that the γ(i1 ,i2 ,i3 ,i4 ,i5 ) only reflects the influence of ionosphere on the ranging accuracy, however, to further analyze its impacts on ambiguity resolution, it should be converted to β(i1 ,i2 ,i3 ,i4 ,i5 ) in the unit of cycle · m−1 , as shown below,



f12 · (i1 f1 + i2 f2 + · · · + i5 f5 ) (11) β(i1 ,i2 ,i3 ,i4 ,i5 ) = c Theoretically, the number of linear combinations is infinite. However, in order to achieve similar performance as IFPPP, the influence factors of wavelength, ionosphere and noise level must be considered comprehensively. In this paper, the following three criteria are used to determine the appropriate IR combination, as follows:

232

Q. Zhao et al.

• Regardless of ambiguity resolution or positioning calculation, the IR observations should have sufficiently small ionosphere delay. Taking the 100m ionosphere delay as an example, ensure that the impact of the ionosphere on ambiguity resolution and ranging accuracy is less than 0.1 cycle and 1 cm respectively, that is, β(i1 ,i2 ,i3 ,i4 ,i5 ) < 0.001 and γ(i1 ,i2 ,i3 ,i4 ,i5 ) < 0.0001 is required. • The IR observations should have small noise level to ensure the reliability of ambiguity resolution and centimeter-level positioning. Considering that the noise amplification factor of dual-frequency IFPPP is usually about 3, so it is required that μ(i1 ,i2 ,i3 ,i4 ,i5 ) should be less than 3.0. • In order to be able to resist the influence of residual geometric errors, the wavelength of IR combination should not be too small. Take the E1/E5a dual-frequency IF combination with equivalent wavelength of about 10.9 cm as a reference, the λ(i1 ,i2 ,i3 ,i4 ,i5 ) is required to be greater than 10.0 cm. For coefficients ik , by traversing each combination within range [-10, 10], the IR combination list in Table 1 is finally determined. For the convenience of comparison, the corresponding parameters of IFPPP is also given in the table. It can be seen from Table 1 that, the IR combination belongs to NL combination with wavelength of about 10.9 cm, and the noise level is slightly better than that of IFPPP. As for ionosphere, the IFPPP can completely eliminate its first-order delay, while the IRPPP is affected by residual ionosphere, but the magnitude is already very small, even for 100 m ionosphere delay, its impact on ranging accuracy is less than 1mm, and its impact on ambiguity resolution is less than 0.1 cycle. In terms of ambiguity resolution, the IF ambiguity lose its integer nature and needs to be decomposed to WL and NL ambiguity first, and then to be fixed separately, while the IRPPP based on integer coefficients guarantee the integer solvability of the combined IR ambiguity, so there is no need to adopt the two-step ambiguity fixing process, and the process is simplified. In addition, the IFPPP requires multiple epoch smoothing process during the ambiguity initialization phase to ensure that the WL ambiguity is reliably fixed, and then the NL ambiguity can be solved. When the WL ambiguity of some satellites cannot be fixed successfully, the number of available NL ambiguity will inevitably decrease, which will undoubtedly degrade the fixing performance. In contrast, this problem does not exist in IRPPP, since it can directly fix the NL ambiguity without resolving the WL ambiguity first, thus resulting more available NL ambiguity in theory. For a better understanding, Fig. 1 further gives a comparison of the ambiguity fixing flowchart of both IFPPP and IRPPP. 3.2 Multi-frequency IRPPP Observation Model Based on Eq. (8) and the selected IR combination, the pseudorange observations are further integrated to build the full-rank IRPPP model. In order to avoid additional code bias introduced by multi-frequency pseudoranges, such as differential code bias (DCB) correction at satellite end and inter-frequency bias (IFB) parameters at receiver end [27], only dual-frequency pseudoranges consistent with satellite clock reference are used for parameter estimation. Similar to IFPPP, in order to remove the correlation between parameters, it is also necessary to adopt the re-parameterization process shown

2.26

IFPPP

−1.26

0

−2

4

i3

i2

i1

E6

E5a

E1

IRPPP

Type 0

i4

E5b −1

i5

E5ab

10.89cm

10.87cm

λ(i1 ,i2 ,i3 ,i4 ,i5 )

0

−0.000085

γ(i1 ,i2 ,i3 ,i4 ,i5 )

0

−0.000784

β(i1 ,i2 ,i3 ,i4 ,i5 )

Table 1. Galileo five-frequency IRPPP and dual-frequency IFPPP combination parameters

2.588

2.478

μ(i1 ,i2 ,i3 ,i4 ,i5 )

Precise Point Positioning Ambiguity Resolution with Multi-frequency … 233

234

Q. Zhao et al. Table 2. FCB Daily STD Statistical Results of IFPPP and IRPPP

FCB Type IFPPP IRPPP

STD [cycle] Max

Min

Average

WL

0.028

0.011

0.018

NL

0.055

0.021

0.036

LC

0.055

0.022

0.036

MW Hatch Filter

Dual-Frequency IFPPP

Multi-Frequency IRPPP

Smoothed Float WL Ambiguity

Float IF Ambiguity

Float IR Ambiguity

Integer Rounding

Float NL Ambiguity

Fixed Integer WL Ambiguity

Fixed Integer NL Ambiguity

Fixed Integer IR Ambiguity

Ambiguity-Fixed Solution Fig. 1. Ambiguity resolution flowchart of IFPPP and IRPPP

in Eq. (3). In addition, since the ionosphere will be ignored in the IRPPP, the dualfrequency pseudorange in the form of IF combination is used for parameter estimation instead of raw pseudorange, and then the IRPPP observation model can be obtained as follows, ⎧ s s ⎨ Pr,IF = ρrs + Trs + c · ˜tr − c · ˜t s + er,IF ⎩ Ls

s s s ˜ ˜s r,(i1 ,i2 ,i3 ,i4 ,i5 ) = ρr + Tr + c · tr − c · t + λ(i1 ,i2 ,i3 ,i4 ,i5 ) · N r,(i1 ,i2 ,i3 ,i4 ,i5 ) + μ(i1 ,i2 ,i3 ,i4 ,i5 ) · ε1

s

(12)

where N r,(i1 ,i2 ,i3 ,i4 ,i5 ) is the combined float IR ambiguity. Based on the above model, the Kalman filter method is used to calculate the float solution and the corresponding variance-covariance matrix. To get ambiguity-fixed solution, the receiver FCB in float ambiguity is first eliminated by means of the between-satellitesingle-difference (BSSD). Then the satellite FCB correction is applied to restore its integer characteristics. Finally, the LAMBDA algorithm with ratio-test is used to determine the optimal ambiguity vector, and the partial ambiguity fixing (PAF) strategy can be used simultaneously to improve the fixing success rate. The satellite FCB used above can

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

235

be estimated with globally distributed stations, and the specific details is not described here, as there have been rich related studies.

4 Experiment Analysis In order to compare the performance of IFPPP and IRPPP, 28 stations distributed around the world are selected for experiments, and the specific station distribution is shown in Fig. 2. The data collection date is DOY245 in 2022, and the sampling rate is 30s. All stations could receive Galileo five-frequency signals. In data processing, the kinematic mode is adopted, and the precise orbit and clock products are provided by Center for Orbit Determination in Europe (CODE). The cutoff elevation is set to 10 degrees, and the elevation dependent stochastic model was adopted with the priori precision of 0.003 m and 0.3 m for raw carrier and pseudorange, respectively. In ambiguity resolution process, for WL ambiguity in IFPPP, the number of smoothing epochs is set to 20, and the integer rounding threshold is set to 0.25 cycle; for the NL ambiguity of the two models, the ratio-test with threshold of 2.0 is used as the criterion for judging whether it is fixed successfully. The PAF strategy with successively increased elevation is adopted, and the initial elevation is set to 10°, and then increased by 5 degrees in each iteration, until the number of satellite is less than 5 or the number of ambiguity is less than 4 [28]. In addition, compared with GPS, the satellite number of Galileo is relatively small. In order to ensure the performance of Galileo-only PPP at different time periods in different regions, the observation data is divide into arcs of 2 h. Only the arcs with the average number of visible satellites not less than 8 is processed, and a total of 135 observation sessions are processed. Then, the results are analyzed from the aspects of the available NL ambiguity number, TTFF and positioning accuracy. The TTFF referred to here is defined as the epoch time when ambiguity-fixed solution is obtained and also keeps fixed in the following 20 epochs. Further, the performance of the two models is evaluated by combining the results from all solved arcs. 4.1 Result Analysis of FCB The satellite FCB with high accuracy is the prerequisite for PPP ambiguity fixing, so the WL/NL FCB in IFPPP and the corresponding linear combination (LC) FCB in IRPPP are first estimated based on globally distributed stations. The specific results are shown in Fig. 3, and for better presentation, the results of some satellites are overall shifted. It can be seen from the figure that whether it is IFPPP or IRPPP, the corresponding daily FCB series has a good stability, and the detailed statistical results are shown in Table 2. The stability of FCB is mainly related to the corresponding wavelength. Therefore, the WL FCB has the best stability. As for the NL FCB of IFPPP and the LC FCB of IRPPP, since the wavelength of both are about 10.9 cm, as listed in Table 1, so their stability is also basically the same. Overall, the average standard deviations (STD) of WL/NL/LC FCB are 0.018 cycle, 0.036 cycle and 0.036 cycle, respectively. 4.2 Performance Comparison of IRPPP and IFPPP Due to space limitation, taking the EUR2 station as an example, Fig. 4 shows the number of available NL ambiguity of the two models during 02:00–04:00 period. It can be

236

Q. Zhao et al.

IFPPP-WL

FCB [cycle]

0.8 0.4 0.0 -0.4 -0.8 0.8 0.4 0.0 -0.4 -0.8

IFPPP -NL

FCB [cycle]

Fig. 2. User station distribution for IRPPP validation

02

07

12 GPST [hh]

17

22

E02 E03 E04 E05 E07 E08 E09 E10 E11

E12 E13 E14 E15 E18 E19 E21 E24 E25

E26 E27 E30 E31 E33 E34 E36

IRPP P-LC

02

07

12 GPST [hh]

17

22

Fig. 3. Time series of daily WL/NL FCB of IFPPP and LC FCB of IRPPP

clearly seen that in the initialization stage, since the IFPPP needs to successfully fix WL ambiguity first, the number of NL ambiguity injected into LAMBDA search process is zero, while the IRPPP does not need to fix WL ambiguity, and can try to fix the NL ambiguity from the initial epoch. In addition, the WL ambiguity in IFPPP is calculated by MW combination, which is greatly affected by pseudorange noise, and sometimes the integer rounding strategy cannot guarantee that all satellites are fixed successfully. In this case, even with multiple epochs smoothing, the number of available NL ambiguity is still less than that of IRPPP, especially for the newly rising satellites or satellites with cycle slip, the smoothing process need to be restarted. According to the statistics, the average number of available NL ambiguity of IFPPP and IRPPP during this period is 6.0 and 8.7 respectively. The IRPPP generally has more available NL ambiguity, and

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

237

this can provide more choices for ambiguity subset selection in subsequent PAF process, which is expected to improve ambiguity resolution performance, such as TTFF.

# of SD Amb

12 9 6 IFPPP IRPPP

3 0 02:00

02:30

03:00 GPST [hh:mm]

03:30

04:00

Fig. 4. Number of available NL ambiguity of IFPPP and IRPPP at station EUR2 during 02:00– 04:00.

U [cm]

E [cm]

N [cm]

Figure 5 further shows the positioning results of the two models during this period. In terms of positioning accuracy, since the noise levels of the two models are very close, as listed in Table 1, the positioning accuracies of the two models are basically the same with only counting the ambiguity-fixed solutions. However, in terms of TTFF, the IFPPP is affected by its smaller number of available NL ambiguity in the convergence phase, the TTFF is 15.0 min. In contrast, the IRPPP achieves a shorter TTFF with only 7.0 min.

20 10 0 -10 -20 20 10 0 -10 -20 40 20 0 -20 -40

IFPPP-Float

IRPPP-Float

IFPPP-Fixed

IRPPP-Fixed

TTFF=15.0min

TTFF=7.0min

RMS=0.7cm

RMS=0.7cm

RMS=0.3cm

RMS=0.3cm

RMS=2.5cm

RMS=2.8cm

02:00 02:30 03:00 03:30 04:00 02:00 02:30 03:00 03:30 04:00 GPST [hh:mm] GPST [hh:mm] Fig. 5. Positioning results of IFPPP and IRPPP at station EUR2 during 02:00–04:00.

238

Q. Zhao et al.

Taking 10 min as the unit interval, Fig. 6 shows the TTFF distribution of all arcs. The IFPPP needs to complete the WL ambiguity resolution first during convergence phase, so the overall TTFF is more than 10 min, and the TTFF of most arcs is concentrated in the range 10 ~ 20 min. In contrast, the IRPPP is cable to fix NL ambiguity from the first epoch, and especially when the number of visible satellites is sufficient, the TTFF of some arcs is less than 10 min. Overall, the average TTFF of IFPPP and IRPPP is 22.8 min and 17.7 min, and the TTFF of IRPPP is 22.6% shorter than that of IFPPP.

Number of Obs Session

80 IFPPP 60

IRPPP

AVE=22.8min AVE=17.7min

40 20 0

10 20 90 0 0 70 80 50 60 30 40 10 20 0~ 10 ~ 20 ~ 30 ~ 40 ~ 50 ~ 60 ~ 70 ~ 80 ~ 0 ~1 0 0~1 1 0~1 9 1 1

TTFF Distribution Interval [min] Fig. 6. TTFF distribution of IFPPP and IRPPP for all arcs.

Figure 7 shows the positioning accuracies of the two models by making statistics of all arcs. Since the noise levels of the two models are basically equivalent, both IFPPP and IRPPP achieve centimeter-level accuracy of 3.4 cm and 3.6 cm, respectively. Probably because the IRPPP is affected by the residual ionosphere, its accuracy is slightly worse than IFPPP. However, given the kinematic positioning mode is adopted, it can be assumed that there is no significant difference in positioning accuracy.

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

239

9 AVE=3.4cm

6

IRPPP AVE=3.6cm

3

0

AC24 AMC4 AREG ARHT BRO1 BRUX BSHM CAS1 CEBR CIBG EUR2 GAMG HARB KAT1 KIR8 KIRI KITG KOUG KRGG MAC1 MAO0 MCHL MET3 OBE4 RGDG SEYG THU2 YEL2

3D RMS [cm]

IFPPP

Fig. 7. Positioning accuracy of IFPPP and IRPPP for all user stations.

5 Conclusion The PPP ambiguity resolution can improve positioning accuracy and reliability. In conventional dual-frequency IFPPP, since the IF ambiguity lose its integer property, it is usually decomposed into WL and NL ambiguity first, and then fixed separately. In order to take full advantages of multi-frequency signals and simplify the ambiguity resolution process, based on multi-frequency IR combination, this paper proposes an IRPPP model, which almost eliminate the effect of ionosphere, while ensures the integer solvability of the combined ambiguity, so that ambiguity-fixed solution comparable to IFPPP can be obtained with only one-step of directly fixing the combined IR ambiguity. Further, the performance of IFPPP and IRPPP is compared through globally distributed stations in terms of the number of available NL ambiguity, TTFF and positioning accuracy. The results show that compared with the IFPPP, the proposed IRPPP does not require the smoothing process for WL ambiguity fixing, and has more available NL ambiguity overall, which provides more choices for PAF, so its TTFF is shorter than that of IFPPP, and the average TTFF is shortened from 22.8 min to 17.7 min, with a reduction of 22.6%. In terms of positioning accuracy, since the noise levels of the two models are very close, the positioning accuracies of the two models are basically the same, and both the IFPPP and IRPPP achieve centimeter-level accuracy of 3.4 cm and 3.6 cm, respectively. Acknowledgments. This work is partially supported by the National Natural Science Foundation of China (Grant No. 42204027), and the Foundation of Laboratory of Science and Technology on Marine Navigation and Control, China State Shipbuilding Corporation (Grant No. 2021010104). The authors sincerely thank IGS and CODE for providing multi-GNSS data and products.

References 1. Zumberge JF, Heflin MB, Jefferson DC et al (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res: Solid Earth 102(B3):5005–5017

240

Q. Zhao et al.

2. Kouba J, Héroux P (2001) Precise point positioning using IGS orbit and clock products. GPS Solut 5(2):12–28 3. Ge M, Gendt G, Rothacher M et al (2008) Resolution of GPS carrier-phase ambiguities in precise point positioning (PPP) with daily observations. J Geodesy 82(7):389–399 4. Gabor MJ, Nerem RS (1999) GPS carrier phase ambiguity resolution using satellite-satellite single differences[C]. In: Proceedings of the 12th international technical meeting of the satellite division of the institute of navigation (ION GPS 1999). 1999, pp 1569–1578 5. Laurichesse D, Mercier F, Berthias JP et al (2009) Integer ambiguity resolution on undifferenced GPS phase measurements and its application to PPP and satellite precise orbit determination. Navigation 56(2):135–149 6. Collins P, Lahaye F, Heroux P et al (2008) Precise point positioning with ambiguity resolution using the decoupled clock model. In: Proceedings of the 21st international technical meeting of the satellite division of the institute of navigation (ION GNSS 2008), pp 1315–1322 7. Li P, Zhang X, Ren X et al (2016) Generating GPS satellite fractional cycle bias for ambiguityfixed precise point positioning. GPS Solut 20(4):771–782 8. Geng J, Meng X, Dodson AH et al (2010) Integer ambiguity resolution in precise point positioning: method comparison. J Geodesy 84(9):569–581 9. Shi J, Gao Y (2014) A comparison of three PPP integer ambiguity resolution methods. GPS Solut 18(4):519–528 10. Laurichesse D, Banville S (2018) Innovation: instantaneous centimeter-level multi-frequency precise point positioning. In: GPS World 4 July 2018 11. Geng J, Bock Y (2013) Triple-frequency GPS precise point positioning with rapid ambiguity resolution. J Geodesy 87(5):449–460 12. Li X, Li X, Liu G et al (2019) Triple-frequency PPP ambiguity resolution with multiconstellation GNSS: BDS and Galileo. J Geodesy 93(8):1105–1122 13. Li X, Liu G, Li X et al (2020) Galileo PPP rapid ambiguity resolution with five-frequency observations. GPS Solut 24(1):1–13 14. Guo J, Xin S (2019) Toward single-epoch 10-centimeter precise point positioning using Galileo E1/E5a and E6 signals. In: Proceedings of the 32nd international technical meeting of the satellite division of the institute of navigation (ION GNSS+ 2019), pp 2870–2887 15. Gu S, Lou Y, Shi C et al (2015) BeiDou phase bias estimation and its application in precise point positioning with triple-frequency observable. J Geodesy 89(10):979–992 16. Li P, Zhang X, Ge M et al (2018) Three-frequency BDS precise point positioning ambiguity resolution based on raw observables. J Geodesy 92(12):1357–1369 17. Geng J, Guo J, Meng X et al (2020) Speeding up PPP ambiguity resolution using triplefrequency GPS/BeiDou/Galileo/QZSS data. J Geodesy 94(1):1–15 18. Zhao Q, Pan S, Gao W et al (2022) Multi-GNSS fast precise point positioning with multifrequency uncombined model and cascading ambiguity resolution. Math Probl Eng 19. Geng J, Guo J, Chang H et al (2019) Toward global instantaneous decimeter-level positioning using tightly coupled multi-constellation and multi-frequency GNSS. J Geodesy 93(7):977– 991 20. Geng J, Guo J (2020) Beyond three frequencies: an extendable model for single-epoch decimeter-level point positioning by exploiting Galileo and BeiDou-3 signals. J Geodesy 94(1):1–15 21. Elsobeiey M (2015) Precise point positioning using triple-frequency GPS measurements. J Navig 68(3):480–492 22. Duong V, Harima K, Choy S et al (2019) An optimal linear combination model to accelerate PPP convergence using multi-frequency multi-GNSS measurements. GPS Solut 23(2):1–15 23. Melbourne WG (1985) The case for ranging in GPS-based geodetic systems. In: Proceedings of the first international symposium on precise positioning with the global positioning system, 1985, Rockville, MD, USA, 15–19 April 1985, pp 373–386

Precise Point Positioning Ambiguity Resolution with Multi-frequency …

241

24. Temiissen JG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. J Geodesy 70:65–82 25. Yang Y, Gao W, Guo S et al (2019) Introduction to BeiDou-3 navigation satellite system. Navigation 66(1):7–18 26. Feng Y (2008) GNSS three carrier ambiguity resolution using ionosphere-reduced virtual signals. J Geodesy 82(12):847–862 27. Guo F, Zhang X, Wang J et al (2016) Modeling and assessment of triple-frequency BDS precise point positioning. J Geodesy 90(11):1223–1235 28. Gao W, Gao C, Pan S (2017) A method of GPS/BDS/GLONASS combined RTK positioning for middle-long baseline with partial ambiguity resolution. Surv Rev 49(354):212–220

Simulation Study on Real-Time Orbit Determination Based on GNSS for LEO Satellite Considering the Effect of Electric Propulsion Jiapeng Wu1 , Wanwei Zhang2(B) , Fuhong Wang2 , Meng Wang1 , and Chengxiang Yin1 1 Space Star Technology Co., Ltd, Beijing, China 2 School of Geodesy and Geomatics, Wuhan University, Wuhan, Hubei, China

[email protected]

Abstract. The traditional GNSS pseudo-range based Real-Time Orbit Determination(RTOD) cannot meet the requirements for high-precision and reliable orbit determination of LEO satellite with electric propulsion in the whole process, this paper conducts mathematical modeling of electric thrust based on the magnitude, direction and satellite attitude of electric thrust, improves the traditional GNSS pseudo-range based RTOD algorithm, and takes into account the impact of inertial system acceleration generated by electric thrust during the Kalman time update process of RTOD, A new RTOD algorithm based on GNSS pseudo-range considering the influence of electric propulsion is proposed. Then, based on GNSS signal simulator and space-borne GPS/BDS receiver, the simulation scenarios of single electric thruster and dual electric thruster are designed and simulation GPS/BDS data are collected. Using the independently developed RTOD software SATODS, offline simulation RTOD data processing tests are conducted on the collected simulation data. The test results show that the new RTOD algorithm based on GNSS pseudo-range considering the influence of electric propulsion proposed in this paper is not affected by the electric thrust, which verifies the adaptability of the improved RTOD algorithm to the orbit determination of orbital maneuver conditions with the electric propulsion. Keywords: Electric propulsion · Low Earth Orbit (LEO) · GNSS · Real-Time Orbit Determination (RTOD) · Pseudo-range

1 Introduction Real-time, accurate and reliable orbital information is essential for LEO satellites in many space science applications, such as high-resolution earth observation, satellite altimetry, GNSS radio occultation, etc. [1–3]. Since the 1990s, space-borne GNSS receiver has become the main technical means of high-precision orbit determination for spacecraft because of its advantages of continuous observation, high accuracy, low cost and power, small size and light weight [4, 5, 5]. At present, the GNSS pseudo-range based Real-Time Orbit Determination(RTOD) accuracy is about 1.0 ~ 2.0 m by using GNSS broadcast © Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 242–255, 2024. https://doi.org/10.1007/978-981-99-6944-9_21

Simulation Study on Real-Time Orbit Determination

243

ephemers [6–8],while the GNSS carrier-phase based RTOD accuracy can reach about 0.3 ~ 0.5 m [9–11], due to the higher ranging precision of GNSS carrier phase. However, because the GNSS carrier-phase based RTOD algorithm involves the processing of carrier phase ambiguity parameter, it requires higher calculation power of the onboard processor. Therefore, the RTOD algorithm based on GNSS pseudo-range,which can meet the requirements of real-time orbit measurement and control of most satellites,is still the mainstream in most LEO applications. However, Near-Earth spacecraft, especially LEO satellites, will gradually deviate from their preset orbit due to the influence of the earth’s non-spherical gravity, atmospheric drag and other perturbation forces. In order to complete the corresponding scientific tasks, satellites often need to maintain a preset orbit; When the orbit of a LEO satellite deviates beyond the preset range, to return to the preset orbit, it is necessary to apply thrust to the satellite. For example, TerraSAR-X satellite orbit maneuvers need to be performed approximately once a week to keep it in orbit [12], so satellite orbit maneuvers are unavoidable. Although the kinematic based RTOD method is not affected by maneuver, its accuracy is difficult to meet the requirements of 1.0 ~ 2.0 m. The traditional GNSS simplified dynamics based RTOD applies only to the normal working condition without orbital maneuver, for orbital maneuver conditions, if not effectively identify and correctly handle the GNSS observations of the orbital maneuver segment, the RTOD accuracy of this segment will be significantly reduced, and the RTOD filter may even diverge in some serious case. Therefore, the traditional GNSS based RTOD cannot guarantee the precision, continuity and reliability of autonomous orbit determination under orbital maneuver conditions. In the case of orbit maneuvering, on-board accelerometer data can usually be used to replace the non-conservative force model. The accuracy of orbit determination is higher than the conventional method of modeling all the perturbation forces, and the accuracy is comparable to that of the empirical force model error compensation method [13, 14]. In addition, the influence of orbit maneuvering can be eliminated to a large extent by using appropriate maneuvering thrust modeling and numerical integration strategy [15]. At present, the main propulsion modes for orbit maneuvering of LEO satellites include chemical propulsion, cooling propulsion and electric propulsion. Ionic electric propulsion has the characteristics of wide range continuous fine adjustment, low noise, long life and high specific impulse, etc., thus it has been used more and more in satellite orbital maneuver. Due to the relatively accurate thrust of ionic electric propulsion, it is possible to conduct thrust modeling in orbit. In this paper, the traditional GNSS pseudo-range RTOD algorithm is im-proved, and a new RTOD algorithm considering the effect of electric propulsion is proposed. In this method, the magnitude, direction and attitude of the electric thrust were used for mathematical modeling, and the effect of the inertial system acceleration caused by the electric propulsion force was introduced into the RTOD time update process of the Kalman filter. Based based on GNSS signal simulator and on board GPS/BDS receiver, the simulation scenarios of single electric thruster and dual electric thruster are designed and simulation GPS/BDS data are collected. Using the independently developed RTOD software SATODS, offline simulation RTOD data processing tests are conducted on the collected simulation data. The RTOD test results are analyzed and evaluated to verify the validity of the improved algorithm in this paper.

244

J. Wu et al.

2 RTOD Algorithm 2.1 GNSS Measurements There is no tropospheric delay error for LEO scenarios, the onboard GPS/BDS single frequency pseudo-range observation equation can be expressed as: ⎧ i i ⎨ PL1 = ρ i + cδtr − cδtsi + δρIon + εP i L1 (1) j j ⎩ P = ρ j + cδtr − cδts + δρ j + ε j B1 Ion P B1

j PB1

i and are the single-frequency pseudo-range observations of the i-th GPS where, PL1 satellite and the j-th BDS satellite, in unit m. ρ i and ρ j are the geometric distances between the onboard GNSS receiver and the i-th GPS satellite and the j-th BDS satellite, j respectively. δtr is the clock bias of GNSS receiver, δtsi and δts represent the clock bias parameters of the i-th GPS satellite and the j-th BDS satellite, respectively; c is the i and δρ j are ionospheric delays, in m. ε speed of light in a vacuum. δρIon i and ε j PL1 Ion PB1 are multipath error and observation noise, respectively.

2.2 Dynamical Model of LEO In the Earth Centered Inertia (ECI) system, the motion equation of the LEO satel-lite can be expressed by the first-order differential equations:  r˙ = v (2) v˙ = a where, r , v is the position and speed of the aircraft respectively; a is the total acceleration of the LEO satellite. Generally, this differential equations can be solved by numerical integration, such as Runge–Kutta method, etc. In the case of electric thrust, the satellite total acceleration can be expressed as: a = ag + ang + af + T · aω

(3)

where, a is the total acceleration of the LEO satellite in the ECI frame; ag represent the acceleration caused by conservative forces, including Earth’s central gravity, nonspherical gravity, N-body gravity, Earth’s solid tides and ocean tides perturbation force, etc.; ang represent the acceleration caused by non-conservative forces, including atmospheric drag, solar pressure, etc.; af is the acceleration caused by electric thrust, and aω is the empirical compensation acceleration, which is used to compensate the influence of the small perturbation force that cannot be modeled or incorrectly modeled. A first-order Gauss-Markov random model is adopted to compensate the dynamics model in three directions: R(Radial), A (Along-track), and C (Cross). T is the transformation matrix from the RAC system to the ECI system. → The acceleration − a f caused by the electric thrust can be calculated as following: −−→ − → b a f = Cbi CSRF f SRF /M

(4)

Simulation Study on Real-Time Orbit Determination

245

In formula (4), Cbi is the transformation matrix from the satellite body coordinate system to the ECI system, which can be calculated from the attitude data of the satellite, b is the transformation matrix such as attitude quaternion or attitude euler angle; CSRF from the Scientific Reference Frame(SRF) of the electric thrust to the satellite body coordinate system, which can be calculated by the specific installation direction of the T −−→  electric thruster in the satellite system. f SRF = fxSRF , fySRF , fzSRF is the electric thrust vector on the three axes of the electric thruster in its SRF coordinate system, and the unit is N, The accuracy of the electric thrust, guaranteed by the satellite propulsion subsystem, is less than 1%; M is the mass of the LEO satellite, and the unit is kg. It can be modified on the ground by up-injection telemetry parameters. 2.3 Extended Kalman Filter Model Combined with the simplified dynamics model of LEO satellite, a extended Kalman filter is used to estimate the satellite position, speed and other state parameters. The state quantity of RTOD filter is selected as: X = ( R1×3 V1×3 bG bC b Cd Cr W1×3 )T

(5)

where, R1×3 and V1×3 are the estimated position and velocity of the LEO satellite in the ECI system, respectively; bG and bC represent the GPS/BDS receiver clock offset, while b are receiver clock drift; Cd and Cr are the atmospheric drag coefficient and solar pressure coefficient to be estimated respectively; W1×3 is the compensation acceleration to be estimated in three directions of RAC. i and P j with The partial derivatives of GPS/BDS pseudo-range observations PL1 B1 j respect to RTOD filter state quantity X are HGi and HC , can be expressed as: ⎧ i ∂PL1 ⎪ → i ⎪ = ( −(U T − e i )T1×3 01×3 1 0 0 0 0 01×3 ) ⎨ HG = ∂X j ⎪ ∂PB1 ⎪ j → ⎩ = ( −(U T − HC = e j )T1×3 01×3 0 1 0 0 0 01×3 ) ∂X

(6)

→ → e i and − e j are where, U T is the conversion matrix from ECI system to ECEF system; − the unit line of sight vectors of the LEO satellite relative to the i-th GPS satellite and the j-th BDS satellite; 01×3 is 1 × 3 dimensional vector, vector elements are all 0. The state equation and observation equation in the RTOD filtering are as follows: Xk = k,k−1 Xk−1 + Wk−1 (7) Zk = Hk Xk−1 + Vk where, Xk and Zk are state vector and observation vector in Extended Kalman filter (EKF),respectively; k,k−1 is state transition matrix and its associated derivation formula refer to [16, 17], Hk is design matrix(also known as observation matrix); Wk−1 and Vk are system noise and observation noise, respectively.

246

J. Wu et al.

3 Simulation Scenarios and RTOD Strategy 3.1 Simulation Scenarios The STK (Satellite Tool Kit) software is used to design the single electric thruster scenes and the dual electric thrusters scenes of LEO satellite. The initial orbit of the LEO satellite with 265 km orbit height is the same in the two simulation scenarios(UTC start time:4 Feb 2021 04:00:00, duration:12 h). The installation diagram of the electric thruster in the two simulation scenarios is shown in Fig. 1. < ;

R WKUXVWHU

R

WKUXVWHU

WKUXVWHU

Fig. 1. Schematic diagram of electric thruster installation in two simulation scenarios.

In the single electric thruster scenes, the satellite attitude is set to zero, and the installation direction of the electric thruster is along the X axis in the XOY plane of the satellite body coordinate system, that is, the direction of the thrust generated is roughly along the direction of the satellite speed. While, in the dual electric thrusters, the satellite attitude is set as 4.75° of yaw, in which the installation direction of electric thruster 1 is the same as that of single electric thrusters scenes, and the angle between electric thruster 2 and electric thruster 1 is 9.5°. Table 1 shows the working sequence of each electric thruster design in the two simulation scenarios. In the single electric thruster scenes, the electric thruster works for 1200s each time, and the electric thrust is 20mN. In the dual electric thruster scenes, the two electric thrusters work at the same time, and each time works for 750s, and each electric thrust is 20mN. The block diagram of GNSS simulation data acquisition and processing is shown in Fig. 2. The satellite reference orbit (including position, velocity, acceleration and attitude) in the two simulation scenes is generated by STK software as previously mentioned, and then the satellite reference orbit is imported into the GNSS signal simulator,which sends the simulated GNSS RF signal and NMEA message to the GPS/BDS receiver and the ground test equipment in real-time,respectively.The ground test equipment parses the NMEA message to obtain the current simulation time, and searches the satellite attitude and thrust data in the attitude and thrust file according to the current simulation time, then sends them to the GPS/BDS receiver to simulate satellite attitude and thrust broadcasting on board. The GNSS receiver collects the original GNSS observations and broadcast ephemeris in real-time and stores them in the computer as Rinex files. In order to simulate actual on-orbit application scenarios as much as possible, the GNSS measurement simulation considers various main error items, including GNSS orbit and clock

Simulation Study on Real-Time Orbit Determination

247

Table 1. Time sequence of electric thruster operation in two simulation scenarios. Scenes

Work start time (UTC)

Work end time (UTC)

Duration(s)

Thrust value (mN)

Single thruster

4 Feb 2021 04:34:38.953

4 Feb 2021 04:54:38.953

1200

20

4 Feb 2021 06:04:18.786

4 Feb 2021 06:24:18.786

1200

20

4 Feb 2021 07:34:05.265

4 Feb 2021 07:54:05.265

1200

20

4 Feb 2021 09:03:50.151

4 Feb 2021 09:23:50.151

1200

20

4 Feb 2021 10:33:40.307

4 Feb 2021 10:53:40.307

1200

20

4 Feb 2021 12:03:29.755

4 Feb 2021 12:23:29.755

1200

20

4 Feb 2021 13:33:13.654

4 Feb 2021 13:53:13.654

1200

20

4 Feb 2021 15:03:02.059

4 Feb 2021 15:23:02.059

1200

20

4 Feb 2021 04:34:01.196

4 Feb 2021 04:46:31.196

750

20

4 Feb 2021 06:03:36.662

4 Feb 2021 06:16:06.662

750

20

4 Feb 2021 07:33:21.842

4 Feb 2021 07:45:51.842

750

20

4 Feb 2021 09:03:07.424

4 Feb 2021 09:15:37.424

750

20

4 Feb 2021 10:33:40.307

4 Feb 2021 10:45:30.159

750

20

4 Feb 2021 12:02:53.635

4 Feb 2021 12:15:23.635

750

20

4 Feb 2021 13:32:40.937

4 Feb 2021 13:45:10.937

750

20

4 Feb 2021 15:02:29.910

4 Feb 2021 15:14:59.910

750

20

Dual thruster

error, ionospheric delay error, measurement noise, receiver clock error, and so on. The independently developed RTOD software SATODS (Spaceborne GNSS Au Tonomous Orbit Determination System) is used to process the simulation data in the offline simulative way. Finally, The RTOD results of SATODS software can be evaluated and analysed by comparing with the satellite reference orbit.

248

J. Wu et al. 5HIHUHQFHRUELW

STK $WWLWXGHDQGWKUXVW

*166VLJQDO VLPXODWRU

10($PHVVDJH

*URXQG7HVW (TXLSPHQW $WWLWXGHDQGWKUXVW

*1665)VLJQDO

*166UHFHLYHU

5HIHUHQFHRUELW

*36%'6observations

SATODS 572'UHVXOWV

2UELW FRPSDULVRQ

Fig. 2. Block diagram of GNSS simulation data acquisition and processing.

3.2 RTOD Strategy Figure 3 shows the processing scheme for RTOD of Electrically Propelled LEO Satellite. First of all, GPS/BDS observations are preprocessed to perform standard single point positioning(SPP)/single point velocity(SPV) solution; Then, at the beginning of filtering, the filtering state vector and its covariance are initialized using information such as SPP/SPV. In the time update step of EKF, a one-step prediction filtering state vector and its covariance are generated. In this process, mathematical modeling of electric thrust is conducted according to the magnitude, direction and satellite attitude of electric thrust. Considering the impact of inertial system acceleration generated by electric thrust, in the measurement update step of EKF, The Kalman filter gain matrix is calculated, and then the estimated filter state vector and its covariance are generated; Finally, the RTOD results are output by using 4th-order Runge-Kutta orbit integral prediction and 5th-order Hermite polynomial orbit interpolation. With the continuous input of observations, the above process will be repeated until the last epoch. It should be noted that the system noise for empirical compensation acceleration aω is preset with different values in the RTN directions, and its system noise covariance matrix is calculated using a first order Gaussian Markov model. Considering that there are errors in calculating thrust acceleration based on broadcast attitude and thrust with errors, this paper adjusts the role of simplified dynamic based RTOD in the entire autonomous navigation system by appropriately amplifying the process noise of the first order Gaussian Markov model for compensating acceleration during orbital maneuvers to absorb perturbed acceleration errors. Although the data is processed offline, the algorithm simulates the real real-time process, that is, only the past and current data are used in data processing and quality control. Table 2 shows the strategies used for offline simulative RTOD solution.

Simulation Study on Real-Time Orbit Determination

249

Fig. 3. Processing processing scheme for RTOD of electrically propelled LEO satellite.

4 Results and Analysis 4.1 Tracking Performance Figure 4 shows the statistical distribution of the percentage of the number of GPS/BDS tracking satellites in the two simulation scenarios. It can be seen that: (1) the statistical results of the number of GPS/BDS tracking satellites in the two simulation scenarios are basically the same. (2) The largest percentage of the number of tracking satellites is 9 and 10 for GPS and BDS, respectively. (3) In the two simulation scenarios, the epochs with the number of GPS and BDS tracking satellites greater than or equal to 5 account for 100% of the total epochs. The GNSS observation conditions of the two simulation scenarios are very good.

4.2 Acceleration Error In order to verify the correctness of the electric thrust modeling, this paper analyzes the total acceleration calculation error of the LEO satellite in the ECI system. As shown in Fig. 5. subgraph(a) is the working sequence diagram of the thruster in the two scenarios. The orbit period of the LEO satellite is about 90 min. In each circle

250

J. Wu et al. Table 2. The strategies of the offline simulative RTOD solution.

Measurement model GPS/BDS Observations

Single frequency code and doppler observations

Elevation cutoff angle

5°

GNSS orbit and clock

Broadcast ephemeris

LEO PCO and PCV

PCO is corrected by the nominal value, PCV is not considered

LEO Satellite attitude

Attitude quaternion

Dynamical model Earth gravity field

EGM 2008 30 × 30,neglect the time-varying

N-body gravitation

Moon and Sun only, low precision analytic method[16]

Earth and ocean tide

Simplified earth model, neglect the ocean tide

Atmosphere drag

Static modified Harris-Priester model (density), fixed effective area

Solar radiation

Cannonball model, fixed effective area

Empirical acceleration

Radial, along-track and cross with a first-order Gauss-Markov model

Numerical integration

4th-order Runge-Kutta method

Reference frame Coordinate system

WGS84/CGCS2000

Precession/nutation

IAU1976/IAU 1980 simplified model

Earth rotation parameter

Rapid Predicted EOP (RPE) in IERS Bulletin A

Estimation Estimator

Extended Kalman-filter

Mode

Sequential processing in real-time with 30s interval measurement update

Estimated parameters

Position, velocity, GPS/BDS receiver clock offsets, receiver clock drift, drag and radiation pressure scale factors, empirical accelerations

of orbit, the thruster starts to work at the satellite perigee, with an interval of about 60 min. The time sequence of electric thruster operation in two scenarios is listed in Table 1 for details. In the dual thruster scenes, since the angle between electric thruster 2 and electric thruster 1 is 9.5°, when the satellite attitude yaw is 4.75°, the combined thrust of the two electric thrusters is about 39.8626mN, and the thrust direction is the same as that of the single electric thruster scenes. Subgraph(b) shows the comparison of satellite orbital heights between the two scenarios. Compared with the single electric thruster scenes, the dual thruster scenes has a short duration but a large combined thrust. After several cycles of work, the orbital

Simulation Study on Real-Time Orbit Determination

251

percentage[%]

40 30

Single thruster

GPS

BDS

20 10 0 5

6

7

8

9

10

11

12

10

11

12

40

GPS

percentage[%]

Dual thruster

BDS

30 20 10 0 5

6

7

8

9

Number of tracked GPS/BDS satellites

Fig. 4. Percentage of the number of tracked GPS/BDS satellites.

height of perigee is lower but apogee is higher, that is, the orbital eccentricity of the dual thruster scenes becomes larger than that of the single thruster scenes. Subgraph(c) shows the calculation error of the total acceleration of LEO satellites in the ECI system for the two simulation scenarios. The error of the total acceleration is calculated as follow: First, calculate the acceleration of the inertial system caused by electric thrust according to formula (4).Then calculate the acceleration in the ECI coordinate system caused by other items according to the dynamic model settings in Table 2.Finally calculate the total acceleration of the satellite in the ECI coordinate system according to formula (3).Taking the total acceleration of the LEO satellite in the ECI coordinate system generated by STK software as a reference, the total acceleration error of the LEO satellite in the ECI coordinate system is analyzed. It can be seen from the subgraph(c) that the total acceleration calculation error is approximately in the range of 5.0e−5 –4.5e−6 m/s2 . The RMS of acceleration calculation error during non orbital maneuvers and orbital maneuvers is about 2.7e−6 m/s2 and 3.2e−6 m/s2 , respectively. So the magnitude of the total acceleration calculation error does not differ significantly due to the presence of thrust, which shows that the total acceleration of the LEO satellite in the ECI coordinate system calculated after considering the thrust modeling in this paper is correct. 4.3 RTOD Results Using the independently developed RTOD software, named SATODS, the offline simulative RTOD data processing tests were conducted on the collected simulation data. And then the offline simulative RTOD results were analyzed and evaluated by comparing with the satellite reference orbit of the two simulation scenarios.

252

J. Wu et al. Single thruster scenes

Dual thruster scenes

Thrust[mN]

(a) Thrust operation sequence comparison of the two scenes 20 10 0 0

2

Height[km]

270

2

6

8

10

12

265

260 0

Acc error[m/s ]

4

(b) Orbit height comparison of the two scenes

2

4

6

8

10

12

(c) Acceleration error comparison of the two scenes

-6

6.0x10

-6

3.0x10

0.0

0

2

4

6

8

10

12

Time[h]

Fig. 5. Total ECI acceleration error of satellites in two simulation scenarios.

Table 3 shows the root mean square (RMS) statistical accuracy of the RTOD position error of the LEO satellite based on single GPS, single BDS and GPS + BDS for the two simulation scenarios in RAC and 3D(3-Dimensional). It can be seen that: (1) The RTOD accuracy based on single GPS and single BDS is basically the same, and the accuracy of the RTOD position error is about 1.5 m (3DRMS). (2) Compared with the single GNSS case, the combined GPS + BDS RTOD accuracy has been improved, but the improvement effect is limited, about 5%. (3) The RTOD position error is the smallest in the R direction, followed by the C direction, and the largest in the A direction, which may be related to the orbit height of the LEO satellite. It is widely known that when the orbit height is lower, the atmospheric drag perturbation magnitude is larger. However, the simplified atmospheric drag model used in the RTOD algorithm proposed in this paper is difficult to accurately de-scribe the atmospheric drag of the LEO especially when the orbit height is lower 300 km, thus causes the orbit error of the A direction to be larger than other directions. The comparison of position error of the LEO satellite based on GPS + BDS in RAC directions for the two simulation scenarios is shown in Fig. 6. It can be seen that: (1) The magnitude of statistical accuracy of 3DRMS for GPS + BDS combined RTD in the two simulation scenarios is equivalent, and the trend of change curve in three directions of RAC is the same, with only minor differences, This may be related to the minor difference in GPS/BDS observation data quality.

Simulation Study on Real-Time Orbit Determination

253

Table 3. Statistics of the offline simulative RTOD results. Scenes

GNSS

R/m

A/m

C/m

Single thruster

GPS

0.506

1.212

0.783

BDS

0.498

1.207

0.771

GPS + BDS

0.484

1.115

0.756

GPS

0.523

1.245

0.804

BDS

0.518

1.238

0.793

GPS + BDS

0.511

1.155

0.784

Dual thruster

Single thruster scenes

Double thruster scenes

4

R[m]

2 0 -2 -4 0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6

8

10

12

6

A[m]

3 0 -3 -6 6

C[m]

3 0 -3 -6

Time[h]

Fig. 6. Comparison of LEO position error based on GPS + BDS in RAC directions.

(2) The trend of the change curve of the two simulation scenarios in the three directions of RAC is independent of whether the thrusters are working. The position error of RTOD when the thruster in the working period is basically equivalent to that in the non working period, that is, the RTOD results are basically not affected by the electric thrust. Therefore, the improved RTOD algorithm proposed in this paper is applicable to orbital maneuver conditions with the electric propulsion.

5 Conclusions In this paper, a new RTOD algorithm based on GNSS considering the effect of electric propulsion is proposed. Based on GNSS signal simulator and on board GPS/BDS receiver, the two simulation scenarios of single and dual electric thruster are designed

254

J. Wu et al.

and simulation data are collected. Using the independently developed RTOD software SATODS, the offline simulative RTOD data processing tests are conducted on the collected simulation data. The offline simulative RTOD results were analyzed and evaluated by comparing with the satellite reference orbit of the two simulation scenarios. The following conclusions can be drawn: (1) The calculation error of satellite total acceleration is independent of whether there is thrust, which shows that it is correct and feasible to carry out mathematical modeling of electric thrust according to the magnitude, direction and satellite attitude of electric thrust. (2) Based on the simulation GPS/BDS data, the offline simulative RTOD results position accuracy is about 1.5 m (3DRMS), of which the Radial error is the smallest, followed by the Along-track error, and the Cross error is the largest. (3) The change curve of RTOD error is basically not affected by the electric thrust. Which verifies the adaptability of the improved RTOD algorithm proposed in this paper to the orbit determination of orbital maneuver conditions with the electric propulsion. Although the improved RTOD algorithm has been tested and verified by simulation data, the RTOD software will be transplanted into GNSS receiver, and its applicability and correctness to real data need to be further tested in the future. Acknowledgments. This study is financially supported by the National Natural Science Foundation of China (Grant 62073044, 62103307), the State Key Laboratory of Satellite Navigation System and Equipment Technology, Shijiazhuang, Hebei, 050081, China (Grant CEPNT2021KF-05).The support of all these institutions is highly appreciated and gratefully acknowledged here.

References 1. van den IJssel J, Encarnacao J, Doornbos E et al (2015) Precise science orbits for the Swarm satellite constellation. Adv Space Res 56(6):1042–1055 2. Jayles C, Chauveau JP, Rozo F (2010).DORIS/Jason-2: better than 10 cm on-board orbits available for near-real-timealtimetry. Adv Space Res 46(12):1497–1512 3. Montenbruck O, Hauschild A, Andres Y et al (2013) (Near-)real-time orbit determination for GNSS radio occultation processing.GPS Solut 17(2):199–209 4. Hart RC, Hartman KR, Long AC, Lee T, Oza DH (1996) Global Positioning System (GPS) Enhanced Orbit Determination Experiment (GEODE) on the small satellite technology initiative (SSTI) lewis spacecraft. In: Proceedings of ION GPS96,Kansas City, Missouri, 20,September 1996 5. Goldstein DB (2000) Real-time autonomous precise satellite orbit determination using the global positioning system. University of Colorado, Boulder, CO, USA 6. Reichert A, Meehan T, Munson T (2002) Toward decimeter-level real-time orbit determination: a demonstration using the SAC-C and CHAMP spacecraft. In: Proceedings of the 15th international technical meeting of the satellite division of the institute of navigation(ION GPS 2002), Portland, 24–27,September 2002 7. Montenbruck O, Ramos-Bosch P (2002) Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solut 12:187–198

Simulation Study on Real-Time Orbit Determination

255

8. Xiong C, Chuanfang L, Zhu J et al (2017) Orbit determination using real tracking data from FY3C-GNOS. Adv Space Res 60:543–556 9. Xuewen G, Wanwei Z, Jizhang S et al (2022) In-flight performance assessment of the singlefrequency GPS/BDS receiver for Yaogan-30 series satellites real-time navigation. IEEE Trans Aerosp Electron Syst 58(5):4495–4505 10. Wang F, Gong X, Sang J et al (2015) A novel method for precise onboard real-time orbit determination with a standalone GPS receiver. Sensors 15(12):30403–30418 11. Gong X, Sang J, Wang F, Li X (2020) LEO onboard real-time orbit determination using GPS/BDS data with an optimal stochastic model. Remote Sens 12:3458 12. Hauschild A, Montenbruck O (2021) Precise real-time navigation of LEO satellites using GNSS broadcast ephemerides.Navigation 1:419–432 13. Jaggi A, Montenbruck O, Moon Y et al (2012) Inter-agency comparison of TanDEM-X baseline solutions. Adv Space Res 50(2):260–271 14. Helleputte TV, Visser P (2008) GPS based orbit determination using accelerometer data. Aerosp Sci Technol 12(6):478–484 15. Chao L, Hongzheng C, Chao H et al (2013) Precision orbit determination for GRACE based on accelerometer measurements(in chinese). In: 4th China satellite navigation conference(CSNC) proceedings on proceedings, pp. 135–139 Wuhan,China 16. Ju B, Gu D, Herring TA et al (2017) Precise orbit and baseline determination for maneuvering low earth orbiters. GPS Solut 21(1):53–64 17. Montenbruck O, Gill E (2000) Satellite orbits: models, methods and applications, 1st edn. Springer, Heidelberg 18. Fuhong W (2010) A kalman filtering algorithmn for precision real-time orbit determination with space-borne GPS measurements. Geomatics Inf Sci Wuhan Univ 35(06):653–656

Analysis of the Performance of Broadcast Ionospheric Model for Anti-disturbance Capability Xianggao Yan1 , Xiaolin Jia2(B) , Yongxing Zhu2 , Jialong Liu3,4 , and Zhichao Zhang1 1 College of Geological Engineering and Geomatics, Changan University, Xian 710054, China 2 Xian Research Institute of Surveying and Mapping, Xian 710054, China

[email protected]

3 Shanghai Astronomical Observatory, CAS, Shanghai 200030, China 4 University of Chinese Academy of Sciences, Beijing 100049, China

Abstract. Magnetic storms and solar flares cause anomalous changes in ionospheric electron content, during which the performance of the broadcast ionospheric model correction decreases. In this paper, we use the CODE GIM (Center for Orbit Determination in Europe global ionosphere maps) product as a reference to evaluate the BDGIM (BeiDou global ionospheric correction model) BDS Klobuchar (BKlob), GPS Klobuchar (GKlob), and NeQuick G models in terms of its correction performance and the adaptability to magnetic storm and flare events. The analysis of the measured data shows that (1) during the magnetic storm, the BDGIM model does not show significant changes at low and mid-latitudes, while the correction rate is less than 20.0% at high latitudes; the BKlob and GKlob models show fluctuations at mid and high latitudes. The fluctuation range is roughly 20.0–40.0% in mid-latitudes and the correction rate is less than 20.0% in high latitudes; the NeQuick G model decreases the correction rate to about 20.0% at mid and high latitudes in the southern hemisphere during the outbreak of higher-grade magnetic storms. (2) During solar flares, the BDGIM model shows no significant change in the correction performance in the Northern Hemisphere and globally, and fluctuations in the Southern Hemisphere; the BKlob and GKlob models show fluctuations in the Southern Hemisphere at mid-latitudes, and the correction rate in the Southern Hemisphere at high latitudes is less than 20.0%; the NeQuick G model shows a decrease in all regions, and the correction rate ranges from 20.0% to 40.0%. (3) The BDGIM model has better adaptation to flares, and the NeQuick G model has better adaptation to magnetic storms; the BKlob and GKlob models have comparable adaptation to both, with differences in individual latitudinal bands. Keywords: Broadcast ionosphere model · Correct performance · Magnetic storm · Solar flare · Ionosphere

© Aerospace Information Research Institute 2024 C. Yang and J. Xie (Eds.): CSNC 2024, LNEE 1094, pp. 256–268, 2024. https://doi.org/10.1007/978-981-99-6944-9_22

Analysis of the Performance of Broadcast Ionospheric Model

257

1 Introduction The ionosphere is a very important part of the solar-terrestrial space atmosphere, and also serves as one of the main sources of error for satellite navigation system services, resulting in ranging errors of up to hundreds of meters [1]. Reasonable and accurate simulation and forecast of ionospheric delay has become one of the important topics in the research of single-frequency user positioning of the Global Navigation Satellite System (GNSS). The Chinese BeiDou (BDS), the European Union (Galileo) and the United States (GPS) have all established their own broadcast ionospheric models, and broadcast model parameters in their broadcast ephemeris. Currently, GPS uses the 8parameter Klobuchar model in the geomagnetic coordinate system to correct ionospheric delays; BDS uses the improved 8-parameter Klobuchar and 9-parameter BDGIM models to correct ionospheric delays, and Galileo uses the 3D ionospheric electron density model NeQuick model to correct ionospheric delays [2–4]. A comprehensive and systematic evaluation and analysis of broadcast ionospheric models can provide reference information for global GNSS single-frequency users using ionospheric delay services. A large number of previous studies have focused on the evaluation of GKlob, BKlob, BDGIM and NeQuick G models during ionospheric calm, and less during ionospheric disturbances [5, 6]. Preliminary evaluation studies on the correction performance of BDGIM, BKlob, GKlob and NeQuick G models during ionospheric disturbances have been conducted by domestic and foreign scholars, and it is found that the BDGIM model has stronger magnetic storm resistance than the Klobuchar model and can better reflect the actual changes in the ionosphere, but the correction accuracy will be reduced by about 30 to 60% [7–9]. Existing studies in the open literature during ionospheric disturbances have mainly focused on single magnetic storm events, while the impact of flares on the performance of multi-system broadcast ionospheric models has been less studied and multi-system comparisons are inadequate. In this paper, we analyze the multi-system broadcast ionospheric model in terms of model correction rate, RMS value, and extreme value of correction rate for three magnetic storms and two solar flare events selected from September 2020 to April 2022 for BKlob, GKlob, BDGIM, and NeQuick G. We obtain the variation of correction performance of the model in different latitude bands, and the adaptation characteristics for flares and magnetic storms.

2 Event Selection and Data Processing Methods 2.1 Experimental Event Selection In this paper, three magnetic storms and two solar flare events occurred between September 2020 and April 2022 were selected for the experiment, and the storms were classified according to the geomagnetic KP index as follows: Earth geomagnetic storms (KP = 7), small and medium geomagnetic storms (KP = 6, 5), as follows. The data of the above solar activity events are obtained from the Space Environment Prediction Center (SEPC) of the Chinese Academy of Sciences Table 1.

258

X. Yan et al. Table 1. Magnetic storms and solar flare events from 2020 to 2022

Year

Doy

Event type

Grade

2020

267–274

Magnetic storm

Small geomagnetic storms

2021

132

Magnetic storm

Earth magnetic storm

2021

308

Magnetic storm

Earth magnetic storm

2022

90

Solar Flares

X1.3 level

2022

107–110

Solar Flares

X1.1 level, X2.2 level

2.2 Assessment Benchmark Selection When evaluating broadcast ionospheric models, GIM products are generally chosen as the benchmark. At present, many ionospheric analysis centers, both at home and abroad, provide global ionospheric grid map GIM products, and some scholars have analyzed the accuracy of GIM products of different analysis centers and found that the accuracy of products of CODE analysis centers is higher globally, so this paper chooses CODE GIM products as the benchmark for evaluating four broadcast ionospheric models [10–12]. 2.3 Data Processing Methods In this paper, the VTEC at the grid point of the GIM product is compared with the corresponding VTEC of the broadcast ionosphere model, using the GIM product provided by the CODE analysis center as a benchmark, and four broadcast ionosphere models are evaluated. The evaluation indexes are mainly the model correction rate, RMS value, and the extreme value of the correction rate. The experiments mainly analyze the effects of magnetic storms and solar flares on the correction performance of BKlob, BDGIM, GKlob and NeQuick G models; first, in order to better observe the perturbation of ionosphere at various latitudes of the globe during the occurrence of magnetic storms and solar flares, MGEX stations at different latitudes are selected to extract the GNSS dual-frequency measured electron content, and the ionospheric electron content transformation characteristics of the global ionosphere at different latitudes during magnetic storms and solar flares are extracted. In order to analyze the impact of magnetic storms and solar flares on the broadcast ionospheric model, the time periods before and after the occurrence of magnetic storms and flares are selected and evaluated by using CODE GIM products as the benchmark, and the correction rates, RMS values and extreme values of the models at different latitude bands and globally are calculated.

3 Analysis of Magnetic Storms and Solar Flare Ionospheric Disturbances To observe the global ionospheric changes at different latitudes during the occurrence of magnetic storms and solar flares, VTEC values were extracted from the corresponding stations at low, mid, and high latitudes in the northern and southern hemispheres, and

Analysis of the Performance of Broadcast Ionospheric Model

259

the selected stations are shown in Fig. 1. Figure 3 shows the VTEC time series of each station during the X1.3 solar flare on March 31, 2022.

Fig. 1. GNSS station distribution

It can be seen from Fig. 2 that the ionospheric disturbances at the selected stations at different latitudes in the northern and southern hemispheres all show an increase in VTEC on the annual cumulative day 132, but the extent of the increase is different, among which the MGUE and OHI3 stations show a significant increase in VTEC on the annual cumulative day 132, with the maximum difference of about 15 TECU. It can be seen that the ionospheric disturbances in the mid and high latitude regions of the southern hemisphere are obvious, and both show an increase in VTEC.

Fig. 2. VTEC during the occurrence of the magnetic storm on May 12, 2021 at the selected measurement station

Both magnetic storms and solar flares have caused ionospheric perturbations, while solar flare ionospheric perturbations occur 1 to 2 days after the occurrence of the flare, and different regions show different characteristics of changes, such as an increase or decrease in VTEC values, while magnetic storms cause ionospheric perturbations that

260

X. Yan et al.

Fig. 3. VTEC during the solar flare occurrence on March 31, 2022 at the selected measurement stations

generally occur on the same day; how the performance of the broadcast ionospheric model correction will perform for abnormal changes in ionospheric electron content is discussed in the next sections. The analysis will be discussed in the next sections.

4 Analysis of Broadcast Ionospheric Model Perturbations During Magnetic Storms 4.1 Analysis of Broadcast Ionospheric Model Performance During the Magnetic Storms from September 23 to September 30, 2020 This magnetic storm showed a continuous geomagnetic disturbance from September 23 to September 30. To analyze the broadcast ionospheric model correction performance, the four broadcast ionospheric models with different latitude bands and global correction rates during the corresponding annual cumulative days 267 to 274 were calculated as shown in Fig. 4. “NM”, “NL”, “SL”, “SM”, “ SH”, which represent the high and mid-low latitudes in the northern hemisphere and the low and mid-high latitudes in the southern hemisphere, respectively. (1) The BDGIM model is not greatly affected by magnetic storms in the mid-latitude and low-latitude regions of the northern and southern hemispheres and globally, but a significant decrease in the annual cumulative day 270 occurs; (2) The BKlob and GKlob models do not show significant changes in the low-latitude and global regions, but the correction accuracy is lower in the mid- and high-latitude regions. 4.2 Analysis of the Performance of Four Broadcast Ionospheric Models During the May 12, 2021 Magnetic Storm The correction rates of each latitude band and global for 2 days before and after the occurrence of the magnetic storm are calculated as in Fig. 5. (1) The BDGIM model has no significant change in the correction accuracy except in the high latitudes of the southern hemisphere, and the correction accuracy is decreasing during the magnetic storm, with a correction rate of 29.27% in the high latitudes of the southern hemisphere on the day of the magnetic storm. The correction accuracy in

Analysis of the Performance of Broadcast Ionospheric Model

261

Fig. 4. Four broadcast ionosphere models for the period of 267–274 annual cumulative days Different latitude bands in the northern and southern hemispheres, global correction rate

the northern hemisphere and the globe showed a significant decrease after the magnetic storms occurred, and the most obvious in the high latitudes of the northern hemisphere, less than 10.0%, while the correction accuracy in all regions of the NeQuick G model did not change significantly; (2) The BDGIM model has poor correction accuracy in the high latitudes of the Southern Hemisphere on the day of the magnetic storm, the BKlob and GKlob models have low correction accuracy in the middle and high latitudes of the Southern Hemisphere, and the NeQuick G model correction accuracy is not significantly affected. 4.3 Performance Analysis of Four Broadcast Ionospheric Models During the Nov. 4, 2021 Magnetic Storm In order to analyze the correction accuracy of the broadcast ionospheric model in different regions, the correction rates of the broadcast ionospheric model in different latitude bands and global regions are calculated in Fig. 6. (1) The BDGIM model showed significant fluctuations at high latitudes in the Northern Hemisphere, and still performed better at mid-latitudes and low latitudes in the Northern Hemisphere and at low latitudes in the Southern Hemisphere. Both the southern hemisphere mid-latitude and high-latitude regions show a significant decrease on the day of the magnetic storm, which is less than 40.0%; (2) both the BKlob and GKlob models perform poorly in the northern hemisphere

262

X. Yan et al.

Fig. 5. Model of four broadcast ionospheres during the Doy: 130–134 magnetic storm different latitude bands in the northern and southern hemispheres, global correction rate

high-latitude regions with significant fluctuations and the lowest correction rate is less than 15.0. In the northern hemisphere mid-latitude regions both models have gradually decreasing correction rates. In the Southern Hemisphere, both mid- and high-latitude regions experienced a decrease on the day of the magnetic storm, with less than 40.0% in mid-latitude regions and less than 20.0% in high-latitude regions. (3) The NeQuick G model showed a significant decrease in the day of magnetic storms in the middle and high latitudes of the southern hemisphere, but no significant change in the other latitudes; the four models showed a certain degree of decrease in the correction rate on the day of magnetic storms in the global region. In summary, the correction performance analysis of the BKlob, GKlob, BDGIM, and NeQuick G models during magnetic storms are analyzed as follows. (1) The BDGIM model can still maintain a good correction performance at low latitudes, and the correction performance at high latitudes is greatly affected, with a correction rate less than 20.0%, which is Beacuse the BDGIM model is an improved spherical harmonic function and updates a set of parameters every two hours, which can better reflect the magnetic storm changes, and the correction performance at high latitudes is poor, probably because the model parameter updates are limited by the domestic distribution of tracking stations [13, 14]; (2) BKlob model and GKlob model have less influence on the correction performance at low latitudes, and the correction rate is less than 10.0% during magnetic storms at high latitudes. The poor correction

Analysis of the Performance of Broadcast Ionospheric Model

263

Fig. 6. Model of four broadcast ionospheres during the Day: 305–313 magnetic storms different latitude bands in the northern and southern hemispheres, global correction rate

performance of BKlob and GKlob models during magnetic storms is mainly limited by the models themselves; because the BKlob model parameters are updated every two hours, the GKlob model parameters are updated once a day, the BKlob model performs slightly better than the GKlob model [15]; (3) The NeQucik G model has a better correction performance during magnetic storms, but the correction rate decreases to some extent in the middle and high latitudes of the southern hemisphere in the case of geomagnetic storms. The better correction performance of the NeQuick G model may be related to the introduction of the Az parameter, which allows the NeQuick G model to broadcast parameters on a daily like the GKlob model and takes into account the effects of solar activity and geomagnetic variations on the ionosphere [16].

5 Analysis of Broadcast Ionospheric Performance During Solar Flares 5.1 Performance Analysis of the X1.3 Level Flare Broadcast Ionosphere Model for March 31, 2022 The four ionospheric model correction rates for different latitudinal belts, the AsiaPacific region and the globe are calculated for the period from March 30 to April 4 as shown in Fig. 7.

264

X. Yan et al.

(1) BDGIM model has no significant changes during the solar flare occurrence except for the high latitudes in the southern hemisphere, and its correction accuracy is good, while the high latitudes in the southern hemisphere show some fluctuations; BKlob and GKlob models have some decreases in the middle and high latitudes in the southern hemisphere, and the correction rates in the middle and high latitudes in the southern hemisphere show large fluctuations, while the correction rates in other latitudinal bands do not show significant changes.The NeQuick G model shows a slowly decreasing trend in the correction rate at middle and high latitudes in the Northern Hemisphere, and its correction rate remains between 20.0% and 30.0% during the flare period, with no significant change in the correction rate at low latitudes in the Northern and Southern Hemispheres, a decreasing trend at mid-latitudes in the Southern Hemisphere, and significant fluctuations at high latitudes; (2) NeQuick G model, whose correction rate decreases significantly in all latitudinal bands, with the correction rate around 20.0%, and the global regional NeQuick G model correction rate decreases significantly.

Fig. 7. Four broadcast ionospheric models during the annual cumulative solar flare of 89–94 Different latitudinal bands in the northern and southern hemispheres, global and Asia-Pacific regional correction rates

Analysis of the Performance of Broadcast Ionospheric Model

265

5.2 Analysis of the Broadcast Ionospheric Model Performance of the X1.1 and X2.2 Level Flares on April 17 and 22, 2022 The four ionospheric model correction rates for different latitudinal belts, the AsiaPacific region and the globe are calculated for the period from April 15 to April 23 as shown in Fig. 8. (1) The BDGIM model shows some fluctuations in the high latitudes of the southern hemisphere. The BKlob and GKlob models show large fluctuations in the correction rates in the middle and high latitudes of the southern hemisphere; (2) The NeQuick G model shows a slow decreasing trend in the correction rates in the middle and high latitudes of the northern hemisphere, and its correction rates remain between 20.0 and 30.0% during the flare period, and the correction rates in the low latitudes of the northern and southern hemispheres There is no significant change in the correction rate at low latitudes in the northern and southern hemispheres, a decreasing trend at mid-latitudes in the southern hemisphere, and a significant fluctuation at high latitudes.

Fig. 8. Four broadcast ionosphere models during the annual cumulative solar flare of 105–113 Correction rates for different latitudinal belts in the northern and southern hemispheres, globally, and in the Asia-Pacific region

Summarizing the results of the BKlob, GKlob, BDGIM, and NeQuick G model correction performance analysis during the flare period, the preliminary analysis is as follows.

266

X. Yan et al.

(1) The correction rate of the BDGIM broadcast ionospheric model is almost unaffected in all regions, but there are certain fluctuations in high latitudes, and the correction performance is poor in high latitudes, probably because the update of model parameters is limited by the domestic distribution of tracking stations; (2) The BKlob and GKlob models are affected in the middle and high latitudes, and the lowest can be reduced to less than 30.0%, while the NeQuick model is affected in almost all regions, and the correction rate is roughly reduced to between 20.0 and 40.0%; for the BKlob and GKlob models, the correction performance is more affected in the southern hemisphere, which is related to its own poor model correction effect; (3) The poor correction performance of the NeQuick G model is probably due to the inability of the Az parameter to respond accurately to the anomalous changes in ionospheric electron content caused by flares, resulting in a decrease in correction performance.

6 Comparison of the Effects of Geomagnetic Storms and Flares on the Performance of Broadcast Ionospheric Models To better analyze the influence of magnetic storms and solar flares on the performance of their broadcast ionospheric models, the polar values of their correction rates during the occurrence of magnetic storms and solar flares were calculated and compared with their fluctuations as shown in Fig. 9. From Fig. 9, the conclusions can be made as follows: In general, geomagnetic storms have a greater impact on the ionosphere than flares, and the BDGIM model has better adaptation during flares, while the NeQuick G model is more adaptable during magnetic storms, e.g., the BDGIM model has a range of extremes from 5.88 to 15.89% during flares, while the NeQuick G model has a range of 4.94 to 18.08% during magnetic storms; each of the two models adapts to different events and has approximately equal impact on them. The BKlob and GKlob models have almost equal adaptability during magnetic storms and flares, with differences in individual latitudinal bands.

7 Conclusion In this paper, the correction performance of the broadcast ionosphere model in different regions during magnetic storms and flares is analyzed with the CODE GIM product as the benchmark, and draw relevant conclusions. (1) During magnetic storms, the NeQuick G model performs the best, and the BDGIM model is basically comparable to the BDGIM model in terms of correction performance at middle and low latitudes, and outperforms the BDGIM model at high latitudes, and the BKlob model is slightly better than the GKlob model in terms of correction performance, but worse than the BDGIM model; (2) During solar flares, the BDGIM model performs slightly better, and the correction performance at low and middle latitudes is During solar flares, the BDGIM model performs slightly better, with stable correction performance at low and mid-latitudes and large fluctuations at high latitudes; the NeQuick G model performs worse, with a decrease in correction performance in all regions; the BKlob and GKlob models outperform the NeQuick G model in the northern hemisphere, with large

Analysis of the Performance of Broadcast Ionospheric Model

267

Fig. 9. Fluctuations under the occurrence of magnetic storms and solar flares for four broadcast ionospheric models

fluctuations in the southern hemisphere; (3) Magnetic storms cause more effects on the BDGIM model, flares cause more effects on the NeQuick G model, while the BKlob and GKlob models both cause comparable effects. The NeQuick G model introduces the Az parameter, which takes into account the influence of geomagnetic changes on the ionosphere, but may not accurately reflect the changes in ionospheric electron content caused by flares. The BDGIM model is an improved spherical harmonic function, which can better reflect the ionospheric changes caused by solar activity and geomagnetic changes at low and middle latitudes, but the update of model parameters is limited by the distribution of domestic tracking stations, resulting in poor correction performance at high latitudes; the BKlob and GKlob models themselves have poor correction effects, so it is difficult to better reflect the ionospheric changes caused by solar activity and geomagnetic changes. In this paper, the effects of four broadcast ionospheric models during magnetic storms and solar flares in the past three years from 2020 to 2022 are evaluated, but the types and numbers of events are relatively small, and subsequent experiments on a large number of events are still needed to obtain more general conclusions and the mechanisms of their effects and their mathematical relationships need further in-depth study.

References 1. Wang N, Yuan Y, Li Z et al (2016) Improvement of Klobuchar model for GNSS singlefrequency ionospheric delay corrections. Adv Space Res 57(7):1555–1569 2. Yunbin Y, Xingliang HUO, Baocheng Z (2017) Research progress of precise models and correction for GNSS ionospheric delay in China over recent years. Acta geodaetica et cartographica sinica 46(10):1364

268

X. Yan et al.

3. Jun-chen XUE, Shu-li SONG, Wen-yao ZHU et al (2011) Reliability study on the measured ionospheric variation and satellite hardware delay of regional GPS network. Acta Astronom Sinica 52(04):310–321. https://doi.org/10.15940/j.cnki.0001-5245.2011.04.010 4. Yongxing ZHU, Shusen TAN, Xia REN, Xiaolin JIA (2020) Accuracy Analysis of GNSS Global Broadcast Ionospheric Model. Geomatics and Information Science of Wuhan University 45(5):768–775. https://doi.org/10.13203/j.whugis20180439 5. Yue M, Yongxing Z, Xiaoyong S (2020) Accuracy analysis of broadcast ionosphere model of global navigation satellite system. jgg 40(9):888–891 6. Jialong LIU, Xiaolin JIA, Qingrui TAO et al Accuracy evaluation of multi-GNSS broadcast ionospheric model. Bull Surv Mapp (1): 66 7. Wang G, Yin Z, Hu Z et al (2021) Analysis of the BDGIM performance in BDS single point positioning. Remote Sensing 13(19):3888 8. Yi J, Chen L, Fangyuan L et al (2020) Analysis of anti-disturbance capability of ionospheric correction for BDS. jgg, 40(9):892–897 9. Zhu Y, Tan S, Zhang Q et al (2019) Accuracy evaluation of the latest BDGIM for BDS-3 satellites. Adv Space Res 64(6):1217–1224 10. Xiude C, Xiaolin J, Yongxing Z et al (2017) Analysis the accuracy of different ionospheric grid products. jgg, 37(8):849–855 11. Zi-Shen LI, Ning-Bo WANG, Min LI et al (2017) Evaluation and analysis of the global ionospheric TEC map in the frame of international GNSS services. Chinese Journal of Geophysics (in Chinese) 60(10):3718–3729. https://doi.org/10.6038/cjg20171003 12. Liu J, Hongzhou C, Liu C et al (2011) Analysis based in code gim of ionospheric tecanomaly before earthquake. jgg, 31(6):39–42 13. Yuan Y et al (2019) The BeiDou global broadcast ionospheric delay correction model (BDGIM) and its preliminary performance evaluation results. Navigation 66(1):55–69 14. Yunbin Y, Min L, Xingliang H et al (2021) Research on performance of BeiDou global broadcast ionospheric delay correction model(BDGIM) of BDS-3. .Acta Geodaetica et Cartographica Sinica. 50(4):436–447. https://doi.org/10.11947/j.AGCS.2021.200421 15. Qiang Z, Qile Z, Hongping Z, Zhigang H, Yue W (2014) Evaluation on the precision of klobuchar model for beidou navigation satellite system. Geomat Inf Sci Wuhan Univ. 39(2):142–146. https://doi.org/10.13203/j.whugis20120716 16. Ningbo W, Yunbin Y, Zishen L et al (2017) Performance analysis of different nequick ionospheric mode lparameters. Acta Geod Cartogr Sin. 46(4):421–429. https://doi.org/10.11947/ j.AGCS.2017.20160400

Global Instantaneous Centimeter-Level Multi-constellation and Multi-frequency Precise Point Positioning with Cascading Ambiguity Resolution Lizhong Qu(B) , Luping Wang, Haoyu Wang, Wei Jiang, and Yiwei Du School of Geomatics and Urban Spatial Informatics, Engineering and Architecture, Beijing University of Civil, 1 Zhanlanguan Road, Beijing 100044, China [email protected]

Abstract. With the progresses of Global Navigation Satellite System (GNSS), a number of satellites transmitting multi-frequency signals contribute precise point positioning (PPP). Global instantaneous or single-epoch centimeter-level PPP may be reached because global single-epoch narrow-lane (NL) ambiguity resolution (AR) may be possible with many wide-lane (WL) ambiguities being fixed instantaneously, which can improve the accuracy of the instant NL ambiguities. In this article, the cascading AR (CAR) method was extended to the GPS, Galileo, and BDS3 all-frequency signals. The performance of instantaneous PPP was investigated with global public stations. The results showed that attributed to the additional frequency observations, the instant positioning accuracy improved substantially. On a global scale, the instant horizontal and up positioning accuracy improved from about 20 and 60 cm, respectively, for the dual-frequency PPP-CAR to about 6 and 20 cm, respectively, for the multi-frequency PPP-CARs. These results are quite encouraging for global autonomous driving cars because better positioning accuracy is expected once the multi-constellation and multi-frequency signals are integrated with inertial sensors. Keywords: Instantaneous centimeter-level · Precise point positioning · Multi-constellation and multi-frequency · Cascading ambiguity resolution

1 Introduction GNSS (Global Navigation Satellite System) precise positioning technology can provide users’ absolute coordinates, which plays an indispensable role in automatic driving [1]. To achieve global instantaneous high-precision positioning, such as centimeter-level accuracy, single-epoch ambiguity resolution (AR) is a prerequisite [2]. GNSS shortbaseline RTK technology eliminates atmosphere or instrumental errors, and single-epoch AR is easy [3]. However, with the increase in the distance between the rover and reference stations, the spatial correlation decreases, which results in the short operating distance ( → | 2 S1/2 , F = 0 > is chosen as the clock transition with the frequency of 12.6 GHz.

(a)

(b) Fig. 1. a Relevant energy level of 171 Yb+ . b Relevant energy level of 113 Cd+

Cadmium ions are cooled and detected by 214.5nm through the cycle transition between | 2 S1/2 , F = 1, mF = 1and | 2 P3/2 , F = 2, mF = 2. The pump light can be obtained by shifting the frequency of the cooling light with acousto optic modulators (AOMs) [2] since the hyperfine splitting frequency of 2 P3/2 is only 800 MHz.

High-Performance Microwave Frequency

283

The ion trap we used is a three-stage linear Paul ion trap composed of four cylindrical electrodes, which is well designed to be close to the ideal four-stage potential ratio [2]. A titanium sublimation pump in addition to the ion pump was used to further improve the vacuum degree, which was always better than 10–9 Pa during the experiment. We measure the hyperfine level transition of the ground state of 171 Yb+ and 113 Cd+ using the Ramsey detection method. Based on Ramsey fringe with high signal-to-noise ratio (SNR), we conducted closed-loop locking operation. Furthermore, systematic frequency shifts and their uncertainties were carefully √ evaluated. The short-term stability the short-term stability of of the 171 Yb+ microwave clock is 8.5 × 10−13 / τ [3]. And √ the 113 Cd+ microwave clock is obtained to be 4.2 × 10−13 / τ [4], nearly two times higher than that of the previous system [2] and closed to the short-term stability limit. The evaluation of the systematic frequency uncertainties of the 171 Yb+ system [3] and 113 Cd+ system [4] is listed in Table 1. The second order Zeeman frequency shifts (SOZS) and the second order Doppler shift (SODS) are the main factors limiting the frequency accuracy of ion microwave frequency standard. Table 1. Estimated systematic frequency uncertainties of the 171 Yb+ and 113 Cd+ microwave ion clocks Shift

Uncertainty (10–15 ) of 171 Yb+ system

Uncertainty (10–15 ) of 113 Cd+ system

SOZS

3

3

SODS