Acta Geodaetica et Cartographica Sinica ›› 2025, Vol. 54 ›› Issue (12): 2153-2167.doi: 10.11947/j.AGCS.2025.20250057
• Geodesy and Navigation • Previous Articles Next Articles
Jian CHEN1,2,3(
), Jiahui WANG1,2,3, Xingwang ZHAO1,2,3, Chao LIU1,2,3(), Chunyang LIU1,2,3, Xuexiang YU1,2,3
Received:2025-02-14
Revised:2025-10-28
Online:2026-01-15
Published:2026-01-15
Contact:
Chao LIU
E-mail:cj_19930815@163.com;chliu1@aust.edu.cn
About author:CHEN Jian (1993—), male, PhD, lecturer, majors in multi-frequency multi-system GNSS satellite precise positioning. E-mail: cj_19930815@163.com
Supported by:CLC Number:
Jian CHEN, Jiahui WANG, Xingwang ZHAO, Chao LIU, Chunyang LIU, Xuexiang YU. Single-epoch RTK positioning optimization method based on BDS-3/Galileo multi-frequency ionosphere-reduced combinations[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(12): 2153-2167.
Tab. 1
Triple-frequency EWL/WL phase combination properties from BDS-3 and Galileo"
| 不同系统 | 组合系数 | λ（3）/m | β（3） | σTL/cycle | |||
|---|---|---|---|---|---|---|---|
| a1 | a2 | a3 | d≤20 km | 20<d≤100 km | |||
| BDS-3（B1C，B2a，B3I） | 0 | －1 | 1 | 3.256 1 | －1.663 1 | 0.063 2 | 0.163 9 |
| 1 | 2 | －3 | 2.442 1 | －0.610 2 | 0.158 8 | 0.175 4 | |
| 1 | 1 | －2 | 1.395 5 | －1.061 4 | 0.113 9 | 0.252 7 | |
| Galileo（E1，E5a，E5b） | 0 | －1 | 1 | 9.768 4 | －1.747 7 | 0.056 9 | 0.077 8 |
| 1 | 3 | －4 | 1.085 4 | －1.157 6 | 0.213 3 | 0.381 4 | |
| 1 | 2 | －3 | 0.976 8 | －1.216 6 | 0.165 8 | 0.404 7 | |
Tab. 2
Quad-frequency EWL/WL phase combination properties from BDS-3 and Galileo"
| 不同系统 | 组合系数 | λ（4）/m | β（4） | σTL/cycle | ||||
|---|---|---|---|---|---|---|---|---|
| a1 | a2 | a3 | a4 | d≤20 km | 20<d≤100 km | |||
| BDS-3（B1C，B1I，B2a，B3I） | 1 | －1 | 0 | 0 | 20.932 3 | －1.009 2 | 0.074 0 | 0.075 4 |
| 0 | 0 | －1 | 1 | 3.256 1 | －1.663 1 | 0.0632 | 0.163 9 | |
| －1 | 2 | 2 | －3 | 3.185 4 | －0.488 7 | 0.189 6 | 0.195 1 | |
| 0 | 1 | 2 | －3 | 2.764 6 | －0.557 5 | 0.158 4 | 0.169 5 | |
| 1 | 0 | 2 | －3 | 2.442 1 | －0.610 2 | 0.158 8 | 0.175 4 | |
| Galileo（E1，E6，E5a，E5b） | 0 | 0 | －1 | 1 | 9.768 4 | －1.747 7 | 0.056 9 | 0.077 8 |
| 0 | 1 | 1 | －2 | 7.326 3 | －1.503 0 | 0.099 8 | 0.116 9 | |
| 0 | 1 | 0 | －1 | 4.186 5 | －1.607 9 | 0.061 8 | 0.129 4 | |
| 1 | －3 | 2 | 0 | 3.256 1 | －0.303 5 | 0.159 1 | 0.161 7 | |
| 1 | －2 | 1 | 0 | 1.542 4 | －1.012 1 | 0.112 7 | 0.225 0 | |
Tab. 3
Penta-frequency EWL/WL phase combination properties from BDS-3 and Galileo"
| 不同系统 | 组合系数 | λ（5）/m | β（5） | σTL/cycle | |||||
|---|---|---|---|---|---|---|---|---|---|
| a1 | a2 | a3 | a4 | a5 | d≤20 km | 20<d≤100 km | |||
| BDS-3（B1C，B1I，B2a，B2b，B3I） | 1 | －1 | 0 | 0 | 0 | 20.932 3 | －1.009 2 | 0.074 0 | 0.075 4 |
| 0 | 0 | －1 | 1 | 0 | 9.768 4 | －1.747 7 | 0.056 9 | 0.077 8 | |
| 0 | 0 | 0 | －1 | 1 | 4.884 2 | －1.620 8 | 0.060 8 | 0.115 5 | |
| 0 | 0 | －1 | 0 | 1 | 3.256 1 | －1.663 1 | 0.063 2 | 0.163 9 | |
| 1 | －1 | －1 | 0 | 1 | 2.817 8 | －1.575 1 | 0.098 0 | 0.192 4 | |
| 0 | 1 | 2 | 0 | －3 | 2.764 6 | －0.557 5 | 0.158 4 | 0.169 5 | |
| 0 | 1 | 1 | 1 | －3 | 2.154 8 | －0.820 0 | 0.149 6 | 0.187 5 | |
| Galileo（E1，E5a，E5b，E5，E6） | 0 | －1 | 0 | 1 | 0 | 19.536 8 | －1.770 2 | 0.056 0 | 0.062 1 |
| 0 | 0 | 1 | －1 | 0 | 19.536 8 | －1.725 2 | 0.056 8 | 0.062 5 | |
| 0 | －1 | 1 | 0 | 0 | 9.768 4 | －1.747 7 | 0.056 9 | 0.077 8 | |
| 0 | 0 | －1 | 0 | 1 | 4.186 5 | －1.607 9 | 0.061 8 | 0.129 4 | |
| 1 | 2 | 0 | 0 | －3 | 3.256 1 | －0.303 5 | 0.159 1 | 0.161 7 | |
| 1 | 1 | 0 | 1 | －3 | 2.791 0 | －0.513 1 | 0.149 5 | 0.159 3 | |
| 1 | 1 | 1 | 0 | －3 | 2.442 1 | －0.664 6 | 0.150 0 | 0.170 5 | |
| 1 | 1 | －1 | 1 | －2 | 1.674 6 | －0.951 0 | 0.125 0 | 0.209 9 | |
| 1 | 1 | 0 | 0 | －2 | 1.542 4 | －1.012 1 | 0.112 7 | 0.225 0 | |
Tab. 4
Properties of NL combinations for BDS-3 and Galileo in triple-quad-penta-frequency observations"
| 不同系统 | 组合系数 | λ（n）/m | β（n） | σTL/cycle | |||||
|---|---|---|---|---|---|---|---|---|---|
| a1 | a2 | a3 | a4 | a5 | σ∇ΔI1=5 cm | σ∇ΔI1=25 cm | |||
| BDS-3 | 1 | 0 | 0 | 0.190 3 | 1 | 0.273 3 | 1.321 6 | ||
| 4 | －3 | 0 | 0.108 1 | －0.009 9 | 0.258 9 | 0.337 9 | |||
| 2 | 2 | －3 | 0 | 0.109 3 | 0.000 5 | 0.210 9 | 0.300 3 | ||
| 2 | 2 | －3 | 0 | 0 | 0.109 3 | 0.000 5 | 0.210 9 | 0.300 3 | |
| 2 | 3 | 0 | －1 | －3 | 0.106 3 | －0.001 9 | 0.249 8 | 0.332 8 | |
| 3 | 1 | －1 | －3 | 1 | 0.108 7 | －0.001 9 | 0.233 0 | 0.317 0 | |
| Galileo | 1 | 0 | 0 | 0.190 3 | 1 | 0.273 3 | 1.321 6 | ||
| 4 | －2 | －1 | 0.109 3 | 0.009 6 | 0.246 5 | 0.326 8 | |||
| 4 | －3 | 0 | 0.108 1 | －0.009 9 | 0.258 9 | 0.337 9 | |||
| 4 | 0 | －2 | －1 | 0.109 3 | 0.009 6 | 0.246 5 | 0.326 8 | ||
| 4 | 0 | －3 | 0 | 0.108 1 | －0.009 9 | 0.258 9 | 0.337 9 | ||
| 4 | －3 | －1 | 1 | 0 | 0.108 7 | －0.000 3 | 0.264 7 | 0.340 9 | |
| 4 | －2 | 0 | －1 | 0 | 0.108 7 | －0.000 1 | 0.246 4 | 0.327 0 | |
| 4 | －1 | 1 | －3 | 0 | 0.108 7 | 0.000 2 | 0.265 3 | 0.341 4 | |
| 4 | 0 | －3 | －1 | 1 | 0.108 3 | －0.002 9 | 0.266 5 | 0.342 9 | |
Fig. 1
Flowchart of single-epoch RTK positioning using multi-frequency IR combinations applied to BDS-3 and Galileo"
Tab. 5
Baseline information"
| 数据集 | 基线 | 年积日 | 长度/km | 观测时长/h | 采样间隔/s | 接收机类型 | 位置 |
|---|---|---|---|---|---|---|---|
| A | STR1-TID1 | 2025-128 | 9.7 | 24 | 30 | SEPT POLARX5 | 澳大利亚 堪培拉 |
| B | LYSK-NJGY | 2025-199 | 29.9 | 24 | 30 | NET20 PLUS/HEMI_DF5r | 中国 南京 |
| C | CEBR-VILL | 2025-14 | 35.3 | 24 | 30 | SEPT POLARX5 | 西班牙 阿维拉•马德里 |
Tab. 6
Scheme for selecting combination observables in BDS-3 and Galileo TCAR"
| 系统 | 方案 | TC-Ⅰ | TC-Ⅱ | TC-Ⅲ | |
|---|---|---|---|---|---|
| BDS-3 | 1 | （0，－1，1） | （1，1，－2） | （1，0，0） | （4，－3，0） |
| 2 | （1，2，－3） | ||||
| Galileo | 3 | （0，－1，1） | （1，2，－3） | （1，0，0） | （4，－2，－1） |
| 4 | （1，3，－4） | ||||
Tab. 7
Ambiguity fixing success rates of different combination for BDS-3 and Galileo TCAR"
| 系统 | 数据集 | 方案 | TC-Ⅰ | TC-Ⅱ | BF | IR |
|---|---|---|---|---|---|---|
| BDS-3 | A | 1 | 100 | 100 | 97.92 | 99.97 |
| 2 | 100 | 100 | 97.92 | 99.97 | ||
| B | 1 | 100 | 100 | 79.93 | 90.69 | |
| 2 | 100 | 100 | 79.93 | 90.69 | ||
| C | 1 | 100 | 99.73 | 73.30 | 80.52 | |
| 2 | 100 | 99.20 | 73.19 | 80.52 | ||
| Galileo | A | 3 | 100 | 90.10 | 99.44 | 99.48 |
| 4 | 100 | 83.58 | 99.27 | 99.41 | ||
| C | 3 | 100 | 64.19 | 73.97 | 81.61 | |
| 4 | 100 | 61.06 | 74.47 | 81.69 |
Tab. 8
Ambiguity fixing success rate of BDS-3/Galileo TCAR"
| 数据集 | TC-Ⅰ | TC-Ⅱ | BF | IR |
|---|---|---|---|---|
| A | 99.97 | 99.97 | 99.97 | 99.97 |
| C | 100 | 97.22 | 95.35 | 97.64 |
Tab. 9
Scheme for selecting combination observables in BDS-3 and Galileo QCAR"
| 系统 | 方案 | QC-Ⅰ | QC-Ⅱ | QC-Ⅲ | QC-Ⅳ | |
|---|---|---|---|---|---|---|
| BDS-3 | 5 | （0，0，－1，1） | （0，1，2，－3） | |||
| 6 | （1，－1，0，0） | （0，0，－1，1） | （1，0，2，－3） | （1，0，0，0） | （2，2，－3，0） | |
| 7 | （0，0，－1，1） | （－1，2，2，－3） | ||||
| Galileo | 8 | （0，1，1，－2） | （1，－3，2，0） | |||
| 9 | （0，0，－1，1） | （0，1，0，－1） | （1，－3，2，0） | （1，0，0，0） | （4，0，－2，－1） | |
| 10 | （0，1，0，－1） | （1，－2，1，0） | ||||
Tab. 10
Ambiguity fixing success rates of different combination schemes for BDS-3 and Galileo QCAR"
| 系统 | 数据集 | 方案 | QC-Ⅰ | QC-Ⅱ | QC-Ⅲ | BF | IR |
|---|---|---|---|---|---|---|---|
| BDS-3 | A | 5 | 100 | 100 | 100 | 99.97 | 99.97 |
| 6 | 100 | 100 | 100 | 99.97 | 99.97 | ||
| 7 | 100 | 100 | 100 | 99.97 | 99.97 | ||
| B | 5 | 100 | 100 | 100 | 93.35 | 98.49 | |
| 6 | 100 | 100 | 100 | 93.35 | 98.49 | ||
| 7 | 100 | 100 | 100 | 93.35 | 98.49 | ||
| C | 5 | 100 | 100 | 99.88 | 82.35 | 86.01 | |
| 6 | 100 | 100 | 99.76 | 82.35 | 86.01 | ||
| 7 | 100 | 100 | 99.92 | 82.35 | 86.01 | ||
| Galileo | A | 8 | 100 | 100 | 99.93 | 99.90 | 99.93 |
| 9 | 100 | 100 | 99.93 | 99.90 | 99.93 | ||
| 10 | 100 | 100 | 100 | 99.90 | 99.93 | ||
| C | 8 | 100 | 100 | 97.44 | 85.38 | 89.79 | |
| 9 | 100 | 100 | 99.22 | 85.56 | 89.36 | ||
| 10 | 100 | 100 | 99.72 | 85.56 | 89.40 |
Tab. 11
Ambiguity fixing success rate of BDS-3/Galileo QCAR"
| 数据集 | QC-Ⅰ | QC-Ⅱ | QC-Ⅲ | BF | IR |
|---|---|---|---|---|---|
| A | 100 | 100 | 100 | 99.97 | 99.97 |
| C | 100 | 100 | 99.24 | 96.35 | 97.74 |
Fig. 2
Positioning errors and RMS values for NL combinations in the database A"
Tab. 12
The improvement ratio of IR combinations over BF in positioning accuracy under three modes"
| 数据集 | 方向 | BDS-3 | Galileo | BDS-3/Galileo |
|---|---|---|---|---|
| A | N | 30.00 | 44.25 | 45.65 |
| E | 37.26 | 31.88 | 30.61 | |
| U | 28.46 | 32.09 | 36.49 | |
| 水平 | 32.33 | 40.63 | 41.99 | |
| 三维 | 29.01 | 33.44 | 37.45 | |
| C | N | 30.85 | 35.63 | 65.14 |
| E | 20.56 | 22.57 | 72.91 | |
| U | 2.22 | 13.04 | 33.49 | |
| 水平 | 24.16 | 27.15 | 70.73 | |
| 三维 | 8.72 | 16.23 | 45.49 |
Tab. 13
Scheme for selecting combination observables in BDS-3 and Galileo PCAR"
| 系统 | 方案 | PC-Ⅰ | PC-Ⅱ | PC-Ⅲ | PC-Ⅳ | PC-Ⅴ |
|---|---|---|---|---|---|---|
| BDS-3 | 11 | （0，0，－1，1，0） | （1，－1，0，0，0） | （0，0，0，－1，1） | （0，1，2，0，－3） | （1，0，0，0，0）（2，2，－3，0，0） |
| 12 | （1，－1，0，0，0） | （0，0，－1，0，1） | （0，1，2，0，－3） | |||
| 13 | （1，－1，0，0，0） | （0，0，0，－1，1） | （0，1，1，1，－3） | |||
| 14 | （0，0，－1，0，1） | （0，1，1，1，－3） | （1，－1，－1，0，1） | |||
| Galileo | 15 | （0，－1，0，1，0） | （0，0，1，－1，0） | （1，1，－1，1，－2） | （1，2，0，0，－3） | （1，0，0，0，0）（4，－2，0，－1，0） |
| 16 | （0，0，－1，0，1） | （1，1，1，0，－3） | （1，1，－1，1，－2） | |||
| 17 | （0，－1，1，0，0） | （0，0，－1，0，1） | （1，1，0，1，－3） | （1，1，0，0，－2） |
Tab. 14
Comparison of ambiguity fixing success rates for BDS-3 PCAR under different schemes"
| 数据集 | 方案 | PC-Ⅰ | PC-Ⅱ | PC-Ⅲ | PC-Ⅳ | BF | IR |
|---|---|---|---|---|---|---|---|
| A | 11 | 100 | 100 | 100 | 100 | 99.51 | 99.55 |
| 12 | 100 | 100 | 100 | 100 | 99.51 | 99.55 | |
| 13 | 100 | 100 | 100 | 100 | 99.51 | 99.55 | |
| 14 | 100 | 100 | 100 | 100 | 99.51 | 99.55 | |
| B | 11 | 100 | 100 | 100 | 100 | 94.96 | 98.42 |
| 12 | 100 | 100 | 100 | 100 | 94.96 | 98.42 | |
| 13 | 100 | 100 | 100 | 100 | 94.96 | 98.42 | |
| 14 | 100 | 100 | 100 | 99.96 | 94.96 | 98.42 | |
| C | 11 | 100 | 100 | 100 | 99.88 | 82.71 | 86.82 |
| 12 | 100 | 100 | 100 | 99.92 | 82.71 | 86.98 | |
| 13 | 100 | 100 | 100 | 99.76 | 82.67 | 86.78 | |
| 14 | 100 | 100 | 99.88 | 99.36 | 82.71 | 86.82 |
Tab. 15
Comparison of ambiguity fixing success rates for Galileo PCAR under different schemes"
| 数据集 | 方案 | PC-Ⅰ | PC-Ⅱ | PC-Ⅲ | PC-Ⅳ | BF | IR |
|---|---|---|---|---|---|---|---|
| A | 15 | 100 | 100 | 100 | 100 | 99.97 | 100 |
| 16 | 100 | 100 | 99.97 | 100 | 99.97 | 100 | |
| 17 | 100 | 100 | 99.93 | 100 | 99.97 | 100 | |
| C | 15 | 100 | 100 | 99.29 | 99.40 | 86.94 | 90.15 |
| 16 | 100 | 100 | 99.18 | 99.86 | 86.94 | 90.07 | |
| 17 | 100 | 100 | 99.22 | 99.75 | 86.98 | 90.11 |
Tab. 16
Comparison of ambiguity fixing success rates for BDS-3/Galileo PCAR under different schemes"
| 数据集 | PC-Ⅰ | PC-Ⅱ | PC-Ⅲ | PC-Ⅳ | BF | IR |
|---|---|---|---|---|---|---|
| A | 100 | 100 | 100 | 100 | 99.93 | 99.97 |
| C | 100 | 100 | 100 | 99.86 | 97.29 | 98.30 |
Fig. 3
Positioning root mean square of the NL combinations under three solution modes"
Fig. 4
The changing trend of the geomagnetic activity index over 30 consecutive days"
Tab. 17
Performance comparison of BDS-3/Galileo PCAR ambi-guity resolution under active ionospheric conditions"
| 数据集 | PC-Ⅰ | PC-Ⅱ | PC-Ⅲ | PC-Ⅳ | BF | IR |
|---|---|---|---|---|---|---|
| A | 100 | 100 | 99.55 | 100 | 99.90 | 100 |
| C | 100 | 100 | 96.42 | 98.72 | 99.76 | 99.83 |
Tab. 18
Three-dimensional positioning root mean square of BDS-3/Galileo PCAR in ionospheric active regions"
| 数值集 | BF/cm | IR/cm | 提升/（%） |
|---|---|---|---|
| A | 4.45 | 1.64 | 63.15 |
| C | 6.54 | 3.51 | 46.33 |
Fig. 5
Positioning availability of three modes in different cutoff elevation angles"
Tab. 19
Single-epoch ambiguity resolution success rates for the three modes across varying cut-off elevation angles"
| 窄巷组合 | 截止高度角 | BDS-3 | Galileo | BDS-3/Galileo |
|---|---|---|---|---|
| BF | 15° | 82.35 | 85.56 | 96.35 |
| 20° | 83.71 | 86.26 | 97.92 | |
| 25° | 82.15 | 86.43 | 96.74 | |
| 30° | 89.44 | 83.66 | 93.75 | |
| IR | 15° | 86.01 | 89.40 | 98.40 |
| 20° | 86.27 | 88.80 | 98.37 | |
| 25° | 86.38 | 89.42 | 98.68 | |
| 30° | 92.96 | 87.41 | 97.57 |
Tab. 20
The RMS values of NL positioning across varying cut-off elevation angles"
| 窄巷组合 | 截止高度角 | BDS-3 | Galileo | BDS-3/Galileo | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| N | E | U | N | E | U | N | E | U | ||
| BF | 15° | 1.88 | 2.48 | 4.51 | 1.74 | 2.26 | 5.06 | 1.75 | 2.99 | 4.18 |
| 20° | 1.87 | 2.40 | 4.37 | 1.82 | 2.34 | 4.98 | 1.79 | 2.85 | 4.49 | |
| 25° | 1.73 | 1.99 | 3.70 | 1.94 | 2.43 | 5.15 | 1.87 | 2.67 | 4.60 | |
| 30° | 1.23 | 1.73 | 3.08 | 2.09 | 2.14 | 4.78 | 1.84 | 2.49 | 4.87 | |
| IR | 15° | 1.30 | 1.97 | 4.41 | 1.12 | 1.75 | 4.40 | 0.61 | 0.81 | 2.78 |
| 20° | 1.18 | 1.84 | 4.14 | 1.10 | 1.79 | 4.49 | 0.55 | 0.56 | 2.55 | |
| 25° | 1.35 | 1.89 | 4.24 | 1.14 | 1.93 | 4.46 | 0.59 | 0.67 | 2.42 | |
| 30° | 1.36 | 1.58 | 3.88 | 1.54 | 1.63 | 3.94 | 0.68 | 0.79 | 2.64 | |
Fig. 6
Average root mean square of dataset C under four cut-off elevation angles"
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