# Analysis of Characteristics for Inter-System Bias on Multi-GNSS Undifferenced and Uncombined Precise Point Positioning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Undifferenced and Uncombined Observation Equations

#### 2.2. ISB Definition

#### 2.3. ISB Parameter Stochastic Model

## 3. Data Sets and Processing Strategies

## 4. Experimental Validation

#### 4.1. Analysis of Short- and Long-Term Time Characteristics of ISB

#### 4.1.1. Analysis of DBD Effect on Time Characteristics of the ISB

#### 4.1.2. Analysis of Short-Term Time Characteristics of the ISB

- (1)
- The $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$ and $IS{B}_{GBM}^{G-sys}$ values estimated are not the same because of the differences in data processing strategies used by different analysis centers. $IS{B}_{AC}^{GE}$, $IS{B}_{AC}^{GC2}$, and $IS{B}_{AC}^{GC3}$ values are different due to time system differences between GNSS systems and receiver hardware delays. Thus, in the short term, the ISB values are correlated with the receiver, GNSS system, the adoption of analysis center products.
- (2)
- For the $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$ results, where FLUC was ±0.25 ns, the monthly average short-term FLUC was ±0.20 ns, even $IS{B}_{AC}^{GE}$ was ±0.10 ns, which can be related to Galileo’s good signal quality. The $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$ short-term TC show similarity in variation for the same stations. Moreover, it is evident that the ISB TC values of the same GNSS system fall within the same magnitude range. Among the three analysis center products, which use GBM product stability as the worst, COM and WUM are comparable.
- (3)
- The short-term FLUC of $IS{B}_{AC}^{GE}$, $IS{B}_{AC}^{GC2}$, and $IS{B}_{AC}^{GC3}$ are not the same, but the TC values in the same magnitude. The $IS{B}_{AC}^{GE}$, with monthly average short-term ISB STD less than 0.02 ns and FLUC within ±0.07 ns, shows the best performance. $IS{B}_{AC}^{GC3}$ performs slightly worse than $IS{B}_{AC}^{GE}$, STD less than 0.03 ns and the FLUC is within ±0.10 ns. The $IS{B}_{AC}^{GC2}$ is the worst.

#### 4.1.3. Analysis of the Long-Term Time Characteristics of ISB

- (1)
- The RMS values of $IS{B}_{AC}^{GE}$, $IS{B}_{AC}^{GC2}$, and $IS{B}_{AC}^{GC3}$ are different, as well as the RMS values of $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$. Thus, the RMS of the ISBs on different stations, such as $IS{B}_{Sta1}^{GE}$, $IS{B}_{Sta2}^{GE}$, and $IS{B}_{Sta3}^{GE}$, indicates that the ISBs are correlated with receivers, GNSS systems, and adoption of analysis center products in the long term.
- (2)
- It is clear that the FLUC of $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$ are not the same, but the TC values between the three are in the same magnitude. The monthly average FLUCs of $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$ were 1.82 ns, 1.69 ns, and 2.58 ns, corresponding to average STDs of 0.59 ns, 0.47 ns, and 0.76 ns, respectively, where $IS{B}_{WUM}^{G-sys}$ performed the best, $IS{B}_{COM}^{G-sys}$ and $IS{B}_{WUM}^{G-sys}$ were comparable.
- (3)
- $IS{B}_{AC}^{GE}$, $IS{B}_{AC}^{GC2}$, and $IS{B}_{AC}^{GC3}$ long-term TC are not the same. Within 31 days, their monthly average FLUCs were 0.88 ns, 2.86 ns, and 2.06 ns, respectively. The overall ISB monthly average FLUC was 2.03 ns, and the corresponding monthly average STDs were 0.28 ns, 0.88 ns, and 0.59 ns, with an overall monthly average STD < 0.61 ns, $IS{B}_{AC}^{GE}$ fluctuating the smallest, $IS{B}_{AC}^{GC3}$ the second, and $IS{B}_{AC}^{GC2}$ performed the worst.

#### 4.2. Receiver and ISB Relationship Analysis

## 5. Conclusions

- (1)
- ISB is associated with the station receiver type, receiver antenna type, various analysis center products, and GNSS systems.
- (2)
- Variations in the short- and long-term TC of $IS{B}_{AC}^{GE}$, $IS{B}_{AC}^{GC2}$, and $IS{B}_{AC}^{GC3}$ are not the same. The short-term TC of $IS{B}_{COM}^{G-sys}$, $IS{B}_{WUM}^{G-sys}$, and $IS{B}_{GBM}^{G-sys}$ are similar, while not for the long-term. The short-term ISB time series performed better than the long-term time series.
- (3)
- The results of the ISB TC show little correlation between receiver type and receiver antenna. DBD effect on ISB can be ignored for the concussive day’s process.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**). The result without DBD correction at 31 stations (DOY 110-40, 2021). (

**b**). The result of DBD correction at 31 stations (DOY 110-40, 2021).

**Figure 3.**(

**a**). Time series of the ISB at stations AJAC, FFMJ, and PTGG (DOY 119, 2021). (

**b**). Daily ISB STD of station AJAC, FFMJ, and PTGG (DOY 110-140, 2021). (

**c**). Daily ISB FLUC of station AJAC, FFMJ, and PTGG (DOY 110-140, 2021).

Manufacturer | Number of Stations (Site Name) | |
---|---|---|

JAVAD | TRE_3 | 2 (POTS, ULAB) |

TRE_3 DELTA | 3 (FFMJ, GODN, WARN) | |

LEICA | GR30 | 2 (GENO, MATE) |

GR50 | 1 (AJAC) | |

TRIMBLE | NTR9 | 2 (TRO1, TLSE) |

ALLOY | 4 (BRST, CHPG, GANP, LMMF) | |

NTR10 | 1 (GOPE) | |

SEPTENTRIO | ASTERX4 | 2 (RIO2, TASH) |

POLARX5 | 8 (DGAR, DJIG, MDO1, MIZU, PTGG, SUTH, TOW2, YAR3) | |

POLARX5TR | 6 (BRUX, CEBR, HARB, KOUG, PARK, WTZS) | |

sum | 31 |

Options | Processing Strategies |
---|---|

Observation | UC observation |

Signal | BDS: B1, B3; GPS: L1, L2; GAL: E1, E5a |

Parameter estimation | EKF |

Observation interval | 30 s |

Weight distribution of observed values | Height angle model |

Elevation | 7° |

Satellite orbit | CODE, WHU, GFZ precise ephemeris |

Satellite clock | CODE, WHU, GFZ precise clock offset |

Phase center correction | IGS14.ATX |

PCV | GPS/Galileo |

Phase windup | Model correction |

Solid earth tide | |

Ocean load | |

Polar motion | |

Relativistic effect | |

Tropospheric delay | Model correction + random walk |

Ionospheric delay | Random walk |

ISB | White noise |

Receiver coordinates | Static, estimated as constants |

Receiver clock | White noise estimation |

Ambiguity | Estimated as float constants for each arc |

RMS/ns | ||||
---|---|---|---|---|

AC | SITE | $\mathit{I}\mathit{S}{\mathit{B}}_{}^{\mathit{G}\mathit{E}}$ | $\mathit{I}\mathit{S}{\mathit{B}}_{}^{\mathit{G}\mathit{C}2}$ | $\mathit{I}\mathit{S}{\mathit{B}}_{}^{\mathit{G}\mathit{C}3}$ |

COM | AJAC | 13.54 | 37.43 | 38.75 |

FFMJ | 6.38 | 16.65 | 7.95 | |

PTGG | 14.70 | 43.12 | 45.71 | |

WUM | AJAC | 15.50 | 42.06 | 45.69 |

FFMJ | 8.39 | 21.50 | 14.22 | |

PTGG | 16.31 | 46.60 | 52.23 | |

GBM | AJAC | 8.22 | 24.74 | 21.17 |

FFMJ | 1.05 | 44.69 | 52.18 | |

PTGG | 9.57 | 18.95 | 13.74 |

ISB STD/ns | ISB FLUC/ns | |||||||
---|---|---|---|---|---|---|---|---|

AC | COM | WUM | GBM | AVG | COM | WUM | GBM | AVG |

$IS{B}_{}^{GE}$ | 0.01 | 0.02 | 0.02 | 0.02 | 0.06 | 0.07 | 0.08 | 0.07 |

$IS{B}_{}^{GC2}$ | 0.04 | 0.03 | 0.04 | 0.04 | 0.15 | 0.12 | 0.13 | 0.13 |

$IS{B}_{}^{GC3}$ | 0.02 | 0.03 | 0.04 | 0.03 | 0.09 | 0.10 | 0.10 | 0.10 |

AVG | 0.03 | 0.03 | 0.04 | 0.03 | 0.11 | 0.11 | 0.13 | 0.11 |

ISB STD/ns | ISB FLUC/ns | |||||||
---|---|---|---|---|---|---|---|---|

AC | COM | WUM | GBM | AVG | COM | WUM | GBM | AVG |

$IS{B}_{}^{GE}$ | 0.14 | 0.12 | 0.57 | 0.28 | 0.45 | 0.45 | 1.74 | 0.88 |

$IS{B}_{}^{GC2}$ | 1.32 | 0.64 | 0.69 | 0.88 | 3.78 | 2.33 | 2.46 | 2.86 |

$IS{B}_{}^{GC3}$ | 0.40 | 0.56 | 0.80 | 0.59 | 1.45 | 2.08 | 2.64 | 2.06 |

AVG | 0.59 | 0.47 | 0.76 | 0.61 | 1.82 | 1.69 | 2.58 | 2.03 |

**Table 6.**Short-term ISB differences between station BRUX and other stations for 31 stations (DOY128, 2021).

ISB STD/ns | ISB FLUC/ns | |||||||
---|---|---|---|---|---|---|---|---|

AC | COM | WUM | GBM | AVG | COM | WUM | GBM | AVG |

$IS{B}_{}^{GE}$ | 0.0208 | 0.0205 | 0.0205 | 0.0206 | 0.1100 | 0.1220 | 0.1260 | 0.1193 |

$IS{B}_{}^{GC2}$ | 0.0351 | 0.0247 | 0.0256 | 0.0285 | 0.1250 | 0.1710 | 0.1840 | 0.1600 |

$IS{B}_{}^{GC3}$ | 0.0241 | 0.0224 | 0.0253 | 0.0239 | 0.0900 | 0.0919 | 0.0939 | 0.0919 |

AVG | 0.0267 | 0.0225 | 0.0238 | 0.0243 | 0.1083 | 0.1283 | 0.1346 | 0.1238 |

**Table 7.**Long-term ISB differences between station BRUX and other stations for 31 stations (DOY110-140, 2021).

ISB STD/ns | ISB FLUC/ns | |||||||
---|---|---|---|---|---|---|---|---|

AC | COM | WUM | GBM | AVG | COM | WUM | GBM | AVG |

$IS{B}_{}^{GE}$ | 0.1383 | 0.2122 | 0.2879 | 0.2128 | 0.5400 | 0.7680 | 0.7310 | 0.6797 |

$IS{B}_{}^{GC2}$ | 0.661 | 0.8643 | 0.9217 | 0.8157 | 1.9260 | 2.2100 | 2.2960 | 2.1440 |

$IS{B}_{}^{GC3}$ | 0.2035 | 0.208 | 0.2913 | 0.2343 | 0.6830 | 0.7610 | 0.8600 | 0.7680 |

AVG | 0.3343 | 0.4282 | 0.5003 | 0.4209 | 1.0497 | 1.2463 | 1.2957 | 1.1972 |

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## Share and Cite

**MDPI and ACS Style**

Lu, Y.; Yang, H.; Li, B.; Li, J.; Xu, A.; Zhang, M.
Analysis of Characteristics for Inter-System Bias on Multi-GNSS Undifferenced and Uncombined Precise Point Positioning. *Remote Sens.* **2023**, *15*, 2252.
https://doi.org/10.3390/rs15092252

**AMA Style**

Lu Y, Yang H, Li B, Li J, Xu A, Zhang M.
Analysis of Characteristics for Inter-System Bias on Multi-GNSS Undifferenced and Uncombined Precise Point Positioning. *Remote Sensing*. 2023; 15(9):2252.
https://doi.org/10.3390/rs15092252

**Chicago/Turabian Style**

Lu, Yangyang, Hu Yang, Bo Li, Jun Li, Aigong Xu, and Mingze Zhang.
2023. "Analysis of Characteristics for Inter-System Bias on Multi-GNSS Undifferenced and Uncombined Precise Point Positioning" *Remote Sensing* 15, no. 9: 2252.
https://doi.org/10.3390/rs15092252