# A Theoretical and Empirical Integrated Method to Select the Optimal Combined Signals for Geometry-Free and Geometry-Based Three-Carrier Ambiguity Resolution

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## Abstract

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## 1. Introduction

## 2. Theoretical Experiment and Result Analysis

#### 2.1. Basic Concept of GF and GB TCAR Using Combined Signals

- Step 1. Fixing the EWL ambiguities.$$\nabla \u25b5{N}_{\text{EWL}}=\frac{1}{{\lambda}_{\text{EWL}}}[\nabla \u25b5{P}_{\text{code}}-\nabla \u25b5{\Phi}_{\text{EWL}}-({\beta}_{\text{code}}+{\beta}_{\text{EWL}})\nabla \u25b5{I}_{1}]$$$$\nabla \u25b5{\widehat{N}}_{\text{EWL}}={[\nabla \u25b5{N}_{\text{EWL}}]}_{\text{round-off}}$$$$\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{EWL}}=\nabla \u25b5{\Phi}_{\text{EWL}}+{\lambda}_{\text{EWL}}\nabla \u25b5{\stackrel{\u02c7}{N}}_{\text{EWL}}$$
- Step 2. Fixing WL ambiguities.$$\nabla \u25b5{N}_{\text{WL}}=\frac{1}{{\lambda}_{\text{WL}}}[\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{EWL}}-\nabla \u25b5{\Phi}_{\text{WL}}+({\beta}_{\text{WL}}-{\beta}_{\text{EWL}})\nabla \u25b5{I}_{1}]$$$$\nabla \u25b5{\stackrel{\u02c7}{N}}_{\text{WL}}={[\nabla \u25b5{N}_{\text{WL}}]}_{\text{round-off}}$$$$\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{WL}}=\nabla \u25b5{\Phi}_{\text{WL}}+{\lambda}_{\text{WL}}\nabla \u25b5{\stackrel{\u02c7}{N}}_{\text{WL}}$$
- Step 3. Fixing the ambiguities of fundamental signals.$$\nabla \u25b5{N}_{\text{single}}=\frac{1}{{\lambda}_{\text{single}}}[\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{WL}}-\nabla \u25b5{\Phi}_{\text{single}}+({\beta}_{\text{single}}-{\beta}_{\text{WL}})\nabla \u25b5{I}_{1}]$$$$\nabla \u25b5{\stackrel{\u02c7}{N}}_{\text{single}}={[\nabla \u25b5{N}_{\text{single}}]}_{\text{round-off}}$$$$\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{single}}=\nabla \u25b5{\Phi}_{\text{single}}+{\lambda}_{\text{single}}\nabla \u25b5{\stackrel{\u02c7}{N}}_{\text{single}}$$

- Step 1. Fixing the EWL ambiguities in a GF model. In this step, instead of using a GB model, a GF model (Equations (10)–(12)) is usually adopted, as there is no difference in the success rate between them and it is more convenient to conduct AR in a GF model [11].
- Step 2. Fixing the ambiguities of the second EWL/WL signals.$$\left[\begin{array}{c}\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\text{EWL}}-\nabla \u25b5{\rho}_{0}\\ \nabla \u25b5{\Phi}_{\mathrm{W}2}-\nabla \u25b5{\rho}_{0}\end{array}\right]=\left[\begin{array}{cc}A& 0\\ A& -I\xb7{\lambda}_{\mathrm{W}2}\end{array}\right]\left[\begin{array}{c}\delta x\\ {N}_{\mathrm{W}2}\end{array}\right]+\left[\begin{array}{c}{\epsilon}_{\nabla \u25b5P}\\ {\epsilon}_{\nabla \u25b5{\Phi}_{\mathrm{W}2}}\end{array}\right]$$
- Step 3. Fixing the ambiguities of the NL signal.$$\left[\begin{array}{c}\nabla \u25b5{\stackrel{\u02c7}{\Phi}}_{\mathrm{W}2}-\nabla \u25b5{\rho}_{0}\\ \nabla \u25b5{\Phi}_{\mathrm{W}3}-\nabla \u25b5{\rho}_{0}\end{array}\right]=\left[\begin{array}{cc}A& 0\\ A& -I\xb7{\lambda}_{\mathrm{W}3}\end{array}\right]\left[\begin{array}{c}\delta x\\ {N}_{\mathrm{W}3}\end{array}\right]+\left[\begin{array}{c}{\epsilon}_{\nabla \u25b5P}\\ {\epsilon}_{\nabla \u25b5{\Phi}_{\mathrm{W}3}}\end{array}\right]$$

#### 2.2. Selection of Combined Signals Based on Theoretical Consultation and Result Analysis

#### 2.2.1. Theoretical Selection of Combined Signals for GF TCAR

#### 2.2.2. Theoretical Selection of Combined Signals for GB TCAR

## 3. Empirical Experiment and Result Analysis

#### 3.1. Basic Information of the Data and the Process Strategy

#### 3.2. Experiment of the Combined Signal Pairs in GF TCAR

#### 3.3. Experiment of the Combined Signals in GB TCAR

## 4. Conclusions

- In GF TCAR, there are no code with EWL signal pair and EWL with WL signal pair which can perform better than the corresponding traditional signal pairs. However, for the WL with single signal pairs, ones working with WL(1, 0, −1) can provide higher success rates by a maximum number of 8% for the baselines whose length are shorter than 200 km or over 400 km, while for other lengths of baselines, ones with WL(1, −2, 1) can perform better compared to the traditional signal pairs, which work with WL(1, −1, 0). The ambiguity corrected EWL carrier phase signals perform better than any combined code signals in AR when fixing ambiguities of WL signals. When working with the EWL corrected signal, all the WL signal pairs proposed by this paper can have the same success rate regardless of the baseline length.
- In GB TCAR, the WL(1, −1, 0) signal selected in this paper has higher probability to provide more reliable ambiguities especially when the baseline length is short, compared to the traditional EWL(1, −6, 5) signal. The combined signal NL(3, 5, −7) performs more stable than the traditional NL(4, 0, −3), while NL(4, −3, 0) always shares the same success rate with NL(4, 0, −3).
- In GF TCAR, although the results obtained by the theoretical section meet the results using the real data, the assumed success rates using pure carrier phase combined signal pairs will be lower than those using the real data, while the success rates using signal pairs involving both carrier phase and code will be higher than those using real data. To the best of our knowledge, this might be because of the slightly incorrect assumptions of the effect of the error sources and the imprecision of the mathematic model used in the theoretical analysis. However, the real reason needs further research.
- In GB TCAR, although the proposed theoretical method can be used to analyze the relationship between the signals and the noise level, there was not showing a strong correlation between the wavelength to total noise ratio using the theoretical model and the success rate using the real data, which means there might be still some unselected signals with a good success rate when only the wavelength to total noise ratio was considered. Therefore, future improvement could be on developing a theoretical method to demonstrate the GB success rate directly in order to discover all the signals with high performance.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The variation trend of the wavelength to total noise ratio and the AR success rates of the signal pairs in Table 1 for GF TCAR. (

**a**) The wavelength to total noise ratio; (

**b**) AR success rates.

**Figure 5.**The variation trend of the wavelength to total noise ratio and the AR success rates of the signal pairs of GF TCAR in Table 4 updates with the DD ionospheric bias on L1. (

**a**) The wavelength to total noise ratio; (

**c**) AR success rate; (

**b**,

**d**) are the amplification of the rectangle in (

**a**,

**c**) respectively.

**Figure 7.**The variation trend of the wavelength to total noise ratio of the selected combined signals in Table 5 updates with the effect percentage of the total noise level from 0 to 100. (

**a**) The EWL, WL and NL combined signals; (

**b**) The NL combined signals only.

**Figure 8.**The variation trend of DD ionospheric bias and DD tropospheric bias updates with the baseline length. (

**a**) The ionospheric bias; (

**b**) The tropospheric bias.

**Figure 9.**The variation trend of AR success rate in GF TCAR. (

**a**) The signal pairs with combined code and EWL carrier phase; (

**b**) The signal pairs with ambiguity corrected EWL carrier phase signal and WL carrier phase signal.

**Figure 10.**The variation trend of AR success rate in GF TCAR. (

**a**) The signal pair with combined code and WL carrier phase; (

**b**–

**d**) The signal pairs with ambiguity corrected WL carrier phase signal and fundamental signals L1, L2 and L5 respectively.

**Figure 11.**The variation trend of AR success rate in GB TCAR. (

**a**) The success rates of all the selected combined signals; (

**b**) The reliable ratio of WL(1, −1, 0) compared to EWL(1, −6, 5).

**Table 1.**The wavelength, ISFs and the measurement noise of the EWL signals working with combined code signals in GF TCAR.

Signal Pairs | ${\mathit{\lambda}}_{\mathbf{(}\mathit{i}\mathbf{,}\mathit{j}\mathbf{,}\mathit{k}\mathbf{)}}$ (m) | ISFs | ${\mathit{\sigma}}_{\mathbf{\text{GF}}}$ (Cycle) |
---|---|---|---|

code(1, 4, 5)-EWL(0, 1, −1) | 5.8610 | −0.0844 | 0.0610 |

code(1, 2, 3)-EWL(0, 1, −1) | 5.8610 | −0.1381 | 0.0589 |

code(1, 2, 5)-EWL(0, 1, −1) | 5.8610 | −0.0876 | 0.0632 |

code(0, 1, 1)-EWL(0, 1, −1) | 5.8610 | 0 | 0.0667 |

code(1, 7, 19)-EWL(0, 1, −1) | 5.8610 | −0.0024 | 0.0690 |

code(1, 6, 19)-EWL(0, 1, −1) | 5.8610 | 0.0003 | 0.0703 |

code(1, 4, 5)-EWL(0, 2, −2) | 2.9305 | −0.0844 | 0.1219 |

code(1, 2, 3)-EWL(1, −6, 5) | 3.2561 | 1.5060 | 0.1845 |

code(1, 2, 3)-EWL(1, −7, 6) | 7.3263 | 3.5612 | 0.1918 |

Signal Pairs | ${\mathit{\lambda}}_{\mathbf{(}\mathit{i}\mathbf{,}\mathit{j}\mathbf{,}\mathit{k}\mathbf{)}}$ (m) | ISFs | ${\mathit{\sigma}}_{\mathbf{\text{GF}}}$ (Cycle) |
---|---|---|---|

EWL(0, 1, −1)-WL(1, −5, 4) | 2.0932 | 1.0570 | 0.1537 |

EWL(0, 1, −1)-WL(1, −4, 3) | 1.5424 | 0.7788 | 0.1499 |

EWL(0, 1, −1)-WL(1, −3, 2) | 1.2211 | 0.6166 | 0.1566 |

EWL(0, 1, −1)-WL(1, −2, 1) | 1.0105 | 0.5103 | 0.1726 |

EWL(0, 1, −1)-WL(1, −1, 0) | 0.8619 | 0.4352 | 0.1957 |

EWL(0, 1, −1)-WL(1, 0, −1) | 0.7514 | 0.3794 | 0.2236 |

Signal Pairs | ${\mathit{\lambda}}_{\mathbf{(}\mathit{i}\mathbf{,}\mathit{j}\mathbf{,}\mathit{k}\mathbf{)}}$ (m) | ISFs | ${\mathit{\sigma}}_{\mathbf{\text{GF}}}$ (Cycle) |
---|---|---|---|

code(1, 0, 0)-WL(1, −5, 4) | 2.0932 | 0.3384 | 0.2727 |

code(9, 3, 1)-WL(1, −4, 3) | 1.5424 | 0.2344 | 0.2711 |

code(7, 2, 1)-WL(1, −4, 3) | 1.5424 | 0.2323 | 0.2720 |

code(3, 1, 1)-WL(1, −4, 3) | 1.5424 | 0.3025 | 0.2510 |

code(4, 2, 1)-WL(1, −3, 2) | 1.2211 | 0.1519 | 0.2934 |

code(8, 5, 4)-WL(1, −2, 1) | 1.0105 | 0.1203 | 0.3164 |

code(8, 6, 5)-WL(1, −1, 0) | 0.8619 | 0.0815 | 0.3547 |

code(1, 1, 0)-WL(1, −1, 0) | 0.8619 | 0 | 0.4147 |

code(1, 1, 1)-WL(1, 0, -1) | 0.7514 | 0.0950 | 0.3890 |

**Table 4.**The ISFs and the measurement noise of the WL signals working with fundamental signals for GF TCAR.

Signal Pairs | ISFs | ${\mathit{\sigma}}_{\mathbf{\text{GF}}}$ (Cycle) | ||||
---|---|---|---|---|---|---|

L1 | L2 | L5 | L1 | L2 | L5 | |

WL(1, −5, 4) | 1.6616 | 2.3085 | 2.4549 | 1.4483 | 1.1286 | 1.0815 |

WL(1, −4, 3) | 1.9397 | 2.5867 | 2.7330 | 0.8452 | 0.6586 | 0.6311 |

WL(1, −3, 2) | 2.1020 | 2.7489 | 2.8953 | 0.4979 | 0.3879 | 0.3718 |

WL(1, −2, 1) | 2.2083 | 2.8552 | 3.0016 | 0.2798 | 0.2180 | 0.2090 |

WL(1, −1, 0) | 2.2833 | 2.9303 | 3.0766 | 0.1531 | 0.1193 | 0.1144 |

WL(1, 0, −1) | 2.3391 | 2.9861 | 3.1324 | 0.1321 | 0.1030 | 0.0987 |

Combined Signals | ${\mathit{\lambda}}_{\mathbf{(}\mathit{i}\mathbf{,}\mathit{j}\mathbf{,}\mathit{k}\mathbf{)}}$ (m) | ISF | ${\mathit{\sigma}}_{\mathbf{GB}}$ (Cycle) |
---|---|---|---|

EWL(1, −6, 5) | 3.2561 | −0.0744 | 0.1594 |

WL(1, −5, 4) | 2.0932 | −0.6616 | 0.1316 |

WL(1, −4, 3) | 1.5424 | −0.9397 | 0.1042 |

WL(1, −2, 1) | 1.0105 | −1.2083 | 0.0525 |

WL(1, −1, 0) | 0.8619 | −1.2833 | 0.0333 |

NL(3, 0, −2) | 0.1263 | 0.2136 | 0.0881 |

NL(2, 6, −7) | 0.1314 | 0.2252 | 0.1916 |

NL(3, 5, −7) | 0.1140 | 0.0256 | 0.1886 |

NL(4, 0, −3) | 0.1081 | −0.0099 | 0.1205 |

NL(4, −3, 0) | 0.1145 | 0.0902 | 0.1217 |

Stations | Location | Antenna Type | Interval | Signals | Data Loss Rate |
---|---|---|---|---|---|

CUT0 | Perth | TRM59800.00 SCIS | 30 s | GPS L1/L2/L5 | 0 |

CUT2 | Perth | TRM59800.00 SCIS | 30 s | GPS L1/L2/L5 | 0 |

PERT | Perth | TRM59800.00 NONE | 30 s | GPS L1/L2/L5 | 0 |

BURA | Burakin | JAVRINGANT_DM SCIS | 30 s | GPS L1/L2/L5 | 0 |

KELN | Kellerberrin | JAVRINGANT_DM SCIS | 30 s | GPS L1/L2/L5 | 0 |

MTMA | Mt. Magnet | LEIAR25.R3 LEIT | 30 s | GPS L1/L2/L5 | 0 |

WAGN | Wagin | LEIAR25.R3 LEIT | 30 s | GPS L1/L2/L5 | 0 |

KALG | Kalgoorlie | JAVRINGANT_DM SCIS | 30 s | GPS L1/L2/L5 | 0 |

RAVN | Ravensthorpe | JAVRINGANT_DM SCIS | 30 s | GPS L1/L2/L5 | 0 |

Stations | Length | Stations | Length | Stations | Length |
---|---|---|---|---|---|

CUT0-CUT2 | 0 km | BURA-PERT | 187 km | BURA-KALG | 411 km |

CUT0-PERT | 22 km | WAGN-PERT | 222 km | MTMA-KALG | 459 km |

CUT0-BURA | 136 km | KELN-RAVN | 312 km | CUT0-KALG | 545 km |

CUT0-KELN | 176 km | KELN-KALG | 369 km |

Item | GF Mode | GB Mode |
---|---|---|

Observations | DD phase and code | DD phase and code |

Combination mode | Combined signal pairs | Combined signals |

Float estimation | Difference within signal pairs | LSQ |

integer estimation | Round off | Integer LSQ |

Weight | Equal weight | Noise value dependent |

Ionospheric bias | Mitigated using combined signal pairs | Mitigated using signal pairs |

Tropospheric bias | ||

Success rate | Statistics | Threshold value and reliable ratio |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zhao, D.; Roberts, G.W.; Lau, L.; Hancock, C.M.; Bai, R.
A Theoretical and Empirical Integrated Method to Select the Optimal Combined Signals for Geometry-Free and Geometry-Based Three-Carrier Ambiguity Resolution. *Sensors* **2016**, *16*, 1929.
https://doi.org/10.3390/s16111929

**AMA Style**

Zhao D, Roberts GW, Lau L, Hancock CM, Bai R.
A Theoretical and Empirical Integrated Method to Select the Optimal Combined Signals for Geometry-Free and Geometry-Based Three-Carrier Ambiguity Resolution. *Sensors*. 2016; 16(11):1929.
https://doi.org/10.3390/s16111929

**Chicago/Turabian Style**

Zhao, Dongsheng, Gethin Wyn Roberts, Lawrence Lau, Craig M. Hancock, and Ruibin Bai.
2016. "A Theoretical and Empirical Integrated Method to Select the Optimal Combined Signals for Geometry-Free and Geometry-Based Three-Carrier Ambiguity Resolution" *Sensors* 16, no. 11: 1929.
https://doi.org/10.3390/s16111929