Improved PPP Ambiguity Resolution with the Assistance of Multiple LEO Constellations and Signals
Round 1
Reviewer 1 Report
This article aims to propose and discuss the multi-frequency LEO-augmented multi-GNSS PPP AR.
The English language is quite adequate and the content is complete: there are few language errors and typos, so I suggest to revise the paper and correct the language and typos by a native English-speaking colleague.
Even if the paper is written in an adequate form, some interesting aspects related to PPP approach without the use of LEO satellites are neglected (especially precision and accuracy obtainable today, considering different constellations). Another interesting aspect is to try to verify (or to justify) how these results change according to the processing software. Have you tried to process these data with another software?
The reference section is adequate but I suggest you to add some relevant contributions related to PPP positioning (in terms of precision, accuracy and convergence time) considering also where these tests have been performed (USA, Europe, Africa, Asia).
Starting from the previous considerations, the paper can not be accepted in the present form but needs a revision.
Author Response
Response to Reviewer 1 Comments
This article aims to propose and discuss the multi-frequency LEO-augmented multi-GNSS PPP AR.
The English language is quite adequate, and the content is complete: there are few language errors and typos, so I suggest to revise the paper and correct the language and typos by a native English-speaking colleague.
Answer: Thanks for your suggestions. The language errors and typos were corrected. And we also invited a native English-speaking colleague to review this paper. We expressed our gratitude for his work in Acknowledgements.
Even if the paper is written in an adequate form, some interesting aspects related to PPP approach without the use of LEO satellites are neglected (especially precision and accuracy obtainable today, considering different constellations). Another interesting aspect is to try to verify (or to justify) how these results change according to the processing software. Have you tried to process these data with another software?
Answer: Thanks for your suggestions. The GNSS PPP solutions with different constellations has been investigated by many researchers and the results (in terms of precision, accuracy and convergence time) were added in the introduction. Supplemented: “To assess the precise positioning performance with current multi-constellation GNSS, observation data of Multi-GNSS Experiment (MGEX) and BeiDou Experimental Tracking Network (BETN) networks are were employed by Li et al. [9-10]. It has been confirmed that the performance of PPP in terms of convergence, accuracy, continuity, and reliability can be significantly improved by the fusion of multi-GNSS [9,11-12]. The convergence time of GPS-only PPP can be shortened by 70% when GLONASS, BDS, and Galileo observations are added while the positioning accuracy is improved by about 25% [9]. For PPPAR, the time to first fix (TTFF) and positioning accuracy can also be improved by the fusion of multi-GNSS. The positioning accuracy of GCRE fixed solution within 10 min is (1.84, 1.11, 1.53) cm while the GPS-only result is (2.25, 1.29, 9.73) cm for the east, north and vertical components [13].”.In the result analysis section, the single- and four-system PPP solutions without the use of LEO constellations were presented to make comparison with the LEO-augmented solutions. The positioning series as well as the time to first fix were provided in Fig. 6 and Tab.1.
In principle, the results shouldn’t change along with the software if the algorithm is correct. We haven’t use other software because it is difficult for us to obtain another software with the capability of LEO-augmented PPP.
The reference section is adequate but I suggest you to add some relevant contributions related to PPP positioning (in terms of precision, accuracy and convergence time) considering also where these tests have been performed (USA, Europe, Africa, Asia).
Answer: Thanks for your comments. We have added some references related to PPP positioning.
Supplemented: “
Zumberge J F, Heflin M B, Jefferson D C, Watkins M M, Webb F H (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. Journal of geophysical research: solid earth, 102(B3), 5005-5017
Witchayangkoon, B. (2000). Elements of GPS precise point positioning. in Spatial Information Science and Engineering, University of Maine.
Shi C, Zhao Q, Li M, Tang W, Hu Z, Lou Y, Liu J. (2012). Precise orbit determination of Beidou Satellites with precise positioning. Science China Earth Sciences, 55(7), 1079-1086.
Cai C, Gao Y (2013). Modeling and assessment of combined GPS/GLONASS precise point positioning. GPS Solut, 17(2):223-236.
Li P, Zhang X (2014). Integrating GPS and GLONASS to accelerate convergence and initialization times of precise point positioning. GPS Solution, 18: 461–471.
Li X, Zhang X, Ren X, Fritsche M, Wickert J, Schuh, H. (2015). Precise positioning with current multi-constellation global navigation satellite systems: GPS, GLONASS, Galileo and BeiDou. Scientific reports, 5, 8328.”.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
I have mixed feelings about the paper by Xin Li et al. The paper suffers of two main problems: First, it is more an engineering report, and written as such, than a scientific paper. Second, the English proficiency is poor, to the point that I do not really understand the meaning of many sentences.
I suggest to regroup all the results in Tables and Figures, to avoid jargon as possible (hundreds of acronyms are present), to add a real discussion, and to have the paper resubmitted. A technical journal should be more appropriate.
One benefit to introduce LEO constellations is the capability to do high resolution (time and space) monitoring of the neutral atmosphere, and especially water vapor. This is not mentioned in the paper.
Author Response
Response to Reviewer 2 Comments
I have mixed feelings about the paper by Xin Li et al. The paper suffers of two main problems: First, it is more an engineering report, and written as such, than a scientific paper. Second, the English proficiency is poor, to the point that I do not really understand the meaning of many sentences.I suggest to regroup all the results in Tables and Figures, to avoid jargon as possible (hundreds of acronyms are present), to add a real discussion, and to have the paper resubmitted. A technical journal should be more appropriate.
Answer: Thanks for your comments. The result analysis section was revised with much stronger focus on key findings and principal interpretations. The redundant figures and tables were deleted, and the results were presented with more concise text in the revision. The acronyms (such as EWL, WL and NL) have been reduced to make the article easy to understand. The discussion section was rewrite as following: “The long initialization time is still a serious problem of PPP, which limits the wider application of PPP in some time-critical applications. The fast motion of LEO satellites contributes to geometry diversity, allowing for the rapid convergence of PPP [21-23]. Since the previous studies focused on the contribution of LEO constellation to dual-frequency GNSS float solutions, in this contribution, we investigated the PPP rapid ambiguity resolution with the augmentation of the LEO constellations and the triple-frequency observation data was fully exploited to further improve the performance of ambiguity resolution. Results demonstrated that the TTFF of GREC fixed solution can be shortened from 7.1 to 4.8, 1.1, and 0.7 min and the positioning accuracy can be improved by about 60%, 80%, and 90% with the augmentation of 60, 192, and 288 LEO satellites, respectively. Moreover, the TTFF of 288-LEO augmented GREC solutions can be shortened to 33 s with the triple-frequency observations. The joint of the third frequency brings more combinations to assist the narrow-lane ambiguity resolution, which can further improve the performance of the PPP AR. The 288-LEO constellation has the capability of achieving high-precision positioning independently with the averaged TTFF of 71.8 s and 55.2 s, in dual-frequency and triple-frequency modes, respectively. Since the LEO constellation is still in the demonstration stage or under construction, the contribution of multiple LEO-constellations and signals to PPP AR was investigated based on the simulation data. With more and more LEO satellites available, it is necessary to assess the augmentation performance of LEO based on the measured data.
The fusion of GNSS and LEO constellation can significantly increase the number of observed satellites, optimize the spatial geometry and improve convergence, accuracy, continuity and reliability of precise positioning. Especially, the multi-frequency signals of LEO constellation, bring new opportunities for rapid ambiguity resolution. In the sequential studies, the observations of different LEO constellation with different obit altitudes and types (such as a combination of equatorial and polar circular orbits) should be fully exploited to further improve the multi-GNSS performance. In addition, the LEO constellation not only enhances precise positioning applications, but also can provide high-resolution (time and space) observable for monitoring of the neutral atmosphere, which might be highly valuable for meteorological applications such as now-casting of severe weather events or regional short-term forecast systems.”.
Finally, we have checked and modified the language carefully and corrected language errors and typos. In addition, we also invited a native English-speaking colleague to review this paper.
One benefit to introduce LEO constellations is the capability to do high resolution (time and space) monitoring of the neutral atmosphere, and especially water vapor. This is not mentioned in the paper.
Answer: Thanks for your suggestions. The benefit of LEO constellation for meteorological applications was added in the discussion section.
Supplemented:
“In the sequential studies, the observations of different LEO constellation with different obit altitudes and types (such as a combination of equatorial and polar circular orbits) should be fully exploited to further improve the multi-GNSS performance. In addition, the LEO constellation not only enhance precise positioning applications, but also can provide high-resolution (time and space) observable for monitoring of the neutral atmosphere, which might be highly valuable for meteorological applications such as now-casting of severe weather events or regional short-term forecast systems.”
Author Response File: Author Response.pdf
Reviewer 3 Report
Authors present the potential advantages of including low earth orbit satellites in the GNSS constellations. The work address the problem respect to the reduction of the time to first fix in Precision Point Positioning Service and the increase in position accuracy. The results have been obtained using a detailed simulation that include all error components in a GNSS system.
The work of the paper is correctly executed and the results are coherent. Paper use established signal phase methods to solve ambiguities in position solution. In this sense, the original contribution is on the use of low earth orbit satellites in GNSS constellation. But it is clear that increasing the number of satellites, with high spatial diversity, a high S/N ratio drives to an increase in accuracy and a more efficient resolution of ambiguities.
In the other side, in spite of the results can be obvious, it can be interesting to quantify the value of the time to first fix reduction and accuracy increase.
If the paper is finally published, the next minor errors should be corrected:
Pg 3 Line 122: P an L are interchanged
Pg 13: In figure 12 DYNG should be YEL2
Pg 14 Line 364 (Figure 15 should be Figure 14)
Author Response
Response to Reviewer 3 Comments
Authors present the potential advantages of including low earth orbit satellites in the GNSS constellations. The work address the problem respect to the reduction of the time to first fix in Precision Point Positioning Service and the increase in position accuracy. The results have been obtained using a detailed simulation that include all error components in a GNSS system.
The work of the paper is correctly executed and the results are coherent. Paper use established signal phase methods to solve ambiguities in position solution. In this sense, the original contribution is on the use of low earth orbit satellites in GNSS constellation. But it is clear that increasing the number of satellites, with high spatial diversity, a high S/N ratio drives to an increase in accuracy and a more efficient resolution of ambiguities.
In the other side, in spite of the results can be obvious, it can be interesting to quantify the value of the time to first fix reduction and accuracy increase.
Answer: Thanks for your suggestions. The value of the time to first fix as well as the reduction were provided in the Tab.1. And the accuracy increase was also described in detail in the revision, such as: “At the 1 min-observation session, the positioning accuracy of GREC-only and 60-LEO augmented PPP fixed solutions are greater than 8 cm in three components while the positioning accuracy of 192-LEO and 288-LEO augmented GNSS fixed solutions can achieve 3-4 cm in the horizontal components, 4-6 cm in the vertical components. At the 5 min-observation session, the positioning accuracy of multi-GNSS fixed solutions was improved by about 60%, 80% and 90% from (3.5, 1.8, 4.9) cm to (1.8, 0.6, 2.6) cm, (0.70, 0.23,1.4) and (0.34, 0.21, 1.34) cm in east, north and up components, with the inclusion of 60, 192 and 288 LEO satellites, respectively. ”.
If the paper is finally published, the next minor errors should be corrected:
Pg 3 Line 122: P an L are interchanged
Pg 13: In figure 12 DYNG should be YEL2
Pg 14 Line 364 (Figure 15 should be Figure 14)
Answer: Corrected.
Author Response File: Author Response.pdf
Reviewer 4 Report
This paper discussed an interesting topic related to the improved PPP-AR performance including LEO satellites. Still, some issues need to be clarified before publication.
Line 19: Shouldn't the GPS L5 signal be at 1176.45 MHz?
Line 27: Maybe use 1.1 instead of 1.09, to keep consistent with others.
Line 46: Please add space between 2020 and [7]
Line 50: Please add space between PPP and AR
Line 53-54: “Toward … more frequencies”: this sentence is not complete
Line 55: “extra frequency” à “extra frequencies”; “is potential to” à “could potentially”
Line 57-61: This sentence is too long. Please reformulate
Line 76: “the numerous” à “numerous”
Line 87-88: “and the benefits …” make it another sentence
Line 89: “… and … and …” à “…, …, and …”
Line 102-103: Please reformulate the first sentence. The logic is not clear.
Line 122: L_{r,j}^s should appear before P_{r,j}^s, as it is “phase and pseudorange observation equations”
Line 136: Is the dry part of tropospheric delays modelled? With which model?
Line 137-138: Maybe introduce the STK software at its first appearance in this paper, i.e., at line 96.
Line 143-145: This sentence is not complete. Please reformulate
Line 148: Please write STDs as standard deviations at its first appearance. It should be 0.003 m, not 0.003 mm.
Line 151: Again, why not 1176.45 MHz, which is the frequency of GPS L5?
Line 165: There should be a comma between d_{r,ewl} and d_{r,wl}.
Line 168: \lambda_{IF12} hasn’t been explained yet.
Line 171: “it temporal characteristic” à “its temporal characteristic”
Line 177: If the NL UPDs are stable on a daily basis, why did you estimate them in real-time mode (line 170)?
Line 178: What do you mean by “the ambiguities will remain few hours”?
Line 185: Were the previous texts in this paragraph not describing triple-frequency PPP AR? How do you deal with the receiver UPDs in Eq. (6)-(8)?
Line 194-195: Please describe how it benefits NL ambiguity resolution.
Author Response
Response to Reviewer 4 Comments
This paper discussed an interesting topic related to the improved PPP-AR performance including LEO satellites. Still, some issues need to be clarified before publication.
Line 19: Shouldn't the GPS L5 signal be at 1176.45 MHz?
Answer: Sorry for misunderstanding. It should be 1176.45 MHz for GPS L5 signals.
Line 27: Maybe use 1.1 instead of 1.09, to keep consistent with others.
Line 46: Please add space between 2020 and [7]
Line 50: Please add space between PPP and AR
Answer: Corrected.
Line 53-54: “Toward … more frequencies”: this sentence is not complete
Answer: The sentence has been corrected as following: “The new-generation GNSS (GPS Block IIF, BDS, and Galileo) are transmitting signals on three or more frequencies.”.
Line 55: “extra frequency” à “extra frequencies”; “is potential to” à “could potentially”
Line 76: “the numerous” à “numerous”
Line 87-88: “and the benefits …” make it another sentence
Line 89: “… and … and …” à “…, …, and …”
Line 122: L_{r,j}^s should appear before P_{r,j}^s, as it is “phase and pseudorange observation equations”
Line 137-138: Maybe introduce the STK software at its first appearance in this paper, i.e., at line 96.
Line 148: Please write STDs as standard deviations at its first appearance. It should be 0.003 m, not 0.003 mm.
Line 151: Again, why not 1176.45 MHz, which is the frequency of GPS L5?
Line 165: There should be a comma between d_{r,ewl} and d_{r,wl}.
Line 171: “it temporal characteristic” à “its temporal characteristic”
Answer: Corrected.
Line 57-61: This sentence is too long. Please reformulate
Answer: This sentence has been reformulated as following: “Gu et al. [15] and Li et al. [16,17] performed triple-frequency PPP AR based on raw observations of GPS, BDS and Galileo, respectively. Their results indicated that the third frequency could lead to an improvement in PPP accuracy during the initialization phase.”
Line 102-103: Please reformulate the first sentence. The logic is not clear.
Answer: The sentence was corrected as: “Apart from the LEO observations, the GNSS observations were also simulated in this paper since the BDS and Galileo haven’t been fully completed.”
Line 136: Is the dry part of tropospheric delays modelled? With which model?
Answer: Supplemented: “The dry part of tropospheric delays can be calculated by empirical models—Saastamoinen model in this study [25].”
Line 143-145: This sentence is not complete. Please reformulate
Answer: This sentence was corrected as: “The ionospheric delay can be computed based on the map function and total electron content (TEC) provided by the CODE’s global ionosphere maps (GIM) [27].”.
Line 168: \lambda_{IF12} hasn’t been explained yet.
Answer: Supplemented: “ denotes ionosphere-free ambiguity and refers to the corresponding wavelength.”
Line 177: If the narrow-lane UPDs are stable on a daily basis, why did you estimate them in real-time mode (line 170)?
Answer: The estimated narrow-lane UPDs also include the biases affected by inaccurate modelling of the observations, which results in the fluctuations of the fractional parts. Recent years, the improvement of model precision makes the narrow-lane UPDs more stable than before with the STD less than 0.1 cycles (Li et al. 2018). However, the narrow-lane UPDs are still not as stable as the wide-lane UPDs. Thus, the time-dependent change has to be considered in order to provide high-quality UPDs, unlike to the use of daily means as corrections for the wide-lane. To avoid the misunderstanding, we deleted the sentence “And it has been demonstrated that the narrow-lane UPDs can keep stable during a daily time.”.
Line 178: What do you mean by “the ambiguities will remain few hours”?
Answer: To avoid the misunderstanding, we rewrite this sentence as following:
“Recent years, the improvement of model precision of observations makes the fractional parts of ambiguities more stable than before. Therefore, in the process of narrow-lane UPD estimation, if the satellite cannot be observed at this epoch at a station, the ambiguity at last epoch will be used for narrow-lane UPD estimation to ensure enough observations.”.
Line 185: Were the previous texts in this paragraph not describing triple-frequency PPP AR? How do you deal with the receiver UPDs in Eq. (6)-(8)?
Answer: To avoid the misunderstanding, we regroup this paragraph, and the receiver UPDs in Eq. (6)-(8) were obtained by averaging fractional parts of ambiguities with satellite UPDs corrected, which was mentioned in this paragraph:
“With the UPD corrections, the multi-frequency LEO-augmented PPP AR was achieved by fixing Ewide-lane, wide-lane and narrow-lane ambiguities step by step. The satellite UPDs can be removed by the UPD products while the receiver UPDs of Ewide-lane and wide-lane ambiguities were obtained by averaging fractional parts of ambiguities with the satellite UPDs corrected. After the removal of the UPDs, the Ewide-lane and wide-lane ambiguities were fixed by round strategy[37].With the long wavelength, the extra-wide-lane and wide-lane ambiguity can be fixed to integer very efficiently, then the integer Ewide-lane and wide-lane ambiguities will formulate an ambiguity-fixed ionosphere-free wide-lane measurement (, [14]), which can be expressed as Eq.6.
(6)
with
(7)
(8)
The ambiguity-fixed ionosphere-free observation formulated by phase observations can be regarded as high-precision code measurement to enable rapid convergence for ambiguity-float solution. Since the narrow-lane ambiguities are derived from the float ionosphere-free ambiguities and integer wide-lane ambiguities, the rapid convergence of float solution can also improve the efficiency of narrow-lane ambiguity resolution. For the narrow-lane ambiguity resolution, the satellite UPDs were corrected based on UPD products while the receiver UPDs were eliminated by the across-satellite single-difference method. Then the LAMDA method was applied to fix the narrow-lane ambiguity [31-32]. The ratio test with the threshold of 2 was used to validate effectiveness of the ambiguity resolution. The inclusion of LEO satellites significantly increases the number of candidate ambiguities; however, it also makes the users difficult to fix all ambiguities. Therefore, the partially fixed integer solution was introduced for rapid ambiguity resolution [38]. ”.
Line 194-195: Please describe how it benefits narrow-lane ambiguity resolution.
Answer: Supplemented: “The ambiguity-fixed ionosphere-free observation formulated by phase observations can be regarded as high-precision code measurement to enable a rapid convergence for ambiguity-float solution. Since the narrow-lane ambiguities are derived from the float ionosphere-free ambiguities and integer wide-lane ambiguities, the rapid convergence of float solution can also improve the efficiency of narrow-lane ambiguity resolution.”.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have properly addressed most of the concerns I had with the previous submission.
So, the paper can be published.
Author Response
Thanks for the reviewer’s great work.
And I have checked the language and result carefully to ensure the correctness.
Reviewer 4 Report
line 142: map function --> mapping function
line 178-179: If the satellite is not observale at this epoch, does it make sense to use its ambiguity at all? The observation of this satellite is missing in the model.
line 184-185: rounding strategy not round strategy
line 193-194: "... to enable rapid convergence for an ambiguity-float solution." Please clarify which ambiguities in Eq. 6-8 were already fixed, and which are to be estimated. If there are still ambiguities to be estimated, how could it be considered as high-precision code observations?
line 195: "float ionosphere-free ambiguities": Please identify these ambiguities in your equations.
line 195-196: "Since the narrow lane ambiguities .... wide-lane ambiguities" Please cite an Equation here, or formulate this equation
line 198: "across-satellite" --> "cross-satellite". So you are fixing between-satellite narrow-lane ambiguities, right? If so, please state this.
Author Response
Response to Reviewer 4 Comments
line 142: map function --> mapping function
Answer: Corrected.
line 178-179: If the satellite is not observable at this epoch, does it make sense to use its ambiguity at all? The observation of this satellite is missing in the model.
Answer: Based on the fractional parts of narrow-lane ambiguities, the narrow-lane UPDs at satellite and receiver sides can be estimated with the least square method [36]. With the improvement of model precision of observations, the fractional parts of narrow-lane ambiguities are relatively stable and can be forecasted for few hours. Thus, in the process of narrow-lane UPD estimation, if the satellite cannot be observed at this epoch at a station, the fractional parts of ambiguities at last epoch will be used for narrow-lane UPD estimation to ensure enough observations.
line 184-185: rounding strategy not round strategy
Answer: Corrected.
line 193-194: "... to enable rapid convergence for an ambiguity-float solution." Please clarify which ambiguities in Eq. 6-8 were already fixed, and which are to be estimated. If there are still ambiguities to be estimated, how could it be considered as high-precision code observations?
Answer: The ambiguities in Eq. 6-8 are all fixed to integers. To avoid the misunderstanding, the sentences were corrected as following: “The ambiguity-fixed ionosphere-free observation formulated by phase observations and fixed extra-wide-lane and wide-lane ambiguities can be regarded as high-precision code measurement to enable rapid convergence for an ambiguity-float solution.”
line 195: "float ionosphere-free ambiguities": Please identify these ambiguities in your equations.
line 195-196: "Since the narrow lane ambiguities .... wide-lane ambiguities" Please cite an Equation here, or formulate this equation
Answer: The Equation 5 has been cited here as following: “As shown in Eq.5, the narrow-lane ambiguities are derived from the float ionosphere-free ambiguities and integer wide-lane ambiguities, …”
line 198: "across-satellite" --> "cross-satellite". So you are fixing between-satellite narrow-lane ambiguities, right? If so, please state this.
Answer: Yes, the between-satellite narrow-lane ambiguities were fixed.
Supplemented: “Then the LAMDA method was applied to fix the between-satellite narrow-lane ambiguities.”
Author Response File: 
Author Response.docx
