Inter-Satellite Single-Difference Ionospheric Delay Interpolation Model for PPP-RTK and Its Positioning Performance Verification
Round 1
Reviewer 1 Report
Review of the article entitled: Inter-Satellite Single-Differ Ionospheric Delay Interpolation
The PPP-RTK model and its verification of positioning performance.
Achieving high-precision, real-time positioning is of particular importance in many areas of the economy. In my opinion, aviation is the most important of these, as the real-time position of an aircraft is the basis of air traffic safety. The main parameter having a great influence on the positioning precision is the atmospheric delay. An interpolation model using the atmospheric delay coefficient to represent the ionospheric delay SD based on the mean ionospheric penetration position (IPP) of a pair of satellites and the central position of the lattice, as proposed by the authors of the article, which I believe to be correct. The hypothesis put forward by the authors was confirmed and confirmed for four scenarios. As the main goal of the paper, the authors chose the task of quickly resolving the total indeterminacy (IAR) by using precise information on atmospheric delays provided by regional networks. The proposed methodology includes DIM (Distance Based Linear Interpolation Method), LSM (Low Order Surface Model), USM and DSM. The authors of the article did not explain the abbreviations used in the text, what they are required to do. I miss the Linear Interpolation Method-LIM and Linear Combination Model-LCM, which are the most popular and previously developed correction interpolation methods.
The authors examined the accuracy of the interpolation for four different scenarios. For research and analysis, they selected CORS data from the largest network of stations providing code and phase observations for precise positioning. The acceptance of only four scenarios was probably dictated by the size of the article.
In the "Discussion" section, the authors showed that the accuracy of DSM interpolation is the best among the tested interpolation methods. They proved that after identifying and correcting mean square ionospheric delay errors for interpolation accuracy, DSM positioning performance was better than that of DIM. This proves that the DSM is the benchmark for using the atmospheric retardation factor to determine the ionospheric retardation SD. The aim of the article has been achieved.
The text is correct and understandable. However, there is no information about the sources of the formulas used in the article.
Author Response
see the revision report.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
- The paper is interesting and treats atmospheric models for PPP-RTK. The interpolation model used by the authors shows improvements when compared with the existing model. Some minor corrections are necessary to improve the manuscript.
- Lines- 75 until 77: The content has already been written in abstract.... avoid repetition
- Line 88: What you wanna mean by "certain application value"? explain...
- What are the parameters Tu and Ti in equations 1 and 2? Would it be representing the troposphere?
- In equations 4 and 5, the ionosphere delay appears with a preceding delta (delta_I), while in eequation6 the ionosphere is represented only by the letter ‘I’.
- Use the same notation both for troposphere and ionosphere and explain if it is really necessary to use delta.
- In equation 8: if you consider cts_ = cts + dif_rs, we suppose that the satellite clock (ts) is being estimated and not unconstrained from IGS products. Can you explain?
- In my opinion, figure 3 is not necessary since the RMS values are presented in Table 2.
- Line 393: "The interpolation accuracy of DSM is better than that of DIM and DSM"... maybe you wanna mean “DSM is better than DIM and USM”
- I suggest an entire revision of the English.
Author Response
see the revision report.
Author Response File: Author Response.pdf
Reviewer 3 Report
This paper proposes a DSM atmospheric interpolation model. And the authors analyzed its performance in PPP-RTK. The model takes into account the relationship between the IPP points of different satellites and the network center point, which effectively increases the accuracy of atmospheric error modeling in PPP-RTK as well as further improves the positioning accuracy of the user end.
However there is a slight upgrade of the English grammar required. So I have added a list of questions and also grammar hints which should be treated in an upgraded version.
1. L82 ' USM' What words does this acronym represent?
2. L85 ' PP-RTK' you mean PPP-RTK?
3. L86 ' DSM' What words does your 'DSM' method acronym represent?
4. L122 ' …the definitions of the other terms are the same as for equation (1) …' The coefficients here should have a different interpretation than in eq. (1), re-phase.
5. L126 ' the interpolation value ' Interpolation coefficients should be related to the atmospheric delay of the reference stations, not interpolate value. You need make it clear here.
6. L126 ' …is not suitable for SD atmospheric delay interpolation. ... ' What puzzles me is why the LSM coefficients aren't suitable for SD atmospheric interpolation.
7. L138 There is no expression for φ and λ_in equations (3) and (4)
8. Line 141-142 There are a total of 6 unknown coefficients in this eq. (4). Why do you need at least four stations? if three will be enough?
9. Line 146 This is my main concern. Please explain why the mean IPP value is used here. When the distance between the IPP of reference satellite and the rover satellite point is large on the ionosphere single layer, could this mean value precisely represent the SD IPP?
10. Line 151 '... the previous equation.' You should clarify which equation you are referring to here
Author Response
see the revision report.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Further comments and Suggestions
1) In Section 2.1, Why the LSM model was introduced? If the LSM is used for tropospheric delay modeling, is the model lacking elevation direction information reasonable? If this model is used for ionospheric modeling, why are the results of this model not available in the ionospheric delay accuracy verification experiments?
2) In Section 2.2, the function model of server-side and user-end positioning is given. As a complete positioning model system, it is recommended to add the corresponding stochastic model.
3) In the experiment of performance analysis of PPP-RTK for Medium-scale networks, the empirical mean squared errors of ionospheric were used, how did the author get the value of 2cm, 4cm for ionospheric delays, and 1.5cm, 2.5cm for tropospheric delays in scenario A and B? Are the values suitable for both the DIM and DSM models? Are the values suitable for all periods? Same question for Large-scale networks.
4) The experiment in this paper only compares the positioning performance at TTFF. What is the difference between the positioning performance of the two models (DIM and DSM) after fixed?
5) The English language is needed to be further polished. For example, in lines 386-387, the sentence is a little bit wordy.
Author Response
Answer: Thanks very much for your comments and suggestions. We answered the question below and revised the article in a revision mode. Moreover, we marked the parts of the paper that need to be modified in yellow.
1) In Section 2.1, Why the LSM model was introduced? If the LSM is used for tropospheric delay modeling, is the model lacking elevation direction information reasonable? If this model is used for ionospheric modeling, why are the results of this model not available in the ionospheric delay accuracy verification experiments?
Answer: Thanks very much for your suggestion. The LSM can consider various forms, such as elevation, to adapt to different application environments, so it was chosen as an interpolation function of tropospheric delay that is sensitive to elevation. So, the tropospheric delay model used the LSM with first-order horizontal and vertical coordinates, which was indicated in section 4.2.5. The LSM bases on the coordinates of the reference station, while the DSM bases on the ionospheric pierce point (IPP) of each satellite pair. Although the accuracy is similar in the hundreds of kilometers of reference network tested, it is more rigorous to interpolate the ionospheric delay of user stations by using the IPP of the reference station and satellite pair for regional ionospheric interpolation. The DSM provides a reference for using the atmospheric delay coefficient based on the IPP to characterize the SD ionosphere delay, which can advance the application of PPP-RTK.
2) In Section 2.2, the function model of server-side and user-end positioning is given. As a complete positioning model system, it is recommended to add the corresponding stochastic model.
Answer: Thanks very much for your suggestion. The code and carrier observations were still modeled randomly based on elevation angle at the service and user end. The empirical value was used to determine the the weights of tropospheric zenith delay and ionospheric delay. We have been supplemented it in Section 3.
3) In the experiment of performance analysis of PPP-RTK for Medium-scale networks, the empirical mean squared errors of ionospheric were used, how did the author get the value of 2cm, 4cm for ionospheric delays, and 1.5cm, 2.5cm for tropospheric delays in scenario A and B? Are the values suitable for both the DIM and DSM models? Are the values suitable for all periods? Same question for Large-scale networks.
Answer: Thanks very much for your suggestion. The covariance matrix of interpolated atmospheric corrections has been usually determined using empirical functions in single baseline RTK. It is common that the STD of the ionospheric corrections is modeled as a linear function of the baseline length [24]
Similarly, the linear function can also be employed to determine the atmospheric stochastic model for network augmented PPP. In the experiment of the article, the average distance in A and B networks is 41 km and 98 km respectively, so the mean squared errors of the undifferenced ionospheric delays were approximately 2cm (2.8cm for SD ionospheric delay) and 4cm (5.6cm for SD ionospheric delay) in scenario A and B, respectively. The average accuracy of SD ionospheric delay in network A is 1.4cm and 1.8cm respectively, and that in network B is 3.2cm and 4.3cm respectively. The actual accuracy is better than the empirical model, but the difference is small. It is because that the quality of the corrections of the surrounding reference stations should be much better than that of the single base station. But considering that too tight constraints will reduce the accuracy of the results, we think the empirical value used is appropriate, but it may not be optimal for all periods. Meantime, the accuracy of ZTD also adopts the empirical value of article [25] based on the baseline length. Figure 1 and Figure 2 showed the accuracy of tropospheric zenith delay interpolation respectively in scenario A and B respectively. The actual accuracy is better than the empirical model for most periods.
On the other hand, the accuracy of the actual atmospheric delay can be calculated by the accuracy of the interpolation model and the accuracy of the corrections calculated by the reference station, but this increases the burden of communication. Therefore, how to balance traffic and available information deserves further study.
The expression of relevant parts in the paper is not appropriate and has been modified.
Figure 1. Time series of tropospheric zenith delay in scenario A.
Figure 2. Time series of tropospheric zenith delay in scenario B.
4) The experiment in this paper only compares the positioning performance at TTFF. What is the difference between the positioning performance of the two models (DIM and DSM) after fixed?
Answer: Thanks very much for your suggestion. Figures 6 and 7 show the RMS value and its distribution of the positioning bias after the successful fixation in each period in scenario A. The average RMS values of the DSM in the three directions (E, N, U) were 0.8, 0.9, and 2.7 cm, which represent improvements of 11.6%, 28.8% and 10.6% compared with DIM, respectively. Figure 9 shows the RMS value of the positioning bias after successfully fixed in all periods in scenario B. The average RMS values in the three directions (E/N/U) of the DSM were 1.0, 1.1, and 4.0 cm, respectively, which represent increases of 8.2%, 29.2% and 11.2% compared with DIM respectively. The statement in the article is not clear and has been modified.
5) The English language is needed to be further polished. For example, in lines 386-387, the sentence is a little bit wordy.
Answer: Thank you for your detailed suggestions. We have improved it in the revision.
Author Response File: Author Response.pdf
Round 3
Reviewer 3 Report
This manuscript has been greatly improved compared to the previous. Mainly corrected some English expressions and key diagrams. To sum up, I suppose that this article is suitable for publication in the Remote Sensing.
Additionally, there are some minor issues here.
1. The description of the method in the context should be as consistent as possible. For example, when describing the DSM method, the “mean position of the IPP” was not mentioned as before in Section 5.
2. The sentence “The interpolation accuracy of DSM is better than that of DIM and DSM” in lines 415-416 should be corrected to “The interpolation accuracy of DSM is better than that of DIM and USM”. There may be similar problems in the paper, please search carefully.
3. The format used in the context should be the same. For example, 0.045m used italic formatting, while 0.042 did not in lines 376.
4. We have noticed that the author has made corrections to some key figures and numbers, please explain why the changes were made.
5. There may still be spelling errors and irregular expressions in the paper, please check it again.
Author Response
Comments and Suggestions for Authors This manuscript has been greatly improved compared to the previous. Mainly corrected some English expressions and key diagrams. To sum up, I suppose that this article is suitable for publication in the Remote Sensing. Additionally, there are some minor issues here.
Answer: We answered the question below and revised the article in a revision mode. Moreover, we marked the parts of the paper that need to be modified in yellow.
- The description of the method in the context should be as consistent as possible. For example, when describing the DSM method, the “mean position of the IPP” was not mentioned as before in Section 5.
Answer: Thanks very much for your suggestion. We have improved it in the revision. - The sentence “The interpolation accuracy of DSM is better than that of DIM and DSM” in lines 415-416 should be corrected to “The interpolation accuracy of DSM is better than that of DIM and USM”. There may be similar problems in the paper, please search carefully.
Answer: Thanks very much for your suggestion. We have corrected it in the revision. - The format used in the context should be the same. For example, 0.045m used italic formatting, while 0.042 did not in lines 376.
Answer: Thanks very much for your suggestion. We have corrected them in the revision. - We have noticed that the author has made corrections to some key figures and numbers, please explain why the changes were made.
Answer: Thank you for your question. Because several groups of results are tested, the statistical examples of DIM in this paper are not consistent with those described in this paper for the large-scale networks, so the relevant results were changed. The results are similar, so they are not explained. - There may still be spelling errors and irregular expressions in the paper, please check it again.
Answer: Thanks very much for your suggestion. We conducted a comprehensive review of the articles and corrected them in the revision.
