A Concise Method for Calibrating the Offset of GPS Precise Satellite Orbit
Round 1
Reviewer 1 Report
[1] Line 2: GPS should be present in the title because GPS data are used in the case study. GPS satellite POD is made with 1-day arc in the study.
[2] Section 1 Introduction: How about the study of GPS satellite POD with 1-day arc? How about the study of GPS satellite POD with multi-day arcs? How about the clock error of GPS satellite? How about the product jumping on consecutive days for 1-day arc POD by comparing with the POD precision?
[3] Lines 93-97: The 11th-order Lagrange interpolation polynomial is used here, but only a total of 8-epoch interval lengths is consisted. Why?
[4] Eq. (5): Omiga should be a frequency, which should be determined based on the time series.
[5] Section 3: How about the statistics of jumping for POD and clock error from all ACs? Many data should be used to make the statistics. What software is used in the study?
[6] Table 2: Here is the strategy of PPP for single-epoch PPP or daily PPP?
[7] Table 3: How many data are used in the table?
[8] Section 4: What software is used in the study?
[9] Figure 6 and table 4: Only one case is shown. Many cases should be statistically made.
[10] Line 207: The sub title is the same as that in line 311.
[11] Table 5: What about the kinematic POD and the dynamic POD or reduced-dynamic POD?
[12] Figure 6 and table 6: Only one case is shown. Many cases should be statistically made.
[13] Figure 9 and table 7: Only one case is shown. Many cases should be statistically made.
[14] In the study, how to process the clock errors for precise ephemerides of GPS satellite? How about the multi-day POD of GPS satellites?
[15] Literatures should be written in normal format.
Author Response
Dear reviewer:
We would like to express our great thanks to you for your help and consideration about our manuscript entitled "A concise method for calibrating the offset of precise satellite orbit and clock products" (Manuscript ID: remotesensing-2085802).
The comments and suggestions provided by the reviewers are all valuable and very helpful for revising and improving our paper. We have studied the comments carefully and have made revisions accordingly. In our revised manuscript, all the changes made to the manuscript are marked up using the "Track Changes" function of MS Word processor. In addition, we have prepared a detailed one-by-one response letter to the reviewers’ comments.
You can also see all the response in the attachment.
Open Review 1
- Line 2: GPS should be present in the title because GPS data are used in the case study. GPS satellite POD is made with 1-day arc in the study.
Reply:
GPS has been present in the title. In our strategies, the arc length of GPS POD includes not only 24 hours but also for 30 hours and 72 hours.
- Section 1 Introduction: How about the study of GPS satellite POD with 1-day arc? How about the study of GPS satellite POD with multi-day arcs? How about the clock error of GPS satellite? How about the product jumping on consecutive days for 1-day arc POD by comparing with the POD precision?
Reply:
Due to factors such as force model defects, DBD exists regardless of the GPS POD strategy, single-day arc or multi-day arc solution. In this study, the precision orbit and clock products released by each analysis center are able to provide centimeter-level positioning accuracy as mentioned in Section 1 and then they are calibrated. Therefore, the status of GPS POD and clock offset is not in consideration.
The jump between consecutive days is internal coincidence accuracy, and the accuracy of orbit is usually evaluated with an external reference. The accuracy comparison between them makes no sense.
- Lines 93-97: The 11th-order Lagrange interpolation polynomial is used here, but only a total of 8-epoch interval lengths is consisted. Why?
Reply:
I apologize that this is a mistake that confuses Lagrange interpolation with numerical integration and it has been corrected in the paper.
For the Lagrangian interpolation, only the 9th order was used in this study. For the current day orbit with a 15-minute interval, the center point of the first interpolation is 7 minutes and 50 seconds, and 5 epoch points are required before and after. Excluding the epoch at the 00:00 moment of the day, the last 4 epochs of the previous day need to be used. The interpolation of the last time period of the current day is the same.
- (5): Omiga should be a frequency, which should be determined based on the time series.
Reply:
Omiga is set as a constant in Eq. (5), But the weights of the data used in the correction phase change in Eq. (3) and Eq. (4) due to the change in time. The time-weighted method in this study is to change the weights of the data according to their distance to the center point of the smoothing window, thus retaining the Omiga constant.
- Section 3: How about the statistics of jumping for POD and clock error from all ACs? Many data should be used to make the statistics. What software is used in the study?
Reply:
This study mainly corrects the jumping of GPS orbit. The correction of clock offset is only to ensure the coupling between orbit and clock offset, so the jump of clock offset product is not concerned. The jump of POD is summarized in Section 1, like ‘the 1D mean daily boundary discontinuity (DBD) of final GPS orbit products is reported to be a dozen millimeters (mm) or even tens of millimeters’, so it is not given all the evaluation in Section 3. In order to save space and aim at the influence of orbit arc length, only AC products with multi-day solution, single-day solution and integrated solution strategy are selected.
All calculations are conducted based on the Positioning And Navigation Data Analyst (PANDA) software as mentioned in ‘Acknowledgments’.
- Table 2: Here is the strategy of PPP for single-epoch PPP or daily PPP.
Reply:
Table 2 shows the strategy for post-processing kinematic PPP and the instruction has been added.
- Table 3: How many data are used in the table?
Reply:
The results in Table 3 are solved by 7-day arcs of data from 17 stations from day 063 to day 100 in 2010. And the instructions for all the data used in this study have been added in Section 3.
- Section 4: What software is used in the study?
Reply:
All calculations are conducted based on the Positioning And Navigation Data Analyst (PANDA) software as mentioned in ‘Acknowledgments’.
- Figure 6 and table 4: Only one case is shown. Many cases should be statistically made.
Reply:
A total of 17 stations from day 63 to day 100 in 2010 were selected for PPP validation in Section 4.1. Since the performance is consistent across stations, only one station per day is selected to show the variation of the data at the boundary and non-boundary in Figure 6.
In contrast, the statistics of the results for all stations for all days are shown in Table 4. The description of Table 4 is also added to the document.
- Line 207: The sub title is the same as that in line 311.
Reply:
This mistake has been corrected.
- Table 5: What about the kinematic POD and the dynamic POD or reduced-dynamic POD?
Reply:
Table 5 contains the observation model and force model. Only the observation model is used for kinematic LEO POD while dynamic LEO POD requires the use of both. And reduced-dynamic LEO POD is not included in this study. Description and Table 5 have been updated in Section 4.2.
- Figure 6 and table 6: Only one case is shown. Many cases should be statistically made.
Reply:
A total of data from day 63 to day 100 in 2010 was selected for dynamic LEO POD validation in section 4.2. Since the performance is consistent across all days, only one day of data is selected to show the variation at the boundary and non-boundary in Figure 7 (I think you are talking about Figure 7).
Comparatively, the statistics of the results for all days are shown in Figure 8 and Table 6. The description of Table 6 is also added to the document.
- Figure 9 and table 7: Only one case is shown. Many cases should be statistically made.
Reply:
A total of data from day 63 to day 100 of 2010 was selected for kinematic LEO POD validation in section 4.3. Since the performance is consistent across all days, only one day of data is selected to show the variation at the boundary and non-boundary in Figure 9.
Comparatively, the statistics of the results for all days are shown in Figure 10 and Table 7. The description of Table 7 is also added to the document.
- In the study, how to process the clock errors for precise ephemerides of GPS satellite? How about the multi-day POD of GPS satellites?
Reply:
The correction of GPS clock offset in this study is only used to absorb the change of orbit to ensure the consistency of orbit and clock, as seen in Eq. (6).
The current status of multi-day solution GPS orbits and the results after calibration are shown in all experiments using COD and JPL products in Sections 3 and 4.
- Literatures should be written in normal format.
Reply:
All the literatures have been corrected with reference to previous RS articles.
Author Response File: 
Author Response.docx
Reviewer 2 Report
This manuscript proposed a method for addressing the day-boundary discontinuity.
The basis of the method is described in eq. 1 and 2, which weight the two data
sets before and after the boundary. The weights are given in eq. 3 and 4. These
weighting functions use a cosine taper. This is not an unreasonable choice, but
it is also not unique. The authors do not describe why they chose this weighting function, and why it is better than other weighting functions that might have been considered.
The change in the radius of the orbit computed by this method is then pushed into the clock solution by eq. 6. This step requires significant justification, and the opposite proposal in which the clock solution is pushed back into the estimate of the orbital radius would also be a possible method. In fact, if the stability of the clock is sufficiently great, this would have been a better choice.
The data used in the study are more than 12 years old (line 142), and a lot
has changed since then. Whatever the merit might have been for the analysis
methods at that time, it is not obvious that it provides any current insight
with respect to more recent data.
Finally, I note that the problem of discontinuities at the day boundaries
has been known for many years, and there is a very large literature on
this subject, especially in the timing community. I invite the authors to
study the literature and to compare the solution that they propose with
the very large number of solutions that have been published.
Author Response
Dear reviewer:
We would like to express our great thanks to you for your help and consideration about our manuscript entitled "A concise method for calibrating the offset of precise satellite orbit and clock products" (Manuscript ID: remotesensing-2085802).
The comments and suggestions provided by the reviewers are all valuable and very helpful for revising and improving our paper. We have studied the comments carefully and have made revisions accordingly. In our revised manuscript, all the changes made to the manuscript are marked up using the "Track Changes" function of MS Word processor. In addition, we have prepared a detailed one-by-one response letter to the reviewers’ comments.
You can also see all the response in the attachment.
Open Review 2
- This manuscript proposed a method for addressing the day-boundary discontinuity. The basis of the method is described in eq. 1 and 2, which weight the two data sets before and after the boundary. The weights are given in eq. 3 and 4. These weighting functions use a cosine taper. This is not an unreasonable choice, but it is also not unique. The authors do not describe why they chose this weighting function, and why it is better than other weighting functions that might have been considered.
Reply:
The purpose of this paper is to use the most concise method to directly process the precision products provided by each analysis center, and to reduce the jumping phenomenon of the products near the daily boundary as much as possible while ensuring comparable accuracy.
We tried the linear function, and the effect is not very satisfactory, probably because of the error of some periodic terms inside the orbit, so we refer to the trigonometric weighting method adopted by JPL.
- The change in the radius of the orbit computed by this method is then pushed into the clock solution by eq. 6. This step requires significant justification, and the opposite proposal in which the clock solution is pushed back into the estimate of the orbital radius would also be a possible method. In fact, if the stability of the clock is sufficiently great, this would have been a better choice.
Reply:
Resolving the solved clock offset after adjusting the orbit is a good method, but it requires the user to rewrite the solved clock offset using the global station data. Since the satellite orbit radial direction and the satellite clock difference are highly coupled (Like mentioned in Section 2, The satellite clock offset can absorb up to 97% of the orbit error in the radial direction for the GPS constellation and is also discussed in the literature 44 and 45), we choose a simpler way to implement it here. Although the method used in this paper is not very perfect, but it is feasible according to the experimental results.
- The data used in the study are more than 12 years old (line 142), and a lot has changed since then. Whatever the merit might have been for the analysis methods at that time, it is not obvious that it provides any current insight with respect to more recent data.
Reply:
DBD exists regardless of the method used for GPS POD due to force model deficiencies and other factors. To verify the reliability of the results, we also compared and evaluated the GPS POD solution strategy and DBD for each analysis center from 2010 to 2022. There is no significant change in their solution strategies and no significant improvement in DBD from the results. Therefore, although the experimental data are relatively old, they have little impact on the analysis of this paper.
- Finally, I note that the problem of discontinuities at the day boundaries has been known for many years, and there is a very large literature on this subject, especially in the timing community. I invite the authors to study the literature and to compare the solution that they propose with the very large number of solutions that have been published.
Reply:
The presence of DBD in GPS precision products has indeed been identified at an early stage, and it has been shown in various papers that DBD has an impact on timing and positioning accuracy at PPP boundaries. Moreover, we have found that DBD has a non-negligible impact on the positioning results when processing long-duration low-orbit data in a kinematic solution.
At present, the methods used mainly focus on the reduction of DBD by improving the mechanical model and error model in the process of GPS POD. It is necessary to recalculate the global data, which is relatively complicated. Therefore, we propose a processing method as simple as possible.
Author Response File: Author Response.docx
Reviewer 3 Report
Paper presents interesting research and very well prepared. Test procedure is clear and justified. Text is written in a logical and thoughtful way, creating a coherent whole. Below are some remarks:
（1）“For the experiment, the final orbit and clock offset from day 063 to day 100 of 2010 were selected.” Why did you choose 2010? Have you analyzed the precise orbit of recent years? Is it possible that the DBD in recent years is so small that it can be ignored.
（2）In Section 4.2, it is necessary to explain which period of time the GRACE satellite data is used for the precise orbit determination experiment.
（3）For the IGS product in Figure 7, please explain why the 1D RMS of INPUT I is better than INPUT II and INPUT III?
(4) Why do sections 4.2 and 4.3 have the same name?
Answer to all remark please place into the corrected text for future readers.
Author Response
Dear reviewer:
We would like to express our great thanks to you for your help and consideration about our manuscript entitled "A concise method for calibrating the offset of precise satellite orbit and clock products" (Manuscript ID: remotesensing-2085802).
The comments and suggestions provided by the reviewers are all valuable and very helpful for revising and improving our paper. We have studied the comments carefully and have made revisions accordingly. In our revised manuscript, all the changes made to the manuscript are marked up using the "Track Changes" function of MS Word processor. In addition, we have prepared a detailed one-by-one response letter to the reviewers’ comments.
You can also see all the response in the attachment.
Open Review 3
- “For the experiment, the final orbit and clock offset from day 063 to day 100 of 2010 were selected.” Why did you choose 2010? Have you analyzed the precise orbit of recent years? Is it possible that the DBD in recent years is so small that it can be ignored.
Reply:
DBD exists regardless of the method used for GPS POD due to force model deficiencies and other factors. To verify the reliability of the results, we also compared and evaluated the GPS POD treatment strategy and DBD for each analysis center from 2010 to 2022. There is no significant change in their solution strategies and no significant improvement in DBD from the results. Therefore, although the experimental data are relatively old, they have little impact on the analysis of this paper.
- In Section 4.2, it is necessary to explain which period of time the GRACE satellite data is used for the precise orbit determination experiment.
Reply:
The time period has been additionally declared in the first appearance of the time description in section 3.
- For the IGS product in Figure 7, please explain why the 1D RMS of INPUT I is better than INPUT II and INPUT III?
Reply:
The purpose of the method in this study is to improve the smoothness of the server-side product as much as possible with the most concise calibration method while ensuring comparable accuracy. Therefore, some constraints and corrections are not taken into account, such as the original product position accuracy constraint and the error in the remaining directions when the clock offset is corrected. Although the accuracy of the results will be reduced, it is still acceptable, as in the case of using IGS products in Figure 7.
- Why do sections 4.2 and 4.3 have the same name?
Reply:
This mistake has been corrected.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Authors have addressed the manuscript based on the precious proposals and comments. Several problems are following:
[1] ‘clock’ is present in the title. How to process the jumping in clock products in detail?
[2] PODs of GPS satellites can be made in 1-day arc, 2-day arc, or 7-day arc. So the offset of precise orbit may mainly be present in 1-day arc products.
[3] The offset is small by comparing with the POD accuracy.
[4] In tables 6 and 7, what accurate position is used for the statistics?
Author Response
Dear reviewer:
We would like to express our great thanks to you for your help and consideration about our manuscript entitled "A concise method for calibrating the offset of precise satellite orbit and clock products" (Manuscript ID: remotesensing-2085802).
The comments and suggestions provided by the reviewers are all valuable and very helpful for revising and improving our paper. We have studied the comments carefully and have made revisions accordingly. In our revised manuscript, all the changes made to the manuscript are marked up using the "Track Changes" function of MS Word processor. In addition, we have prepared a detailed one-by-one response letter to the reviewers’ comments.
You can also seen the response in the attachment.
Open Review 1
[1] ‘clock’ is present in the title. How to process the jumping in clock products in detail?
Reply:
The jumping in clock products in this study is not processed, and the correction of clock offset is only to ensure the coupling with the orbit. To avoid vagueness, the ‘clock’ has been removed from the title.
[2] PODs of GPS satellites can be made in 1-day arc, 2-day arc, or 7-day arc. So, the offset of precise orbit may mainly be present in 1-day arc products.
Reply:
Due to the deficiencies of the non-conservative force models, adding arc length of pod can only partly reduce the impact of DBD. For example, the DBD of CODE and JPL products reach 18 mm and 23 mm respectively. Moreover, most ACs provide daily orbits. Based on the current processing strategy, we found that the DBD could still influence out kinematic position results. The meaning of our manuscript maybe not clear and we hence add some description in the first part.
[3] The offset is small by comparing with the POD accuracy.
Reply:
Compared with orbit accuracy, the offset is small and it has no influence on daily solution. However, it affects continuous data processing which is demonstrated in our manuscript, which will be important for some certain applications, for example the earth gravity parameter estimation using kinematic LEO results.
[4] In tables 6 and 7, what accurate position is used for the statistics?
Reply:
The results in both Table 6 and 7 are compared with the scientific orbits provided by JPL This description has been added in Section 4.
Author Response File: Author Response.docx
