Reduced-Dynamic Orbit Determination of Low-Orbit Satellites Taking into Account GNSS Attitude Errors
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIt is an interesting paper. However, I have several concerns/comments. Namely, that comparing the effect of nominal and quaternion GPS attitude orientation is an extreme case, equivalent to completely neglecting GPS eclipsing. In reality, GPS eclipsing is relatively well documented and corresponding models are widely available and used in PODs and PPPs. Had GPS eclipsing models been used, little or no improvements would have been obtained. Nevertheless, using quaternion attitude should still be safer than the models, since it ensures the consistency with the GPS orbits/clocks POD.
The second comments concerns the reduce dynamic approach used. More information would be useful about the specific approach used here, as the approaches in the given references differ significantly.
Finally, it should be specified how the LEO accuracy were derived. The accuracy is illusive and implies the knowledge of the ground truth, which generally is not known or available for real applications and even more so for LEO ones.
The paper could be improved as suggested in the enhancements and corrections listed below
------detailed suggestions/corrections -------
l. 29: … during eclipse seasons (note: when non eclipsing quaternions agree with the nominal orientation)
l. 31: … during eclipsing …
l. 39: .. the impact of neglecting eclipsing attitude models..
l. 42: .. corrections, caused by neglecting eclipsing attitude models. Based on
l. 59: As a result, the satellite has to use an eclipsing attitude model in order to resume the nominal attitude orientation when outside the eclipsing region. GPS satellites, depending on Block types employ different eclipsing attitude models, published or determined a posteriori. Neglecting or using a wrong eclipsing model causes attitude anomalies [3,4].
l. 62: … may employ different eclipsing attitude models …
l. 72: … based on the GYM model. What is GYM? Perhaps “ based on published and experimental data”: would be better here.
L, 86-87: a 360 deg. error cause zero PCO error. Suggest to use here ‘ a difference up to 360 deg in yaw angles ..”
l. 92: … cause a maximum difference of about 1 week in integer recovery clocks … ?? not sure what is meant here
l. 111: GNSS attitude quaternions
l. 115, 116: quaternion yaw angle
l. 140: where ATAN2 is the FORTRAN function of tan-1, giving signed angles between (-,)
Eq. 3: shout use u T (transpose) or spell-out the vector u: (u1, u2, u2)
l 149: … and u (u1,u2, u3) sin … is the vector part …
l. 156-159: due to different sun angle calculations??? a bit confusing and may be even incorrect
Note: For small negative beta angles (|beta|< 1 deg. ), some GPS IIF satellites were observed to rotate in an opposite (wrong) direction than the nominal one, which may or may not be accounted for by a POD software. Or alternatively, when beta angle changes the sign during eclipsing, the POD software may use a wrong rotation direction. This is why it is important, in order to be consistent with GNSS POD, to use the corresponding quaternions, which may differ from the nominal, or even from the actual yaw rotations in both magnitude and direction. Using consistent (quaternion) yaw rotations greatly reduces errors caused by yaw eclipsing model errors.
Tab. 1: legend should explain GREC (G-GPS, R-GLONASS, E-Galileo, C-BDS)
l. 209: .. 220 km and are equipped with GNSS receiver observing GPS only.
l. 213: GNSS orbit positions and clock ..
Tab2: what is the arc length? Any stochastic orbit parameters? More details would be useful here
Figs 3-5: should show the year (2023)
Why Figs 4-5: (BDS2-3) when only GPS is used? Are they necessary here?
l. 231-233: this makes a little or no sense.
In general, the noon maneuvers are shorter and happen when the required nominal yaw rate exceeds the maximum hardware rate. The midnight maneuvers are usually longer since they are required to bridge the Earth’s shadow when the sun sensor direction is not available.
l. 235: .. GPS satellite eclipse seasons have a semiannual ..
l. 238: 13.5 deg rather 10 deg would be a better shadow limit for GPS
l. 243-246: is this BDS2,3 necessary here?
Figs 6-8: should distinguish between night (shadow ) and noon (not shaded) crossing, i.e., the noon crossings should not be shaded. (since often different eclipsing maneuvers are used for noon than for night crossings)
Also, at times either the indicated eclipsing or the nominal attitude are wrong, see e.g., the Fig. 6 at ~12h, Fig. 7 at ~24h, etc please check/verify/correct. Furthermore, why quaternion angles are not shown outside eclipsings? Since outside eclipsings they are the same as the nominal and cannot be seen like in Fig. 8?
l. 259: ..quaternion GPS satellite yaw angles..
l. 265: .. quaternion yaw angle..
Figs 9, 10 right : (for G18) are wrong, since 1) G18 (Block IIIA has nearly zero x,y PCO offsets (< 2 cm), so PCO errors must be << 2 cm
2) the nominal phase wind up should show a curve similar to the left side (is it due to a wrong or insufficient interval shown?
Fig. 10 units (IF cycles?)
l. 290: IF cycles
Figs 11, 12: should show units (mm)
l. 301: .. fixed orbit phases ..? not sure what is meant here
Fig. 15: should use the same colours as the previous Figs, i.e. blue - nominal, red -quaternions
Also: The day 93 nominal accuracy are rather poor/suspicious and should be investigated/explain, it may be due to eclipsing model problems of some plane the A satellites?
l. 338: Fig. 15 shows the three-dimensional orbital accuracies ..
How the accuracy was determined? What was the ground truth? This should be explained/specified
Tab. 4: How the accuracy was determined? This should be explained/specified.
Suggest to used here improvements, rather than “promote”. Also when and if the anomalous day 93 is explained/corrected the improvements will be significantly smaller
l. 357: .. maximum correction error caused is weeks ???
l. 366: quaternion and nominal
Author Response
Comment 1: l.29 … during eclipse seasons (note: when non eclipsing quaternions agree with the nominal orientation)
Response 1: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 2: l. 31: … during eclipsing …
Response 2: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment3: l. 39: .. the impact of neglecting eclipsing attitude models..
Response3: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment4:l. 42: .. corrections, caused by neglecting eclipsing attitude models. Based on
Response4: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment5:l. 59: As a result, the satellite has to use an eclipsing attitude model in order to resume the nominal attitude orientation when outside the eclipsing region. GPS satellites, depending on Block types employ different eclipsing attitude models, published or determined a posteriori. Neglecting or using a wrong eclipsing model causes attitude anomalies [3,4].
Response5: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment6: l. 62: … may employ different eclipsing attitude models …
Response6: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment7: l. 72: … based on the GYM model. What is GYM? Perhaps “ based on published and experimental data”: would be better here.
Response7: Thank you for pointing that.I agree with this comment.Defined GYM as the JPL GPS Yaw Model
Comment8: L, 86-87: a 360 deg. error cause zero PCO error. Suggest to use here ‘ a difference up to 360 deg in yaw angles ..”
Response8: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment9: l. 92: … cause a maximum difference of about 1 week in integer recovery clocks … ?? not sure what is meant here
Response9: Thank you for pointing that.I agree with this comment.We revised the text to avoid the ambiguous “1-week” statement and instead state more generally that inconsistent attitude handling can introduce significant discontinuities/offsets in ambiguity-resolved clock/phase-bias products if not treated consistently.
Comment10: l. 111: GNSS attitude quaternions
Response10: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment11: l. 115, 116: quaternion yaw angle
Response11: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment12: l. 140: where ATAN2 is the FORTRAN function of tan-1, giving signed angles between (-,)
Response12: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment13: Eq. 3: shout use u T (transpose) or spell-out the vector u: (u1, u2, u2);l 149: … and u (u1,u2, u3) sin … is the vector part …
Response13: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment14: l. 156-159: due to different sun angle calculations??? a bit confusing and may be even incorrect
Note: For small negative beta angles (|beta|< 1 deg. ), some GPS IIF satellites were observed to rotate in an opposite (wrong) direction than the nominal one, which may or may not be accounted for by a POD software. Or alternatively, when beta angle changes the sign during eclipsing, the POD software may use a wrong rotation direction. This is why it is important, in order to be consistent with GNSS POD, to use the corresponding quaternions, which may differ from the nominal, or even from the actual yaw rotations in both magnitude and direction. Using consistent (quaternion) yaw rotations greatly reduces errors caused by yaw eclipsing model errors.
Response14: Thank you for pointing that.We sincerely appreciate the reviewer's insightful comments. We have revised the relevant content in the manuscript according to your suggestions. The original statement about the possible opposite rotation of the quaternion yaw angle due to different sun angle calculations by server - side and client - side software was rather confusing. We have now replaced it with a more detailed and accurate description, as shown above. This new version clearly presents the observed phenomenon of GPS IIF satellites rotating in the wrong direction under specific conditions, the potential issues with the POD software, and the importance of using corresponding quaternions to reduce errors. We believe these modifications will significantly improve the clarity and scientific rigor of our manuscript.
Comment14: Tab. 1: legend should explain GREC (G-GPS, R-GLONASS, E-Galileo, C-BDS)
Response14: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment15: l. 209: .. 220 km and are equipped with GNSS receiver observing GPS only.
Response15: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment16: l. 213: GNSS orbit positions and clock ..
Response16: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment17: Tab2: what is the arc length? Any stochastic orbit parameters? More details would be useful here
Response17: Thank you for pointing that.I agree with this comment.The arc length is 90 min.
Comment18: Figs 3-5: should show the year (2023)
Response18: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment19: Why Figs 4-5: (BDS2-3) when only GPS is used? Are they necessary here?
Response19: Thank you for pointing that.I agree with this comment.Clarified in Section 3.2 and Section 3.1 that GRACE-FO tracks GPS only; BeiDou plots are shown for completeness to illustrate that similar eclipse-season behavior exists for other GNSS constellations, and multi-constellation POD is left for future work.
Comment20: l. 231-233: this makes a little or no sense.
In general, the noon maneuvers are shorter and happen when the required nominal yaw rate exceeds the maximum hardware rate. The midnight maneuvers are usually longer since they are required to bridge the Earth’s shadow when the sun sensor direction is not available.
Response20: Thank you for pointing that.I agree with this comment.In general, noon turns are relatively short and occur when the required nominal yaw rate exceeds the spacecraft’s maximum allowable yaw rate. Night turns are typically longer because they have to bridge the eclipse interval, during which the Sun sensor direction is unavailable for attitude determination.
Comment21: l. 235: .. GPS satellite eclipse seasons have a semiannual ..
Response21: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment22: l. 238: 13.5 deg rather 10 deg would be a better shadow limit for GPS
Response22: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment23: l. 243-246: is this BDS2,3 necessary here?
Response23: Thank you for pointing that.I agree with this comment.BeiDou plots are shown for completeness to illustrate that similar eclipse-season behavior exists for other GNSS constellations
Comment24: Figs 6-8: should distinguish between night (shadow ) and noon (not shaded) crossing, i.e., the noon crossings should not be shaded. (since often different eclipsing maneuvers are used for noon than for night crossings)
Also, at times either the indicated eclipsing or the nominal attitude are wrong, see e.g., the Fig. 6 at ~12h, Fig. 7 at ~24h, etc please check/verify/correct. Furthermore, why quaternion angles are not shown outside eclipsings? Since outside eclipsings they are the same as the nominal and cannot be seen like in Fig. 8?
Response24: Thank you for pointing that.I agree with this comment.We sincerely thank the reviewer for the careful examination of the attitude maneuver visualization in Figures 6–8. After a thorough check, our response to this specific comment is as follows. In the current figures, shading is not used to distinguish noon (sunlit) maneuvers from midnight (eclipse) maneuvers, mainly for the following reason: the primary purpose of Figures 6–8 is to illustrate the temporal distribution of attitude-control events. In addition, Figures 6–8 only show the attitude states at epochs when GRACE-C could receive signals from the corresponding GNSS satellite; the missing segments occur because GRACE-C did not track that satellite at those times.
Comment25: l. 259: ..quaternion GPS satellite yaw angles..
Response25: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment26: l. 265: .. quaternion yaw angle..
Response26: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment27: Figs 9, 10 right : (for G18) are wrong, since 1) G18 (Block IIIA has nearly zero x,y PCO offsets (< 2 cm), so PCO errors must be << 2 cm
2) the nominal phase wind up should show a curve similar to the left side (is it due to a wrong or insufficient interval shown?
Response27: Thank you for pointing that.I agree with this comment.1）
We used the igs20.atx file. The entry for the GPS G18 satellite at the L1 frequency is:
“G01 START OF FREQUENCY
-64.42 19.48 1188.42 NORTH / EAST / UP”
From this entry, the UP component of the PCO is 1188.42 mm. The PCO values shown in the figure were computed according to Equation (8) in the manuscript.
2）Figure 10 shows the phase wind-up corrections for two satellites on DOY 106 and DOY 105 of 2023 over different time intervals. All values in the figure were computed using the corresponding equation in the manuscript. For clarity, we present the phase wind-up corrections over the entire day.
Phase wind-up correction for G18 on DOY 105, 2023.
Phase wind-up correction for G06 on DOY 106, 2023.
Comment28: Fig. 10 units (IF cycles?)
Response28: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment29: l. 290: IF cycles
Response29: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment30: Figs 11, 12: should show units (mm)
Response30: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment31: l. 301: .. fixed orbit phases ..? not sure what is meant here
Response31: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment32: Fig. 15: should use the same colours as the previous Figs, i.e. blue - nominal, red -quaternions
Also: The day 93 nominal accuracy are rather poor/suspicious and should be investigated/explain, it may be due to eclipsing model problems of some plane the A satellites?
Response32: Thank you for pointing that.I agree with this comment.
The figure shows the time series of position errors for GRACE-D on DOY 93. It can be seen that the poorer performance on this day is mainly caused by a bias in the along-track (A) component around 05:00. To rule out potential operational or configuration mistakes, we repeated the processing, and the results remained the same as shown. The only difference between the two processing strategies is the choice of the attitude model.
Comment33: l. 338: Fig. 15 shows the three-dimensional orbital accuracies ..
How the accuracy was determined? What was the ground truth? This should be explained/specified
Response33: Thank you for pointing that.I agree with this comment.
The 3D orbit residual is defined by taking a third-party scientific orbit product as the reference (“truth”) and computing the root-sum-square of the coordinate differences between the estimated precise orbit solution and the reference orbit in the radial, along-track, and cross-track components. It is computed as:
where ΔR, ΔA, and ΔC denote the coordinate differences between the estimated precise orbit and the selected third-party orbit product in the radial, along-track, and cross-track directions, respectively
Comment34: Tab. 4: How the accuracy was determined? This should be explained/specified.
Suggest to used here improvements, rather than “promote”. Also when and if the anomalous day 93 is explained/corrected the improvements will be significantly smaller
Response34: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment35: l. 357: .. maximum correction error caused is weeks ???
Response35: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment36: l. 366: quaternion and nominal
Response36: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Reviewer 2 Report
Comments and Suggestions for Authors- Supplement specific data sources and time range details for model validation in the abstract to improve completeness.
- Supplement the setting basis of the solar elevation angle threshold in Figures 3-5 to enhance scientific rigor.
- Add quantitative comparison data of interpolation errors of attitude quaternion products from different analysis centers.
- Refine the specific implementation steps and parameter settings of the residual method in LEO satellite PCV estimation.
- Increase the analysis of attitude impacts at different stages of the eclipse season (e.g., mid-term/terminal stage).
- Supplement the comparative experimental results of attitude error impacts between Beidou satellites and GPS satellites.
- Clarify the physical meanings and value range annotations of each parameter in the phase wind-up correction formula.
- Add a sensitivity analysis of the impact of attitude quaternion sampling rate on orbit accuracy.
Author Response
Comment 1: Supplement specific data sources and time range details for model validation in the abstract to improve completeness.
Response 1: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 2: Supplement the setting basis of the solar elevation angle threshold in Figures 3-5 to enhance scientific rigor.
Response 2: Thank you for pointing that.I agree with this comment.
The critical condition for a satellite to enter the Earth’s shadow is that the absolute value of the Sun elevation angle ? satisfies:
where ? is the Earth’s radius, and ℎ is the satellite orbit altitude.
Comment 3: Add quantitative comparison data of interpolation errors of attitude quaternion products from different analysis centers.
Response 3: Thank you for pointing that.I agree with this comment.The figure presents a comparison of products from two analysis centers for DOY 100 of 2023. The comparison method is based on the following equation：
Comparison of CNES and WUM attitude quaternion products (G18) on DOY 100, 2023.
Comparison of CNES and WUM attitude quaternion products (G07) on DOY 100, 2023.
Comment 4: Refine the specific implementation steps and parameter settings of the residual method in LEO satellite PCV estimation.
Response 4: Thank you for pointing that.I agree with this comment.
A residual-based approach was adopted to model the receiver antenna PCV for the LEO satellite. The procedure is as follows:
-
The antenna sky space is discretized into a 5°X5°grid, where the xx- and yy-axes represent elevation and azimuth, respectively. The carrier-phase residuals obtained from the orbit determination are assigned to the corresponding grid cell according to their azimuth and elevation angles.
-
The antenna sky space is further discretized into a finer 2.5°X2.5°grid. For each fine-grid cell, the residuals falling within the cell are averaged, and the resulting mean value is taken as the PCV correction for the corresponding 5°X5°grid node.
-
The PCV corrections estimated in Step (2) are applied in the precise orbit determination. Using the updated carrier-phase residuals, the above modeling and application procedure is repeated to refine the PCV corrections.
Comment 5: Increase the analysis of attitude impacts at different stages of the eclipse season (e.g., mid-term/terminal stage).
Response 5: Thank you for pointing that.I agree with this comment.Thank you for your interest in our satellite attitude analysis methodology. In response to the suggestion of conducting a phase-by-phase analysis, we would like to provide additional clarification considering the specific characteristics of the attitude quaternion data source. In this study, we use the IGS-provided attitude quaternion products, whose key difference from the nominal attitude model lies in their handling during Earth-shadow (eclipse) periods. Therefore, the phase-dependent impact is inherently captured by these quaternion products.
Comment 6: Supplement the comparative experimental results of attitude error impacts between Beidou satellites and GPS satellites.
Response 6: Thank you for pointing that.I agree with this comment.
These are related experiments conducted by our team to investigate the impact of attitude errors on BDS processing. However, because there are currently no publicly available open-source LEO datasets that include BDS observations, we are unable to add the corresponding comparative POD experiments in this manuscript. The figure above illustrates the effects on BDS satellites when using the nominal attitude model versus the quaternion-based attitude model.
Comment 7: Clarify the physical meanings and value range annotations of each parameter in the phase wind-up correction formula.
Response 7: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 8: Add a sensitivity analysis of the impact of attitude quaternion sampling rate on orbit accuracy.
Response 8: Thank you for pointing that.I agree with this comment.An insufficient attitude sampling rate can delay updates of the torque (attitude-dependent) model, leading to biased acceleration computations in orbit propagation; after long-term accumulation, this can significantly degrade orbit accuracy.
Reviewer 3 Report
Comments and Suggestions for AuthorsSatellite attitude is a critical factor in high-precision orbit determination. This study focuses on the influence of attitude modeling on precise orbit determination for Low Earth Orbit (LEO) satellites, addressing a topic of both practical and theoretical relevance. The work examines the effects of attitude anomalies during eclipse seasons on the accuracy of LEO satellite orbit solutions. Through comparison of different attitude methods, including the nominal attitude model and the attitude quaternion approach, the study analyzes how attitude errors propagate into key observational corrections such as antenna phase center offset and phase wind-up.There are also some detailed comments and suggestions for the paper as follows：
1.Please verify the consistency between all figure/table captions and their descriptions in the main text, with particular attention to distinguishing between " phase residuals " and "orbit accuracy." For example, line 337 Fig. 14 as " three-dimensional orbital accuracies," whereas its actual content refers to phase residuals. Please unify such terminology throughout the manuscript.
2.It is recommended to standardize terminology throughout the text by replacing imprecise expressions such as "receiver clock difference" (e.g., line 176) with established terms like "receiver clock offset" to enhance terminological accuracy and consistency.
3.Fig. 15 shows that orbit accuracies results based on attitude quaternions exhibit lower accuracy compared to those based on nominal attitude on certain dates. Please provide possible explanations for this observed discrepancy.
4.Please supplement the " Data and processing strategy " section with a clear description of the specific source of the reference orbit truth data used, ensuring the completeness of the accuracy evaluation methodology.
Author Response
Comment 1: Please verify the consistency between all figure/table captions and their descriptions in the main text, with particular attention to distinguishing between " phase residuals " and "orbit accuracy." For example, line 337 Fig. 14 as " three-dimensional orbital accuracies," whereas its actual content refers to phase residuals. Please unify such terminology throughout the manuscript.
Response 1: Thank you for your careful review of the rigor of our manuscript. I agree with this comment.We have thoroughly checked the consistency between all figure/table captions and the main text, confirmed that the following terminology inconsistencies existed, and have completed systematic corrections accordingly.
Comment 2: It is recommended to standardize terminology throughout the text by replacing imprecise expressions such as "receiver clock difference" (e.g., line 176) with established terms like "receiver clock offset" to enhance terminological accuracy and consistency.
Response 2: Thank you for your careful review of the rigor of our manuscript. I agree with this comment.We have thoroughly checked the consistency between all figure/table captions and the main text, confirmed that the following terminology inconsistencies existed, and have completed systematic corrections accordingly.
Comment 3:Fig. 15 shows that orbit accuracies results based on attitude quaternions exhibit lower accuracy compared to those based on nominal attitude on certain dates. Please provide possible explanations for this observed discrepancy.
Response 3: Thank you for your valuable suggestion. We carefully compared the processing results and found that, consistent with the figure, the quaternion-based POD solution indeed performs worse than the nominal-attitude solution on certain days. In a typical POD workflow, various error sources are corrected and dynamical parameters are estimated first, after which the orbit is propagated to obtain the satellite coordinates. In our quaternion-based processing, however, the attitude quaternions are transitioned using spherical linear interpolation (SLERP). SLERP is inherently an approximate method, as it assumes that the attitude varies linearly along a great-circle arc on the unit quaternion sphere between two known attitude samples. In reality, attitude changes are often not strictly linear; thus, this approximation may not represent the true attitude evolution precisely, which can lead to a loss of accuracy.
Comment 4: Please supplement the " Data and processing strategy " section with a clear description of the specific source of the reference orbit truth data used, ensuring the completeness of the accuracy evaluation methodology.
Response 4: Thank you for pointing that.I agree with this comment.We have added this part in the revised manuscript.
Reviewer 4 Report
Comments and Suggestions for Authors- What are the attitude measurement devices carried by GNSS and low orbit satellites, respectively?
- The attitude of GNSS satellites changes, and the attitude of low orbit satellites also changes. How do they affect the precise orbit determination of LEO satellites?
- How does the attitude measurement accuracy of GNSS and LEO satellites change in eclipse seasons?
- The reduced-dynamic POD method is an important technology for precise orbit determination of LEO satellites using onboard GNSS. What is the novelty of using simplified reduced-dynamics POD method in this manuscript?
- Does the attitude of GNSS satellites affect the presence of PCO in GNSS satellite antennas?
- What factors affect the presence of PCO and phase wind-up in LEO satellite antennas?
- Satellite attitude includes roll, pitch, yaw, etc. The focus of this manuscript is on yaw. How do other attitudes affect on POD?
- The POD methods for LEO satellites mainly include dynamic method, kinematic methods, and reduced-dynamic methods. Section introduction in the article only mentions the latter two types.
- All variables in all equations must be explained clearly.
- Abbreviations used for the first time in the main text should be given their full names first.
- The influence of phase wind-up is not considered in Eq. (7). Is the impact of PCO on pseudorange and phase the same?
- Section 2 does not provide a detailed simplified reduced-dynamic POD algorithm.
- Are DOYs 90 to 109 of 2023 the eclipse seasons?
- EIGEN6C model is used in the study. How many degrees and orders are the gravity field model up to?
- What is unit used in Figs. (11) and (12)?
- What is RES in Figs. 13 and 14?
- How to evaluate the precision orbit determination of LEO satellite? In general, SLR verification is considered the most reliable result.
- The manuscript should further strengthen the citation and analysis of references.
Author Response
Comment 1: What are the attitude measurement devices carried by GNSS and low orbit satellites, respectively?
Response 1: Thank you for pointing that.I agree with this comment.GPS satellites use a combination of sun sensors, star trackers, and gyroscopes for attitude determination, with sun sensors providing primary reference in sunlight regions and star trackers ensuring high precision attitude knowledge.GRACE-FO use Star Tracker .
Comment 2: The attitude of GNSS satellites changes, and the attitude of low orbit satellites also changes. How do they affect the precise orbit determination of LEO satellites?
Response 2: Thank you for pointing that.I agree with this comment.Satellite yaw attitude describes the transformation from the satellite body-fixed frame to the inertial frame or the Earth-fixed frame. It can affect the orbit determination accuracy of navigation satellites in two main ways: (1) by influencing observation-related corrections, including the satellite antenna phase center offset (PCO) correction and the phase wind-up correction; and (2) by affecting the computation of attitude-dependent perturbing forces, such as solar radiation pressure and Earth albedo radiation pressure.
Comment 3: How does the attitude measurement accuracy of GNSS and LEO satellites change in eclipse seasons?
Response 3: Thank you for pointing that.I agree with this comment.For GNSS satellites, the study “GNSS satellite attitude characteristics during eclipse season” explicitly states that, under different ?-angle conditions, the yaw-angle estimation accuracy for GPS satellites is approximately ±3°. However, due to attitude-control mode switching during Earth-shadow periods and insufficient adaptability of certain attitude models, differences in model parameters, processing strategies, and observation-data quality among analysis centers can lead to discrepancies in the estimated yaw angles, with the maximum difference reaching up to 180°.
Comment 4: The reduced-dynamic POD method is an important technology for precise orbit determination of LEO satellites using onboard GNSS. What is the novelty of using simplified reduced-dynamics POD method in this manuscript?
Response 4: Thank you for pointing that.I agree with this comment.Incorrect satellite attitude modeling can trigger a series of problems, such as errors in analysis-center orbit and clock products when the attitude model used by the analysis center differs from that adopted by users, increased modeling errors during Earth-shadow periods, or even the exclusion of satellites during processing due to attitude-model inconsistencies. Replacing the traditional nominal attitude model with the attitude quaternion products provided by the analysis centers ensures full consistency between user-side processing and analysis-center product generation, and effectively mitigates the negative impact of conventional attitude-model errors on orbit determination accuracy.
Comment 5: Does the attitude of GNSS satellites affect the presence of PCO in GNSS satellite antennas?
Response 5: Thank you for pointing that.I agree with this comment.The antenna phase center offset (PCO) is the vector difference between the satellite antenna phase center and the satellite center of mass. Its correction requires satellite attitude information to transform the PCO vector from the body-fixed frame to the inertial frame. If the attitude measurement/modeling error is large, the resulting PCO correction error will propagate directly into the positioning results.
Comment 6: What factors affect the presence of PCO and phase wind-up in LEO satellite antennas?
Response 6: Thank you for pointing that.I agree with this comment.Attitude variations change the projection direction of the PCO vector in the inertial frame and therefore require real-time correction. In addition, the relative rotation of the receiving antenna and the transmitting antenna (on the GNSS satellite) about the polarization axis changes the carrier phase and can reach up to one carrier cycle (e.g., about 19 cm for the GPS L1 signal).
Comment 7: Satellite attitude includes roll, pitch, yaw, etc. The focus of this manuscript is on yaw. How do other attitudes affect on POD?
Response 7: Thank you for pointing that.I agree with this comment.Because the satellite body-fixed Z-axis remains fixed, GNSS satellite attitude control is achieved primarily through yaw steering. The yaw angle is defined as the angle between the satellite body-fixed X-axis and the satellite flight (along-track) direction.
Comment 8: The POD methods for LEO satellites mainly include dynamic method, kinematic methods, and reduced-dynamic methods. Section introduction in the article only mentions the latter two types.
Response 8: Thank you for the reviewer’s meticulous comments. Regarding your valuable remark that “the introduction of the LEO POD methodology section only mentions the kinematic and simplified reduced-dynamic approaches, while the dynamic approach is not covered,” we provide the following response and additional clarification.
The dynamic approach propagates the orbit by establishing the satellite’s equations of motion, including relevant perturbing forces (e.g., Earth’s nonspherical gravity field, atmospheric drag, and solar radiation pressure), and numerically integrating these equations using estimated initial orbit parameters. In this study, we focus on orbit determination based on onboard GNSS observations, where high-precision GNSS pseudorange and carrier-phase measurements are used to improve the LEO orbit solution. Therefore, a full dynamic orbit determination approach was not included in the introduction.
Comment 9: All variables in all equations must be explained clearly.
Response 9: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 10: Abbreviations used for the first time in the main text should be given their full names first.
Response 10: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 11: The influence of phase wind-up is not considered in Eq. (7). Is the impact of PCO on pseudorange and phase the same?
Response 11: Thank you for pointing that.I agree with this comment.The impact of the phase center offset (PCO) on pseudorange and carrier-phase observations is essentially the same.
Comment 12: Section 2 does not provide a detailed simplified reduced-dynamic POD algorithm.
Response 12: Thank you for pointing that.I agree with this comment.We have added this part in the revised manuscript.
Comment 13: Are DOYs 90 to 109 of 2023 the eclipse seasons?
Response 13: Thank you for pointing that.I agree with this comment.Eclipse intervals for different orbital planes must be determined based on the specific orbital parameters and the date. For each GPS orbital plane, eclipse periods occur periodically, but they are not fixed; they vary with the relative motion of the Earth and the Sun. During DOY 90–109, some GPS satellites are in eclipse.
Comment 14: EIGEN6C model is used in the study. How many degrees and orders are the gravity field model up to?
Response 14: Thank you for pointing that.I agree with this comment.We have added the model degree and order in the processing strategy table (120 × 120).
Comment 15: What is unit used in Figs. (11) and (12)?
Response 15: Thank you for pointing that.I agree with this comment.We have added this part in the revised manuscript.
Comment 16: What is RES in Figs. 13 and 14?
Response 16: Thank you for pointing that.I agree with this comment.This is a wrong thing.It is RMS（Root Mean Square）actually.Sorry about that.
Comment 17: How to evaluate the precision orbit determination of LEO satellite? In general, SLR verification is considered the most reliable result.
Response 17: Thank you for pointing that.I agree with this comment.
The 3D orbit residual is defined by taking a third-party scientific orbit product as the reference (“truth”) and computing the root-sum-square of the coordinate differences between the estimated precise orbit solution and the reference orbit in the radial, along-track, and cross-track components. It is computed as:
where ΔR, ΔA, and ΔC denote the coordinate differences between the estimated precise orbit and the selected third-party orbit product in the radial, along-track, and cross-track directions, respectively
Comment 18: The manuscript should further strengthen the citation and analysis of references.
Response 18: Thank you for pointing that.I agree with this comment.In this study, we will systematically broaden the literature coverage by adding highly cited papers from the past five years, as well as classic cross-disciplinary references, to the core section on the simplified reduced-dynamic orbit determination method.
Reviewer 5 Report
Comments and Suggestions for AuthorsSuggested revisions for manuscript ID: remotesensing-4043018
This manuscript focuses on a simplified dynamic orbit determination method for Low Earth Orbit (LEO) satellites considering Global Navigation Satellite System (GNSS) attitude errors. By comparing the nominal attitude and quaternion attitude of GNSS satellites during the eclipse season, the study analyzes their impacts on Antenna Phase Center Offset (PCO) and phase wrap correction, and constructs a simplified dynamic orbit determination model using quaternion attitude.The data from GRACE-C and GRACE-D satellites are employed for validation. The results indicate that compared with the nominal attitude strategy, the quaternion strategy reduces phase residuals by 3.6% and 3.9%, and improves three-dimensional (3D) orbit accuracy by 7.3% and 4.5%, respectively. This demonstrates that attitude consistency is crucial for centimeter-level orbit accuracy. The experimental results hold high practical reference value; however, the following revisions or supplements are required prior to publication:- The manuscript does not elaborate on why the quaternion attitude is basically consistent with the nominal attitude for certain GNSS satellites (e.g., Block IIR) during the eclipse season, while significant differences exist for Blocks IIF and IIIA. It is recommended to supplement the analysis of relevant physical mechanisms or control logic.
- The manuscript only uses GRACE-FO data from Days 90–109 of 2023. It is suggested to extend the data to multiple years or multiple satellite missions to enhance the generalizability of the conclusions.
- The impact of attitude errors on other systems (e.g., Galileo, BeiDou Navigation Satellite System (BDS)) has not been thoroughly analyzed. It is recommended to supplement multi-system comparative experiments.
- Formulas (1) and (2) on Page 4 have duplicate numbering. The consistency of formula numbering throughout the manuscript should be verified and corrected.
- Please address the issue of table page breaks in the full text, specifically for Table 1 on Page 5.
Author Response
Comment 1: The manuscript does not elaborate on why the quaternion attitude is basically consistent with the nominal attitude for certain GNSS satellites (e.g., Block IIR) during the eclipse season, while significant differences exist for Blocks IIF and IIIA. It is recommended to supplement the analysis of relevant physical mechanisms or control logic.
Response 1: Thank you for the reviewer’s valuable comments. The issue you raised is indeed crucial. The original manuscript mainly focused on presenting the observed phenomena and did not sufficiently discuss the underlying physical and attitude-control mechanisms that lead to different eclipse-season attitude behaviors among GPS satellites. For GPS Block IIR satellites, the body-fixed X-axis points away from the Sun. A key characteristic of this block is that it can effectively maintain the nominal attitude during Earth-shadow periods; therefore, it does not perform “eclipse maneuvers” in the strict sense, but only exhibits noon turns and night turns.
Comment 2: The manuscript only uses GRACE-FO data from Days 90–109 of 2023. It is suggested to extend the data to multiple years or multiple satellite missions to enhance the generalizability of the conclusions.
Response 2: Thank you for the reviewer’s comment. You are right that using only GRACE-FO data from DOY 90–109 provides a limited sample. In our future work, we will extend the experiments to multiple months of data to further test and validate the method, thereby improving the generality and robustness of our conclusions.
Comment 3: The impact of attitude errors on other systems (e.g., Galileo, BeiDou Navigation Satellite System (BDS)) has not been thoroughly analyzed. It is recommended to supplement multi-system comparative experiments.
Response 3: Thank you for pointing out this critical issue. You are absolutely right. In this preliminary study, our processing was indeed based only on GPS observations. The original intention of showing the Sun elevation angle of BDS orbital planes in Section 3.2 was to provide a comparative background with the GPS system and to highlight potential differences in attitude behavior arising from different constellation geometries.
However, without using actual BDS observations for quantitative analysis, this discussion becomes insufficiently supported and somewhat off-topic. To ensure strict consistency among the methodology, results, and discussion, we propose two options:
-
Remove the BDS-related figures and discussion in Section 3.2 in the revised manuscript. We will fully focus the analysis on the GPS system, making the manuscript more concise and rigorous. We appreciate the reviewer’s suggestion, which has helped us improve the overall quality of the paper.
-
Strengthen the BDS-related discussion in Section 3.2 by elaborating on the potential mechanisms by which similar attitude anomalies in BDS satellites could affect PCO and phase-center-related corrections. Nevertheless, without BDS-based LEO POD results, it remains difficult to provide a detailed quantitative assessment in terms of orbit performance.
Comment 4: Formulas (1) and (2) on Page 4 have duplicate numbering. The consistency of formula numbering throughout the manuscript should be verified and corrected.
Response 4: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 5: Please address the issue of table page breaks in the full text, specifically for Table 1 on Page 5.
Response 5: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Reviewer 6 Report
Comments and Suggestions for AuthorsIt is of great value, especially for precise orbit determination of LEO satellites, to study the difference between nominal yaw angles and attitude quaternion yaw angles of GNSS satellites during eclipse. I have several comments concerning the contents and the presentation of the manuscript which may help improve the publication.
- In Eq(1), the first formula should be used to calculate e_z other than e_x. According to the description, Z-axis points toward the Earth’s center, and r_sat is the unit vector of satellite position.
- It would be better to use symbols of different fonts to represent scalars and vectors (such as e, u and r), respectively.
- It seems that only GPS observables were used in the data processing. However, the solar elevation angle in the orbital planes of BDS is displayed and discussed in Section 3.2. Would the authors please provide some explanation to show the necessity of doing so?
- Could the authors explain why the time series in Figs. 6-8 are piecewise rather than continuous?
Comments on the Quality of English Language
The language should be checked and improved all through the paper to make it easier for the readers to exactly understand the meaning, such as
(1) the sentence in Line 191-192;
(2) the sentence in Line 193-194;
(3) the sentence in Line 310-311;
(4) the sentence in Line 336-338, which can be divided into at least two sentences.
Author Response
Comment 1: In Eq(1), the first formula should be used to calculate e_z other than e_x. According to the description, Z-axis points toward the Earth’s center, and r_sat is the unit vector of satellite position.
Response 1: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 2: t would be better to use symbols of different fonts to represent scalars and vectors (such as e, u and r), respectively.
Response 2: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Comment 3:It seems that only GPS observables were used in the data processing. However, the solar elevation angle in the orbital planes of BDS is displayed and discussed in Section 3.2. Would the authors please provide some explanation to show the necessity of doing so?
Response 3: Thank you for pointing out this critical issue. You are absolutely right. In this preliminary study, our data processing indeed relies only on GPS observations. The original intention of presenting the Sun elevation angle (β) plots for BDS orbital planes in Section 3.2 was to provide a comparative background with the GPS system and to highlight the potential differences in attitude behavior arising from different constellation geometries.
However, without using actual BDS observations for a quantitative analysis, this part of the discussion becomes weakly supported and somewhat off-topic. To maintain strict consistency among the methodology, results, and discussion, we propose two options:
-
Remove the BDS-related figures and discussion in Section 3.2 in the revised manuscript. We will focus entirely on the GPS-based analysis, which will make the manuscript more concise, focused, and rigorous. We appreciate the reviewer’s suggestion, which has helped us improve the overall quality of the paper.
-
Strengthen the BDS-related analysis in Section 3.2 by discussing the potential mechanisms through which similar attitude anomalies in BDS satellites could affect PCO and phase-center-related corrections. Nevertheless, in the absence of BDS-based LEO POD results, it remains difficult to provide a detailed assessment in terms of orbit-performance metrics.
Comment 4: Could the authors explain why the time series in Figs. 6-8 are piecewise rather than continuous?
Response 4: Thank you for pointing that.I agree with this comment.Figures 6–8 only show the attitude information at epochs when GRACE-C could receive signals from the corresponding GNSS satellite. The missing segments occur because GRACE-C did not track that satellite at those times.
Comment 5:
The language should be checked and improved all through the paper to make it easier for the readers to exactly understand the meaning, such as
(1) the sentence in Line 191-192;
(2) the sentence in Line 193-194;
(3) the sentence in Line 310-311;
(4) the sentence in Line 336-338, which can be divided into at least two sentences.
Response 5: Thank you for pointing that.I agree with this comment.We have corrected this part in the revised manuscript.
Round 2
Reviewer 4 Report
Comments and Suggestions for AuthorsThe authors have addressed all comments and proposals. I recommend the manuscritp should be accepted for publication in Remote Sensing.
Reviewer 5 Report
Comments and Suggestions for AuthorsMy concerns have been basically clarified. Thank you for the revisions, and I have no further comments.
