Analysis of the Effect of Tilted Corner Cube Reflector Arrays on Lunar Laser Ranging

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

(1)       This article elaborates, through theoretical analysis, modeling, and simulation experiments, that for an array-type laser ranging retroreflector like APOLLO, achieving a sufficiently narrow laser pulse could possibly distinguish reflected laser pulse from different CCRs in specific directions. This in turn enhances ranging precision. The study emphasizes the limitations of the laser reflector array and holds potential practical significance.

(2)       Even if the ranging accuracy can be improved to distinguish adjacent rows of CCRs, there appear to be few incident angles that meet this condition. Therefore, adding an analysis of suitable incident angles could enhance the practicality of the work.

(3)       Reference to Formula (10) in the radar equation appears incorrect. It is recommended that the original literature be verified.

(4)       Section 2.2 of the article comprehensively deduces and models the detection probability under consideration of noise false alarm conditions. It is suggested to provide experimental validation of this mathematical model and clarify its application within the paper's work.

(5)       "Ps" in Formula (16) is not found in the text. It is inferred that "Pt" might have been mistakenly written as "Ps."

(6)       Despite using different fonts for subscripts in Formulas (12) and (15) to distinguish "Nnoise," it still has the potential to confuse readers. It is recommended to replace the characters of one of them.

(7)       Section 2.3 of the article emphasizes the influence of laser pulse width on the envelope distinguishability of reflected pulses. However, it seems to fall short of analyzing the overall precision of the laser ranging process, which has greater practical significance.

(8)       Was the simulation in Figure 9 based on the detection model in Section 2.2? Were all the factors affecting precision beyond the laser pulse width considered? If both yes, no further suggestions.

(9)       The sentence— "Currently, Geodetic Observatory Wettzell conducts LLR using 10ps lasers. Analysis of their ranging data shows multiple peaks in the echo, but due to the limited number of data points, it cannot fully reflect the tilted state of the CCR array" —indicates that the assumption made in this paper, that optimizing the laser pulse width to 10ps will potentially enable the CCRs of APOLLO reflector array to be distinguished, already has some observational verification. It is recommended to present the relevant data analysis in the paper.

Author Response

Comments 1: This article elaborates, through theoretical analysis, modeling, and simulation experiments, that for an array-type laser ranging retroreflector like APOLLO, achieving a sufficiently narrow laser pulse could possibly distinguish reflected laser pulse from different CCRs in specific directions. This in turn enhances ranging precision. The study emphasizes the limitations of the laser reflector array and holds potential practical significance.

 

Response 1: Thank you for your careful review of our manuscript.

 ​

Comments 2:  Even if the ranging accuracy can be improved to distinguish adjacent rows of CCRs, there appear to be few incident angles that meet this condition. Therefore, adding an analysis of suitable incident angles could enhance the practicality of the work.

 

Response 2: Thank you for your kind comments. In our manuscript, we primarily discussed special incidence angles to more vividly illustrate our points. However, the tilt state of the CCRs can still be inverted from the corresponding echo histograms at other incidence angles; see Figure 8 in the manuscript.

 

Comments 3:  Reference to Formula (10) in the radar equation appears incorrect. It is recommended that the original literature be verified.

 

Response 3: Thank you for pointing out this issue. The previous LiDAR equation had some modifications based on the equations from the literature, but we did not provide a detailed explanation. In the revised manuscript, we have adjusted the LiDAR equation to be consistent with the literature. Additionally, several equations were added afterward to provide a detailed explanation of each parameter in Equation 10. Equations 11-14 have been added to the revised manuscript to provide a detailed explanation of the parameters in Equation 10.

 

Comments 4:  Section 2.2 of the article comprehensively deduces and models the detection probability under consideration of noise false alarm conditions. It is suggested to provide experimental validation of this mathematical model and clarify its application within the paper's work.

 

Response 4: Thank you for your valuable feedback. In Sections 2.2 and 2.3, noise and other factors were introduced into the simulation system. In the revised manuscript, we have added a detailed description of this part, as outlined in Section 2.3. However, due to time constraints, experimental verification cannot be conducted in the short term. We will continue to work in this area and publish papers to present our progress in a timely manner.

 

Comments 5:  "Ps" in Formula (16) is not found in the text. It is inferred that "Pt" might have been mistakenly written as "Ps." 

 

Response 5: Thank you for reminding us. The term "Ps" in Equation 16 should indeed be "Pt". In the revised manuscript, for Equation 18.

 

Comments 6:  Despite using different fonts for subscripts in Formulas (12) and (15) to distinguish "Nnoise," it still has the potential to confuse readers. It is recommended to replace the characters of one of them

 

Response 6: Thank you for reminding us. Thank you for your suggestion. We have changed "N_noise" to "V_noise" in Equation 12. In the revised manuscript, for Equation 16.

 

Comments 7:  Section 2.3 of the article emphasizes the influence of laser pulse width on the envelope distinguishability of reflected pulses. However, it seems to fall short of analyzing the overall precision of the laser ranging process, which has greater practical significance.

 

Response 7: Thank you for your suggestion. Based on your advice, we have added an analysis of other factors affecting the precision of LLR in Section 2.3. However, to highlight the significant impact of the tilt of the CCR array on the precision of LLR, we have kept the values for other factors relatively small. Additionally, we have improved the simulation system and presented the new results using this updated system in the revised manuscript. During the improvement of the simulation system, we discovered some minor issues in the previous program, which led to lower precision in single shot measurements. After promptly correcting the program, the new results are consistent with the previous conclusions, but the specific values obtained are different.

 

Comments 8: Was the simulation in Figure 9 based on the detection model in Section 2.2? Were all the factors affecting precision beyond the laser pulse width considered? If both yes, no further suggestions.

 

Response 8: Thank you for your suggestions. The simulation in Figure 9 is based on the detection model developed in Section 2.2, considering not only the laser pulse width envelope but also other factors affecting precision, such as noise.

 

Comments 9: The sentence— "Currently, Geodetic Observatory Wettzell conducts LLR using 10ps lasers. Analysis of their ranging data shows multiple peaks in the echo, but due to the limited number of data points, it cannot fully reflect the tilted state of the CCR array" —indicates that the assumption made in this paper, that optimizing the laser pulse width to 10ps will potentially enable the CCRs of APOLLO reflector array to be distinguished, already has some observational verification. It is recommended to present the relevant data analysis in the paper.

 

Response 9: Thank you for your suggestions. In our manuscript, the data in Table 2 is from Wettzell. Based on your suggestion, we have added an analysis of the Geodetic Observatory Wettzell measured data in the revised manuscript, which can be found in the Discussion section at the end of the document.

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

2.1. Use capital letter for "effect".

Line 134: "When the ranging laser is incident on the CCR array...", please improve the wording. 

Figure 6: the same letter d is used for the separation between individual reflectors instead of the separation of the array as in Figure 5.  Please clarify using different notation. 

Equation (3) is wrong: the dimentions does not match.  The R matrix shall be transposed, and the result of the multiplication on the right hand side of (3) shall be transposed again to obtain R' with proper size. 

Equation (4) is also wrong by the same arguments, please follow previous reccomendation. 

Please compute vector F in detail and check considering previous reccomendation.  Notice that with alpha=0 and beta=0, F=(0,0,0), which is wrong. 

Line 173: Some symbol is expected between L and [0,0,-1). 

Equation (5) can not be a projection (i.e. an symmetric idempotent operator), but uses a rotation matrix instead. Moreover, the vector can not be multipled by the matrix as written.  Please review all the algebraic presentation as it is very confusing: the development is obscure as the equations are wrong. 

The authors should work on the method explanation in order to make easier to understand the idea for any reader.  

 

 

 

Comments on the Quality of English Language

In general the English is good, some comments are given on the review. 

Author Response

Comments 1: Use capital letter for "effect".

 

Response 1: Thank you. In the revised manuscript, 'Effect' in the title has been capitalized.

 ​

Comments 2: line 134: "When the ranging laser is incident on the CCR array...", please improve the wording.

 

Response 2: Thanks! Here is a more refined version of the sentence: “When a laser pulse is incident on a CCR array with an inclination angle of φ, the distance difference between the laser pulse traveling to the nearer and farther CCRs is ΔL .”

 

Comments 3: the same letter d is used for the separation between individual reflectors instead of the separation of the array as in Figure 5.  Please clarify using different notation.

 

Response 3: Thank you for your suggestions. In the revised manuscript, we have updated the spacing between the rows of reflectors in Figure 5 to be represented as d_n.

 

Comments 4: Equation (3) is wrong: the dimentions does not match.  The R matrix shall be transposed, and the result of the multiplication on the right hand side of (3) shall be transposed again to obtain R' with proper size. Equation (4) is also wrong by the same arguments, please follow previous reccomendation.

 

Response 4: Thank you for your guidance. Equations 3 and 4 were indeed incorrect. We have revised both equations in the revised manuscript according to your suggestions. We transposed the initial R matrix, and the current R matrix is now a 3-row by 100-column matrix.

 

Comments 5: Please compute vector F in detail and check considering previous reccomendation.  Notice that with alpha=0 and beta=0, F=(0,0,0), which is wrong.

 

Response 5: The normal vector of the CCR array after rotation is denoted as F=（cosαsinβ，sinα, cosαcosβ). When both alpha and beta are 0, meaning the CCR array is directly facing the Earth, the normal vector is denoted as F=(0,0,1). In this case, F is exactly parallel to the incident laser direction vector L, but in the opposite direction.

 

Comments 6: Some symbol is expected between L and (0,0,-1).

 

Response 6: Thank you ! This should be L=(0,0,-1). There is a missing "=" sign in the middle.

 

Comments 7: Equation (5) can not be a projection (i.e. an symmetric idempotent operator), but uses a rotation matrix instead. Moreover, the vector can not be multipled by the matrix as written.  Please review all the algebraic presentation as it is very confusing: the development is obscure as the equations are wrong.

 

Response 7: Thank you for pointing out this error. After transposing the previous R matrix and calculating using Equations 3 and 4, the calculations here should be correct.

 

Comments 8: The authors should work on the method explanation in order to make easier to understand the idea for any reader.

 

Response 8: Thank you for your valuable feedback. It is possible that the R matrix was not transposed in the previous version of the manuscript, which prevented subsequent calculations and made it difficult for you to understand the methods described in the paper. We sincerely apologize for this oversight and appreciate you pointing out the error. We understand the importance of clearly explaining the methodology to ensure that readers can fully grasp the core ideas of the paper. Although the current version of the manuscript emphasizes the methodology throughout, we have added some equations for explanatory parameters in the revised manus

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper addresses a persistent problem in Lunar Laser Ranging, namely exact extraction of the proper range measurement from optical ranging. For that purpose one has to make the distinction between "accuracy" and "precision". While the former describes the bias free distance, the latter establishes the inherent scatter of the repeated measurements. It is important to keep these two quality markers distinctly apart. Unfortunately this has not been done in this paper consistently. Therefore, I encourage the authors to make this unambiguously clear in the manuscript. It does not help to improve the precision, if the accuracy of the measurement is not improved at the same time.

The paper concentrates on the increased precision obtained by the reduction of the laser pulse width. While the pulse width is certainly an important contributor, there are more critical components in the ranging process, which are influencing the measurement process. These are the timer resolution and stability and the detection triggering process. In order to assert an improvement, it is necessary to look at these components as well. How does a shorter pulse width act on the trigger process? Does it introduce an additional offset? Shorter pulses usually come with lower energy. What is the intensity related effect on the ranging results, both in terms of timer and trigger dispersion. To make ground target measurements to investigate the removal of the tilt problem is a good approach. How well are these simulations matched to the real LLR operation? From the histograms I see hundreds and thousands of echoes over a few minutes. This does not correspond to typical LLR operations (and if it does, the LLR range measurements are most likely significantly biased, because of the intensity dependance of the detector response, which degrades accuracy.   

The expectation raised by this manuscript is an improved precision, if the effect of reflector panel tilt can be resolved by ultrashort laser pulses. This tilt effect is neither static nor acting only on one axis. Earlier publications by  the LLR group in Grasse and the Apache Point, include tilt from the flat panel in the convolved reflector response function together with lunar libration. This process has run into difficulties, as these perturbations are not well enough known. From this point of view this paper here has merit, but it does not explain how to deal with simultaneous tilt in two axes in particular when the direction of tilt does not align with the rows of the reflector panel. This point requires clarification. 

Table 6 is incomplete. What is the number of photons in the link budget of the Yunnan staton? That is important for the remainder of the manuscript. There should be a comparison between the link budget for lunar ranging and the link budget on the target simulation. 

Figer 15 - 18 are composed out of 10 very similar diagrams. I recommend to reduce them considerably in order to make the manuscript more concise and shorter.  

Comments on the Quality of English Language

The paper is written in good English, but a few very minor issues are still present. One example from row 298: What does "higher-precision foundational data" mean?

Is it "Yunnan Observatory" or is there more than one, so that "Yunnan Observatories" is correct?

Author Response

Response Letter to ‘Remote Sensing’ Submission

 

Paper ID: 3095608

Paper Title: Analysis of the Effect of Tilted Corner Cube Reflector Array on Lunar Laser ranging

Reviewer #3

 

Comments 1: The paper addresses a persistent problem in Lunar Laser Ranging, namely exact extraction of the proper range measurement from optical ranging. For that purpose one has to make the distinction between "accuracy" and "precision". While the former describes the bias free distance, the latter establishes the inherent scatter of the repeated measurements. It is important to keep these two quality markers distinctly apart. Unfortunately this has not been done in this paper consistently. Therefore, I encourage the authors to make this unambiguously clear in the manuscript. It does not help to improve the precision, if the accuracy of the measurement is not improved at the same time.

 

Response 1: Thank you. In the analysis of the simulation results, we qualitatively assessed that our method not only improves the single-shot ranging precision but also enhances the accuracy of the ranging data. A figure was added after Figure 16 to illustrate how the tilt of the corner reflector can affect the accuracy of the ranging measurements. This qualitative analysis was also conducted in the evaluation of the experimental results. In the Results and Discussion sections, we also addressed both ‘precision’ and ‘accuracy’. However, we have not yet performed a quantitative analysis of the improvement in ranging accuracy. We may explore this aspect in our future work.

 

Comments 2: The paper concentrates on the increased precision obtained by the reduction of the laser pulse width. While the pulse width is certainly an important contributor, there are more critical components in the ranging process, which are influencing the measurement process. These are the timer resolution and stability and the detection triggering process. In order to assert an improvement, it is necessary to look at these components as well. How does a shorter pulse width act on the trigger process? Does it introduce an additional offset? Shorter pulses usually come with lower energy. What is the intensity related effect on the ranging results, both in terms of timer and trigger dispersion. To make ground target measurements to investigate the removal of the tilt problem is a good approach. How well are these simulations matched to the real LLR operation? From the histograms I see hundreds and thousands of echoes over a few minutes. This does not correspond to typical LLR operations (and if it does, the LLR range measurements are most likely significantly biased, because of the intensity dependance of the detector response, which degrades accuracy.

 

Response 2: Thank you for your professional suggestions. In the revised manuscript, we have focused on revising the simulation experiment section, incorporating the effects of the timer, detector, noise and other factors on the echo. We conducted a more detailed examination of the parameters, ensuring that the echo closely matches the actual LLR measurements to better validate our conclusions. In the revised manuscript, based on your suggestions, we have made significant updates in Section 2.3.

 

Comments 3: The expectation raised by this manuscript is an improved precision, if the effect of reflector panel tilt can be resolved by ultrashort laser pulses. This tilt effect is neither static nor acting only on one axis. Earlier publications by the LLR group in Grasse and the Apache Point, include tilt from the flat panel in the convolved reflector response function together with lunar libration. This process has run into difficulties, as these perturbations are not well enough known. From this point of view this paper here has merit, but it does not explain how to deal with simultaneous tilt in two axes in particular when the direction of tilt does not align with the rows of the reflector panel. This point requires clarification.

 

Response 3: Thanks. In the earlier theoretical model analysis, we considered the case where both axes have an inclination, as shown in Figure 8. It is possible to determine the tilt state of the CCR array in the same way. However, we did not present this case in the experimental results because the CCR array used in our experiment had a small number of rows and columns when a sufficient spacing was maintained, making it difficult to draw clear conclusions. To present our viewpoint more clearly, we only included the special case with an inclination on a single axis. In future work, we plan to improve the experiment further by purchasing a laser with a narrower pulse width and creating CCR arrays that match the dimensions of the CCR arrays on the moon.

 

Comments 4: Table 6 is incomplete. What is the number of photons in the link budget of the Yunnan staton? That is important for the remainder of the manuscript. There should be a comparison between the link budget for lunar ranging and the link budget on the target simulation.

 

Response 4: Yes, it’s right. In the revised manuscript, we have updated the parameter table according to the revised LiDAR equation and also included the echo photon budget.

 

Comments 5: Figer 15 - 18 are composed out of 10 very similar diagrams. I recommend to reduce them considerably in order to make the manuscript more concise and shorter.

 

Response 5: Thank you for your suggestion. In the Experimental Results section, although Figures 15-18 may look somewhat similar, they address different issues. Figure 15 presents the ranging experiments with different numbers of corner reflectors. Its results demonstrate that our ground-based experiments' adjustments for laser normal incidence are effective and that the experimental ranging system is stable, laying a foundation for subsequent experiments. We have changed the style of Figure 16, and its results indicate that tilting the reflector array does indeed reduce the accuracy and precision of laser ranging, underscoring the necessity of this research. Additionally, we have added a new figure (Figure 17) to illustrate how the tilt of the corner reflector array affects the accuracy of the ranging results. The results in Figure 18 which was Figure 17 in the previous manuscript, indicate that it is possible to determine which corner reflectors the echo peaks originate from by analyzing the echo signals. Figure 19 which was Figure 18 in the previous manuscript, shows that selecting the middle peak as the signal for processing not only improves the ranging accuracy but also enhances the precision.

 

Comments 6: Comments on the Quality of English Language：

The paper is written in good English, but a few very minor issues are still present. One example from row 298: What does "higher-precision foundational data" mean?

Is it "Yunnan Observatory" or is there more than one, so that "Yunnan Observatories" is correct?

 

Response 6: Thank you for your careful reading of this manuscript. The sentence on row 298 of the previous manuscript has been revised to the following sentence, now located on row 307 in the revised manuscript. “Long-term acquisition of such LLR data can be used to infer changes in the tilt state of the CCR array, thereby validating the theory of lunar libration. This provides higher precision data for the study of Earth-Moon models.”

Yunnan Astronomical Observatories comprises multiple observation stations, hence the official name uses 'observatories'

 

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This work investigates the effect of the tilt of corner cube reflector (CCR) arrays on lunar laser ranging (LLR). A mathematical model was established, and the experiment also was conducted to evaluate the mathematical model based on the 1.2-meter laser ranging system at Yunnan Observatories, China. In the last, the study provides a solution to reduce random errors caused by the tilt of the CCR array. The results can be a reference for upgrading ground-based equipment for laser ranging.

Some specific issues should be addressed before it can be published.

Specific comments

1)  Lines 448-456, the single-shot ranging precision is improved from 25.64 cm to 1.1 cm, which is consistent with that in the text in Lines 311-313. Lines 11-12, the precision is improved from 25.64 cm to 0.985cm. That’s inconsistent.

2)   In Table 2, Universal Time is used, but the readers cannot know the local time inferred from Table 2 and the text. The longitude can be added in the text.

Moreover, the time of experiment in Yunnan should be introduced.

The precision may depend on local time, due to the water vapour in the atmosphere may vary with local time.

3)  The conclusion is verbose, such as Lines 433-436 can be deleted. It’s better to shorten the conclusion, which summarize the work and emphasis the results of this study.

Author Response

Comments 1: This work investigates the effect of the tilt of corner cube reflector (CCR) arrays on lunar laser ranging (LLR). A mathematical model was established, and the experiment also was conducted to evaluate the mathematical model based on the 1.2-meter laser ranging system at Yunnan Observatories, China. In the last, the study provides a solution to reduce random errors caused by the tilt of the CCR array. The results can be a reference for upgrading ground-based equipment for laser ranging.

 

Response 1: Thank you for your comments on our manuscript

 

Comments 2: Lines 448-456, the single-shot ranging precision is improved from 25.64 cm to 1.1 cm, which is consistent with that in the text in Lines 311-313. Lines 11-12, the precision is improved from 25.64 cm to 0.985cm. That’s inconsistent.

 

Response 2: Thank you for your careful review and for pointing out our errors. Random noise was introduced into the simulation system, which led to minor inconsistencies in the results of each run. Therefore, the results used in the abstract and the main text of the previous manuscript may not have been from the same run. We have added an analysis of other factors affecting the precision of LLR in Section 2.3. However, to highlight the significant impact of the tilt of the CCR array on the precision of LLR, we have kept the values for other factors relatively small. Additionally, we have improved the simulation system and presented the new results using this updated system in the revised manuscript. During the improvement of the simulation system, we discovered some minor issues in the previous program, which led to lower precision in single shot measurements. After promptly correcting the program, the new results are consistent with the previous conclusions, but the specific values obtained are different. The revised manuscript now ensures consistency between the data in the abstract and the main text.

 

Comments 3: In Table 2, Universal Time is used, but the readers cannot know the local time inferred from Table 2 and the text. The longitude can be added in the text. Moreover, the time of experiment in Yunnan should be introduced. The precision may depend on local time, due to the water vapour in the atmosphere may vary with local time.

 

Response 3: Thank you for your suggestions. In the revised manuscript, Table 2 has been updated to include the latitude and longitude coordinates of the observation station. Additionally, we have also provided information on the timing of the ground-based experiments conducted at the Yunnan Observatories.

 

Comments 4: The conclusion is verbose, such as Lines 433-436 can be deleted. It’s better to shorten the conclusion, which summarize the work and emphasis the results of this study.

 

Response 4: Thank you for your suggestions. In response to your and the journal editor's comments, we have separated the 'Results' and 'Discussion' sections of the manuscript and shortened their length.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

There are still some minor issues to correct, as follows: 

Equation (11) The parenthesis (R_E+h_t) does have the exponent as subindex instear of superindex. 

Definition in equation (21) is only valid for certain relation between t_1 and t_2, please clarify. 

Comments on the Quality of English Language

Ok

Author Response

Comments 1: Equation (11) The parenthesis (R_E+h_t) does have the exponent as subindex instear of superindex.

 

Response 1: Thank you for pointing out my mistake. I must have made an error while typing this equation. The correction has already been made in the revised manuscript.

 ​

Comments 2: Definition in equation (21) is only valid for certain relation between t_1 and t_2, please clarify.

 

Response 2: Thank you for your suggestion. Based on your advice, I have added the corresponding explanation in the revised manuscript. The times t₁ and t₂ here should be any two moments within the detector's response time interval，and it is required that t_1< t_2.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This is a revised version of a paper that intends to introduce a correction for the effect of libration in LLR observations. In the previous review I indicated that a substantial shortening of the paper presentation would be beneficial. Now I notice that it has even grown longer. Since a large number of generally known facts from LLR are reiterated here, I would strongly recommend to take them out and refer the reader to one or two references. 

 

The novel part deals with the distinction of geometrical and physical libration, but it does not explain the difference between the two. This should be explained. 

 

The paper assumes tilt effects about the central position of the flat reflector arrays (this is how I interpret section 2.2.1, but is this really the case? I do not think so. The CCR arrays are moving sideways with the surface of the moon over an angular range of nearly 8 degrees. That means that the point about which it tilts is way outside the area of the CCR array, so a reduction to the center is not an adequate representation. This is what I was asking for in the last review: Please make a clear distinction between precision and accuracy. For the test of theories it is only accuracy that matters. 

 

Equation 10 in section 2.2.2 is wrong. Instead of the reduced Planck constant there should be the Planck constant h itself. Furthermore, the authors talk about number of photons, which is also incorrect. This equation describes the number of generated photo-electrons in the detector. That is a different thing. 

 

Section 2.2.3 describes detection probabilities as function of time, but does not make use of it subsequently. The simulated return pulse rates are all normalized to 1 for each row of reflectors, which does not reflect reality as the experimental data in figure 18 clearly shows. This is not consistent. The pulse spreading from a tilted flat panel is not caused by random errors as asserted in row 317 and 318. It is rather causing a systematic bias and that is affecting the accuracy of the measurements. 

 

In the previous report I have asked what other error sources are contributing to the dispersion of the measurements. Ok, the answer is the timer contributes by 5 ps, but what is the dependence on the intensity, usually in this community referred to as the “timewalk”. This is strongly detector dependent and can be as large as 100 ps or even worse. Since the authors compare local ground target measurements with lunar echoes, this can cause a large discrepancy.  

 

The ground target measurements presented in section 3.2 are not epoch sensitive, so the presentation of the dates of the conducted experiments is not required. I do not believe that the signal levels between ground target and LLR measurements are well matched. In the lower part of figure 15, there are more than 30 000 returns presented in only 200 seconds. This return rate appears to be much to high. 

 

The result discussed for fig. 16 and 17 are a bit obscure to me. What are the units of the y axis in fig. 17? How was the array rotated? If it was rotated about the middle of the array, as indicated in figure 11, then there is a different problem. The rotation around the center should only change the spread but not the bias. To me it seems that this bias is introduced by the intensity variation. As the reflectors become less and less sensitive due to the large tilt, the intensity drops and a timewalk bias is introduced, which would make figure 17 plausible. 

 

In general, I would strongly recommend to clearly present the results that reduce the systematic bias effects in the LLR measurements. How much of that is captured by existing models and how much is missing from the data treatment since the experienced tilt is not a rotation about the center of the array.  

 

Finally, I am a bit irritated by the paragraph between the line 260 and 267. Why should there be the necessity for a fitting procedure on a static ground target. It appears to me that this kind of treatment rather adds than removes errors.

 

 

Section 2.1.2 above table 3: round-trip flight time -> round trip flight time difference.

 

 

 

 

 

Comments on the Quality of English Language

The language is fine with very few small issues.

Author Response

Comments 1: This is a revised version of a paper that intends to introduce a correction for the effect of libration in LLR observations. In the previous review I indicated that a substantial shortening of the paper presentation would be beneficial. Now I notice that it has even grown longer. Since a large number of generally known facts from LLR are reiterated here, I would strongly recommend to take them out and refer the reader to one or two references.

 

Response 1: Thank you. We originally agreed with your comment. But,in the first round of review comments, one of the reviewers suggested that we explain parts of the LIDAR equation and add an analysis of the measured data from the German station. Another reviewer recommended making the manuscript as detailed as possible in order to make easier to understand the idea for any reader. As a result, the revised manuscript is longer than the original one.

Regarding the peer review of this article, there were a total of four reviewers, and some of the revision suggestions were somewhat inconsistent, which has put us in a difficult position. We sincerely hope for your understanding.

 

Comments 2: The novel part deals with the distinction of geometrical and physical libration, but it does not explain the difference between the two. This should be explained.

 

Response 2: Thank you for your professional suggestions. Based on your suggestion, we have added content in the revised manuscript to explain geometric libration and physical libration.

 

Comments 3: The paper assumes tilt effects about the central position of the flat reflector arrays (this is how I interpret section 2.2.1, but is this really the case? I do not think so. The CCR arrays are moving sideways with the surface of the moon over an angular range of nearly 8 degrees. That means that the point about which it tilts is way outside the area of the CCR array, so a reduction to the center is not an adequate representation. This is what I was asking for in the last review: Please make a clear distinction between precision and accuracy. For the test of theories it is only accuracy that matters.

 

Response 3: Thank you for your comments, we believe your point of view is correct. We wish to make an effort to explain. Our approach should be quite similar to your understanding. As shown in the figure1 below, the CCR array placed on the lunar surface is intended to point towards the center of the Earth. However, due to the lunar libration, from the perspective of the Moon, the normal of the CCR array deviates by an angle φ from the direction pointing to the Earth's center. Therefore, we consider that the CCR array is affected by lunar libration, similar to a rotation around the x and y axes in the coordinate system, resulting in a certain tilt angle. We are studying the echo pulse broadening by treating each CCR in the array as an individual reflection point. In contrast, the two papers referenced later treat the CCR array as a whole in their studies. ([1] Samain, E., J. F. Mangin, C. Veillet, J. M. Torre, P. Fridelance, J. E. Chabaudie, D. Feraudy et al. "Millimetric lunar laser ranging at OCA (Observatoire de la Côte d'Azur)." Astronomy and Astrophysics Supplement Series 130, no. 2 (1998): 235-244.); [2]Turyshev, Slava G., James G. Williams, William M. Folkner, Gary M. Gutt, Richard T. Baran, Randall C. Hein, Ruwan P. Somawardhana, John A. Lipa, and Suwen Wang. "Corner-cube retro-reflector instrument for advanced lunar laser ranging."Experimental Astronomy 36 (2013): 105-135.)

In LLR experiments, we believe that the echoes come from the CCR array. So the measured values represent the distance from the station to the CCR array. As for the bias caused by the changes in the central point position due to lunar libration, corrections for these deviations have been considered in the forecast of the ranging experiment.

Figure 1 Schematic diagram showing the deviation of the lunar CCR array's orientation towards Earth due to lunar libration

 

Figure 2 Retroreflector orientation[1] This figure is from the aforementioned reference 1, and it clearly also represents the CCR array as rotating relative to the Earth’s direction.

Comments 4: Equation 10 in section 2.2.2 is wrong. Instead of the reduced Planck constant there should be the Planck constant h itself. Furthermore, the authors talk about number of photons, which is also incorrect. This equation describes the number of generated photo-electrons in the detector. That is a different thing.

 

Response 4: Thank you for pointing out our mistakes. Indeed, an incorrect letter was used in Equation 10. In the revised manuscript, the Planck constant is now denoted by the letter ℎ, and the description of ?_e in the equation has been corrected to represent the mean number of photoelectrons recorded by the detector.

 

Comments 5: Section 2.2.3 describes detection probabilities as function of time, but does not make use of it subsequently. The simulated return pulse rates are all normalized to 1 for each row of reflectors, which does not reflect reality as the experimental data in figure 18 clearly shows. This is not consistent. The pulse spreading from a tilted flat panel is not caused by random errors as asserted in row 317 and 318. It is rather causing a systematic bias and that is affecting the accuracy of the measurements.

 

Response 5: Thank you! In Figures 7 and 8, we indeed used normalized plots of the superimposed echo pulses without considering the impact of detection probability. These two figures primarily compare the waveform of the emitted laser pulse with the shape of the echo envelope. Without normalization, the waveform values of the echo relative to the emitted pulse are very small, making it difficult to visually compare them on the same graph. This was done to make the theoretical analysis clearer.

 However, in the simulation results, the effect of detection probability has been included in the echo histogram in Figure 9. The phenomenon is not as pronounced as in the ground target experiment shown in Figure 18. The reason is the difference in the magnitude of the number of echo photons. In the lunar simulation experiment, the echo is relatively weaker, making this effect less noticeable. In contrast, in the ground target experiment, to fully present the experimental results, a larger number of echo photons was used, making the earlier arriving photons more likely to be detected, thus making the phenomenon more apparent.

We apologize if our expression was not accurate and caused any misunderstanding. Lines 317 and 318 describe the random errors due to the timing and spatial positions of the echo photons arriving at the detector's target surface. We have made revisions to this in the revised manuscript.

 

Comments 6: In the previous report I have asked what other error sources are contributing to the dispersion of the measurements. Ok, the answer is the timer contributes by 5 ps, but what is the dependence on the intensity, usually in this community referred to as the “timewalk”. This is strongly detector dependent and can be as large as 100 ps or even worse. Since the authors compare local ground target measurements with lunar echoes, this can cause a large discrepancy.

 

Response 6: Thank you for your careful reading of this manuscript. Please allow us to explain the revisions made in the manuscript. In the revised manuscript, we added sources of errors such as the detector and timer. However, our primary focus is on the effect of the tilt of the CCR array on LLR, so we used smaller values for these error sources in the simulations. The uncertainty caused by "timewalk" is considered in this part of the detector, with values ranging from 10 to 50 ps.

For the ground target experiment, we used a kilohertz laser ranging system, which resulted in stronger echoes compared to actual LLR echoes. However, this does not directly compare with LLR data but rather uses the ground target experiment data to demonstrate that narrow pulse width laser ranging can reflect the tilt of the CCR array.

 

Comments 7: The ground target measurements presented in section 3.2 are not epoch sensitive, so the presentation of the dates of the conducted experiments is not required. I do not believe that the signal levels between ground target and LLR measurements are well matched. In the lower part of figure 15, there are more than 30 000 returns presented in only 200 seconds. This return rate appears to be much to high.

 

Response 7: Thank you.We agree with you that the dates of the conducted experiments is not required. But,in the previous review comments, one of the reviewers raised a question: “In Table 2, Universal Time is used, but the readers cannot know the local time inferred from Table 2 and the text. The longitude can be added in the text. Moreover, the time of experiment in Yunnan should be introduced. The precision may depend on local time, due to the water vapour in the atmosphere may vary with local time.” In response to this suggestion and to ensure the completeness of the experimental information, we have added the timing details of the experiment in the revised manuscript.

Regarding the issue of the number of echo photons in the ground target experiment, we used a kilohertz laser system for this experiment, which resulted in a higher number of echo photons. This certainly differs from the actual LLR scenario(Currently, the laser ranging systems actually used at various stations operate at 10 Hz or 20 Hz.). Our purpose in conducting the experiment this way was solely to verify the theoretical analysis presented earlier.

 

Comments 8: The result discussed for fig. 16 and 17 are a bit obscure to me. What are the units of the y axis in fig. 17? How was the array rotated? If it was rotated about the middle of the array, as indicated in figure 11, then there is a different problem. The rotation around the center should only change the spread but not the bias. To me it seems that this bias is introduced by the intensity variation. As the reflectors become less and less sensitive due to the large tilt, the intensity drops and a timewalk bias is introduced, which would make figure 17 plausible. In general, I would strongly recommend to clearly present the results that reduce the systematic bias effects in the LLR measurements. How much of that is captured by existing models and how much is missing from the data treatment since the experienced tilt is not a rotation about the center of the array.

 

Response 8: We believe your point of view is correct. The y-axis of Figure 17 represents the peak coordinates of the histogram of ranging echoes under different tilt angles. The CCR is not tilted in the latitude direction but has varying tilt angles in the longitude direction. Figure 17 is intended to demonstrate that as the tilt angle of the CCR array increases, the peak point of the histogram of ranging echoes shifts forward, which can lead to bias in the ranging distance calculation and affect the accuracy of the ranging results. Due to the reduced effective reflective area of the CCR, the echo intensity weakens, which indeed introduces some bias due to "timewalk". However, echo energy is also affected by various factors, and the resulting system bias are topics that need further investigation. Improving the accuracy of LLR will also be a major focus in our upcoming studies. Currently, this paper focuses only on the effect of the tilt of the CCR array on the precision of lunar laser ranging.

 

Comments 9:Finally, I am a bit irritated by the paragraph between the line 260 and 267. Why should there be the necessity for a fitting procedure on a static ground target. It appears to me that this kind of treatment rather adds than removes errors.

 

Response 9: Thank you for pointing out our mistake. We used polynomial fitting when processing the SLR or LLR measured data and writing the simulation program, but we did not apply fitting to the data during the processing of the ground target ranging data. The previous manuscript had an incorrect expression, and we have revised this in the resubmitted version.

 

Comments 10:Section 2.1.2 above table 3: round-trip flight time -> round trip flight time difference.

Response 10: Thank you. In the revised manuscript, we have corrected it to the expression you suggested.

 

Author Response File: Author Response.pdf