A Mathematical Method for Estimating the Critical Slope Angle of Sheet Erosion

Round 1

Reviewer 1 Report

see attached pdf

Comments for author File: Comments.pdf

Text will need another thorough edit by an experienced author before resubmission

Author Response

 

                                                   List of response to the comments                  

Manuscript Title: “A Mathematical Method for Estimating the Critical Slope Angle of Sheet Erosion”

We sincerely thank the reviewers for offering careful, helpful and insightful comments that have allowed us to greatly improve the quality of our manuscript. We have incorporated all the reviewer’s specific suggestions and hope that we have fully addressed their general comments. Our responses to each of the reviewer’s comments are addressed below.

 

COMMENTS TO THE AUTHOR(S)

Main comments：

1. In fact, what might be useful is an appendix containing a list of all symbols and acronyms to help readers interpret the ideas as they are presented. This would help alleviate some of the lack of clarity in the current version

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We added the symbols and acronyms in appendix 2.

 

2. Abstract: seems reasonable. The word ‘promptly’ (l.13) does not make sense – do you mean ‘properly’, or ‘conveniently’?

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We revised abstract to better convey the novelty and contribution of this study.

3. Section 2.1 further explanation is required of the reasons for using SHPL and SIL. Readers may not know the terminology or why it is being used here.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We included appendix 1 to enhance the comprehension of the derivation process for the formula.

4. l.77 you seem to imply that rainfall intensity remains constant during rainfall – is this what you mean?

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. Previous studies had made the assumption that there was no temporal variation in rainfall intensity during precipitation events.

5. 142 is confusing. Could you say instead: ‘By equating the Left Hand Side to zero’.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion and fully adopted this sentence.

6. Abstract and conclusions need to be rewritten by revealing the novelty and contribution of this study.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We rewrite abstract and conclusions sections.

7. In Figs.2 (b) and 3 (b) there are only a few points and so the use of RSME seems inappropriate – it may be better to display this information differently as a table for instance

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion and fully adopted this suggestion. We deleted the RSME, and the distinction in comparison was illustrated in Figure a.

Specific comments：

Finally, use of superscripts and subscripts – this needs consistency, please check this throughout. See lines 148, 150, 218, 219, 223, 224, 303, 304 etc…..

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The entire article had been thoroughly reviewed and necessary adjustments had been made to rectify formatting errors and improve sentence structure.

 

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Authors,

After carefully reading your manuscript, I have the following suggestions that might improve journal readers' comprehension of the text:

1.The methodology proposed by you, as well as the WEPP model, uses the GA method, however, when comparing the results of both models, they are different. Could you explain why they are different and justify why your model might be a better approach to explain this phenomenon?

2. Please explain and develop mathematically how is equation 15 obtained? (line 209)

3. Include a more in-depth discussion of the differences observed in Figure 2 and write down some thoughts about the limitations of the results of this study. 8the limitations could be included in discussion section as well as in the conclusions.

4. please review all the units used because thera are some errors like kg m-3 instead kg m^-3 (line 179)

5. Sections 3.3 and 3.4 have the same title, it must be an error. Please correct it.

 

 

The English of the manuscript needs a proofreading and style revision process.

Author Response

 

                                                   List of response to the comments                  

Manuscript Title: “A Mathematical Method for Estimating the Critical Slope Angle of Sheet Erosion”

We sincerely thank the reviewers for offering careful, helpful and insightful comments that have allowed us to greatly improve the quality of our manuscript. We have incorporated all the reviewer’s specific suggestions and hope that we have fully addressed their general comments. Our responses to each of the reviewer’s comments are addressed below.

 

COMMENTS TO THE AUTHOR(S)

1. The methodology proposed by you, as well as the WEPP model, uses the GA method, however, when comparing the results of both models, they are different. Could you explain why they are different and justify why your model might be a better approach to explain this phenomenon?

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. Compared to the GA model (used in Eq. 4), the modified GA model (used in WEPP) considered the influence of slope angle on soil infiltration. However, there was little difference between the GA model and the modified GA model when the rainfall intensity was small (e.g., I/f< 2), Hence, there exists a slight disparity between the outcomes obtained from the WEEP model and Equation (4). Equation (4) needs two parameters (rainfall intensity and soil infiltration rate) to estimate the CSA, whereas the WEPP model requires seven parameters. Thus, the mathematical equations derived in this study are simpler and more efficient to apply.

2. Please explain and develop mathematically how is equation 15 obtained? (line 209)

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The intensity of heavy rainfall is typically inversely proportional to its duration, thus the rainstorm function can be assumed to exhibit an inverse relationship. The intensity of normal rainfall usually exhibits a quadratic pattern, characterized by an initial increase followed by a subsequent decrease over time. The soil infiltration rate exhibits its highest capacity at the beginning of rainfall, gradually decreasing with duration until it eventually stabilizes. Therefore, soil infiltration rate can be described by an exponential function.

3. Include a more in-depth discussion of the differences observed in Figure 2 and write down some thoughts about the limitations of the results of this study. 8the limitations could be included in discussion section as well as in the conclusions. Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. Due to data source limitations, we rediscovered field observation data for validation the derived mathematical equations, and subsequently provided an explanation for the potential factors contributing to the observed discrepancies.
4. l.77 you seem to imply that rainfall intensity remains constant during rainfall – is this what you mean?

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. Revisions were made to rectify formatting errors.

5. Sections 3.3 and 3.4 have the same title, it must be an error. Please correct it.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The title of Section 3.4 has been revised to "Enhancements in Mathematical Equations".

Author Response File: Author Response.docx

Reviewer 3 Report

The novelty of this manuscript has to be clarified.

 

The literature review is not critical and is incomplete. What are the weak points of the existing models? What is the necessity of this study?

 

Why does one need to use such a mathematical model? Estimation of some variables such as ? is not easy in field.

The authors should mention the assumptions to use this model. Many factors affect shear stress estimation for the erosion process which have not been considered in this study. Erosion depends on the grain size, repose angle which have to be discussed in the text.

 

How did the authors collect data for this model? The reliability of data should be explained.  

Where is the source of equation (2)?

Equation (1) uses ta on the left side and n on the right side. Both ta and n use the same variables. So, this may present a spurious equation. Calibration and validation of this equation can show whether this equation is spurious or not.

Development of τ_A and τ’_A is not clear in the text. In addition, no statistics are presented for reliability of these equations as well as (9) and (11).

Figures were not explained well.  Fig. 2 shows that the comparison of the observed data with equation (9) is not good.

 

There are many limitations in this research which should be explained.

It is very difficult to draw any conclusion based on the results, therefore, the results are specific and may not be easily extended.

Conclusion should be rewritten showing better contribution of this research. The authors present three objectives in lines 94-98, however, they do not reply to these objectives

More Physical and theoretical interpretation of the results should be presented in the text.

 

 

In summary:

The superiority of this study and methodology is not clear in comparison to existing ones.

The literature review is not critical and pertinent regarding the existing methods.

 

The authors should present more profound investigation around their methodology, comparing their results with existing studies.

What is the novelty and contribution of this study?

 

 The reliability of data and the presence of outliers in data series should be explained.

The figures were not described well. Almost all parts of the manuscript are presented qualitatively rather than quantitatively.

The results section is purely a report rather than any scientific achievement. 

 

Abstract and conclusions need to be rewritten by revealing the novelty and contribution of this study.

How can the authors manage the limitations of this study?

More theoretical explanation and comparison is required in the results and discussion

 

 

Author Response

                                                   List of response to the comments                  

Manuscript Title: “A Mathematical Method for Estimating the Critical Slope Angle of Sheet Erosion”

We sincerely thank the reviewers for offering careful, helpful and insightful comments that have allowed us to greatly improve the quality of our manuscript. We have incorporated all the reviewer’s specific suggestions and hope that we have fully addressed their general comments. Our responses to each of the reviewer’s comments are addressed below.

 

COMMENTS TO THE AUTHOR(S)

1. The superiority of this study and methodology is not clear in comparison to existing ones. The literature review is not critical and pertinent regarding the existing methods. The authors should present more profound investigation around their methodology, comparing their results with existing studies.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion and rewrite the introduction.

2. What is the novelty and contribution of this study?

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. This study developed mathematical equations considering rainfall intensity and soil infiltration to efficiently estimate both instantaneous (a given instant during rainfall) and cumulative CSAs. Additionally, this study provides valuable insights into the impact of soil bulk density, rainfall intensity and duration on the CSA dynamics.

3. The reliability of data and the presence of outliers in data series should be explained.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We rediscovered field observation data for validation the derived mathematical equations, and we provided an explanation for the potential factors contributing to the discrepancies.

4. The figures were not described well. Almost all parts of the manuscript are presented qualitatively rather than quantitatively. The results section is purely a report rather than any scientific achievement.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We have revised the results and discussion section.

5. Abstract and conclusions need to be rewritten by revealing the novelty and contribution of this study.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We rewrite abstract and conclusions sections.

6. Abstract and conclusions need to be rewritten by revealing the novelty and contribution of this study.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We added the section of limitations.

7. How did the authors collect data for this model? The reliability of data should be explained.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. Relevant data for validation are extracted from peer-reviewed publications.

8. Where is the source of equation (2)? Equation (1) uses ta on the left side and n on the right side. Both ta and n use the same variables. So, this may present a spurious equation. Calibration and validation of this equation can show whether this equation is spurious or not.Development of τA and τ’A is not clear in the text.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. We have included an appendix detailing the derivation of the equations.

 

Author Response File: Author Response.docx

Reviewer 4 Report

his article establishes mathematical equations for instantaneous CSA and cumulative CSA through a method of mathematical equation derivation. Furthermore, the accuracy of the calculated values from the equations was validated through the comparison with actual observational data and mathematical simulation results. The article analyzes the trends in instantaneous CSA and cumulative CSA with the ratio of I/f. The various rainfall events were set as scenarios, allowing for the analysis of discrepancies between instantaneous CSA, cumulative CSA, and actual observational data. This analysis provided theoretical support for the practical applications of instantaneous CSA and cumulative CSA. However, the article still exhibits several issues.Some comments are listed as followings:

 (1) The focus on rainfall intensity and soil infiltration as the sole factors for elucidating the physical mechanisms of CSA in this article is overly limited. It is crucial not to overlook the significant impact of the physical changes in soil structure caused by soil permeability on sheet erosion.

(2) The derivation process of Equation (8) should be reviewed again to ensure its accuracy since it forms the theoretical foundation of this study's analysis.

(3) The figures used in the manuscript should be further improved to ensure their accuracy and facilitate a better understanding of the research content presented in this article.

(4) In terms of writing details, it is recommended to thoroughly review the entire manuscript, correct any major expression errors, and examine language expression, and present logical relationships more clearly.

The language expression is basically clear, but further refinement is still needed

Author Response

 

                                                   List of response to the comments                  

Manuscript Title: “A Mathematical Method for Estimating the Critical Slope Angle of Sheet Erosion”

We sincerely thank the reviewers for offering careful, helpful and insightful comments that have allowed us to greatly improve the quality of our manuscript. We have incorporated all the reviewer’s specific suggestions and hope that we have fully addressed their general comments. Our responses to each of the reviewer’s comments are addressed below.

 

COMMENTS TO THE AUTHOR(S)

1. The focus on rainfall intensity and soil infiltration as the sole factors for elucidating the physical mechanisms of CSA in this article is overly limited. It is crucial not to overlook the significant impact of the physical changes in soil structure caused by soil permeability on sheet erosion.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The limitations of the equation were elucidated in section 3.5 to explicate its inherent flaws.

2. The derivation process of Equation (8) should be reviewed again to ensure its accuracy since it forms the theoretical foundation of this study's analysis.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The accuracy of the derivation in Eq. 8 was re-verified to ensure its accuracy.

3. The figures used in the manuscript should be further improved to ensure their accuracy and facilitate a better understanding of the research content presented in this article

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The remaking of certain figures was undertaken, for instance, Figure 2.

4. In terms of writing details, it is recommended to thoroughly review the entire manuscript, correct any major expression errors, and examine language expression, and present logical relationships more clearly.

Authors’ Response: We sincerely appreciate the reviewer’s helpful suggestion. The entire article had been thoroughly reviewed and necessary adjustments had been made to rectify formatting errors and improve sentence structure.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I have scrutinised the revised version of the submission and am content that all my objections have been addressed satisfactorily.

Reviewer 2 Report

Dear authors,

Thank you very much for the responses to all the suggestions made about your manuscript. I accept your answers, but I am surprised to see different results in various calculations in your new manuscript version. I hope now your results are verified and well written.

It is worth mentioning that it is absolutely the responsibility of the authors to ensure that the calculations were carried out correctly. Furthermore, I must emphasize that the referee does not have sufficient information to recalculate the different pre-established processes.

 

Reviewer 3 Report

Thank you for replying to my comments.  It is now ready to be accepted.