Uncertainty-Based Scale Identification and Process–Topography Interaction Analysis via Bootstrap: Application to Grit Blasting

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please open the attachment file

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English needs to be updated

Author Response

1. English needs to be updated.

Thank you very much for your comment that give us the opportunity to improve our work. The English has been thoroughly reviewed and corrected to ensure optimal reading clarity.

 

2. Strengthening the theoretical explanation is necessary.

We have expanded the sections related to theoretical explanations. First, we added the formula for calculating the Sdr parameter on lines 148–149. The theoretical explanations of the physical phenomenon of blasting have been added on lines 116–137.

3. Other factors like temperature and humidity that may influence surface morphology should be considered by researchers.

 

We conducted additional research regarding the potential influence of humidity and temperature on the surface state. First, during the blasting process, the temperature is too low to alter the structure of the TA6V alloy. Second, this alloy is stainless and is even used for biocompatible bone implants. Third, while it is true that humidity and temperature can influence measurement results, the measurement room where the samples were analysed is a ventilated and temperature-controlled environment. This minimizes any potential impact of these factors on the interferometry measurements (line 600-607) .

4. The statistical methods for smoothing and regression should be carefully studied by researchers.

 

We have taken your comments regarding the use of the Gaussian filter and statistical methods into account, and we have added sections in both the introduction and the discussion. Filtering methods are widely used in roughness analysis, providing a reliable and simple way to process 2D or 3D signals depending on how the surface profile is studied. Common methods include Gaussian [33–35], robust [36,37], and spline (line 606 filters. Gaussian filters, standardized under ISO 16610-21[38], are the most commonly used method [39]. (line 144-148)  Regarding the use of Gaussian filtering, two remarks can be made. This surface filtering method is the most common in surface processing, it is well-documented, and its limitations are well known. However, we acknowledge that for fractal surfaces, such as those generated by blasting, distortions may be introduced by the Gaussian filter on the sharpest features. The chapter on the use of filters in the book by Blunt and Jiang [36] clearly explains the difference between robust and Gaussian filters on a plateau honed surface. However, the robust filter requires longer computation time, which explains our choice in this study. A comparison between both is necessary. (Line 606-613) The statistical methods were discussed in your seventh commentary also.

 

5. Its possible that the finding won’t hold true for other materials or surface treatment methods.

The results of this study may encourage further research to continue determining the reference scale of complex surfaces created by different processes that allow for varying intensity of surface modification. Thus, it is possible that the results could be compared with acid etching, electric discharge machining, and shot peening (i.e., steel beads blasting) on titanium alloys like in the study. By varying parameters such as the type of filament used and the electrical intensity [48] for electric discharge machining, it is possible to modify the surface topography and study the relevant scale for this process. The same applies to soaking time in acid during chemical etching [49]. Some studies on functional optimization already exist and aim to identify a relevant scale for process analysis. For instance, one study highlights the sensitivity of hMSC cells to topographical features at different scales (Ra, Sm) on surfaces generated by EDM (electric discharge machining) [50].( Ligne 645-656. )

6. The authors ought to contrast the study blasting with the body of knowledge already available on the subject.

We were able to compare this study with others, which we presented in the introduction and discussion:

The use of the Sa parameter is documented in the literature for characterizing the influence of blasting on materials [10]. However, the method of calculating the relative area proposed in this study is considered more suitable because the relative area captures surface irregularities at different scales and is sensitive to large-scale features such as deep recesses and significant local variations.

The Sdq parameter is discussed in the study by Ho et al. [15]. Impact craters are generated by SiC particles striking the surface of the substrate. The repetitive impacts can lead to the development of a plastically deformed layer and the formation of additional craters on top of existing ones. This process results in the creation of a series of small peaks and valleys with sharp edges, whose slopes can be characterized using Sdq.

This study, which is very similar to ours, provides a scale of relevance for characterizing the relationship between the value of the multiscale Sdq parameter and the blasting pressure. The filtering is a low-pass filter with a cut-off of 120 µm, which also corresponds to our relevance scale with the Sdr method. This result is interesting because the pressures used in their study vary slightly, ranging from 1 to 7 bar, whereas our samples were blasted from 2 to 8 bar. This comparison supports the generalization of our results (lines 580–595).

7. The authors ought to mention any possible flaws and upcoming research projects.

We have indeed discussed the study's limitations and potential future studies in the discussion section. Specifically, the relationship between blasting pressure and developed area may not be entirely linear across all scales. For this reason, we address the following paragraph in the discussion:

The results indicate that the relationship between pressure and roughness evolution is not strictly linear for some blasting media, such as C300, as initially hypothesized. At lower pressures (e.g., 2 bar), roughness increases are moderate and uniform, reflecting gradual material erosion. In contrast, at higher pressures (e.g., 8 bar), the changes become pronounced, with deeper asperities and irregular textures, likely due to enhanced impact energy and its effects on asperity deformation and void formation.

Given these results, a nonlinear model such as Y=aP^b+c may better capture the relationship between pressure and roughness, where Y represents a roughness parameter, P is the pressure, and a,b,c, are the model parameters. Notably, the model converges to the linear approximation Y=aP+c, as discussed earlier in this article, when bbb is statistically equal to 1. Significant deviations from b≈1  could reveal transitions in the dominant mechanisms of surface modification under different materials and process conditions.

To test if b < 1,b=1 or b>1, bootstrapping techniques proposed in this paper (simple, paired, or residual bootstraps) could assess the statistical significance of parameters a, b and c  providing insights into the interplay between pressure, media type, and substrate response. However, implementing such a nonlinear model would require solving a regression problem through iterative optimization algorithms, ensuring convergence to a unique global minimum. This added complexity would extend beyond the current study's scope. We plan to apply this framework to further research on the nonlinear effects of contact pressure and media type on surface topography. (line 615-635)

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Referee Report

On the paper “ Uncertainty-Based Scale Identification and Process-Topography Interaction Analysis via Bootstrap: Application to Sandblasting “ (fractalfract-3340097) by the authors François Berkmans, Julie Lemesle, Robin Guibert, Michal Wieczorowski, Christopher Brown, Maxence Bigerelle submitted to the Fractal Fract

 

This is interesting paper. It reports various methods to determine a pertinent scale for evaluating the relationship between relative area and grit blasting pressure. The findings obtained contribute a scale of 10,000 µm² for the Patchwork method and a 120 µm cut-off length for the Sdr method, derived from bootstrapping on residual regression across all media. At the relevant scale, every value of R2 inferior to 0.83 is not significant with the threshold of 5% for the two methods of calculation of the relative area. The obtained results are reliable without any doubts. However, I have some addition only after implementation of which the paper can be recommended to publication:

1.     The authors should provide in the 1. Introduction concrete examples of materials promising for the production of wireless devices for transmitting digital information and analyzing it:

(1). H. de O. Barros, R.F. Abreu, T.O. Abreu, W.V. de Sousa, F.E.A. Nogueira, F.F. do Carmo, J.E.V. de Morais, J.P.C. do Nascimento, M.A.S. da Silva, R.S. da Silva, S.V. Trukhanov, D. Zhou, C. Singh, A.S.B. Sombra, High thermal stability of the microwave dielectric properties of ZnNb₂O₆ with CaTiO₃ addition, Phys. B 695 (2024) 416547. https://doi.org/10.1016/j.physb.2024.416547.

2.     The authors should provide in the 1. Introduction concrete examples of modeling and solving optimization problems:

(2). Y. Guo, D. Zhou, D. Li, W. Zhao, Y. Wang, L. Pang, Z. Shi, T. Zhou, S. Sun, C. Singh, S. Trukhanov, A.S.B. Sombra, G. Chen, Improved energy storage performance of sandwich-structured P(VDF-HFP)-based nanocomposites by the additions of inorganic nanoparticles, J. Mat. Chem. C 11 (2023) 6999-7009. https://doi.org/10.1039/D3TC00979C.

3.     The proposed 2 papers should be inserted in References.

 

The paper should be sent to me for the second analysis after the major revisions.

Comments for author File: Comments.pdf

Author Response

Referee Report

 

This is interesting paper. It reports various methods to determine a pertinent scale for evaluating the relationship between relative area and grit blasting pressure. The findings obtained contribute a scale of 10,000 μm² for the Patchwork method and a 120 μm cut-off length for the Sdr method, derived from bootstrapping on residual regression across all media. At the relevant scale, every value of R2 inferior to 0.83 is not significant with the threshold of 5% for the two methods of calculation of the relative area. The obtained results are reliable without any doubts. However, I have some addition only after implementation of which the paper can be recommended to publication:

 

1. The authors should provide in the 1. Introduction concrete examples of materials promising for the

production of wireless devices for transmitting digital information and analyzing it:

(1). H. de O. Barros, R.F. Abreu, T.O. Abreu, W.V. de Sousa, F.E.A. Nogueira, F.F. do Carmo, J.E.V. de Morais,

J.P.C. do Nascimento, M.A.S. da Silva, R.S. da Silva, S.V. Trukhanov, D. Zhou, C. Singh, A.S.B. Sombra, High

thermal stability of the microwave dielectric properties of ZnNb2O6 with CaTiO3 addition, Phys. B 695 (2024) 416547.

https://doi.org/10.1016/j.physb.2024.416547.

 

2. The authors should provide in the 1. Introduction concrete examples of modeling and solving

optimization problems:

(2). Y. Guo, D. Zhou, D. Li, W. Zhao, Y. Wang, L. Pang, Z. Shi, T. Zhou, S. Sun, C. Singh, S. Trukhanov,

A.S.B. Sombra, G. Chen, Improved energy storage performance of sandwich-structured P(VDF-HFP)-based

nanocomposites by the additions of inorganic nanoparticles, J. Mat. Chem. C 11 (2023) 6999-7009.

https://doi.org/10.1039/D3TC00979C.

 

3. The proposed 2 papers should be inserted in References.

The paper should be sent to me for the second analysis after the major revisions.

 

Response:

We thank the reviewer for their thorough review and comments, which have greatly improved our work. We have added the references suggested by the reviewer to the introduction of our study, lines 100 to 113.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript presents a comprehensive study on the relationship between grit blasting pressure and surface roughness, using various bootstrapping methods to determine the most relevant scale for surface characterization. The authors effectively compare three bootstrapping techniques (simple, paired, and residual) alongside two relative area calculation methods (Patchwork and Sdr) to assess their ability to capture the influence of pressure on surface morphology. The findings suggest that the residual bootstrap method provides the most reliable results for identifying the relevant scale, highlighting the importance of this approach in surface treatment analysis. Overall, the study offers valuable insights into grit blasting processes and contributes to more effective material processing strategies.

1.      The manuscript compares multiple bootstrapping methods for determining the relevant scale. Could the authors provide more clarity on why the residual bootstrap method was ultimately chosen as the most promising approach over the other methods, particularly the paired bootstrap method?

2.      The results show variability in the surface roughness across different media (e.g., G 100, G 250, C 300). Could the authors discuss in more detail how the intrinsic properties of each medium might influence the observed results, particularly the differences in slope and intercept distributions?

3.      In Figure 9, the R² values for the Patchwork and Sdr methods show significant differences at the relevant scale. Could the authors elaborate on the potential implications of these differences for practical applications in grit blasting?

4.      The study uses a low-pass Gaussian filter with a 120 μm cut-off. Could the authors discuss the choice of this specific cut-off value in more detail, and how sensitive the results might be to variations in this parameter?

5.      Regarding the tile size in the Patchwork method, how was the specific tile size of 10,000 μm² chosen, and what is the expected impact of selecting different tile sizes on the results of the regression analysis?

6.      In the bootstrap analysis, the threshold R² values for different bootstrapping methods were found to be 0.59 (simple bootstrap), 0.91 (paired bootstrap), and 0.83 (residual bootstrap). How sensitive are these threshold values to changes in the bootstrapping procedure or the choice of media?

7.      How do the results compare for grit blasting at pressures of 2 bar, 4 bar, and 8 bar? What specific differences were observed in the surface morphology across these different pressures, particularly for the C 300 medium?

8.      In Figure 8, the distributions of intercepts and slopes at the relevant scale were analyzed. What trends were observed for G 100 (fine glass beads) and C 300 (angular media) in terms of their relationship between slope stability and the initial roughness?

9.      In terms of pressure variation, how do the results from the study, which uses pressures from 2 to 8 bar, compare with similar studies that use a pressure range of 1 to 7 bar? How might these differences in pressure affect the study's conclusions?

10.   The authors mention that the surface topographies of TA6V samples were grit blasted at 2 bar, 4 bar, and 8 bar using different media. However, the paper findings show that the influence of pressure on surface roughness might not be as linear as suggested, with noticeable variations in roughness even at lower pressures.

11.   The authors state that the surface morphology of the C 300 medium exhibits higher roughness at smaller scales. However, the test results indicate that this higher roughness at smaller scales is not solely due to pressure but also influenced by the intrinsic properties of the angular media.

Author Response

1. The manuscript compares multiple bootstrapping methods for determining the relevant scale. Could the authors provide more clarity on why the residual bootstrap method was ultimately chosen as the most promising approach over the other methods, particularly the paired bootstrap method?

We thank the reviewer for this pertinent question. The decision was based on three key reasons: (1) The residual bootstrap preserves the original regression structure by resampling the residuals, maintaining the relationship between the independent variable (sanding pressure) and the dependent variable (relative area), whereas the paired bootstrap may disrupt this relationship by resampling entire [X,Y] pairs, potentially amplifying noise or introducing artificial trends. (2) By focusing on residuals, the residual bootstrap better captures the variability stemming from model errors without distorting the original data structure, a critical factor in our study to ensure subtle effects, such as roughness variations due to sanding, were not overshadowed by amplified noise. (3) The residual bootstrap yielded narrower confidence intervals and more stable estimates of R², which aligned closely with physical observations, providing more robust statistical inferences. These advantages confirmed the residual bootstrap as the most appropriate method for identifying relevant scales in our analysis. (implemented in the text lines 490-496)

2. The results show variability in the surface roughness across different media (e.g., G 100, G 250, C 300). Could the authors discuss in more detail how the intrinsic properties of each medium might influence the observed results, particularly the differences in slope and intercept distributions?

Thanks for this remark. The variability in surface roughness across the different media (e.g., G 100, G 250, C 300) can be attributed to the intrinsic properties of each medium, including particle hardness, size, and morphology. For instance, the corundum-based C 300 medium, characterized by its higher hardness and angular morphology, induces more aggressive impacts on the surface, resulting in a higher intercept at small scales. It indicates a rougher initial surface even at lower pressures compared to the softer and more spherical glass beads (e.g., G 100 and G 250). In contrast, the glass beads, due to their lower hardness and smoother shape, tend to cause more gradual surface roughening, leading to lower initial roughness and a more pronounced dependency on pressure, as seen in their slope distributions. The differences in particle size also play a role; larger particles (e.g., G 250) can create broader indentations, contributing to different scaling behaviors compared to smaller particles like G 100. These intrinsic properties directly influence the interaction mechanisms between the blasting media and the surface, which are captured in the variations of slope and intercept distributions, reflecting the distinct roughness evolution patterns for each medium. (implemented in text line 549 to 563)

3. In Figure 9, the R² values for the Patchwork and Sdr methods show significant differences at the relevant scale. Could the authors elaborate on the potential implications of these differences for practical applications in grit blasting?

Thank you for your observation. While both methods yield high R² values, there are notable differences between them. The Patchwork method produces smoother histograms, which may be advantageous for certain analyses. However, in the case of the triple repetition for the same medium (G 250), the results differ between the two methods, making it challenging to establish a definitive strategy for selecting one method over the other. To address this, we propose a hybrid approach: for the analysis of a specific medium, the method yielding the highest correlation (best R² value) could be prioritized. This strategy combines the strengths of both methods while ensuring the most robust correlation for each specific case. (Implemented in text line 518 to 525)

4. The study uses a low-pass Gaussian filter with a 120 μm cut-off. Could the authors discuss the choice of this specific cut-off value in more detail, and how sensitive the results might be to variations in this parameter? :

The choice of this specific cut-off is directly tied to the analysis in Figure 6, which presents the R² distributions as a function of the calculation scale for relative area under hypotheses H1 (a) and H0 (b), across the three bootstrapping methods: simple bootstrap (i), bootstrap based on pairs (ii), and bootstrap based on residuals (iii). In this figure, the tile size of the Patchwork method (in µm²) is set to half the square of the Sdr method’s cut-off length, ensuring consistency in the scale of analysis between the two methods. The box plots included in these graphs allow us to directly observe the variation in R² values with changes in the cut-off parameter. This visualization provides insights into the stability and sensitivity of the results, illustrating how variations in the cut-off influence the statistical robustness of the correlations. Consequently, the 120 μm cut-off was chosen as a balance between capturing relevant surface features and minimizing noise, while ensuring comparability between the two methods. (implemented in text line 446 to 451)

 

5. Regarding the tile size in the Patchwork method, how was the specific tile size of 10,000 μm² chosen, and what is the expected impact of selecting different tile sizes on the results of the regression analysis?

The triangular tile size of 10,000 μm² was selected based on its ability to yield the highest R² values, as shown in our analysis. The variability around this value is represented by the median of the box plot, which reflects the spread of the R² values at this scale. This selection also accounts for the uncertainties associated with the median values, ensuring that the chosen tile size offers the best balance between data consistency and accuracy. If different tile sizes were selected, we would expect to see changes in the R² values and potentially increased variability in the results, particularly if the tile size is too small or too large for the scale of the surface features being analyzed.

 

6. In the bootstrap analysis, the threshold R² values for different bootstrapping methods were found to be 0.59 (simple bootstrap), 0.91 (paired bootstrap), and 0.83 (residual bootstrap). How sensitive are these threshold values to changes in the bootstrapping procedure or the choice of media?

We have provided a detailed explanation of these results in the appendix of our manuscript, where you will find a table displaying the R² values corresponding to each bootstrapping method applied to different media. To interpret the table, please focus on the columns listing the R² values for the simple bootstrap, paired bootstrap, and residual bootstrap methods. These columns show the threshold values of R² for each method, allowing for a comparison across different experimental conditions. By examining the data, you will see how these thresholds change in response to the choice of media and the type of bootstrapping procedure used. We hope this clarification helps in understanding the sensitivity of the R² thresholds to the variations in both the bootstrapping method and media choice. Please refer to the table from Appendix B for a more in-depth analysis, and do not hesitate to reach out if further clarification is needed.

 

7. How do the results compare for grit blasting at pressures of 2 bar, 4 bar, and 8 bar? What specific differences were observed in the surface morphology across these different pressures, particularly for the C 300 medium?

As shown in the experimental results, the surface morphology exhibits distinct variations with increasing sandblasting pressures. At 2 bar, the surface is characterized by relatively low roughness, with a smoother appearance and smaller-scale features. As the pressure increases to 4 bar, we observe a more pronounced surface roughening, with larger and more defined features forming on the surface. Finally, at 8 bar, the surface becomes significantly more irregular, with deeper indentations and a more aggressive texturing pattern, reflecting the higher impact energy at this pressure. For the C 300 medium, which is known for its specific granule size and hardness, the differences between pressures are even more noticeable. At 2 bar, the C 300 medium results in a relatively uniform surface with mild roughness, while at 4 bar and 8 bar, the increased energy leads to more distinct changes in the morphology, such as larger pits and valleys, particularly at the higher pressure. This change in texture at higher pressures contributes to an increase in the surface area, as measured by the Sdr parameter, as well as a shift in the surface’s functional properties. We believe that these differences in morphology are a direct consequence of the pressure variations during the grit blasting process, with higher pressures resulting in deeper and more extensive alterations to the surface texture. The full details of these observations, including the associated surface roughness parameters, can be found in the results section and summarized in the tables. We hope this helps clarify the specific differences observed across the various pressures for the C 300 medium. ( An appendix A was implemented for with more surface topographies to assess the difference of features depending on the media or pressure )

 In Figure 8, the distributions of intercepts and slopes at the relevant scale were analyzed. What trends were observed for G 100 (fine glass beads) and C 300 (angular media) in terms of their relationship between slope stability and the initial roughness?

Upon examining the data for G 100 (fine glass beads) and C 300 (angular media), we observed distinct trends in their relationship between slope stability and initial roughness: For G 100 (fine glass beads): The data show a relatively stable relationship between slope stability and initial roughness. At lower initial roughness values, the slope tends to remain more consistent, indicating that the surface morphology generated by fine glass beads is less sensitive to changes in initial roughness. As the initial roughness increases, the slopes display moderate variability, but the overall trend suggests that fine glass beads create smoother, more stable surfaces compared to the angular media. The slope distributions for G 100 tend to show lower variability, reflecting the smoother and more homogeneous texture of surfaces processed with these beads. For C 300 (angular media): In contrast, the relationship between slope stability and initial roughness for C 300 shows greater variability. As the initial roughness increases, the slope values become more spread out, indicating that the angular nature of C 300 media leads to a more variable surface morphology. The presence of sharp edges and irregularities in the media results in surfaces that are less stable, with more pronounced fluctuations in the slope as initial roughness increases. These surfaces exhibit higher slope values at higher initial roughness levels, reflecting the increased irregularity and texture caused by angular particles. (implemented in text line 599 to 503)

 

 

9. In terms of pressure variation, how do the results from the study, which uses pressures from 2 to 8 bar, compare with similar studies that use a pressure range of 1 to 7 bar? How might these differences in pressure affect the study's conclusions?

Studies employing a pressure range of 1 to 7 bar typically report a gradual increase in roughness as pressure increases, with 1 bar generally resulting in limited material removal and a relatively uniform morphology. In our study, the 2 to 8 bar range overlaps with the higher end of this spectrum but excludes the 1 bar condition. At 2 bar, we observed minimal roughening with good uniformity, while at 8 bar, significant roughness development occurred, characterized by deeper asperities and increased texture irregularity. In preliminary experiments, we included 1 bar but encountered challenges in achieving a consistent and uniform surface roughness, leading us to exclude it from the main study. These challenges may stem from insufficient kinetic energy at low pressures, non-uniform media flow, and the dominance of pre-existing surface irregularities, which collectively limited reproducibility. At 2 to 8 bar, our results showed consistent trends: surface roughness parameters like Sdr (developed surface area ratio), reflecting pronounced textural changes at higher pressures. The addition of 8 bar in our study, compared to other studies that cap at 7 bar, provided deeper insights into extreme pressure effects, such as significant material removal and void volume increase, which are critical for understanding nonlinear texture evolution. While the exclusion of the 1 bar condition limits the scope at the lower pressure range, this decision ensured robust and reproducible sandblasting conditions. Therefore, the presented results at 2 to 8 bar are both aligned with existing studies and enriched by the inclusion of extreme pressure effects, providing a comprehensive understanding of roughness evolution under these conditions. ( implemented in text line 197 to 199)

 

10. The authors mention that the surface topographies of TA6V samples were grit blasted at 2 bar, 4 bar, and 8 bar using different media. However, the paper findings show that the influence of pressure on surface roughness might not be as linear as suggested, with noticeable variations in roughness even at lower pressures.

Thank you for your observation regarding the influence of pressure on surface roughness. Indeed, while we initially hypothesized a linear relationship between pressure and roughness evolution, our findings suggest a more nuanced trend. As highlighted in the results section, the impact of pressure on roughness parameters, such as Sdr (developed surface area ratio) varies depending on the media used and the specific pressure levels. At lower pressures, such as 2 bar, the roughness increases are moderate and relatively uniform, reflecting a gradual erosion of the material. However, at higher pressures, particularly at 8 bar, the roughness changes become more pronounced and nonlinear, characterized by deeper asperities and irregular texture development. This nonlinearity is likely due to the combined effects of media dynamics and substrate behavior. At lower pressures, the kinetic energy of the grit particles may be insufficient to significantly alter pre-existing surface features, resulting in a less dramatic change in roughness. Conversely, at higher pressures, the increased impact energy leads to more aggressive material removal and a shift in the dominant roughness mechanisms, such as asperity deformation and void formation. Additionally, media type plays a critical role, as observed with the contrasting behaviors of G 100 (fine glass beads) and C 300 (angular media), where the latter exhibited more pronounced changes at higher pressures due to its angular geometry and increased cutting action. This insight also underscores the importance of analyzing the pressure effect at a finer resolution to better understand the nonlinearities observed in the roughness evolution. While our current analysis assumes a linear relationship between pressure and relative area at a given scale, we recognize that this relationship might be more appropriately captured with a nonlinear model. For example, a model of the form Y=aP^b+c could offer a more comprehensive perspective, converging to a linear relationship when b≈1. The bootstrap techniques we employed—simple, paired, and residual—could similarly be applied to this nonlinear model to assess its statistical validity and quantify the parameters b and c. Specifically, bootstrapping could help determine whether b<1, b=1, or b>1, providing a deeper understanding of the pressure's impact on surface damage. For instance, deviations from b=1 or significant values of b could serve as signatures of damage mechanisms specific to the surface morphology. However, implementing such a nonlinear model would require solving a regression problem through iterative optimization algorithms, ensuring convergence to a unique global minimum. This added complexity would extend beyond the current study's scope. Nevertheless, we plan to incorporate this approach in future work to analyze the nonlinear effects of contact pressure on developed surface area for various media. This would allow us to explore potential nonlinearities in greater detail, leveraging the same bootstrap techniques for robust statistical analysis. We greatly appreciate your suggestion and will implement this nonlinear framework in a forthcoming publication, focusing on the implications of nonlinearity in surface development as a function of pressure and media type. (Implemented in text line 615 to 635)

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

After the authors' modifications, the paper is ready for publication. I recommend publishing the paper.

Comments on the Quality of English Language

The English language is fine

Reviewer 2 Report

Comments and Suggestions for Authors

Referee Report

On the paper “ Uncertainty-Based Scale Identification and Process-Topography Interaction Analysis via Bootstrap: Application to Sandblasting “ (fractalfract-3340097-v2) by the authors François Berkmans, Julie Lemesle, Robin Guibert, Michal Wieczorowski, Christopher Brown, Maxence Bigerelle submitted to the Fractal Fract

 

This paper has been well corrected and it can be recommended.

Comments for author File: Comments.doc

Reviewer 3 Report

Comments and Suggestions for Authors

The author's team has conducted well responses to the Reviewer' comments.