An Inversion Algorithm for the Dynamic Modulus of Concrete Pavement Structures Based on a Convolutional Neural Network
Round 1
Reviewer 1 Report
This paper proposed an inversion algorithm for predicting the dynamic modulus based on a convolutional neural network. The paper is well organised, and small suggestions are provided as follows:
· Figure 3. Why did the authors select a general pavement in Quebec, Canada? Please give the explanations.
· Lines 383-386: Please illustrate the reasons for the bigger relative error of the second layer than the first layer.
· Please add the limitations of the work and recommendations for further work.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Attach please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Author Response File: 
Author Response.docx
Reviewer 2 Report
In the reviewed manuscript, the authors are attempting to evaluate elastic properties of a laminate pavement structure employing convolution neural networks with structural transient response serving as an input. Although some promising numerical results are presented, the overall impression from the paper is not positive. The main drawback is the total lack of novelty. The latter relates both to the forward modeling of pavement structures and to the inversion method employed. In its current state, the paper makes an impression of being just a numerical exercise rather than a comprehensive research.
In particular,
1) The description of theoretical derivations is rather common and could be found elsewhere. However, nothing is said about how the boundary conditions between sublayers are satisfied, how infinite half-space is treated with SEM and how the wavenumbers of the whole structure are evaluated.
2) The algorithm for dynamic modulus evaluation was "proposed" by the authors was already discussed in papers https://doi.org/10.1109/ICPHM49022.2020.9187057 and https://doi.org/10.1007/978-3-030-64908-1_8. The fact that instead of a laminate anisotropic composite, a layered isotropic structure is considered could not be treated as a sigh of novelty.
To ensure sufficient level of originality, the authors must either perform sufficiently more rigorous parametric analysis or, alternatively, provide experimental verification of the developed algorithm.
Some other comments:
1) Lines 192-193: What is the precision of such replacement? Are only real-valued wavenumbers considered?
2) Line 202: Could the authors provide any references regarding this "evidence"?
3) From all the derivations provided in Section 2, it is not clear what are their particular benefits from the computational point of view? Why not to use Green's matrix formalism together with residue theory? (e.g., papers http://dx.doi.org/10.1016/j.soildyn.2013.01.008 + http://dx.doi.org/10.1016/j.ijsolstr.2012.06.022)
4) Line 229: Why such huge number for M is necessary?
5) In the performed study, only Young's modulus of each sublayer are considered being unknown, while as it might be judged, density, Poisson's ratio and thickness are prescribed values. To what extent such simplification might be relevant to real-case applications? How would the proposed technique perform if these parameters are also unknown? Was any sensitivity analysis performed?
6) Lines 402-406: This statement is not confirmed by any comparison between the mentioned approaches.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Attach please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Author Response File: Author Response.docx
Reviewer 3 Report
In this paper, a dynamic modulus inversion approach for concrete pavement constructions is developed using the spectral element method (SEM) and a convolutional neural network (CNN). The manuscript is scientifically clear and sound.
To further illustrate the accuracy of the proposed algorithm for the inversion of the dynamic modulus of the pavement structure, beyond RMSE, other statistical parameters, including coefficient of determination (R2), mean absolute percentage error (MAPE), and mean absolute error (MAE) should be carried out.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Attach please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Author Response File: Author Response.docx
Reviewer 4 Report
This paper reports that the dynamic behavior of elastic moduli E1, E2, and E3 of pavement layers was predicted using a convolution neural network (CNN). The 1D-CNN was trained time-series data corresponding to moduli E1, E2, E3, and the regression metrics evaluated by test data were good. The dataset was generated by the theoretical model which the authors presented. The results of the paper are appealing, but there are some unclear things to be mentioned before acceptance.
Can the trained model be used for practical tests? Also, what the limitation of the trained CNN?
It seems that the method of train-validation-test splitting is unclear. Why is the total data size (ex. table 2) equal to the number of training and validation (or test set)? Does it mean validation and test set are the same? (But it would be a wrong splitting.)
The authors presented validation accuracy at line 358. However, accuracy in the common meaning cannot be defined in the regression task. What does the validation accuracy mean in this case?
In line 381, how was relative error calculated?
In Figure 9, I suggest changing the dotted plot into a line plot or scatter plot. I was about to misunderstand that each dot represents the result of each iteration.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Attach please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
Disregard the introduced improvements, the paper is still not worth for being accepted and published.
The main issues have remained the same:
1) The lack of novelty – although the authors are claiming that the combination of CNN for inverse evaluations and SEM for the forward problem is novel, I could not see any novelty in such a combination. The SEM for multilayered was proposed in https://doi.org/10.1016/S0020-7683(00)00112-8 and advanced in Ref. 10, the concept of CNN application for material properties inversion is already well established (see https://doi.org/10.1080/15376494.2021.1982090 as an example) and the combination of ANN and SEM (or analogous concepts) was already discussed in https://doi.org/10.1016/j.advengsoft.2005.10.001 (more than 16 years ago)
2) The authors claim that their algorithm is “efficient and accurate”. In the paper, there is no evidence to what extent is the proposed approach efficient. Its accuracy is also very questionable. In more detail: as an input for the CNN, the authors are using deflection time history without any noise. And even within such totally ideal conditions relative error for Young’s modulus estimation might be as high as 10% (although the authors claim that “in the inversion calculation of the pavement structure parameters is solved by the proposed algorithm in engineering accuracy requirements” this statement might be acceptable if such errors are achieved for real-world experimental data). Moreover, if an attempt is made to recover thickness along with Young’s moduli, the average relative error is drastically increasing and becomes unacceptable for thickness evaluation.
3) In one of the responses the authors mention that “The problem of non-unique problem in the inversion process is overcome”. I could not agree with this statement, especially with respect to the numerical results provided in the paper. To assure such overcoming, the authors should either provide some theoretical proof or, alternatively, perform extensive numerical analysis with respect to the sensitivity of deflection curves to the changes in elastic properties and thickness of each of the sublayers (e.g. as it is done in papers https://doi.org/10.1080/15376494.2021.1982090 and https://doi.org/10.1016/j.compstruct.2020.112569 for dispersion curves). The reviewer’s experience suggests, that, for example, due to the higher stiffness of the upper layer the influence of other sublayers as well as of the half-space on deflection curves could be rather moderate.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Attach please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Author Response File: Author Response.docx
Round 3
Reviewer 2 Report
In general, the authors have adequately responded to the reviewer's comments.
The authors are suggested to extend the third point of the Conclusion. It should be clearly stated that, if along with Young's modulus of each sublayer its thickness is also considered as an unknown value, the precision of the proposed reconstruction procedure is not very high. The latter is clearly visible from Figures 12 (c) and 13 (b, c).
The authors should also clearly state in the Conclusion that all the obtained results are purely numerical and that the effect of noise was not taken into account. The latter could be added as an additional statement to the last paragraph of the paper where its limitations are disclosed.
After introducing these improvements, the paper could be accepted.
Author Response
The co-author and I would like to thank you for all the important comments/suggestion, which has been very helpful in revising the original version of our manuscript. As such, we are very grateful to you. Following please find our detailed replies to each comment, with the corresponding modifications being highlighted in the revised manuscript.
Response to Reviewer #2:
Q1) The authors are suggested to extend the third point of the Conclusion. It should be clearly stated that, if along with Young's modulus of each sublayer its thickness is also considered as an unknown value, the precision of the proposed reconstruction procedure is not very high. The latter is clearly visible from Figures 12 (c) and 13 (b, c). The authors should also clearly state in the Conclusion that all the obtained results are purely numerical and that the effect of noise was not taken into account. The latter could be added as an additional statement to the last paragraph of the paper where its limitations are disclosed.
Response: Thanks for the comments. The Conclusions were revised as suggested.
Revision: Lines 485 and 492, “The algorithm can also be used for the inversion of the thickness and dynamic modulus of the pavement structure layer. As the number of inversion variables increases, the inversion accuracy will decline to a certain extent. The algorithm in this paper is of a high solution accuracy for the inversion of the dynamic elastic modulus of the pavement structure surface, the average relative error of the in-version results is basically controlled within 8%, while the inversion accuracy for the thickness is relatively low, the average relative error of the inversion results is basically controlled within 12%, and the inversion results have good numerical stability.”
Lines 493 and 502, “The limitation of this study is that the contact boundary conditions between the pavement structure and the subgrade are not considered, and all the results obtained are purely simulation results without considering the impact of noise. Previous research shows that the interface between the pavement structure and the subgrade will have a certain degree of relative slip, which has a certain impact on the dynamic response of the pavement surface. Therefore, the inversion accuracy of pavement structure dynamic modulus will also be affected. Hereafter, the contact boundary conditions will be considered in future research, and the proposed algorithm will be applied to the actual tests and engineering practice.”
Author Response File: Author Response.docx
