A New Algorithm for Predicting Dam Deformation Using Grey Wolf-Optimized Variational Mode Long Short-Term Neural Network

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Reviewer 's expert opinion :

1.    All variable symbols must be explained after their first appearance, without missing items. Please the author must complete the variable explanation, carefully modify ! Letters of vectors, vectors, tensors and matrices are checked in black italics and other variables in white italics. (including the letters in the picture)

2.    In the introduction part, the discussion is not detailed enough. It is suggested to supplement the content of dam deformation monitoring, and the innovation points of this paper are expounded in the introduction part. In the reference part, there are few references related to dam deformation, and some references related to the research content of this paper can be replaced accordingly.

3.    In lines 176-177 of the article, the article mentions ' more detailed information on the noise model estimation criteria '.What is the logical relationship between this sentence and the above introduction of the LSTM model Please explain.

4.    What is the significance of the module above the tanh module in the first part of Fig.2 ( a ) ; the whole Figure 2 ( a ) is not clear enough, and please explain the meaning of each letter in the figure.

5.    For formula (10) α (j) and a (j) meaning is the same, if not the same, please add the definition of a (j).

6.    In Formula (11), MPE and IMF are both positive in the text. Please check and unify the positive italics of the formula and the text.

7.    In Figure 3, the first judgment box in the left and right YES and NO correspond to a process respectively, and the judgment box is directly connected to a ' gray wolf position update '. Please check whether the logic of the flow chart is reasonable.

8.    For the evaluation indexes MAE, RMSE and R² used in this paper, the three are all mathematical evaluation indexes. Why is there only R² in this paper, and the rest is positive, please check the positive italics of the full text.

9.    Line 261, this paper uses the GWO algorithm to iterate the process line is Figure 4, and Figure 4 should not be the technical roadmap of the entire paper?

10.  In Figure 5 to Figure 8, the units of each axis are missing, please supplement their units.

11.  Figure 5 is the convergence diagram of GWO algorithm fitness value in three directions of 001 station. Only from Figure 5, the advantages of GWO algorithm can not be seen intuitively. It is suggested to compare the fitness with other algorithms (such as WOA algorithm, SSA algorithm, etc.), so as to verify the advantages of GWO algorithm in this paper.

12.  In the explanation section of Figure 6, the fluctuation of K is between 5 and 10, while the fluctuation of α is 200~1000 in E and U, 200~4000 in N direction. What are the reasons for the inconsistency of the fluctuation and how do you choose the parameters.

13.  In this paper, after VMD decomposition, multi-scale permutation entropy (MPE) screening is used. However, through the narrative part of Figure 8 and Table 1, only MPE is used to analyze each IMF component, and no screening is seen. Please supplement the screening results. At the same time, this paper proposes that MPE greater than 0.6 is the noise part, less than 0.6 is the low-frequency signal part, and 0.6 is set as whether the threshold meets all engineering cases, or whether other values can be set, please elaborate and prove.

14.  In the narrative section of Figure 10, it is mentioned that the R² of VMDGRU in the N direction is only 0.03, and the R² of VMDANN is 0.85, while the prediction accuracy of VMDGRU is not seen from the prediction comparison diagram of Figure 9. Poor, please explain this phenomenon ; and the GRU model should be better than the LSTM and ANN models. Why VMDGRU is worse than VMDANN, please give the corresponding reasons in the narrative part of the article.

15.  This article mentions the test of robustness, but there is no analysis method of robustness in the article. Please refer to other literature and supplement the analysis content of robustness.

For other questions, please refer to the comments in PDF.

Comments for author File: Comments.pdf

Author Response

1. All variable symbols must be explained after their first appearance, without missing items. Please the author must complete the variable explanation, carefully modify! Letters of vectors, vectors, tensors and matrices are checked in black italics and other variables in white italics. (including the letters in the picture)

Reply: Thank you for your comments. We have completed the inspection of the variable explanation, letters of vectors, vectors, tensors and matrices, including the letters in the picture.

2. In the introduction part, the discussion is not detailed enough. It is suggested to supplement the content of dam deformation monitoring, and the innovation points of this paper are expounded in the introduction part. In the reference part, there are few references related to dam deformation, and some references related to the research content of this paper can be replaced accordingly.

Reply: Thank you for your comments. In the introduction section, we have added content on dam deformation monitoring, listed the innovative points of this article, and updated and added corresponding references.

Additional content：

Additional references：

1. Dal Sasso, S. F., Sole, A., Pascale, S., Sdao, F., Bateman Pinzón, A., & Medina, V. Assessment methodology for the prediction of landslide dam hazard. Nat Hazard Earth Sys.2014,14(3), 557-567.
2. Su, H., Li, X., Yang, B., & Wen, Z. Wavelet support vector machine-based prediction model of dam deformation. Mech Syst Signal Pr.2018, 110, 412-427.
3. Ranković, V., Grujović, N., Divac, D., & Milivojević, N. Development of support vector regression identification model for prediction of dam structural behaviour. Struct Saf. 2014,48, 33-39.
4. Kang, F., Liu, J., Li, J., & Li, S. Concrete dam deformation prediction model for health monitoring based on extreme learning machine. Struct Control Hlth.2017. 24(10), e1997.
5. Qu, X., Yang, J., & Chang, M. A Deep Learning Model for Concrete Dam Deformation Prediction Based on RS‐LSTM. J Sensors, 2019, (1), 4581672.
6. In lines 176-177 of the article, the article mentions ' more detailed information on the noise model estimation criteria '. What is the logical relationship between this sentence and the above introduction of the LSTM model Please explain.

Reply: Thank you very much for discovering the description error in LSTM. We modify it: More detailed information about LSTM can be obtained at Schmidhuber(1997), Chen(2023)and Hochreiter(2001) [31,33].

Related References：

36. Schmidhuber J, Hochreiter S. Long short-term memory. Neural Comput. 1997, 9(8), 1735-1780.
37. Chen, H., Lu, T., Huang, J., He, X., & Sun, X. An Improved VMD–EEMD–LSTM Time Series Hybrid Predic-tion Model for Sea Surface Height Derived from Satellite Altimetry Data. Journal of Marine Science and Engi-neering.2023, 11(12), 2386.
38. Hochreiter, S., Bengio, Y., Frasconi, P., & Schmidhuber, J. Gradient flow in recurrent nets: the difficulty of learning long-term dependencies. 2001.
39. What is the significance of the module above the tanh module in the first part of Fig.2 ( a ) ; the whole Figure 2 ( a ) is not clear enough, and please explain the meaning of each letter in the figure.

Reply: Thank you for your comments. The tanh module is an activation function used to help regulate the values flowing through the network to always be limited between -1 and 1. The tanh layer is used to generate new candidate cell states, which helps control the flow and storage of information.  is the logistic sigmoid function, h_t is the current cell output h_t-1 is the input of the previous time, x_t is the input of the current time, and f_t is the forget gate of the moment t. The forget gate combines the input h_t-1 of the previous time with the input x_t of the current time to selectively forget the content. i_t is the input gate,  is a candidate vector,  is the current cell state, and is the previous cell state. is the output gate. We have provided a simplified introduction to LSTM, with a more detailed description in references 29-31.

Related References：

34. Gers F A, Schmidhuber J, Cummins F. Learning to forget: Continual prediction with LSTM. Neural computation. 2000, 12(10), 2451-2471.
35. Graves, A. Long Short-term Memory. Supervised Sequence Labelling with Recurrent Neural Networks.2012, 37–45.
36. Schmidhuber J, Hochreiter S. Long short-term memory. Neural Comput. 1997, 9(8), 1735-1780.
37. For formula (10) α (j) and a (j) meaning is the same, if not the same, please add the definition of a (j).

Reply: Thank you for your comments. In formula 10, it should be ,We have modified the parameters. Related References：

39. Tang, G. J., & Wang, X. L. Application of parameter optimization variational modal decomposition method in early fault diagnosis of rolling bearing. Journal of Xi'an JiaoTong University. 2015,49(5), 73-81.
40. In Formula (11), MPE and IMF are both positive in the text. Please check and unify the positive italics of the formula and the text.

Reply: Thank you for your comments. In Formula (11), MPE is multiscale permutation entropy, IMF is intrinsic mode functions, MPE and IMF are positive. We corrected Formula (11) to .

7. In Figure 3, the first judgment box in the left and right YES and NO correspond to a process respectively, and the judgment box is directly connected to a ' gray wolf position update '. Please check whether the logic of the flow chart is reasonable.

Reply: Thank you for your comments. After updating the manuscript, Figure 3 should be Figure 4，and the manuscript has been updated with Figure 4.

Related References：

32. Lu T., He J., He X., & Tao R. GNSS Coordinate Time Series Denoising Method Based on Parameter-optimized Variational Mode Decomposition. Geomatics and Information Science of Wuhan University. 2023,1-15.
33. For the evaluation indexes MAE, RMSE and R² used in this paper, the three are all mathematical evaluation indexes. Why is there only R2 in this paper, and the rest is positive, please check the positive italics of the full text.

Reply: Thank you for your comments. The coefficient of determination R² is positive, and the formula has been modified.

Related References：

50. Nakagawa, S., Johnson, P. C., & Schielzeth, H. The coefficient of determination R² and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of the Royal Society Interface.2017, 14(134), 20170213.
51. Piepho, H. P. An adjusted coefficient of determination (R²) for generalized linear mixed models in one go. Biometrical Journal,2023, 65(7), 2200290.
52. Line 261, this paper uses the GWO algorithm to iterate the process line is Figure 4, and Figure 4 should not be the technical roadmap of the entire paper?

Reply: Thank you for your comments. Figure 4 should be Figure 5. Figure 5 belongs to section 2.25 “Construction of a New GVLSTM Model Construction”, which shows the flowchart of the newly proposed GVLSTM in the manuscript and is also the technical roadmap that runs through the entire manuscript. We can see that Figure 5 is divided into four steps. The first step is GWO optimization of VMD, the second step is MPE judgment VMD of reconstructed signal, the third step is LSTM prediction, and the fourth step is model evaluation.

10. In Figure 5 to Figure 8, the units of each axis are missing, please supplement their units.

Reply: Thank you for your comments. Figure 5 to Figure 8 should be Figure 6 to Figure 9. Figures 6 and 8 show the optimized parameter values without units, which can be confirmed in reference 27. The horizontal axis of Figure 8 represents time in days, and the vertical axis represents amplitude in millimeters. Figure 9 shows the MPE values without units, which can be confirmed in references 27 and 38.

Related References：

32. Lu T., He J., He X., & Tao R. GNSS Coordinate Time Series Denoising Method Based on Parameter-optimized Variational Mode Decomposition. Geomatics and Information Science of Wuhan University. 2023,1-15.
33. Lu T, Xie J. EEMD-Multiscale Permutation Entropy Noise Reduction Method for GPS Elevation Time Series. Journal of Geodesy and Geodynamics,2021,41(02):111-115.
34. Figure 5 is the convergence diagram of GWO algorithm fitness value in three directions of 001 station. Only from Figure 5, the advantages of GWO algorithm can not be seen intuitively. It is suggested to compare the fitness with other algorithms (such as WOA algorithm, SSA algorithm, etc.), so as to verify the advantages of GWO algorithm in this paper.

Reply: Thank you for your comments. Figure 5 should be Figure 6. During GWO optimization of VMD from Figure 6, convergence occurs after 2 iterations, demonstrating the speed of GWO optimization of VMD. The main advantages of GWO compared to WOA (Whale Optimization Algorithm) and SSA (Sparrow Search Algorithm) are its simple structure, fewer parameters, easy implementation, and ability to achieve a balance between local optimization and global search, thus exhibiting good performance in solving accuracy and convergence speed. ‌

The Grey Wolf Optimization Algorithm (GWO) is inspired by the hunting behavior of grey wolf populations and achieves optimization by simulating the cooperative mechanism of grey wolf populations. This algorithm has the following characteristics:

Simple structure: The GWO algorithm has a simple structure, requires fewer parameters, and is easy to implement.

Adaptive adjustment: The algorithm includes convergence factors and information feedback mechanisms for adaptive adjustment, which can achieve a balance between local optimization and global search.

Good performance: It has good performance in terms of solving accuracy and convergence speed for problems.

Whale Optimization Algorithm (WOA) and Sparrow Search Algorithm (SSA) also have their own characteristics and advantages:

WOA: Inspired by the predatory behavior of whales, we searched for the global optimal solution by simulating the bubble net predation strategy of whales. This algorithm exhibits good performance in dealing with multidimensional and nonlinear problems.

SSA: Inspired by the foraging and anti-predation behavior of sparrows, optimized by simulating their swarm intelligence. This algorithm has certain robustness and effectiveness in dealing with complex optimization problems.

In summary, the main advantages of GWO over WOA and SSA are its simple structure, fewer parameters, ease of implementation, and the ability to achieve a balance between local optimization and global search, resulting in better performance in terms of solution accuracy and convergence speed.

Related References：

24. Faris, H., Aljarah, I., Al-Betar, M. A., & Mirjalili, S..Grey wolf optimizer: a review of recent variants and ap-plications. Neural computing and applications.2018, 30, 413-435.
25. Emary, E., Zawbaa, H. M., & Hassanien, A. E. Binary grey wolf optimization approaches for feature selec-tion. Neurocomputing.2016, 172, 371-381.
26. Mittal, N., Singh, U., & Sohi, B. S. Modified grey wolf optimizer for global engineering optimization. Applied Computational Intelligence and Soft Computing.2016, 2016(1), 7950348.
27. Teng, Z. J., Lv, J. L., & Guo, L. W. An improved hybrid grey wolf optimization algorithm. Soft compu-ting.2019, 23, 6617-6631.
28. Hou, Y., Gao, H., Wang, Z., & Du, C. Improved grey wolf optimization algorithm and application. Sensors. 2022, 22(10), 3810.
29. In the explanation section of Figure 6, the fluctuation of K is between 5 and 10, while the fluctuation of α is 200~1000 in E and U, 200~4000 in N direction. What are the reasons for the inconsistency of the fluctuation and how do you choose the parameters.

Reply: Thank you for your comments. Figure 6 should be Figure 7. Figure 7 shows the distribution of VMD parameters for the six selected sites in the manuscript, and  are parameters obtained by GWO optimization of VMD. Through displacement calculation of 6 stations, the minimum value of  is 5 and the maximum value is 10, so the range of  values is 5-10, and the same applies to  value.

Because each station has displacement in the N, E, and U directions, the range of the calculated  value varies, resulting in different fluctuations. Through GWO-VMD, we can accurately calculate the  value (As shown in the figure).

13. In this paper, after VMD decomposition, multi-scale permutation entropy (MPE) screening is used. However, through the narrative part of Figure 8 and Table 1, only MPE is used to analyze each IMF component, and no screening is seen. Please supplement the screening results. At the same time, this paper proposes that MPE greater than 0.6 is the noise part, less than 0.6 is the low-frequency signal part, and 0.6 is set as whether the threshold meets all engineering cases, or whether other values can be set, please elaborate and prove.

Reply: Thank you for your comments. Figure 8 should be Figure 9. The modal components of VMD decomposition contain a lot of noise and complex signals. After optimizing the VMD parameters, GWO incorporates Mirjalili's concept so that multiscale permutation entropy (MPE) may be used as the criterion for judging noise and signals. After the multiscale permutation entropy (MPE) of each IMF component in VMD decomposition is calculated, low-frequency signals and high-frequency noise may be determined by setting the MPE threshold. When there is less noise in the IMF component, the signal is more regular, and the MPE value is smaller. Conversely, when there is more noise in the IMF component, the MPE value is larger. After multiple experiments, the MPE threshold set in Lu’s (2023) research can effectively filter out noise in deformation monitoring data. After multiple experiments, we found that the MPE threshold used in Lu’s (2023) research could effectively filter out noise in deformation monitoring data. After multiple tests, then, the MPE value was set to 0.6, and low-frequency IMF components below the MPE threshold were reconstructed into new signals to optimize GWO for VMD.

Related References：

32. Lu T., He J., He X., & Tao R. GNSS Coordinate Time Series Denoising Method Based on Parameter-optimized Variational Mode Decomposition. Geomatics and Information Science of Wuhan University. 2023,1-15.
33. Jabloun, M., Ravier, P., & Buttelli, O. On the Genuine Relevance of the Data-Driven Signal Decomposition-Based Multiscale Permutation Entropy. Entropy.2022, 24(10), 1343.
34. Choi, Y. S. Improved multiscale permutation entropy measure for analysis of brain waves. International Journal of Fuzzy Logic and Intelligent Systems.2017,17(3), 194-201.
35. Lu T, Xie J. EEMD-Multiscale Permutation Entropy Noise Reduction Method for GPS Elevation Time Series. Journal of Geodesy and Geodynamics,2021,41(02):111-115.
36. In the narrative section of Figure 10, it is mentioned that the R² of VMDGRU in the N direction is only 0.03, and the R2 of VMDANN is 0.85, while the prediction accuracy of VMDGRU is not seen from the prediction comparison diagram of Figure 9. Poor, please explain this phenomenon; and the GRU model should be better than the LSTM and ANN models. Why VMDGRU is worse than VMDANN, please give the corresponding reasons in the narrative part of the article.

Reply: Thank you for your comments. Figure 9 should be Figure 10 and Figure 10 should be Figure 11. The VMDGRU in Figure 11 is a blue curve, and the distribution of VMDGRU curves in three directions can be seen. The original data curve is a black curve, and the fitting effect between the two is not good. The manuscript compares GVLSTM with VMDGRU and VMDANN, not a single GRU 、ANN and LSTM model. The prediction accuracy after VMD decomposition varies. Based on Figure 10、Figure 11 and Table 2, the RMSE and MAE predicted by VMDGRU are the highest, while the R² value is the lowest. Therefore, the prediction performance of VMDGRU is not good. We have added an explanation after Figure 11.

15. This article mentions the test of robustness, but there is no analysis method of robustness in the article. Please refer to other literature and supplement the analysis content of robustness.

Reply: Thank you for your comments. Robustness refers to the ability of a system to maintain its performance and functionality stability in the face of internal structural and external environmental changes. The robustness in the manuscript should be changed to reliability, and section 3.2 in the text is a verification of reliability.

For other questions, please refer to the comments in PDF.

Reply: Thank you for your comments. The issues in the PDF have been fully addressed in the manuscript.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Please check the comment file.

Comments for author File: Comments.pdf

Author Response

1. In 82~92: The manuscript uses 6 sites, and Figure 1 can be used to supplement the site map.

Reply: Thank you for your comments. We have added site images to the manuscript Figure 1.

2. The introduction section of the Grey Wolf Algorithm uses parameters to represent the four types of wolves. The spacing between rows in the manuscript should be consistent, and the formula parameters should be introduced.

Reply: Thank you for your comments. The formulas in the manuscript are edited using Mathtype, and the line spacing has been modified in the manuscript.

parameters introduced:

Related References：

24. Faris, H., Aljarah, I., Al-Betar, M. A., & Mirjalili, S..Grey wolf optimizer: a review of recent variants and ap-plications. Neural computing and applications.2018, 30, 413-435.
25. Emary, E., Zawbaa, H. M., & Hassanien, A. E. Binary grey wolf optimization approaches for feature selec-tion. Neurocomputing.2016, 172, 371-381.
26. Mittal, N., Singh, U., & Sohi, B. S. Modified grey wolf optimizer for global engineering optimization. Applied Computational Intelligence and Soft Computing.2016, 2016(1), 7950348.
27. Teng, Z. J., Lv, J. L., & Guo, L. W. An improved hybrid grey wolf optimization algorithm. Soft compu-ting.2019, 23, 6617-6631.
28. Hou, Y., Gao, H., Wang, Z., & Du, C. Improved grey wolf optimization algorithm and application. Sensors. 2022, 22(10), 3810.
29. In 208~209: How to determine the threshold of MPE when using MPE to determine the value of IMF in the manuscript?

Reply: Thank you for your comments. After the multiscale permutation entropy (MPE) of each IMF component in VMD decomposition is calculated, low-frequency signals and high-frequency noise may be determined by setting the MPE threshold. When there is less noise in the IMF component, the signal is more regular, and the MPE value is smaller. Conversely, when there is more noise in the IMF component, the MPE value is larger. After multiple experiments, the MPE threshold set in Lu’s (2023) research can effectively filter out noise in deformation monitoring data. After multiple experiments, we found that the MPE threshold used in Lu’s (2023) research could effectively filter out noise in deformation monitoring data. After multiple tests, then, the MPE value was set to 0.6, and low-frequency IMF components below the MPE threshold were reconstructed into new signals to optimize GWO for VMD.

Related References：

32. Lu T., He J., He X., & Tao R. GNSS Coordinate Time Series Denoising Method Based on Parameter-optimized Variational Mode Decomposition. Geomatics and Information Science of Wuhan University. 2023,1-15.
33. Jabloun, M., Ravier, P., & Buttelli, O. On the Genuine Relevance of the Data-Driven Signal Decomposition-Based Multiscale Permutation Entropy. Entropy.2022, 24(10), 1343.
34. Choi, Y. S. Improved multiscale permutation entropy measure for analysis of brain waves. International Journal of Fuzzy Logic and Intelligent Systems.2017,17(3), 194-201.
35. Lu T, Xie J. EEMD-Multiscale Permutation Entropy Noise Reduction Method for GPS Elevation Time Series. Journal of Geodesy and Geodynamics,2021,41(02):111-115.
36. The selection of VMD parameters for GWO optimization is introduced in detail. Are the values of k and α shown in the manuscript.

Reply: Thank you for your comments. In“3.1. GWO optimization of VMD parameter selection”，I introduced GWO optimization of VMD to obtain parameter[,] values, and the results are shown in Figure 7.

5. For Table 2, we can bold the values of GVLSTM without the need for underscores.

Reply: Thank you for your comments. We have removed the underline in Table 2.

6. When comparing GVLSTM with VMDLSTM, the manuscript only used station 001 as an example. Did it consider whether the prediction results of other stations were consistent with those of station 001?

Reply: Thank you for your suggestion. During the experiment, we predicted all 6 stations separately in the manuscript, we only used station 001 for display. The 6 stations experimental results showed that GVLSTM had higher prediction accuracy than VMDLSTM among the six stations. The evaluation of the prediction results of GVLSTM and VMDLSTM in different directions is shown in Figure 13, and the improvement of GVLSTM relative to VMDLSTM is shown in Table 3.

7. The prediction of unstable nonlinear dam time series is very meaningful research, and the manuscript's study is of great reference value. In the future accurate prediction of dam time series, whether LSTM is the optimal deep learning prediction method requires further experimental research.

Reply: Thank you for your comments. In the prediction of dam displacement time series, LSTM is a very effective method, but it is not recommended to use LSTM single model prediction, such as the new method GVLSTM proposed in the manuscript. GVLSTM has obvious advantages in dam deformation prediction compared with other methods, and the prediction results of GVLSTM prediction model after original sequence decomposition and reconstruction have higher accuracy and precision, which provides reliable engineering application data for the research on intelligent prediction of dam deformation.

8. Please check the reference format.

Reply: Thank you for your comments. All reference formats have been checked and modified.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Dear authors,

Only a few minor corrections are needed:

1. Line 100: “…he optimal solution…”

In fact: “….the optimal solution…”

 

2. Line 131: “….the position of wolf d is updated….”

Who is the wolf d ?

 

3. Subsection 2.2.6 Why other criteria were not used or at least mentioned: the Nash-Sutcliffe criterion, Akaike, etc.?

 

4. Line 274: “…as shown in Figure 6. It is evident in Figure 6 that….”

Probably Figure 7 instead of Figure 6.

 

5. Relationship (14): The denominator must be a power of 2.

Author Response

1. Line 100: “…he optimal solution…”

In fact: “….the optimal solution…”

Reply: Thank you for your comments.  We have changed the word 'he' in the manuscript to 'the'.

2. Line 131: “….the position of wolf d is updated….”

Who is the wolf d ?

Reply: Thank you for your comments. We have changed the word 'd' in the manuscript to ''.

3. Subsection 2.2.6 Why other criteria were not used or at least mentioned: the Nash-Sutcliffe criterion, Akaike, etc.?

Reply: Thank you for your comments. The use of MAE and RMSE evaluation models is because they are applicable to different scenarios and can effectively measure the prediction accuracy and stability of the models. The Nash Sutcliffe Efficiency (NSE) is commonly used to quantify the predictive accuracy of simulation models, such as hydrological models. The Akaike Information Criterion (AIC) is a standard used to evaluate the goodness of fit of statistical models, proposed by Japanese statistician Hiroji Akaike. AIC aims to balance the complexity of the model and its fit to the data, selecting the optimal model by balancing the complexity and goodness of fit of the model. ‌ We have referred to references 47-49 and other literature, all of which use RMSE and MAE for model evaluation. However, your suggestion is very good. In future research, we will consider using more methods such as Nash Sutcliffe criteria and Akaike as evaluation indicators for experiments.

4. Line 274: “…as shown in Figure 6. It is evident in Figure 6 that….”. Probably Figure 7 instead of Figure 6.

Reply: Thank you for your comments. After the manuscript was revised based on feedback, the order of figure numbers has changed. The original figure 6 should be figure 8. We have already made the necessary modifications.

5. Relationship (14): The denominator must be a power of 2.

Reply: Thank you for your comments. We have made modifications it in the manuscript.

Author Response File: Author Response.docx