Time–Frequency Domain Analysis of the Ground Vibration of an Elevated Railway and Study on the Elliptic Polarization Dispersion Characteristics of Rayleigh Waves

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper provides useful knowledge about countermeasures for ground vibration caused by high-speed elevated railways, and I believe it is worthy of publication.

Please respond to the following questions.

 

Chapter 1

-Is the “vibration isolation ditch” a new technology in the world? It would be a good idea to thoroughly research existing research around the world.

 

-Regarding the issue of ground vibration caused by high-speed railways, only existing research in China has been investigated. Why not also investigate existing research outside of China?

 

-Why do you use "ditch" and "trench" differently? If they have the same meaning, why not just use "ditch" or "trench"?

 

Chapter 3

- Please provide a geological column diagram.

- Please provide structural specifications such as the viaduct's structural type, girder height, span, and pier specifications.

 

3.1

- Please indicate the vehicle specifications. Wheelbase, vehicle length, etc. have a significant effect on ground vibration.

 

3.1.1

- Please show us the measurement results near the bridge pier. The periodic vibration of the vehicle is clearly visible.

- Why did you evaluate it based on displacement rather than acceleration or speed?

 

Figures 4 and 6

- The numbers are too small to read, please make them readable.

- To compare Figure 4 and Figure 6, the vertical scale and color should be the same, which is easier for readers to understand.

 

Chapters 3 and 4

- Chapter 4 is an analysis after the installation of the vibration isolation trench, and Chapter 3 is actual measurements without the vibration isolation trench? Because of Figure 2, it is easy to misunderstand that Chapter 3 is also based on conditions with the vibration isolation trench

 

Figure 7

- In the paper, the length of the vibration isolation trench is 10m, but I think that the actual construction will be much longer. Is the plan to actually install it along the entire viaduct, along the entire railway line? (This is a simple question, and does not need to be stated in the paper.)

 

5.4

- I think it is inappropriate to describe the ground vibrations caused by a train passing as “earthquake” or “seismic”.

 

5.4.1 Elliptical polarization ⇒ 5.4.2

Author Response

Dear Editor,

We are very grateful for the comments and questions from the reviewer. And the related modifications and reply responses are given as below following the reviewer comments indicated and all the modifications have been indicated by red characters in the manuscript:

This paper provides useful knowledge about countermeasures for ground vibration caused by high-speed elevated railways, and I believe it is worthy of publication.

Please respond to the following questions.

Chapter 1

-Is the “vibration isolation ditch” a new technology in the world? It would be a good idea to thoroughly research existing research around the world.

Response: We sincerely appreciate the reviewer's thoughtful feedback on our manuscript.“vibration isolation ditch” are widely used on highways, more accurately described as a newly introduced technology in high-speed rail applications , but they can indeed be elevated to thoroughly research existing research around the world.

-Regarding the issue of ground vibration caused by high-speed railways, only existing research in China has been investigated. Why not also investigate existing research outside of China?

Response: We are grateful for the reviewer's valuable insights and suggestions.

Your suggest is very good, but we don't have enough funding support. Let's first thoroughly study the domestic research.

-Why do you use "ditch" and "trench" differently? If they have the same meaning, why not just use "ditch" or "trench"?

Response:Thank you, reviewer, for your meticulous attention to our paper. Your are right.We have unified of "trench".

Chapter 3

- Please provide a geological column diagram.

Response: Your discerning observations have greatly assisted in refining our manuscript.We have added this section, at chapter 3 and indicated by red characters in the manuscript. Such as the test site was located at the elevated section near K988+848 of Weinan North Station on the Zhengzhou-Xi’an High-Speed Railway. The strata at the test site are predominantly quaternary, with a total thickness of approximately 200 m. The lithology from top to bottom consists of approximately 19 m of self-weight collapsible loess from the Upper Pleistocene, 28 m of non-collapsible loess from the Upper Pleistocene, and a stable sand layer from the Middle Pleistocene.

- Please provide structural specifications such as the viaduct's structural type, girder height, span, and pier specifications.

Response: We appreciate the reviewer's careful reading and insightful suggestions.

We have added this section, at chapter 3 and  indicated by red characters in the manuscript. Such as The main beam of the elevated bridge in the experimental area is made of C50 concrete, with a beam top width of 11.4m, a beam height of 2 meters, and a beam bottom width of 5.4m. The pier columns are simply supported beams with equal heights and are made of C40 concrete. The length of the pier bottom in the longitudinal direction of the bridge is 2.2m, and the length in the transverse direction of the bridge is 2.8m.

3.1

- Please indicate the vehicle specifications. Wheelbase, vehicle length, etc. have a significant effect on ground vibration.

Response: The reviewer's feedback has been invaluable in shaping our manuscript.

We have added this section, at chapter 3 and  indicated by red characters in the manuscript. The high-speed train operated by the Zhengxi High speed Railway is the "Harmony" CRH2C high-speed train, and the characteristic length of the test train is shown in Figure 2.

 

Figure 2. Test the characteristic length of the train

3.1.1

- Please show us the measurement results near the bridge pier. The periodic vibration of the vehicle is clearly visible.

Response:Thank you for the reviewer's thorough examination and helpful comments.The measurement results near the bridge pier were disorganized due to the influence of various waves, and did not reflect the characteristics of surface wave propagation, which is evident in Figures 14, 15, and 16(we have add some figures so the serial number has changed). Therefore, data collection began at a distance of 15 meters from the bridge pier. It can be used for future research within the range of 0-15 meters.

- Why did you evaluate it based on displacement rather than acceleration or speed?

Response: We are grateful for the reviewer's valuable insights and suggestions. Because the data collected by the selected equipment is in displacement form, both numerical simulation and measured data analysis use displacement

Figures 4 and 6 The numbers are too small to read, please make them readable.To compare Figure 4 and Figure 6, the vertical scale and color should be the same, which is easier for readers to understand.

Response: Thank you for the reviewer's thorough examination and helpful comments.We have made modifications and made the vertical scale and color the same .we have add some figures so the serial number has changed.such as

 

Figure 6. Vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h.

 

Figure 8. Ground vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h with the implementation of the vibration isolation trench.

Chapters 3 and 4

- Chapter 4 is an analysis after the installation of the vibration isolation trench, and Chapter 3 is actual measurements without the vibration isolation trench? Because of Figure 2, it is easy to misunderstand that Chapter 3 is also based on conditions with the vibration isolation trench.

Response: We are grateful for the reviewer's comprehensive review and helpful recommendations.yes, you are right.The two situations are only separated by vibration isolation trench, so wehave marked by red colour in Figure 2. We think it is not necessary to repeat the picture again.

Figure 7

- In the paper, the length of the vibration isolation trench is 10m, but I think that the actual construction will be much longer. Is the plan to actually install it along the entire viaduct, along the entire railway line? (This is a simple question, and does not need to be stated in the paper.)

Response:The reviewer's suggestions have significantly contributed to the improvement of our research.The "10-meter" dimension referenced in this study denotes depth rather than horizontal length, a scale that does not provide effective vibration isolation in practical engineering scenarios. Considering cost optimization, targeted implementation near residential zones is recommended instead of full-length deployment along the entire railway line.

5.4 I think it is inappropriate to describe the ground vibrations caused by a train passing as “earthquake” or “seismic”.

Response: The reviewer's suggestions have significantly contributed to the improvement of our research.I understand what you mean, perhaps using vibration would be more appropriate. If we understand it as distinguishing artificial earthquakes from natural earthquakes, it is also possible.

5.4.1 Elliptical polarization ⇒ 5.4.2

Response:We are thankful for the reviewer's time and effort in providing constructive feedback. We have made modifications.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. The general problem is stated, but the specific gap is not. Authors can consider explicitly mentioning the lack of research on elevated railways and their mitigation in loess soil.

2. Providing some concluding remarks to the introduction section with a clear statement of objectives can provide a clear roadmap for the reader.

3. Instead of a general definition, authors should consider linking the method directly to the study's goals.

4. In this sentence please explicitly state what you are looking for in the spectrum: "The vibration spectrum effectively reflects the frequency components of the vibration and the distribution of the vibration energy."

5. Providing enough detail for another researcher to replicate the work is crucial. The data is digital, so the analysis almost certainly used a Fast Fourier Transform (FFT) algorithm, not a continuous integral. Authors are expected to clarify this point.

6. Authors must consider adding a brief introductory sentence after each title before giving the next subtitle.

7. "A three-component EPS portable digital seismograph" is a good to be mentioned, but the specific model number should be provided.

8. The instrument is described as "three-component," and the results are later presented for X, Y, and Z directions. This orientation should be explicitly defined in the methodology section.

9. Replacing qualitative descriptions like "significantly reduced" with specific numerical values by a table comparing peak amplitudes (Dmax) would be highly effective.

10. Please, briefly explain why the trench is more effective for high frequencies, linking the observation to the physics of wave propagation and diffraction.

11. Authors should explicitly validate the numerical model by comparing its "no trench" output to the field data from Section 3.

12. Defining the elliptic polarisation rate with an equation or clear description will provide scientific clarity and reproducibility.

13. Are the y-axis labels in the polarisation profiles (Figures 11, 12, 14) frequency, pseudo-depth, or something else?

Author Response

Dear Editor,

We are very grateful for the comments and questions from the reviewer. And the related modifications and reply responses are given as below following the reviewer comments indicated and all the modifications have been indicated by blue characters in the manuscript:

1. The general problem is stated, but the specific gap is not. Authors can consider explicitly mentioning the lack of research on elevated railways and their mitigation in loess soil.

Response:We are grateful for the reviewer's comprehensive review and helpful recommendations.Within the abstract, we have incorporated a discussion addressing ground vibrations induced by high-speed trains traversing viaducts in loess soil deposits, alongside noting the relative scarcity of research on this phenomenon. The introduction has been expanded to delineate applications of isolation trenches in relevant engineering domains. These additions are highlighted in blue throughout the manuscript.

Similarly, considerable vibration levels are transmitted to the subgrade and surrounding structures when trains operate on viaducts within the Loess Plateau region. However, research on mitigating these vibration effects remains relatively scarce.

Isolation trenches find predominant application in highway engineering, while their implementation in railway systems—particularly viaduct railways—remains relatively limited. This gap is exemplified by the scarcity of experimental investigations, such as those conducted by (Zhang et al. 2017; Cao et al. 2012), examining the vibration isolation efficacy of trenches on subgrade and underlying soil strata.

2. Providing some concluding remarks to the introduction section with a clear statement of objectives can provide a clear roadmap for the reader.

Response:The reviewer's suggestions have significantly contributed to the improvement of our research.We have added corresponding content in the introduction.This study first analyzed the properties of ground vibrations in the time-frequency domain, utilizing three-component geophones to examine displacement variations in each direction. This included analyses of velocity dispersion characteristics and ellipticity dispersion characteristics. Numerical simulations were subsequently conducted using a large-scale nonlinear finite element analysis software package. The vibration displacement patterns were compared both before and after the incorporation of a 10-meter-deep vibration isolation trench to evaluate the effectiveness of the open trench as a vibration mitigation measure.

3. Instead of a general definition, authors should consider linking the method directly to the study's goals.

Response: Your keen insights have been instrumental in elevating the quality of our paper.We have added a discussion on the relationship between methods and research objectives in the basic theory section.

Time-domain analysis reveals localized concentration characteristics in the vibration response across measurement points, alongside discernible trends in ground vibration displacement amplitude and vibration duration with increasing distance from the source. Frequency-domain (spectral) analysis extracts patterns not statistically discernible in the time domain, enabling the identification of dominant frequency distributions at ground measurement points. This analysis also elucidates the frequency characteristics of vibrations induced by the periodic loading of high-speed trains, and reveals the vibration spectra across different measurement points, directions, and distances resulting from trains operating at varying speeds. Consequently, time-domain analysis and frequency-domain analysis serve as the primary diagnostic tools in this study.

4. 4. In this sentence please explicitly state what you are looking for in the spectrum: "The vibration spectrum effectively reflects the frequency components of the vibration and the distribution of the vibration energy."

Response: We appreciate the reviewer's careful reading and insightful suggestions.After careful consideration, we have revised this sentence to state that The vibration spectrum effectively reflects the frequency components of vibration and the distribution of vibrational energy, allowing for intuitive observation of frequency and energy concentration regions.These additions are highlighted in blue throughout the manuscript.

5. Providing enough detail for another researcher to replicate the work is crucial. The data is digital, so the analysis almost certainly used a Fast Fourier Transform (FFT) algorithm, not a continuous integral. Authors are expected to clarify this point.

Response: We extend our sincere thanks to the reviewer for their insightful and constructive comments.We have already made modifications.such as:by applying the Fast Fourier Transform (FFT) to the time domain data sequence obtained from the tests, the vibration spectrum for ground measurement points at various distances and in various directions caused by trains traveling at different speeds was obtained according to the following equation (Krylov et al. 1994).the modifications have been indicated by blue characters in the manuscript.

6. Authors must consider adding a brief introductory sentence after each title before giving the next subtitle.

Response: The reviewer's feedback has been invaluable in shaping our manuscript.We have adding the brief introductory sentence.such as:

2.Basic method  

This chapter mainly introduces the basic theories and methods used in the article, as well as their relationship with the research content.

3.Data acquisition and processing

‌This chapter details the data acquisition methodology, including the experimental scheme, site selection, and instrumentation. Furthermore, it presents the analysis of ground vibration experimental results obtained from a representative viaduct track segment without vibration isolation trenches.

4.Empty trench vibration isolation effect test

‌This chapter presents the experimental time-frequency domain results for ground vibrations measured at a viaduct track segment equipped with vibration isolation trenches.

5.Analysis of elliptical polarization characteristics of empty trench vibration isolation

‌This chapter primarily investigates the synthetic seismic records of elevated railway tracks both with and without vibration isolation trenches, focusing on the dispersion characteristics of elliptical polarization ratios.

7. "A three-component EPS portable digital seismograph" is a good to be mentioned, but the specific model number should be provided.

Response:  Thank you, reviewer, for your meticulous attention to our paper.We have made modifications .And maeked by blue characters in the manuscript:

The test instrument was a three-component EPS portable digital seismograph(Fig. 4), The EPS portable digital seismometer incorporates an internal data logger and a high-sensitivity three-component sensor. With a frequency bandwidth of 0.2–200 Hz, it encompasses the 1–80 Hz range typically required for ambient vibration measurements. Configurable sampling rates include 1000, 500, 250, 100, and 50 sps.and it was ensured that the bottom of the sensor was in close contact with the ground during the setting up of the instrument.

 

Figure 4. EPS portable digital seismometer

8. The instrument is described as "three-component," and the results are later presented for X, Y, and Z directions. This orientation should be explicitly defined in the methodology section.

Response:Your keen insights have been instrumental in elevating the quality of our paper.In the chapter on data collection and processing, we have added the determination of X, Y, and Z directions.

To ensure consistency between experimental testing and numerical simulations, the following coordinate system is defined: the X-direction aligns with the railway track’s forward travel direction, the Y-direction is perpendicular to the survey line, and the Z-direction runs parallel to the survey line.

9. Replacing qualitative descriptions like "significantly reduced" with specific numerical values by a table comparing peak amplitudes (Dmax) would be highly effective.

Response:The reviewer's comments have provided us with a clearer direction for our research.The omission of comparative peak amplitude /Dmax/ tabulation in favor of qualitative descriptors denoting significant reduction stems from waveform processing—particularly in synthetic seismic record analysis—where bandpass filtering and other treatments alter peak values. Utilization of raw data risks introducing distorted measurements, potentially skewing research outcomes. Consequently, qualitative terminology emphasizing marked attenuation was adopted‌.

10. Please, briefly explain why the trench is more effective for high frequencies, linking the observation to the physics of wave propagation and diffraction.

Response:We are thankful for the reviewer's expertise and the time taken to review our work.the trench is more effective for high frequencies.

Mainly based on ,The velocity of the waves generated by vibration in loess is constant. As the frequency f increases, the corresponding wavelength λ decreases. Since the predominant energy of Rayleigh surface waves is concentrated within approximately one-half wavelength (λ/2), their energy progressively attenuates with wave propagation. When this half-wavelength becomes smaller than the trench depth, the trench effectively damps the transmission of high-frequency vibrations.

11. Authors should explicitly validate the numerical model by comparing its "no trench" output to the field data from Section 3.

Response:The reviewer's comments have been most beneficial in enhancing our work.‌Section 5.2 has been expanded to include comparative validation of the numerical model.‌ This section primarily details the analysis and processing of synthetic seismograms. Inspection of Figure 6 reveals the displacement time history curve of ground vibration induced by a single measurement point. Figure 6presents the vibration spectra obtained from 12 measurement points along different azimuths. Figure 13 demonstrates the comparability of synthetic seismograms recorded at 14 points along the main survey line. Analysis of Figure 11 indicates that in the absence of an isolation trench, surface wave energy is relatively concentrated, as evidenced by the propagation characteristics of the DP wave. This concentration is particularly apparent within the high-frequency region (depicted by the red-shaded area in Figure 8), which corresponds to the waveform following the direct P-wave arrival in Figure 13. ‌These observations collectively validate the numerical model.‌ Furthermore, Figure 5 and Figure 13 illustrate greater waveform instability with increasing propagation distance.the modifications have been indicated by blue characters in the manuscript

12. Defining the elliptic polarisation rate with an equation or clear description will provide scientific clarity and reproducibility.

Response:We appreciate the reviewer's careful reading and insightful suggestions.Subsection 2.3, "Elliptic Polarization Rate," has been added to Section 2 (Basic method) to enhance the structural completeness and theoretical rigor of the paper.

2.3. Elliptic Polarization rate

The elliptic polarization rate exhibits constant behavior within homogeneous elastic half-space media, whereas in layered media, it exhibits frequency-dependent behavior. (Haskell )Haskell proposed that this parameter can be derived by calculating the spectral ratio between the horizontal and vertical component spectra, denoted as H(f) and V(f) respectively, following Fourier transformation. This approach yields the defining equation:

                         (3)

Equation (3) reveals the frequency dependence of the elliptic polarization rate. This dependence constitutes a fundamental characteristic common to all modes of Rayleigh surface waves and is designated as elliptic polarization dispersion.

13. Are the y-axis labels in the polarisation profiles (Figures 11, 12, 14) frequency, pseudo-depth, or something else?

Response: We are thankful for the reviewer's expertise and the time taken to review our work.you are right,the y-axis labels in the polarisation profiles (Figures 13, 14, 16)is pseudo-depth We have added explanations.we have add some figures so the serial number has changed.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The article relates to the impacts of vibrations appearing during train passage on surrounding buildings and stratum structures and the effectiveness of a vibration isolation ditch in mitigating these effects. The authors’ time-frequency domain analysis of ground vibrations revealed that the characteristic frequency of the train induced pulse excitation in the track structure had a pronounced peak in the spectrum curve. They proved that the introduction of a vibration isolation ditch effectively blocks the propagation of vibration waves in the soil, reduces soil vibration, and significantly lowers the peak value in the spectrum. They also performed numerical simulations and analyzed the elliptical polarization dispersion characteristics of surface wave propagation with the vibration isolation ditch in place. Their results confirmed the effective damping performance of the ditch.

My comments are as follows:

1.There are two Subsections 5.4.1. - mistake.

2.Equation (3) – please use spaces and description as for Equations (1) and (2).

3.Equation (1) and its application, please explain assuming discrete data and seeking discrete results. The reader has no idea how to apply this formula.

4.Define all parameters (variables) used in Equation (2) and in the description below.

5.The scope of the tests should be specified – bottom of Page 2, and perhaps the test objectives should be described in more detail (not only vibrations but also displacements in various directions. A workflow would be useful.

6.Letters too small in the graphs in Figure 4.

7.It would be useful to compare the diagrams presented in Figures 3 and 6. Similarly, in Figures 4.6. Therefore, graphs of the differences between the above quantities are justified. Similarly, Figures 11 and 12.

8.The accuracy of computer modeling is not reported or illustrated.

9.How accurate are the results?

10.Did the authors repeat the tests and what differences occurred?

11.Statistical analysis seems to be helpful.

Author Response

Dear Editor,

We are very grateful for the comments and questions from the reviewer. And the related modifications and reply responses are given as below following the reviewer comments indicated and all the modifications have been indicated by red characters in the manuscript: We are very grateful for the comments and questions from the reviewer 2. And the related modifications and reply responses are given as below following the reviewer comments indicated and all the modifications have been indicated by green characters in the manuscript:

The article relates to the impacts of vibrations appearing during train passage on surrounding buildings and stratum structures and the effectiveness of a vibration isolation ditch in mitigating these effects. The authors’ time-frequency domain analysis of ground vibrations revealed that the characteristic frequency of the train induced pulse excitation in the track structure had a pronounced peak in the spectrum curve. They proved that the introduction of a vibration isolation ditch effectively blocks the propagation of vibration waves in the soil, reduces soil vibration, and significantly lowers the peak value in the spectrum. They also performed numerical simulations and analyzed the elliptical polarization dispersion characteristics of surface wave propagation with the vibration isolation ditch in place. Their results confirmed the effective damping performance of the ditch.

My comments are as follows:

1. There are two Subsections 5.4.1. - mistake.

Response:Thank you, reviewer, for your meticulous attention to our paper.We have made modifications and marked them in green font. 5.4.2. Elliptical polarization

2. Equation (3) – please use spaces and description as for Equations (1) and (2).

Response: We sincerely appreciate the reviewer's thoughtful feedback on our manuscript.We have made modifications .

                              (1)

                     (2)

                    (3)

3. Equation (1) and its application, please explain assuming discrete data and seeking discrete results. The reader has no idea how to apply this formula.

Response:We are indebted to the reviewer for their valuable and detailed feedback.

We have provided explanations for the variables in the formula.Higher train speeds induce greater vibrations, as evidenced by the variations in peak displacement values across different directions at various monitoring locations with distance. The attenuation patterns in all three directions at these locations are similar. Specifically, for a train speed of  within 0-75 m from the embankment, vibrations decay rapidly, with peak displacements generally exhibiting Y > Z > X magnitudes; beyond 75 m to 180 m, attenuation becomes more gradual, and peak displacement values in all three directions become comparable.

4. Define all parameters (variables) used in Equation (2) and in the description belo

Response:The reviewer's suggestions have significantly contributed to the improvement of our research.We have made the modifications according to your  suggestions.

where A(t) is a time domain data sequence;  is a sequence of spectral functions in the frequency domain; and ω = 2πf.

5. The scope of the tests should be specified – bottom of Page 2, and perhaps the test objectives should be described in more detail (not only vibrations but also displacements in various directions. A workflow would be useful.

Response:Thank you, reviewer, for your meticulous attention to our paper.We have made modifications .such as:

This study first analyzed the properties of ground vibrations in the time-frequency domain, utilizing three-component geophones to examine displacement variations in each direction. This included analyses of velocity dispersion characteristics and ellipticity dispersion characteristics. Numerical simulations were subsequently conducted using a large-scale nonlinear finite element analysis software package. The vibration displacement patterns were compared both before and after the incorporation of a 10-meter-deep vibration isolation trench to evaluate the effectiveness of the open trench as a vibration mitigation measure.the modifications have been indicated by green characters in the manuscript

6. Letters too small in the graphs in Figure 4.

Response: Your keen insights have been instrumental in elevating the quality of our paper.We have made modifications .Make the letters in the picture clear and visible.we have add some figures so the serial number has changed.

such as

 

Figure 6. Vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h.

 

Figure 8. Ground vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h with the implementation of the vibration isolation trench.

7. It would be useful to compare the diagrams presented in Figures 3 and 6. Similarly, in Figures 4.6. Therefore, graphs of the differences between the above quantities are justified. Similarly, Figures 11 and 12.

Response:We are grateful for the reviewer's valuable insights and suggestions.We have added comparative content in Figure 8 and Figure 14, and marked it in yellow.

Comparison with Figure 6 reveals that the incorporation of the isolation trench significantly enhanced high-frequency attenuation, with particularly pronounced effects observed within the 40–50 Hz frequency band.

However, by comparing Fig 11, Fig12, Fig13, and Fig14, it is found thatthe elliptic polarization rate exhibited greater sensitivity than synthetic seismic records, allowing for more immediate detection of anomalies.

the modifications have been indicated by green characters in the manuscript.

 

8. The accuracy of computer modeling is not reported or illustrated.

Response: Thank you, reviewer, for your meticulous attention to our paper.Computer modeling employed scaled models, incorporating proportional representations of key structural elements such as viaduct height and width. Model parameters were configured to approximate real-world conditions as closely as feasible; however, the inherent complexities of terrain topography preclude perfect replication. Consequently, the analysis focused on minimizing extraneous influences, such as acoustic interference. Future studies will incorporate additional physical variables to enhance rigor. For enhanced methodological clarity, the term "scaled modeling" was explicitly referenced in the numerical simulation methodology description.

9. How accurate are the results?

Response:The reviewer's comments have provided us with a clearer direction for our research. We are sorry,Mathematical modeling and measured data are analogical results, so the issue of accuracy cannot be quantified using data.

10. Did the authors repeat the tests and what differences occurred?

Response:Your feedback has been a valuable asset in the revision of our paper.

Due to funding constraints, repeated field measurements could not be performed. Although repeated numerical simulations were conducted by varying key parameters, such as soil layer thicknesses and associated coefficients, the core findings remained consistent.

11.Statistical analysis seems to be helpful.

Response:The reviewer's suggestions have significantly contributed to the improvement of our research.There are many working conditions that need to be further studied. In the later stage, we will adopt your opinions and use statistical analysis to obtain more perfect research conclusions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I have confirmed that the issues I pointed out have been carefully corrected. Thank you very much.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors clarified my concerns satisfactorily. They responded to my comments. They improved the clarity and solidity of the article. The manuscript is ready for publication.

Comments on the Quality of English Language

Quality of English Language is correct.