# Integration of Wavelet Denoising and HHT Applied to the Analysis of Bridge Dynamic Characteristics

*Reviewer 1:*Anonymous

*Reviewer 2:*Samir Khatir

*Reviewer 3:*Anonymous

**Round 1**

*Reviewer 1 Report*

The paper is full of language flaws which make it hard to read and understand.

The subject is of interest and the dataset quite novel.

The method is simple and not innovative, as it is simply based on the application of wavelet denoising and EMD one after the other, but in the literature, these seem to have never been used together, so, as far as I know, it can be considered novel.

Nevertheless, the paper is overall very poor in explanations and analyses.

The proposed method is not even tested on simulated data (on which you can add noise by purpose on a true signal so that the computation of RMSE and SNR can be done based on this true signal). Furthermore, the comparison in section 4.1 is useless, as the performance evaluation parameters defined in eq. 17/18 are not suitable for assessing the true denoising ability of the algorithms.

A single test on two channels sampled at 10Hz for 3hours cannot be enough for stating that a denoising method is better than another.

Detailed revision:

Line 20-23.

“When Hilbert spectrum analysis is applied to the denoised data, the vibration frequency of the bridge structure could be identified clearly, which is consistent with the theoretical calculation, and determines effectively the natural vibration characteristic of the bridge structure.”

Rephrase improving English and improving “vibration frequency”. In Structural Health Monitoring it is common to perform diagnostics based on modal decomposition. Hence, it is common to speak about natural frequency, which is NOT the same as just “vibration frequency”. What do you mean with “vibration frequency”? Is there a link to SHM modal analysis? Or are you just looking for a “signature” in the vibration spectrum? Clarify.

Line 30. “… various conditions, is …” remove the comma.

Line 33. “long-term continuous high-frequency dynamic” separate with commas, otherwise it is unreadable. “measurement information” select one of the two.

Line 35. “structure safety status monitoring” researchers refer to it as Structural Health Monitoring (SHM).

Line 39. “mine the rich structural health information” Rephrase. Data is mined to extract useful information.

Line 40-42. It is just a repetition of lines 37-40.

Line 45-49. “non-stationary signal processing methods, such as STFT …” Short Time Fourier Transform (STFT) is defined as a non-stationary signal processing method, but later you say that it can lead to the phenomenon of false components. What do you mean? Could you add references? Also, I do not find good to compare STFT window and WT mother wavelet.

Line53-55. “The method carries out adaptive decomposition according to the signal itself, and separates some Intrinsic Mode Functions (IMF) according to the frequency (from high frequency to low frequency) from the non-stationary signal” The method carries out BLIND adaptive decomposition

according to the signal itself, and separates THE NON-STATIONARY SIGNAL INTO some Intrinsic Mode Functions (IMF) according to the frequency CONTENT (from high frequency to low frequency).

Line 57. “noise information” Just NOISE.

“modal aliasing” is it always the same Mode-Mixing problem?

Line 57-60.“However, due to the influence of noise information and signal discontinuity, modal aliasing often occurs when EMD is applied to the decomposition of non-stationary noisy-containing signals, which may result in interference on EMD decomposition results and the subsequent results of Hilbert spectrum analysis, then the effective physical information may not be accurately determined” Too long. Rephrase for clarity.

Line 61. “monitoring data processing” Select what you want to say. E.g., “The GNSS DATA of the …” is more than enough.

Line 66. “in many times” remove it or rephrase.

Line 69. “the pseudo-components” you refer to them as you already introduced them. Is it always the same Mode-Mixing problem?

It is not good to refer to the same issue with several different names without explanation. Reading and understanding the paper becomes harder.

Line 71-73. “Viewing these, concerning the issues of remarkable effect by noise when HHT method is applied to non-stationary data, taking account of the excellent time-frequency localization properties in the field of filtering and denoising” Rephrase. It’s too long and the meaning is not clear.

Line 74-76. “a time-frequency analysis method based on combination of wavelet threshold denoising and HHT is proposed, for the object of GNSS dynamic monitoring data on the Sutong Bridge, for the purpose of identifying the vibration frequency of bridge structure, and the implementation process is given in this study ” Even this half of the phrase is too long. Rephrase it.

Line 77-79. “The method firstly carries out noise reduction preprocessing on the dynamic monitoring data to obtain the lowered decomposition layers in the EMD decomposition process and reduce the effect of marginal effects on the quality of useful signal decomposition;” Unclear. Rephrase it.

Line 106. “the essence of EMD model is to stabilize the original signal” Really? What do you mean by “stabilize”?

Line 111. “The parsing signal is constructed” isn’t this commonly called “analytic signal”?

Line 113. “can be recorded as” what do you mean by “recorded”? Recorded seems incorrect. Expressed? Retrieved?

Eq.(9). Why do you use not the “x” in the formula? Is this a simple product? This is not coherent with the rest of the paper.

Eq.(10/11). I really cannot understand the “RP” notation. Correct it. If we are neglecting r_n then use a “hat” on the x(t) as commonly done. I don’t see the need of adding RP before the summation.

Eq.(12). Isn’t a 1/T normalization missing in the definition?

Line 129. “itself, and but” probably “and” should be removed.

Line 130/131. “Therefore, the signal-to-noise ratio of the signal is relatively low, it greatly affects the recognition accuracy of the vibration frequency.” rephrase.

Line 136/7. “denoising preprocessing for the aiming of the accuracy and timeliness improvement of signal feature extraction” rephrase.

Line 146/7. “Wavelet transform analyzes the signal PSI(t) by scaling of scale and shifting of position” isn’t PSI the wavelet function? Calling it “signal” can be misleading. Also, “scaling of scale” and “shifting of position” is meaningless. Remove “of scale” and “of position”.

Line 156/160. “The threshold function has continuity and high-order conductivity, which overcomes the disadvantages of discontinuity and oscillation of the signal reconstructed from the hard threshold function, also overcomes the deficiency that although the soft threshold function is continuous, the denoising results all need to be subtracted from the threshold and there is systematic deviation if the coefficient exceeds the threshold.” Too long, rephrase for clarity.

Line 163. “may often not eliminate the interference of noise enough” may often not COMPLETELY eliminate the interference of noise enough.

Line 165. “to do some denoising post-processing appropriately,” to do some denoising APPROPRIATE post-processing appropriately DENOISING

Line 166/168. “The denoising method based on wavelet threshold denoising and empirical mode decomposition is called Wavelet-EMD method, whose specific steps are as follows:” is the Wavelet-EMD the novel method you are proposing? It is not clear from this phrase. Highlight this aspect.

Line 171/172. “according to the energy of the decomposed frequency band, the denoising signal which eliminates the interference noise is obtained.” This description is not clear enough. Rephrase.

Line 175. “from the wavelet threshold” remove this. It is not needed and make the phrase unclear.

Line 185. “correlation between each component and the original signal” you say that RHOi is computed between the i-th component and the original signal, but in Eq.(15) you use Si(t) which is NOT the original signal, but a residual. Correct the error.

Line 237-241. “The accuracy analysis of the observation sequence in Fig. 4 shows that the horizontal and vertical mean square errors in this whole time period are mx*, my*, which infers that there is no larger gross error among the observation value, and the accuracy of GNSS dynamic observation data is reliable and suitable for dynamic frequency extraction.” The meaning of “accuracy analysis” is not explained. You speak about Mean Square Error MSE, but it is not clear which is the error under analysis. In Eq.(17), the Root MSE is based on the error of the noise containing signal and the denoised signal. What do you mean here for MSE? Actually, it seems that your MSE here is just the variance of your acquisition, but I hope it is not so as it would be completely useless for assessing the reliability of the data. Clarify.

Table 1. and Lines 272-287. I have the feeling that the selection of the parameters according to which you set up the comparison are not so effective for comparing the denoising methods. According to Eq.(17/18) (your definition of SNR and RMSE + correlation R (undefined!)) you state that Wavelet-EMD is better because it shows the largest SNR and R while RMSE is the smallest. BUT, what if you take a y(i) almost equal to x(i) (y -> x, corresponding to no denoising at all)? According to your definitions, SNR -> infinity, RMS -> 0 R->1.

So having the largest SNR and R and a lower RMS is not proving that Wavelet-EMD is performing better denoising.

This does not mean that your method is not good (actually, by visual inspection, it seems much better than H and S Wavelet), but just that you are not proving it is better than the others in the whole section 4.1.

Table 2. You speak of Main frequency. Is this then corresponding to the first flexural mode of the bridge? Or you look just for a signature in the vibration spectra? As I asked at the beginning, clarify this aspect.

*Author Response*

Author Response File: Author Response.docx

*Reviewer 2 Report*

The paper presents an Integration of wavelet denoising and HHT applied to 3 analysis of the bridge dynamic characteristic. Before considering the paper for publication the authors should revise it based on the following comments:

- The abstract is so long, it should be reduced and reformulated to show the main objective of this study and novelties.
- The parameters should be identified under each equation.
- The experimental analysis should be revised including more information such as :

3.1. Experimental real bridge.

3.2. The position of accelerometers and which axes (*x*,*y*, or *z*).

- What about for other seasons?
- Figs 7 and 8 are not clear should be revised with higher resolution.
- The noise should be considered for each method to make clear the accuracy
- Some relevant works based on experimental modal analysis of different structures can be included:

https://doi.org/10.3390/s20051271

https://doi.org/10.1016/j.jsv.2020.115315

https://doi.org/10.3390/pr8040440

https://doi.org/10.5755/j01.mech.23.4.15254

https://doi.org/10.3390/mi10090608

https://doi.org/10.1016/j.compstruct.2017.12.058

https://doi.org/10.3390/app6070199

https://doi.org/10.1016/j.jsv.2019.02.017

8. Please double-check the equations.

9. English should be revised.

*Author Response*

Dear Members of the Editorial Team and reviewers,

We appreciate the opportunity to modify our paper according to the critical comments of three reviewers. Many thanks for the insightful comments and suggestions of the referees.

Affected by the Corona Virus Disease 2019 (#COVID19) epidemic, the members of the research group have been isolated in different areas far away from the common office places, and there is poor network signal in some places, which causes a lag in communication between the Lab Associates, so the revision of this research paper could not be carried out quickly. We apologies to the editors and reviewers of this journal.

In response to the comments made by both reviewers, we have added analog signal analysis, revised and re-written some sections of the manuscript, eliminating theoretical ambiguity to previously statement. Finally, we invited native English speakers to polish the manuscript. Please find our detailed replies to the reviewers’ specific comments below. Words in red are the changes we have made in the revised manuscript.

We tried our best to improve the manuscript and made some changes in it. These changes will not influence the content and framework of the study. And we did not list all the changes but marked in red in the revised paper. We believe the issues raised improve the quality of the meta-analysis. Thanks for your re-consideration. We have reviewed the final version of the manuscript and approved it for publication. We appreciate for the Editors/Reviewers’ warm work earnestly, and hope that the correction will meet with approval.

To the best of our knowledge and belief, this manuscript neither has been published in whole or in part nor is it being considered for publication elsewhere. We state that there is no conflict of interest and ethical adherence in this study. Kind regards,

Sincerely,

Shengxiang Huang

2020.05.06

Author Response File: Author Response.docx

*Reviewer 3 Report*

Review of the manuscript entitled "Integration of wavelet denoising and HHT applied to analysis of the bridge dynamic characteristic"

The manuscript presents application of Discrete Wavelet Analysis and Hilbert-Huang Transform for filtering bridge displacements obtained with aid of Global Navigation Satellite System (GNSS). This approach is compared with three other filtering techniques, namely: hard, soft and half threshold denoising. The proposed methodology is a currently investigated topic of great importance for bridge engineers dealing with Structural Health Monitoring, however the manuscript suffers from a number of shortcomings which have to be addressed before considering it for possible publication.

Introduction:

1) The first two sentences of paragraph starting from lines 37-39 are repeated in lines 40-43. Please remove redundant sentences.

2) The introductory part of the manuscript presents very limited overview of existing techniques for bridge health monitoring, not mentioning even other methods for displacement measurements in bridges. I think that the Authors should enrich literature review and refer to some recent works on that topic, examples are:

- Yu et al. (2020) Global Navigation Satellite System‐based positioning technology for structural health monitoring: a review. Struct Control Health Monit. 2020; 27:e2467. https://doi.org/10.1002/stc.2467

- An et al. (2019) Recent progress and future trends on damage identification methods for bridge structures. Struct Control Health Monit. 2019; 26:e2416. https://doi.org/10.1002/stc.2416

- Ogundipe et al. (2014) Wavelet De-noising of GNSS Based Bridge Health Monitoring Data, Journal of Applied Geodesy, 8(4), 273-282. https://doi.org/10.1515/jag-2014-0011

3) Section 2. recalls fundamental concepts taken from Empirical Mode Decomposition method. This is standard material which can be found elsewhere. Therefore I recommend to shorten this section and focus on novel aspects of this study avoiding repetition of the basic knowledge.

4) In line 207 it is written "the greater the SNR of the acoustic emission signal after denoising is ... the better the denoising effect is gained". I thought that the present manuscript deals with displacement signals coming from GNSS not from acoustic emission? Please clarify!

5) The proposed methodology is validated only on the real data acquired on Sutong Bridge. This is a little problematic since actual model of this bridge is not known and conclusions draw only on this example can be misleading. I strongly recommend to verify the proposed method also on simple simulation model (containing a few degrees of freedom). Obviously, measurement noise should be then also simulated numerically. Such an example should demonstrated how the proposed methodology captured time-varying frequencies or other system parameters.

6) Finally, It would be also interesting to more thoroughly investigate the discrepancy between finite element model of the analyzed cable-stayed bridge and its real response. In particular, is the predicted value of the first horizontal natural frequency = 0.145 Hz equal to mean value of the oscillating natural frequency presented in Fig.11? Please address this question.

Comments for author File: Comments.pdf

*Author Response*

Dear Members of the Editorial Team and reviewers,

We appreciate the opportunity to modify our paper according to the critical comments of three reviewers. Many thanks for the insightful comments and suggestions of the referees.

Affected by the Corona Virus Disease 2019 (#COVID19) epidemic, the members of the research group have been isolated in different areas far away from the common office places, and there is poor network signal in some places, which causes a lag in communication between the Lab Associates, so the revision of this research paper could not be carried out quickly. We apologies to the editors and reviewers of this journal.

In response to the comments made by both reviewers, we have added analog signal analysis, revised and re-written some sections of the manuscript, eliminating theoretical ambiguity to previously statement. Finally, we invited native English speakers to polish the manuscript. Please find our detailed replies to the reviewers’ specific comments below. Words in red are the changes we have made in the revised manuscript.

We tried our best to improve the manuscript and made some changes in it. These changes will not influence the content and framework of the study. And we did not list all the changes but marked in red in the revised paper. We believe the issues raised improve the quality of the meta-analysis. Thanks for your re-consideration. We have reviewed the final version of the manuscript and approved it for publication. We appreciate for the Editors/Reviewers’ warm work earnestly, and hope that the correction will meet with approval.

To the best of our knowledge and belief, this manuscript neither has been published in whole or in part nor is it being considered for publication elsewhere. We state that there is no conflict of interest and ethical adherence in this study.

Kind regards,

Sincerely,

Shengxiang Huang

2020.05.06

Author Response File: Author Response.docx

**Round 2**

*Reviewer 1 Report*

The revised paper is substantially improved in terms of language and performance evaluation of the proposed method.

The added Section 3.3 now correctly verifies the effectiveness of the proposed algorithm with model signals.

Nevertheless, I can still find at least 3 major conceptual flaws:

- The paper's purpose is clearly that of denoising the GNSS data for estimating the first bending natural frequencies (in horizontal and vertical directions) of the bridge. Nevertheless, the correct name “first natural frequency” never appears. The work only refers to a generic “vibration frequency” which has no meaning. This must be fixed. At least “main vibration frequency” should be used, adding in the text that the main vibration frequency corresponds to the first natural frequency.
- The paper clearly pertains to the field of SHM as the aim is to monitor the state of health of a bridge structure. This must be highlighted and the proper name “SHM” needs to be added.
- In the text, it is common to read “RMSE or MSE of a signal” WITHOUT any description about the reference from which the error is computed. This is an important flaw that must be fixed. MSE and similar parameters need a signal AND a reference to be completely specified. A lack of clarity makes the analysis superficial and the results unsupported. For example, the original analysis in section 4.1 seems meaningless. This must be corrected throughout the whole paper.

Detailed notes can be found hereinafter:

- 1
^{st}Rev unresolved issues

**[**

**Question:** Eq.(10/11). I really cannot understand the “RP” notation. Correct it. If we are neglecting r_n then use a “hat” on the x(t) as commonly done. I don’t see the need of adding RP before the summation.

**Answer:** We feel sorry for the inconvenience brought to the reviewer. After Fourier or Hilbert transform, the result is often a (series of) complex number. The complex number includes a real part and an imaginary part. Here ‘RP’ represents the real part.

**]**

**Comment:**

You still should correct it for the readers. In line 126 it is still written: “where *RP *is the actual part of the original signal”. Fix it writing “where RP is the REAL part of the original signal”. Anyway, the common mathematical notation for real part extraction of a complex value z is Re(z), not RP (https://en.wikipedia.org/wiki/Complex_number#Notation). It would be better to conform to the notation Re( ).

** **

**[**

**Question *: **

“When Hilbert spectrum analysis is applied to the denoised data, the vibration frequency of the bridge structure could be identified clearly, which is consistent with the theoretical calculation, and determines effectively the natural vibration characteristic of the bridge structure.”

Rephrase improving English and improving “vibration frequency”. In Structural Health Monitoring it is common to perform diagnostics based on modal decomposition. Hence, it is common to speak about natural frequency, which is NOT the same as just “vibration frequency”. What do you mean with “vibration frequency”? Is there a link to SHM modal analysis? Or are you just looking for a

“signature” in the vibration spectrum? Clarify.

**Answer *: **We gratefully thanks for the precious time the reviewer spent making constructive remarks. Within the scope of my subject knowledge, the construction engineering involving surveying engineering is divided into three stages: survey design, engineering construction and operation management stage. Structural Health Monitoring always relates to the deformation monitoring in operation management stage of a large building (structure). In fact, the monitoring data were collected during the engineering construction stage of Sutong Bridge. It looks like natural frequency in Structural Health Monitoring, but the difference still exists.

The purpose is really to identify the frequency characteristics. Then the characteristics serve as key factor of structural modal analysis

**Question****:** Table 2. You speak of Main frequency. Is this then corresponding to the first flexural mode of the bridge? Or you look just for a signature in the vibration spectra? As I asked at the beginning, clarify this aspect.

**Answer:** We feel sorry for the inconvenience brought to the reviewer.

Main frequency is corresponding to the first flexural mode of the bridge here. And the question can be answered in similar manner of Answer # (Corresponds to Question #).

**]**

**Comment:**

Given that the purpose of your analysis is to identify the first flexural natural frequency of the bridge structure, why don’t you use its proper name instead of the meaningless “vibration frequency”?

A single “vibration frequency” could exist only for a 1dof model. For a complex continuous structure, the response will always be a sum of infinite modes + noise, leading to a wide-range spectral response. The use vibration frequency is then not only meaningless but also wrong. I could accept “main vibration frequency” if you explain in the text that you are interested in the estimation of the first natural frequency which corresponds to the main frequency (i.e., largest amplitude) in the spectrum.

Correct it throughout the whole paper, starting from the abstract (lines 18 and 22), and add clarification for the correspondence of “main vibration frequency” to the “first natural bending frequency”.

**[**

**Question:** Line 35. “structure safety status monitoring” researchers refer to it as Structural Health Monitoring (SHM).

**Answer:** We totally understand the reviewer’s concern. Similar to the answer of Question *, ‘structure safety status monitoring’ may include the structure safety status monitoring during the engineering construction stage, but Structural Health Monitoring may not. The data in this study is collected during the engineering construction stage of Sutong Bridge. “structure safety status monitoring” may mean more general in our opinion.

**]**

**Comment:**

“structure safety status monitoring” is meaningless in English, in fact it does not appear in the revised version, but it would be good for you to add a framework for your analysis.

I suggest you again to add that the aim of your work is to monitor the health state of a structure so that it falls in the field of Structural Health Monitoring.

Even if SHM is commonly applied to operation management of structures, it is NOT ONLY valid in operation. You can specify this inside the paper.

**[**

**Question:** Line 237-241. “The accuracy analysis of the observation sequence in Fig. 4 shows that the horizontal and vertical mean square errors in this whole time period are mx, my, which infers that there is no larger gross error among the observation value, and the accuracy of GNSS dynamic observation data is reliable and suitable for dynamic frequency extraction.” The meaning of “accuracy analysis” is not explained. You speak about Mean Square Error MSE, but it is not clear which is the error under analysis. In Eq.(17), the Root MSE is based on the error of the noise containing signal and the denoised signal. What do you mean here for MSE? Actually, it seems that your MSE here is just the variance of your acquisition, but I hope it is not so as it would be completely useless for assessing the reliability of the data. Clarify.

**Answer:** We feel sorry for the inconvenience brought to the reviewer.

Firstly, the course ‘Surveying Adjustment’ is one of the most important, rich-basic-theories and data processing techniques in Geomatics and topography. The subject is first adopted and developed by German mathematician Gaussian in the triangulation adjustment of the Hannover radian measurement from 1821 to 1823. ‘accuracy analysis’ is an important basic concept and method of this course. Thus, it is not necessary to introduce the concept of accuracy analysis in our opinion, because the members of the research group of this study are all teachers or students of this discipline. If you are interested in understanding, please refer to the textbook: *Error Theory and Foundation of Surveying Adjustment, by Benzao Tao, Weining Qiu, Yibin Yao, et al., Wuhan University Press*.

Secondly, Nakamura [14] used GNSS technology in 1998 to monitor the dynamic deformation of a suspension bridge with a main span of 720 m and verified that the vibration displacement and main frequency of the main beam under wind load are consistent with the results of the wind tunnel experiments and finite element calculations. Although the GNSS measurement technology and data processing methods at that time were not as advanced and reliable as they are today. So it is unnecessary to worry about the accuracy of GNSS measurements nowadays.

Last but not least, exploring the abundant structural health information hidden in GNSS monitoring signals is crucial due to the fact that the data collected by GNSS monitoring are reliable and contain a certain level of noise.

**]**

**Comment:**

First, the “Prince of Mathematicians” last name is Gauss, as you could have verified before writing the wrong name in this answer.

Second, as I now clearly understand, “accuracy analysis” is not a particular analysis you used, but just an analysis of the accuracy of the GNSS measurements.

So, I repeat again, and I hope this is for the last time.

When you write Mean Square Error (MSE) you mean the error of measurement (in this case the GNSS measurements) from a reference which can be a known quantity (e.g. in section 3.3 you correctly used the true noise-free analog signal as reference) OR a measurement from a better instrument, as done by Nakamura [14], who compared GNSS to accelerometers.

So, I expect you to ALWAYS write in the text “MSE of signal y from reference x”, or to give the right formula you used.

Unfortunately, I was not able to find your reference [16], but in the abstract Huang et al. state that “The monitoring system indicates superiority compared with conventional methods”: starting from the abstract they already declare their reference.

Since you NEVER speak about a reference nor you clarify what error is MSE referred to, what you write in lines 292-296 has NO VALUE AT ALL. You HAVE TO specify a reference for the computation, referring to [16] is not enough.

If you do not give a reference for error computation, the only thing I can think about is that you used the mean value as a reference, so that MSE becomes the variance, but again, I hope it is not the case, as it would be meaningless for the sake of GNSS measurements validation.

Could you please clarify to me and in the text how mx and my are computed (i.e., what is the reference measurement you adopted)?

Last and Least, I hope you put way more care in teaching your ‘Surveying Adjustment’ course. The original text before revision was very superficial, and not suitable for publication in any international Journal. It is my opinion that your revision is now acceptable, but only if you fix the still superficial aspects present in this version.

- Detailed text revision:

Line 37-38: “[sensors] can monitor their local point-like or linear natural frequency characteristics”

It is not clear what do you mean by “linear natural frequency”. Maybe you wanted to state that [sensors] recording local point-like measurements can monitor global characteristics of the bridge such as the structure’s natural frequencies?

Line 218: RMSE wrongly spelled.

Fig 4-6: add an indication of the 4db SNR.

Table 1: Remove it. Table 1 is contained in Table 2 so it is a nonsense to repeat the same information.

Line 321: “The *SNR *and *RMSE *of the denoised signal and the linear correlation coefficient between the denoised signal and the original observation sequence are calculated …” -> The SNR, the RMSE and the linear correlation coefficient R of the denoised signal and the original sequence are calculated …

N.B. YOU ALWAYS HAVE TO SAY WHICH IS THE REFERENCE FOR THE ERROR COMPUTATION.

If I have understood correctly, in this case, you call “noise” the difference of the original GNSS measurement (x) and the denoised version (y), which corresponds to the “error” in RMSE. Error=Noise= x-y. Anyway, I repeat, if this is what you did, this DOES NOT ALLOW YOU TO PROVE THAT A DENOISING METHOD IS BETTER THAN ANOTHER ONE. For example, if you use y=x (NO DENOISING) in fact, the error goes to 0, leading to RMSE ->0, SNR-> infinity, R->1. But this is not the best denoising, this is NO denoising at all.

So what you write in Line 333-334: “The method has a lower *RMSE *of the denoised signal and a higher *SNR *than the three methods […] This result proves that wavelet-EMD achieves the best denoising effect amongst the four methods” is WRONG.

If I have understood correctly then, you have either to remove table 3 or correct it by changing your performance evaluation parameters specification, because the so specified SNR, RMSE and R (i.e., error computed as the difference of noise affected measured signal and denoised signal) is not suitable for a meaningful comparison.

These were suitable for the analysis in section 3.3, which is good, because the reference is the TRUE SIGNAL x (WITHOUT NOISE!) and this is compared against the denoised y. But in this section 4.1 you seem to have changed the true signal with the measured, noise affected, signal in the definition of SNR and RMSE. If this is the case, the analysis is WRONG and MEANINGLESS.

On the contrary, if I have misunderstood, it means that not enough information was given, so please either add the missing information or correct/remove the analysis.

Line 423: you speak about a theoretical and empirical computation of the first flexural mode but the methodology is not highlighted, only reference [33] is given. Personally, I was not able to access this paper. Could you please add information about the computation? Is this from a MDOF model, FEM model, analytic model etc.? Or are the natural frequencies learned from data? Was it computed in reference [33] or did you compute it following reference [33] indications? Which is the accuracy of the result from the used model? Was the model validated? Is it reliable?

Clarify these aspects in the text.

*Author Response*

Please see the attachment.

Author Response File: Author Response.docx

*Reviewer 2 Report*

Decision Accept

Revision required please double-check the references.

[3] S. Khatir, I. Belaidi, T. Khatir, Tawfiq Khatir, A. Hamrani, Y. L. Zhou, M. A. Wahab. Multiple damage

detection in unidirectional graphite-epoxy composite beams using particle swarm optimization and genetic

algorithm [J]. Mechanika, 2017, 23(4): 514-521.

Remark: The third author mentioned 2 times.

[6] S. Khatir, M. A. Wahab, B. Djilali, T. Khati. Structural health monitoring using modal strain energy damage

indicator coupled with teaching-learning-based optimization algorithm and isogoemetric analysis [J].

Journal of Sound and Vibration, 2019, 448: 230-246.

Remark: T. Khatir instead T.Khat

*Author Response*

Please see the attachment.

Author Response File: Author Response.docx

*Reviewer 3 Report*

Re-review of the manuscript entitled "Integration of wavelet denoising and HHT applied to the analysis of bridge dynamic characteristics"

The Authors addressed my major concerns. A new numerical section appeared in the reviews manuscript showing effectiveness of the proposed methodology on four test signals.

Only few minor remarks left:

- line 106: There is a mistake in the cited reference [2626]

- Eq.3: Meaning of the symbol h_1k(t) is missing. It is probably value of h_1 after k-th iteration

- Figs.15-16 and 17-18 showing Hilbert time and marginal spectra for horizontal and vertical oscillations, respectively, could be improved by focusing on frequency range from 0 to 0.4-0.5 Hz

- Finally, I think that the term "Main frequency" appearing in Table 4. could be replaced with "Measured frequency"

After improving above shortcomings I recommend the manuscript for publication.

*Author Response*

Please see the attachmen.

Author Response File: Author Response.docx