Modeling and Characterization of Complex Dynamical Properties of Railway Ballast

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript is interesting and well-written. In the article, the authors propose a new dynamic model of the railway ballast system, which is a sensitive and dominant component of track stiffness, and the dynamic stiffness of the ballast layer is comprehensively modeled with many parameters varying in time and space.

Modeling the complex dynamic properties of railway ballast is a complex and difficult area of ​​research. Many researchers have simplified models of a rail vehicle or railway track.

The authors proposed a new approach to effective non-destructive testing and condition monitoring of railway ballast systems.

The presented research and its results constitute an appropriate source for further research. The following comments are suggested for the authors to improve the article:

1. There are no references to publication no. in the text of the article. 19-22 and 45.

2. How can the ballast model be used in reality, different elastic coefficients (k1, k2) and damping coefficients (c1, c2) - Figure 1?

3. What is the measurement accuracy and what are the measurement uncertainties with the GPR method?

4. Has GPR testing been carried out for contaminated railway ballast?

5. The conclusions should be slightly modified based on the results obtained.

Author Response

Response: We deeply appreciate your insightful comment and advice, which have significantly contributed to the refinement of our manuscript. Your suggestions have been incorporated, and we have reorganized and rewritten the manuscript to ensure clarity.

1. We have meticulously revised the article's references to align with your feedback. The revised manuscript now refers to publications 15, 26, 31-32, and 85, which we believe provide robust support for our arguments.
2. The different elastic coefficients (k1, k2) and damping coefficients (c1, c2) in the model are identified based on the nonlinearity data such as frequency dependence of stiffness, [24-25], data available: https://eprints.soton.ac.uk/435759/, [61-62], etc. The multiple parameter identification methods are used and implemented using Matlab, which has a toolbox/function to find a minimum of constrained nonlinear multivariable, such as x = fmincon(fun, k1,c1,k2,c2).
3. The general discussion of the measurement accuracy and measurement uncertainties with the GPR method has been widely discussed in the community [63-82]. As a case study to compare the proposed new method with conventional methods, we selected the GPR data (Fig.7) from a clean ballast. Very similar traces of clean ballast can be found in many pieces of literature as benchmarks to evaluate wet or contaminated ballast.
4. We are grateful for your suggestion for future work. As you recommended, we are considering comparing the new method AOKTFR with conventional methods (STFT, Wavelet) using GPR data tested from contaminated railway ballast.
5. Your input is invaluable to us, and we modified our conclusions by pinpointing the novelty of the results obtained.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper consists of 3 independent parts, in which the background, the assumption for modelling and some experimental results are described. The point is, that the proposed model has not been used in any way to present some results that could be compared or verified. So, the paper looks like a conference paper with some initial work in terms of ideas, with relatively good literature review (this however should be extended, if the main novelty of the paper is nonlinear analytical modelling).

Simply, the paper does not contain any novelty, but some theoretical considerations not supported by computational examples. The developed new dynamic model of railway ballast system remains as presumption only.

Computational examples related to the proposed model must be shown, leading to comparative study with at least existing results and/or experimental measurements. The model remains unjustified.

Fig. 4 looks promising, if obtained based on Eq. 7. Therefore I recommend major revision hoping that the Authors are able to extend their analysis by the desired comparative study.

Comments on the Quality of English Language

Another careful reading would be appreciated before sending the revised version of the manuscript, no crucial errors recognized.

Author Response

Response: Thank you for your review and comment. We revised the manuscript by including and quoting all related publications as evidence to elaborate the claims.

As the reviewer indicated, the paper consists of 3 independent parts: 1. background, 2. new modeling, and 3. new method to process experimental result:

1. The background reviewed all related research (we included all relevant 99 references in the revised manuscript) and pinpointed the state of the art in this subarea:
2. No existing ballast stiffness model comprehensively quantified the widely reported complex nonlinearity of ballast (frequency-dependent, amplitude-dependent, hysteresis, time-/space-varying) [12-46].
3. STFT and Wavelet methods have been widely used to quantify time-varying properties (including to infer stiffness).

Our proposed new modeling is a significant step forward. It enables the comprehensive quantification of the complex features of ballast stiffness, including frequency-dependent, amplitude-dependent, hysteresis, and time/space-varying. In principle, it is a generalized rheological model that characterizes the nonlinear elastoplastic-viscoelastic properties of ballasts. The new method, AOKTFR, is designed to process time-varying ballast GPS signals. It outperforms STFT in this specific task, demonstrating its relevance and superiority. One of the key strengths of our proposed model is its adaptability. It features several parameters that can be adjusted and optimized to fit all tested ballast stiffness data, regardless of their frequency-dependent, amplitude-dependent, hysteresis, and time/space-varying features. As a benchmark example, Figures 3-4 demonstrate the modeled stiffness with frequency-dependent, amplitude-dependent features, and the hysteresis feature. These features and the variation ranges are widely reported in experimental papers and engineering reports, as cited references in the manuscript.

Building on the extensive literature ([21-46] in the revised manuscript) on the nonlinear modeling of ballast stiffness, our manuscript stands out for its novelty. For the first time, it comprehensively quantifies the frequency-dependent, amplitude-dependent, hysteresis, and time/space-varying features of stiffness. This is a significant advancement, as previous research papers only covered part of these features.

Again, the computational example related to the proposed model, Figures 3-4, gives a modeled stiffness with frequency-dependent, amplitude-dependent, and hysteresis features based on typically tested data range (such as stiffness of ballast 300-500 MN/m). It is noted that the model has several adjustable parameters that could be identified according to other tested stiffness data, such as real stiffness ranging from 50 to 1500MN/m, with frequency ranging from 10 to 1000Hz, etc.

Reviewer 3 Report

Comments and Suggestions for Authors

The problem of modeling the performance of the ballast layer of railway tracks is undoubtedly relevant and not yet definitively resolved, despite the large number of various mathematical models. Therefore, such work should have scientific value.

The formatting of this manuscript does not meet the journal's requirements. The text of this manuscript needs improvement and supplementation. The reviewer offers comments for this.

Section 1.

1. The introduction addresses the issue rather superficially. For example, the authors do not mention such modern mathematical models of the ballast layer performance of railway tracks that are based on the modeling of elastic wave propagation.

2. The authors refer to paper [41] as the primary source of the AOKTFR method, but this is not the case.

3. It is advisable to conclude this section with a clear definition of the objective of this paper.

Section 2.

4. The differential equation for rail deflection (4) is used in many works in various variations. It would be advisable to add references to some of these papers.

5. The manuscript does not explain how exactly to derive the combination of equations (7) and (5) into a mathematical model that can be used for practical calculations. The absence of such solutions is one of the problems in considering the non-uniform stiffness of the ballast in most mathematical models.

6. The results shown in Fig. 3 are general and do not allow for assessing the adequacy of the model and comparing it with other models.

Section 4.

7. Equations (8) and (9) require additional explanations, particularly the quantities involved in them.

8. The authors show that "The results show that the output of conventional spectrogram time-frequency analysis is different from that of AOKTFR analysis. By comparing Figs. 10 with 11, it can be seen the salient difference." But how to prove that it is AOKTFR that provides the correct result?

Section 5.

9. The conclusions are written in general terms. Unfortunately, they do not contain either a justification of the scope of application of the new approach or an assessment of the improvement in results it provides in comparison with other methods.

10. The work is perceived as two separate articles. The authors do not explain the cause-and-effect relationship between Modeling of Complex Dynamical Properties of Railway Track and Ballast (Section 2) and the application of the AOKTFR method for decoding georadar signals (Section 4).

Author Response

Response: We deeply appreciate your valuable comment and advice. Your insights have been instrumental in shaping the manuscript, and we have taken significant steps to enhance it based on your recommendations. Thank you for your invaluable contribution.

1. As we specified in the introduction, the new analytical stiffness model is proposed to cover the nonlinear viscoelastic and elastoplastic properties of ballast comprehensively. We noted that the community has pushed to increase the theoretical understanding of aspects such as elasticity, viscosity, plasticity rheology, etc. In the elasticity aspect, we added reference [17] (D Kurhan et al. Determining the Deformation Characteristics of Railway Ballast by Mathematical Modeling of Elastic Wave Propagation, Appl. Mech. 4, 803–815, 2023). Our proposed model is consistent with all kinds of elastic wave theory, and the model parameters can be adjusted to fit different elastic stress-strain patterns.
2. Thank you for pointing out this; we corrected the manuscript.
3. Thank you for pointing this out. We thoroughly revised the manuscript to elaborate on its objective and novelty.
4. We revised the manuscript by adding references [2-3, 7-8, 18, 25-27, 45] just before Eq (2) as the references for Eq. (2-5). Many of these references directly cover Eq. (4).
5. As we explained in the manuscript, on the one hand, we reviewed the advance of the nonlinear elastoplastic model/viscoelastic models/rheological models with multiple nonlinear spring elements, viscous damping elements, and frictional elements; on the other hand, we summarized experimental observations of nonlinear ballast stiffness features such as frequency dependance, amplitude dependence, hysteresis, etc. Then, we proposed using a modified-generalized Maxwell model with multiple parameters Eq. (6-7), and finally, we used an example to elaborate on the model results (Fig 3 and Fig 4). It is noted that the proposed model is a kind of generalized Maxwell model, which inherently consists of nonlinear elastoplastic-viscoelastic rheological features; on the other hand, it can be readily to model complex ballast stiffness features-frequency dependence, amplitude dependence, hysteresis as shown in Fig3 and 4). Moreover, we added time/space varying terms, which could be used to quantify ballast time/space varying features.
6. Actually, the results shown in Fig. 3 are not general; they were derived/identified for specific ballast stiffness data (as Fig.3 shows: ballast stiffness varying from 300 to 500 MN/m vs frequency varying from 50 to 500 Hz). In principle, the model could be used to quantify complex ballast stiffness: based on specific ballast stiffness data, the parameters in the model can be identified.
7. Thank you for your advice on this. We added several references about this, including reference [99] published several months ago. Interested readers could refer to the related references for details.
8. In principle, AOKTFR was theoretically proved to have a higher resolution than STFT and a better performance in suppressing cross-term errors that exist in STFT.
9. Thank you for your advice on this. We revised the conclusions by specifying the paper’s novel contributions. The model could be used to quantify complex ballast stiffness: based on specific ballast stiffness data, the parameters in the model can be identified. Compared with other similar models, the proposed model comprehensively quantified the widely reported complex nonlinearity of ballast (frequency-dependent, amplitude-dependent, hysteresis, time-/space-varying).
10. This is a good question. Historically, the community has used the GPR signal of ballast to indirectly infer or correlate with ballast stiffness. Many references reported about ballast stiffness/dynamic feature characterization used GPR signal for correlation - see references [10, 22, 60-61, 68, 71, 77-78, 80, 82]. For example, [22] developed a new ballast nonlinear model and used GPS signals for indirect correlations. Following that tradition, we tried to correlate our proposed new model with the characteristics of the ballast GPR signal. The existing research demonstrated that ballast GPR signals exhibit time-frequency dynamic features. The proposed model is consistent with those observations by including a time-varying term. Moreover, we found that AOKTFR can improve the conventional time-frequency analysis of ballast GPR signals; as such, we added it as an extra section relevant to the ballast stiffness/dynamic feature characterization.

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript introduces a complex dynamic stiffness model for railway ballast. In the model, multiple properties including time-varying, nonlinear viscoelastic and elastoplastic properties, are considered together. It is good to have such ambition, but it is debatable whether this model is efficient in applications.

 

1. the proposed model is expected to give more accurate estimations of vehicle-track dynamical responses, but in the whole manuscript, I cannot find any comparison between results from the proposed model and any simplified model.

2. how the data in Figure 3 and 4 are obtained? From measurements or simulations?

3. please list the parameters of the proposed model and if possible, please include their reasonable values or ranges?

4. how to validate and calibrate the proposed model?

5. the result in Figure 10 is totally wrong or inaccurate?

6. how to understand the result in Figure 11? Please elaborate a little bit.  

Comments on the Quality of English Language

readable, but strongly suggest to be sent out for professional editing

Author Response

Response: Thank you very much for your valuable comment and advice. We revised the manuscript, incorporating your suggestions.

In the model, multiple properties, including time-varying, nonlinear viscoelastic, and elastoplastic properties, are considered together. As an application example, Figs 3-4 show the case in which the model is used to quantify a ballast stiffness that has features of frequency-dependent, amplitude-dependent, and hysteresis. The model parameters are identified using the real stiffness/load-deformation data. 

1. a.) The novelty of our proposed model lies in its comprehensive inclusion of time-varying, nonlinear viscoelastic, and elastoplastic properties, a feature not found in existing models as reviewed in the paper,

b.) In application, the case study (Fig.3-4) illustrated that the proposed model can well model ballast stiffness with frequency-dependent, amplitude-dependent, and hysteresis features, a capability that is not present in existing nonlinear models as reviewed in the paper.

2. Thank you for your question. We revised the manuscript by adding the identified parameters of the model associated with Figures 3 and 4. Figures 3 and 4 are case study results of the model based on ballast stiffness data with the famous features of frequency-dependent, amplitude-dependent, and hysteresis (which can be found in cited references). The model parameters are identified simply by using Matlab/parameter optimization functions.
3. Thank you for your advice. We revised manuscript by including the identified model parameters k1,c1,… for the case (Fig3-4)
4. It is a good question about how to validate and calibrate the proposed model: a. in principle, the proposed model is the most comprehensive (time-varying, nonlinear viscoelastic and elastoplastic properties) by comparing with existing ones in all literature; b. the model can be used for real tested ballast: based on the tested ballast stiffness/load/deformation features (such as frequency dependance, amplitude dependence, hysteresis, etc.) the model parameters can be identified by using Matlab optimization toolbox with criteria of minimizing gap/error between model results and data.
5. The result in Figure 10 is from the standard Matlab toolbox function/Spectrogram. It could be due to inaccuracy. It is noted that conventional time-frequency methods like spectrogram have cross-term error issues. AOKTFR was once mathematically proved to have a higher resolution and better performance in suppressing cross terms.

6. The general principles are widely reported in [60-84]. Figure 11 shows the precise pattern of the time-varying frequency of the ballast GPR signal, reflecting the electrical properties of the base ballast material. This pattern can be used to infer its strength characteristics, modulus, and ballast stiffness by extra experimental calibrations. Moreover, GPR can be used efficiently and effectively to survey long distances/sections of railways, offering a promising tool for predicting ballast deformation properties and time/space-varying ballast stiffness.

Reviewer 5 Report

Comments and Suggestions for Authors

Title: Modelling and Characterization of Complex Dynamical Properties of Railway Ballast

Comments of the Reviewer

The authors present a new dynamical stiffness model of railway ballast by incorporating time-varying, non-linear viscoelastic and elastoplastic properties. The developed work may be novel, but it lacks some solidity and depth.

The figures have acceptable quality, and the paper may be improved in terms of language. Thus, the paper may represent a useful contribution, but it is not ready to be published yet. The reviewer has a number of major comments that could usefully be addressed. Thus, considering the scope of the paper, some comments are addressed to the authors:

1)     The Introduction is too long and must be shortened. There are topics referred to in the Introduction that have no relevance to the ballast properties such as the modelling of the vehicle system.

2)     The sentence “…was deeply discussed in lots of previous research” must be improved using a more colloquial language. Moreover, the authors should support this sentence with some references.

3)     The authors should present in the manuscript the differences between an analysis with the ballast modelled with linear elastic model and the ballast modellled with the developed model. The authors should quantify these differences. The authors must support the numerical results with experimental ones.

4)     From the numerical point of view, why use a complicated model if it is difficult to characterize in real analyses and conditions? Are the magnitude of the stresses and strains significant to not consider the linear elastic domain? Please clarify it in the manuscript.

5)     The reviewer understands that the adopted model of the track is a 2D model. Why use a complex model to characterize the ballast if the railway structure is analysed in the 2D domain? The 2D domain is not enough to correctly and accurately obtain the response of the track.

6)     The sentence “Most vehicle dynamics commercial programs have limited track components such as ballast” is not true. What about ANSYS and ABAQUS?

7)     Is the following sentence true? “Commonly the stiffness of different components of the track structure is nonlinear”. Please add a reference. From the reviewer’s point of view, at the foundation level, the strain levels are so low that the response is linear. Even at the sub-ballast level.

 

Author Response

Response: Thank you very much for your valuable comment and advice. We reorganized the manuscript by incorporating your suggestions. The manuscript presents a new stiffness model of railway ballast by incorporating time-varying, nonlinear viscoelastic, and elastoplastic properties of ballast, which have been widely reported in experimental research. As an example, to illustrate the model’s solidity and application in a complex ballast system, for the ballast with specific stiffness features such as frequency-dependent, amplitude-dependent, and hysteresis, the model parameters are identified, and the specific model is established as shown in Fig. 3-4.

1)  We appreciate your suggestion and have revised the manuscript, including the abstract. In response to the diverse comments from reviewers, we have significantly increased the number of cited references. This expansion is intended to provide a more comprehensive evaluation of the manuscript's unique points.

2)   We thank you for your suggestion and have incorporated it into the manuscript. Specifically, we have added reference numbers after each sentence to facilitate easy cross-referencing and to support the specific claims made in the manuscript.

3)   We have meticulously revised the manuscript, incorporating all references related to nonlinear ballast stiffness modeling [21-46]. Our proposed multiple-parameter models provide a comprehensive understanding of the time-varying, nonlinear viscoelastic, and elastoplastic properties of ballast, surpassing the coverage of existing models [21-46]. As a practical application, we have used the proposed model to quantify the ballast stiffness, which exhibits frequency-dependent, amplitude-dependent, and hysteresis features. The model parameters have been identified based on numerous experimental data [22,24-25, 6-62].  

4)   We revised the manuscript by including all references about nonlinear ballast stiffness modeling [21-46] in addition to the classic references about linear models [1-20].  These references have comprehensively discussed the applications of simplified and complex models. The complex model may accurately quantify complicated applications such as contaminated ballast, degraded ballast, railway monitoring, etc.  The conventional literature has detailed discussions about various stresses and strains’ status and the complicated effects such as hysteresis. In complicated ballast applications, features such as viscoelastic and elastoplastic properties exist.

5)   The reviewer raised an important point. In the context of 2D car body/railway vehicle-railway vertical vibration analysis, the choice between a conventional linear stiffness model of ballast and a complex nonlinear stiffness model of ballast depends on the specific application conditions, as demonstrated in [22]. However, the nonlinear ballast stiffness model and the test-date-based model identification play a crucial role in defect detection and condition monitoring.

6) This manuscript limited the discussion of the technical details within this subcommunity reflected by references [1-99]. In this sense, the discussion is limited to the specific train vehicle-track dynamics commercial programs for multi-body simulations of vehicle-track systems such as Vampire, NUCARS, and VI-Rail [47-51]. Many general software programs, such as ANSYS, ABAQUS, COMSO, and SIMULIA, could be used. But that discussion is beyond the scope of this manuscript.

7) This is a good question. According to the cited references [1-99], the used ballast widely exhibits nonlinearity due to vehicle overload/impact, ballast contamination, wetting, crashed particles, degradation, etc.

“Commonly the stiffness of different components of the track structure is nonlinear” has been a quote for some researchers in the community; for example, the following widely cited paper by Woodward et al. indicated: “The stiffness of different components of the track structure is nonlinear primarily, such as the rail pads and subgrade, and…

P.K. Woodward, J. Kennedy, O. Laghrouche, D.P. Connolly, G. Medero, Study of railway track stiffness modification by polyurethane reinforcement of the ballast, Transportation Geotechnics, Volume 1, Issue 4,2014, Pages 214–224.

Regarding the foundation level specified by the reviewer, the following paper is dedicated to the nonlinearity of the foundation:

TranLH., Hoang, T., Foret, G. et al. Calculation of dynamic responses of railway sleepers on a nonlinear foundation. Nonlinear Dyn., 112, 443–458 (2024). https://doi.org/10.1007/s11071-023-09070-w

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Unfortunately, I must sustain my previous comments and the decision. The Authors did not response to my questions, especially related to computational examples and comparative study requested in the first review. The only improvement is based on extended literature review.

I ask once more to present additional computational examples justifying the proposed model.

Also a set of co-authors has been changed. Removing one of co-authors at this stage of preparation is strange and incomprehensible, compared to potential extension of the team, if somebody did effort to help in the improvements of the paper.

Author Response

Reviewer 2:

Unfortunately, I must sustain my previous comments and the decision. The Authors did not response to my questions, especially related to computational examples and comparative study requested in the first review. The only improvement is based on extended literature review.

Response: Thank you for your review and remarks.

As we elaborated before:

1). based on existing fundamental theoretical and experimental research over last 30 years (some references are cited in manuscript), ballast exhibit elastic-viscous-plastic rheological properties.

2). according to modern rheology principle, the proposed multiple parameter model is an elastic-viscous-plastic rheological model;

3). the existing ballast stiffness models only considered part of the complex rheological properties of ballasts [22, 24–26, 29, 43];

4).to demonstrate the application of the model, we selected widely recognized ballast stiffness data to identify the model parameters (such as k1, c1, k2, c3, Ff, x2). The used ballast stiffness data (varying 300-500 MN/m) covers three features: frequency-dependent (varying from 50 to 500 Hz), amplitude-dependent (varying 1-3 mm) and hysteresis loop (y-intercept/y-peak=0.2).

5). It is noted that the similar ballast stiffness data as the above have been widely published in literatures, just to name a few (with frequency-dependent stiffness, amplitude-dependent, hysteresis loop):

• Rail Safety & Standards Board: Review of the Effects of Track Stiffness on Track Performance, United King­dom, 2005
• Berggren, E.: Dynamic Track Stiffness Measurement –Thesis, Royal Institute of Technology KTH, Stockholm, Sweden, 2005
• Traffic&Transportation, Vol. 24, 2012, No. 5, 405-412
• Transportation Geotechnics, 1 (4). pp. 214-224. ISSN 2214-3912,2014
• …
• Int J. of Geosynthetics and Ground Eng. (2023) 9:67, https://doi.org/10.1007/s40891-023-00486-3
• Sound Vib. 2020, 468, 115081.
• Journal of Sound and Vibration 547 (2023) 117533
• Construction and Building Materials 321 (2022) 126413
• Transportation Geotechnics 46 (2024) 101251

6). The critical point is that the proposed model has several parameters which could be adjusted/optimized to fit for tested ballast stiffness data with frequency dependent, amplitude-dependent, hysteresis and time/space-varying features. As a benchmark example, Figures 3-4 show the modelled stiffness based on the ballast stiffness with the specific frequency dependent, amplitude-dependent features and the hysteresis loop. It is noted that the above used frequency/amplitude/hysteresis loop range fall in the widely published data as listed above.

 

I ask once more to present additional computational examples justifying the proposed model.

Response: Thank you for your review and remarks. As elaborated above, Fig.3-4 is the computed example of the model based on stiffness data with practical range: frequency-dependent (varying from 50 to 500 Hz), amplitude-dependent (varying 1-3 mm) and hysteresis loop (y-intercept/y-peak=0.2). The model parameters (such as k1, c1, k2, c3, Ff, x2) are identified by using Matlab optimization program.

 

Also a set of co-authors has been changed. Removing one of co-authors at this stage of preparation is strange and incomprehensible, compared to potential extension of the team, if somebody did effort to help in the improvements of the paper.

Response: Thanks for your suggestion. We have included the removed co-author: Xiaotian Xu.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors provided comprehensive answers to my questions and followed my recommendations. Some responses leave room for discussion but are not incorrect, and therefore do not affect the quality of the paper. The article has been revised, and I recommend it for publication in its current form.

Author Response

Thank you for your valueable comment!

Reviewer 4 Report

Comments and Suggestions for Authors

The proposed model is not prepared, because there is no validation by tests data. 

Comments on the Quality of English Language

Fine

Author Response

Reviewer 4:

The proposed model is not prepared, because there is no validation by tests data. 

Response: Thank you for your review and remarks.

1). The proposed model is elastic-viscous-plastic rheological model with multiple parameters

2). The model is validated by modeling three ballast stiffness features in applications: frequency-dependent, amplitude-dependent and hysteresis loop, and the model parameters (k1, c1, k2, c3, Ff, x2) are identified.

3). The used ballast stiffness data (varying 300-500 MN/m) has three features: frequency-dependent (varying from 50 to 500 Hz), amplitude-dependent (varying 1-3 mm) and hysteresis loop, which are consistent with the ballast stiffness data existed in many publications. 

 

Thank you all for your valuable comment again! We look forward to hearing from you in due time regarding our resubmission and to responding to any further questions and comments you may have.

 

Sincerely,

 

Authors