**3. Conclusions**

In this paper, two frameworks are proposed for estimating the track modulus average, and the standard deviation over the different track section lengths. The frameworks employed *Yrel* data (a relative rail vertical deflection measured using the MRail system) for the track modulus estimations. The relationship between the statistical properties of the track modulus and the *Yrel* data were investigated using artificial neural networks (ANNs). Datasets from FEMs are used to train the ANNs in which their outputs are either the track modulus average or standard deviations. Both statistical and frequency analyses were conducted to identify the optimized inputs for the ANNs from the *Yrel* data. From the results, the track modulus average over a track section length of 10 m or longer is accurately estimated from the average and standard deviation of the *Yrel* data within the corresponding section length. Additionally, the standard deviation of the track modulus over a section length of 25 m or longer is estimated with an acceptable level of accuracy. It is also shown that the trained ANNs work very well for the track modulus estimations even when the *Yrel* values as the ANN inputs are affected by noise. The proposed ANNs are only applicable to a specific rail type and loading condition. Hence, a similar procedure should be followed to train the ANNs for different ranges of rail sections and loading types.

**Author Contributions:** Conceptualization, N.T.D., M.G. and S.F.N.; methodology, N.T.D.; software, N.T.D., S.F.N.; validation, N.T.D., M.G. and S.F.N.; formal analysis, N.T.D., M.G.; writing—original draft preparation, N.T.D.; writing—review and editing, N.T.D., M.G. and S.F.N.; supervision, M.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** The study is funded by IC-IMPACTS (the India-Canada Centre for Innovative Multidisciplinary Partnerships to Accelerate Community Transformation and Sustainability), established through the Networks of Centres of Excellence of Canada.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Selig, E.T.; Li, D. Track modulus: Its meaning and factors influencing it. *Transp. Res. Rec.* **1994**, *1470*, 47–54.


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