## 5 Conclusions  
This paper proposes the use of LFWD-based inverse analysis to estimate the resilient modulus of the track substructure of the existing HAL railway systems. The resilient modulus for each layer of the substructure are set to the elastic parameters of a 3D FE model to perform a dynamic response analysis. In addition, a parametric study is performed to investigate the effects of \(v\) and \(F\) on the track characteristics. Accordingly, based on the numerical results, the necessary conditions to ensure long-term stability of the existing HAL railway are discussed. The following conclusions can be drawn from this work:  
1. The LFWD, which has been proven to be a versatile and reliable non-destructive testing device for road pavements, can be a valuable tool to realize the evaluation of China's HAL railway system. An agreement is noted between the 3D FE model and the deflection data, which further demonstrates that the proposed approach can be used to determine the resilient modulus with a satisfactory accuracy.
2. The effects of various factors on the dynamic response are notable. The vertical peak acceleration (A\({}_{\text{max}}\)) at the bottom layer reduced by approximately 30-50% compared to that at the top layer, whereas the dynamic stress reduced by approximately 95%. Furthermore, the A\({}_{\text{max}}\) and dynamic stress decayed with the depth exponentially and hyperbolically, respectively.
3. The method of service performance evaluation is established based on the results of the parametric study on the effects of varying \(v\) and \(F\). Based on the recommendation that the critical stress ratio should be less 0.2, an endurance limit of \(v\) and \(F\) is assigned to ensure the long-term stability of the existing HAL railways. It was noted that \(F\) can be up to 260 kN, whereas \(v\) must be less than 60 km/h. However, if \(v\) needs to be increased, the corresponding \(F\) must be reduced to ensure service safety.  
It must be noted that the findings in this work were derived considering the case of the HAL train C80, running between Baotou and Shenmu. It is expected that further analogous parametric studies performed considering trains with different configurations or subgrades with various resilient moduli, may lead to different results, which can help identify the different critical regions to improve the current study. Additionally, future research will need to be carried out to examine the vibration velocity of the subgrade which could not be achieved in this study due to the limitation of the current testing apparatus. Furthermore, various multi-body vehicle models will also be used to model the train load.  
Conceptualization and methodology, S.T. and X.L.; software, S.T.; validation, T.L. and I.-R.G.; writing--original draft preparation, S.T.; writing--review and editing, A.C.; funding acquisition, T.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Project of China (grant no. 2018 YFC1505305), the National Natural Science Foundation of China (grant no. 42102311), the State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology (grant number: SKLGDUEK2103), the fellowship of China Postdoctoral Science Foundation (grant no. 2021M701014), and the Postdoctoral Fellowship of Heilongjiang Province (grant no. LBH-Z21062).  
Not applicable.  
The authors declare no conflict of interest.