## 4 Numerical Analysis of the Viaduct and Cutting  
For both the viaduct and cutting environments, the common parameters for all the scenarios are \(v_{R}=200\) km/h and \(f_{c}=41\) GHz. The Doppler frequency will be obtained using the formula \(f_{max}=f_{c}\times\frac{v_{R}}{c}\) with respect to the \(v_{R}\), and the MRS antenna is assumed to be placed at the rooftop of the train. The number of effective scatterers \(N_{1,1}\),\(N_{1,2}=50\). The angle parameter is \(\alpha_{R}=\beta_{T}=\alpha_{T}=\beta_{R}=45^{\circ}\), while for the first and second tap, the angle parameters are \(\alpha_{R0}^{(1)}=25^{\circ}\), \(\beta_{R0}^{(1)}=10^{\circ}\) and \(\alpha_{R0}^{(2)}=85^{\circ}\), \(\beta_{R0}^{(2)}=15^{\circ}\). The structural parameters used from [18,19] of scenario 1 were considered to obtain the k-factor for the evaluation of stationarity in both the viaduct and cutting environment, respectively, as seen in Tables 1 and 2. From the measurements, \(L=15\) m, \(H_{viad}=20\) m, \(H_{BS}=45\) m, \(H_{MRS}=30\) cm, \(H_{train}=3.8\) m, \(D_{s}(t)=\in[0,2502]\) m, \(D_{BP}=400\) m, \(R_{R}=15\) m and \(\gamma_{R}=0\). The viaduct parameters related to power, which are \(\eta_{SB_{1,1}}\) and \(\eta_{SB_{1,2}}\), are 0.4 and 0.6 for the first and second tap, respectively. With respect to the structural dimension parameters of \(H\) and \(h\), the viaduct environment will be categorized into three scenarios. For a cutting environment, the parameters used from the measurement of [19] are \(L=15\) m, \(H_{cut}=7.1\) m, \(H_{BS}=28\) m, \(H_{MRS}=30\) cm, and \(H_{train}=3.8\) m. With respect to the structural dimension parameters of \(W_{up}\) and \(W_{down}\), the cutting environment will be categorized into three scenarios. By using these parameters in the formula \(R_{R}=\frac{W_{up}+W_{down}}{2}\)m, the radius of the cell will be obtained. Here, \(D_{s}(t)=\in[0,1410]\) m, \(D_{BP}=200\)m, and \(\gamma_{R}=0\). The parameters related to power are \(\eta_{SB_{1,1}}\) and \(\eta_{SB_{1,2}}\), which are 0.75 and 0.25 for the first and second tap, respectively. Figure 4 shows how stationarity distance is affected at mmWave frequencies in the two environments when the proposed model is used. These results have been compared with the measurement campaign in [22] for ITU. The parameters used for the viaduct environment were \(v_{R}=285\) km/h, \(L=15\) m, \(H_{viad}=20\) m, \(HB_{S}=20\) m, \(H_{MRS}=30\) cm, and \(H_{train}=3.8\) m whereas for the cutting environment, the parameters used were \(v_{R}=200\) km/h, \(L=20\) m, \(H_{cut}=5\) m, \(H_{BS}=28\) m, \(H_{MRS}=30\) cm, \(H_{train}=3.8\) m, \(W_{up}=40\) m, and \(W_{down}=16\) m. The average stationary interval for the cutting environment obtained was 2.81 m, while the average stationary interval for the viaduct was 3.1 m. If compared with the stationary interval range(2.8 m-4.2 m) as reported in [22], the values obtained match within the range. The absolute values of the time-variant autocorrelation function of the proposed model for the cutting environment are shown in Figure 5. In this case, the effect of the breakpoint distance is evaluated by considering the two scenarios. The time autocorrelation function of \(D_{s}(t)\leq D_{BP}\) is lesser than \(D_{s}(t)>D_{BP}\) because, apart from the structural parameters of the cutting when the MS is very close to the BS, there is a shadowing effect which causes the direct ray from the directional antenna to be very weak, thus resulting in a small k-factor value. This can be related to Equation (3). The results are based on the cutting dimension measurements made in [19]. In this case, only 2 scenarios are considered, where cutting in scenario 2 has a wider dimension than cutting in scenario 1. This is further reflected in the results where scenario 2 has a higher correlation than scenario 1. This is due to the higher Rician k value from a stronger line-of-sight component. Figure 6 depicts the time-variant autocorrelation function of the cutting by comparing the three scenarios.