## 2 Materials and Methods  
### Setup  
A measurement run was carried out with a test train from zb Zentralbahn AG between Stansstad and Engelberg. The line was a meter-gauge railway. Table 1 shows an overview of the route characteristics.  
The environmental conditions during the measurements were rain and snowfall. The rail surface outside the tunnel was mostly wet. The conditions of a dry rail surface were used for measurements in the tunnel section. Four Micro-Epsilon optoNCDT 2300-10 LL laser triangulation sensors were used for the optical measurement of the distance between the rail surface and the sensor lens. The sensors provide a measuring range of 10 mm, a resolution of 0.15 \(\upmu\)m and a reference distance of 35 mm. All four sensors were fixed on a steel plate. The distance values were recorded analogously via an NI 9222 module in a cRIO-9045 from National Instruments with a constant sampling frequency \(f_{S}\) of 30 kHz. No encoder was used to trigger the measurement. The setup, including a protective box for the sensors, was clamped to the bogie of the train between the wheelsets. The sensors were protected from impacts and direct water sprays. The shielded cables were routed along the train into the passenger compartment. The lateral position of the measurement setup was set via clamping to the bogie frame. The measuring points were set to the center of the running surface. This setting was not changed any more during the measuring run. Due to  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Kilometer** & **Location** \\ \hline
2.69 & Stansstad (Start) \\
11.28 & Reference Section A \\
11.40 & Reference Section B \\
18.60 & Start Toothed Rack (Tunnel) \\
21.73 & End Toothed Rack (Tunnel) \\
24.78 & Engelberg (End) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Listing of the route for the measurements, including the reference sections and the toothed rack section inside the tunnel.  
the mounting on the bogie, the lateral adjustment over the running surface could not be changed, as no compensation mechanism was used. The distance between the sensor lenses and the rail surface was set to 35 mm at standstill, which corresponded to the center of the measuring range of the sensor used at this sampling rate. The setup is shown in Figure 1.  
The distances between the sensors were the following: 168 mm (sensor 1 to 2), 113 mm (sensor 2 to 3) and 105 mm (sensor 3 to 4). The identical longitudinal distances between the sensors on the sensor plate were used in the laboratory experiments as described by [PERSON] et al. [18]. Four chord methods were evaluated, including their combination hereafter referenced as the optimized method. The acoustic roughness was calculated according to EN 15610 [20] and the supplements of [PERSON] et al. [21]. In the following, the limit spectrum of acoustic roughness according to EN ISO 3095 [22] is indicated for each comparison. The acoustic roughness was evaluated in the wavelength range between 3 mm and 63 mm in order to match the wavelength range of the reference measurement for comparison. The absolute deviation from the reference measurement was calculated for each one-third octave band at the corresponding wavelength \(\lambda\). The mean deviation corresponded to an averaging over the entire wavelength range evaluated.