## 2 Experimental Approach and Optimization
### Chord Methods & Optimization  
The transfer function of the chord method based on the definition of [PERSON] [21], is given as:  
\[H(\omega)=e^{(a-1)\cdot L\cdot j\cdot\omega}+(a-1)\cdot e^{-L\cdot j\cdot \omega}-a \tag{2}\]  
The transfer function describes the relationship between the measured sensor signals (input) and the estimation of the longitudinal profile (output). \(L\) corresponds to the total length of the chord, and \(a\) gives information about the position of the middle sensor, and varies between 0 and 1. If \(a=0.5\), the middle sensor is located centrally between the other two sensors and the arrangement is symmetrical. If \(a\
eq 0.5\) applies, the arrangementis referred to as asymmetrical. [PERSON] [21] stated that the transfer function of the chord method has undesirable properties. The amplification factor of this function varies greatly depending on the wavelength. Some wavelengths cannot be measured or are distorted. To avoid these effects, a \"decoloring\" process can be applied as described in EN 13848-1 [28]. An example of this can be the correction of the measurement result using the inverse of the known transfer function of the chord method. This was not applied in context of these investigations, as the transfer function of the described chords have multiple values close to zero, and consequently the inverse causes a deterioration of the results. Ideally, the amplification factor should be one and independent of the wavelength. Since four sensors are integrated in the setup, the longitudinal profile can be recorded with four different chord lengths simultaneously, which makes it possible to perform an optimization of the measurement result. The individual chord methods differ in their total chord length \(L\) and their asymmetry \(a\). These are the two design parameters of the chord method that can be varied. The parameters of the individual chord methods are listed in Table 1.  
Figure 3 shows the sensor arrangement as it is mounted onto the sensor plate. The individual chord methods and the position of the middle sensor are also illustrated. In addition to the evaluated chords, which are each based on three sensor values, it would be possible to evaluate six further chord methods based on two sensor values with this setup. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Designation** & \(\mathbf{L_{1}}\)=\(\mathbf{a\cdot L}\) **[mm]** & \(\mathbf{L_{2}}\)=\(\mathbf{L-a\cdot L}\) **[mm]** & **Chord Length \(L\) [mm]** & \(\mathbf{a\) [-]} \\ \hline
3 AP1 (Three-Point Asymmetrical 1) & 168 & 113 & 281 & 0.598 \\
3 AP2 (Three-Point Asymmetrical 2) & 168 & 218 & 386 & 0.435 \\
3 AP3 (Three-Point Asymmetrical 3) & 281 & 105 & 386 & 0.728 \\
3 AP4 (Three-Point Asymmetrical 4) & 113 & 105 & 218 & 0.518 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chord parameters for the methods integrated into the setup. Figure 2: Water and dust on the rail surface. For each one-third octave band, it is possible to choose the optimal chord method with an amplification factor as close to one as possible. The selection of the appropriate chord method for the respective one-third octave band was performed based on the following quality criterion:  
\[Q_{Band}=\sqrt{\frac{1}{N}\cdot\sum_{k=0}^{N-1}(|H(\omega_{k})|-1)^{2}} \tag{3}\]  
\(N\) represents the number of equally spaced evaluation points set within a one-third octave band. The number of evaluation points was set to 1000. If \(Q_{Band}\) is close to zero, the transfer function shows the desired behavior in the evaluated one-third octave band. The spectrum of acoustic roughness determined in this way is consequently composed of the four individual methods, and is referred to in the following as the optimized acoustic roughness. Figure 4 shows the transfer functions with parameters defined in Table 1 for the wavelength range of between 80 mm and 100 mm. Figure 4: The transfer function for the different chord methods in the wavelength range between 80 mm and 100 mm.