## 2 Method for Measuring Rail Surface Roughness
### Chord-Based Measurement with Two or Three Sensors  
The recently proposed multi-sensor-based chord offset measurement uses two or more sensors in consideration of actual platform length (\(L\)) and main wavelength range to be measured to supplement the amplification or attenuation of individual measured values at a specific wavelength. It also increases measurement accuracy by minimizing the excessive amplification or attenuation that occurs at specific wavelengths [14]. Although the method may display different compensated values according to the relative distance between the sensors, it has the advantage of obtaining more accurate measurements throughout the wavelength ranges on the rail surface and easily analyzing rail cracks or unevenness than the conventional analysis method. However, at least two sensors are necessary to synchronize the values of the sensors used in different positions with a value measured in the same position and to increase measurement accuracy. Figure 3 shows the multiple sensor-based chord offset measurement model, which calculates measurement value and phase difference using multiple sensors. This model can calculate optimal sensor positions on a platform with a specific wheelbase (\(L\)) and wheel diameter (\(D\)). The optimal position of each of three sensors can be determined by calculating the measurement size \((Z(\lambda))\) with Equations (3) and (4) first and obtaining the average value of the magnitudes of the calculated measurements. Finally, the sensor position with the minimal final value can be obtained by varying the three sensor positions. With the measuring system manufactured for the test, which had a wheelbase of 716 mm (L = 716 mm) and wheel diameter of 100 mm (D = 100 mm), the optimized positions of the three sensors are calculated as \(d_{1}\) = 222 mm, \(d_{2}\) = 327 mm, and \(d_{3}\) = 432 mm from the rear wheel [14]. Although using the multiple sensors can improve accuracy of the measured value and remove the impact of periodic external disturbance, as shown in the spectral analysis of rail surface roughness, it requires at least three sensors on the platform. Therefore, it is possible to perform an optimization check to verify the results obtained after reducing the number of sensors to two. Although more sensors make for a more accurate measurement of various wavelength ranges of a rail surface, it is possible to optimize positions of two sensors by limiting measurement range to rolling noise of the rail or main wavelength ranges. By using the transfer function, \(H(\lambda)=Z(\lambda)/Y(\lambda)\), of the sensor measurement to the rail surface roughness value, the following equation is obtained in accordance with changes to each sensor position and calculating the sum of squared residuals (SSR), where residual(\(r\)) is \(H(\lambda)-\widehat{H(\lambda)}\) and amplification ratio is considered as \(H_{0}(\lambda)\). Figure 3: Multiple sensor-based chord offset measurement model, where \(\alpha=d_{i}/L_{i}\), \(d_{i}\) is the distance between \(x_{r}\) to \(i\)th sensor (\(i=1,2,3\)) (Adapted and modified from [14]). \[SSR=\sum_{i=1}^{n}r_{i}^{2}=\sum_{i=1}^{n}\Bigl{(}H_{i}(\lambda)-\widehat{H_{i}( \lambda)}\Bigr{)}^{2}=\sum_{i=1}^{n}(H_{i}(\lambda)-H_{0}(\lambda))^{2} \tag{5}\]  
where \(H_{i}(\lambda)\) is the observed response value and \(\widehat{H_{i}(\lambda)}\) is the fitted response value. Optimal sensor positions can be obtained by changing the positions of the two sensors to be installed and using calculations to minimize the sum of the residuals to minimize the amplification ratio of the measured values. Therefore, the positions of two sensors were calculated using the distance between front and rear platform wheels (wheelbase L = 716 mm) with the minimum \(SSR\) value in the main wavelength region (5-500 mm) where rolling noise occurs.