## 4 Discussion
### Comparison of GBR and Strain Sensors  
The approach is able to determine natural frequencies from GBR displacement measurements as well as from strain measurements. Generally, the approach benefits from measurements with high SNR resulting in a smaller variation of the frequency estimations and fewer outliers. In the case of the strain sensors, the highest SNR is achieved by averaging all measurements in the _mean approach_. Since the sensors have a similar noise level and the noise is uncorrelated, averaging reduces the noise power by a factor of \(N\), where \(N\) is the number of sensors. In the case of GBR, the _best approach_ uses the measurement of the target with the highest SNR, which results in the lowest standard deviation and fewest outliers for the frequency estimations. Averaging multiple targets does not increase the SNR compared to the target with the best SNR. One cause for this is the significant difference in signal reflection for the GBR targets, which leads to dissimilar SNRs (see Section 2.1). Secondly, the sensor noise as part of the overall noise level may be correlated between targets. A comparison of the two sensors shows very similar results for the linear regression. After correcting the linear relationship, the GBR achieves a smaller standard deviation than the strain sensors. This difference can be explained by the higher SNR for  
\begin{table}
\begin{tabular}{l l c c c c} \hline \hline \multirow{2}{*}{**Sensor**} & \multirow{2}{*}{**Vehicle Type**} & \multirow{2}{*}{\begin{tabular}{c} **Frequency Change** \\ **in Hz per 10 °C** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Mean** \\ **in Hz** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Standard Deviation** \\ **in Hz** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Number of** \\ **Estimations** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Number of** \\ **Outliers** \\ \end{tabular} } \\ \hline \multirow{3}{*}{GBR} & All vehicles & \(-0.072\) & 3.664 & 0.027 & 1709 & 123 \\  & Heavy vehicles & \(-0.070\) & 3.659 & 0.028 & 721 & 26 \\  & Light vehicles & \(-0.074\) & 3.668 & 0.026 & 988 & 97 \\  & All vehicles & \(-0.073\) & 3.660 & 0.046 & 1709 & 159 \\  & Heavy vehicles & \(-0.075\) & 3.654 & 0.062 & 741 & 19 \\  & Light vehicles & \(-0.072\) & 3.666 & 0.022 & 968 & 140 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of vehicle types for natural frequency estimation with GBR measurements analysed by the _best approach_. In the case of GBR, vehicles are characterised by the absolute of the peak displacement: heavy vehicles > 0.5 mm, light vehicles < 0.5 mm. In the case of the strain sensors, vehicles are characterised by the peak strain: heavy vehicles > 6 μm m\({}^{-1}\), light vehicles < 6 μm m\({}^{-1}\).  
the GBR, even though the noise level for the strain sensors is improved by averaging. The strain sensors' result still has some undetected outliers influencing the standard deviation. Choosing a threshold for the goodness of fit is always a compromise between the number of false positives and false negatives. A threshold of \(R^{2}=50\%\) reduces the number of false negatives to improve the reliability of the linear regression. The resulting higher number of false positives is acceptable since the remaining number of frequency estimations is still very high. We use the same threshold for both sensors to achieve a direct comparison. However, the strain sensors can reach a comparable standard deviation to the GBR if a higher threshold of \(R^{2}=70\%\) is used. Generally, the number of outliers for both sensors shows the importance of the test for goodness of fit, which is necessary to achieve a reliable linear regression.