# Train Speed Estimation from Track Structure Vibration Measurements
## 2 Algorithm of Train Speed Estimation  
The typical dynamic response waveforms of track structure vertical acceleration and displacement are shown in Figure 1. When the rail wheels pass through the track, the track structure undergoes significant deformation, and the signal waveform of vertical acceleration and displacement produce pronounced peaks. Therefore, the train speed can be estimated by combining the intervals of the pulse signals and the geometrical parameters of the train with a procedure similar to the optical sensors.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Test Location** & **Advantage** & **Shortcoming** & **Error** \\ \hline Tachometers & On-board &
\begin{tabular}{c} No additional \\ measurement \\ \end{tabular} & Using inside the train & \(\sim\)5\% \\ GPS & On-board & Cheap & Using inside the train & 5-10\% \\ Optical sensors & Ground & Very accurate & Affected by weather especially rain & \(<\)1\% \\ Wheel counters & Ground & Very accurate & Stationary application & \(<\)1\% \\ Eddy current & Ground & Accurate & Affecting by rail structure state & \(\sim\)1\% \\ sensors & Ground & Easy to use & Accuracy depending the position & 1–5\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Existing methods for train speed estimation.  
During field tests, how to identify and remove the variability of track structures is the main difficulty of estimating train speed. Two main factors influence the track structure dynamic response signals. On the one hand, the dynamic response is influenced by the train system that is coupled with the track system. On the other hand, on-site testing is affected by environmental conditions, structural conditions, equipment defects, and other factors. The original signals can be described with Equation (1):  
\[f(t,y)=f_{\rm{T}}(t,y)+n(t,y) \tag{1}\]  
where \(f(t,y)\) is the original ground testing signals, \(f_{\rm{T}}(t,y)\) denotes the true dynamic response signals, and \(n(t,y)\) is the sum of noise and disturbance in the signals. To achieve accurate train speed estimation, it is necessary to reliably remove the interference and extract the characteristic signals.  
There are three steps in the train speed estimation algorithm (Figure 2). The first step is to improve the signal to noise ratio of the original ground measurement signals with a low-pass filter or wavelet transform and image treatment according to different signal types. The second step is to propose the threshold conditions and time difference conditions to identify the time of the train bogies passing through the sensors. Finally, the train speed is calculated by combining the characteristics of the vibration signals and the geometric dimension of the train. Furthermore, the Grubbs criterion is selected to ensure the validity of the estimated train speed.  
In this paper, three different types of vibration measurements (rail acceleration measurement, slab acceleration measurement, and slab displacement measurement) are employed to estimate the train speed. The signal processing and train speed estimation algorithm are presented and tested in real field tests.  
Figure 1: The typical dynamic response waveform: (**a**) displacement; (**b**) acceleration.