## 4 Experimental Results
compare in [47] displacements detected by a GPS system to the displacements detected by a vision-based system. Both systems correlate well  
Figure 16: Position E2 (a) \(x\)-, (b) \(y\)- and (c) \(z\)-displacements over time as measured by the GNSS sensor. Figure 15: (a) Photo of a GNSS antenna attached to the side of the steel bridge by magnets. (b) Photo of the relay node on top of the first pillar. with each other, but [PERSON] et al. observed additional low-frequency oscillations in the GPS measurements that were not present in the vision-based measurements. Keeping this in mind, it becomes obvious that for professional and efficient vibration monitoring, samples are best taken during nighttime, when there is almost no traffic on the bridge, or the bridge should be closed during this measurement campaign. However, to demonstrate the usability of the developed GNSS sensor network for dynamic structural monitoring, these long-wave effects are simply eliminated from the data samples before applying frequency analysis. In order to eliminate the long-wave signals from the time series of each coordinate component of the point positions \(x\), \(y\) and \(z\), we introduced a cubic spline approximation [48] as also shown in Figures 16-18. The spline is generated by \(n\) cubic polynomials \(p_{n}\) that partially approximate the point position \(x\), \(y\) and \(z\) over time as shown in Equation (5):  
\[y_{t(k)}+\
u=f(\hat{p}), \tag{5}\]  
where vector \(\hat{p}\) comprises the \(n\) sets of polynomial parameters \(p_{n,0}\), \(p_{n,1}\), \(p_{n,2}\) and \(p_{n,3}\). These parameters are determined within a least squares adjustment of the coordinates of the point position at the respective point of time, \(x_{t(k)}\), \(y_{t(k)}\) and \(z_{t(k)}\). For the y-coordinate component, for example, the system of observation equations reads as Equation (6):  
\[ \begin{split} y_{t(k)}+\
u&=\hat{p}_{i,0}+\hat{p}_ {i,1}\Delta t_{k}+\frac{1}{2}\hat{p}_{i,2}\Delta t_{k}^{2}+\frac{1}{6}\hat{p} _{i,3}\Delta t_{k}^{3}\\ 0+\
u&=C(p)\end{split}  \tag{6}\]  
With \(\Delta t_{k}=t(k)-t(0)\). The index \(i\) denotes the respective cubic polynomial \(p_{i}\) depending on the point of time. The length of each of the \(n\) polynomials can be predefined by using a time interval \(\Delta t\) that determines the length of each polynomial. In the following example, \(\Delta t\) was set to 30 s. Therefore, each of the \(n\)-polynomials in the generated spline represents the point position during this period. Furthermore, the quantity \(\
u\) denotes the estimated residual for the introduced observation \(y\). For the residual, \(\
u\) impliesthat its expected value is equal to zero, that is \(E(\
u)=0\). Figure 17: Position E3 (a) \(x_{\succ}\), (b) \(y\)- and (c) \(z\)-displacements over time as measured by the GNSS sensor. Figure 18: Position E4 (a) \(x_{\succ}\), (b) \(y\)- and (c) \(z\)-displacements over time as measured by the GNSS sensor. With \(C(p)\), we introduce a set of C-continuity conditions between two neighboring polynomials \(p_{i}\) and \(p_{i+1}\). For the data presented in the following, we applied \(C_{0}\), \(C_{1}\) and \(C_{2}\) continuity at the transition of each polynomial \(p_{i}\) to the polynomial \(p_{i+1}\). \(C_{0}\) continuity hereby forces identical value of the polynomial representation, that is \(f(p_{i})\equiv f(p_{i+1})\). \(C_{1}\) continuity forces the identical value of the first derivative in time, that is \(\frac{3}{\partial t}f(p_{i})\equiv\frac{3}{\partial t}f(p_{i+1})\), and \(C_{2}\) continuity forces the same value for the second derivative in time as \(\frac{\partial^{2}}{\partial t}f(p_{i})\equiv\frac{\partial^{2}}{\partial t ^{2}}f(p_{i+1})\). Applying spline approximation of the time series of the point positions leads to a smooth representation over time. The estimated spline parameters \(p\) are carrying the long-wave components of the recorded signals.