# Monitoring of Structures and Infrastructures by Low-Cost GNSS Receivers
## 3 The Experience
### Results of Relative Motion  
In Figures 5 and 6, we present the daily solutions of the positioning of C1 and C2 with respect to C3 for a time-span of about 4 years. Note that the first day's solutions are taken as the origin, i.e., only variations are plotted. The first remark is that there is quite a clear separation between the \"slow\" mean motion, interpolated by cubic splines and represented by the red curves in the figures, and the essentially random oscillations of the daily solutions around the former. The r.m.s. of the residuals is summarized in the following table. The \(\sigma X\) figure in Table 2 is reduced to 0.5 mm by using a 3-day spline that better describes a shorter period motion. A second comment is that there is a substantial difference between the behavior of the mean motion for the the bases:  
* C3-C1: very little motion of at most \(\sim\)1 cm over the 4 years in all the coordinates, a long-term very small negative trend in \(X\) and \(Y\);
* C3-C2: marked seasonal signal with an amplitude of \(\sim\)2 cm in \(X\) (longitudinal) with clear annual periodicity;
* C3-C2: very small negative trend in \(Y\) up to March 2020, followed by a descent during springtime, a period in which ASPI had been working on the bridge. Although it is not covered by the displayed data, it is possible to prove stabilization of the descent after about 3 months;
* C3-C2: essential stability in \(Z\) with a pattern very similar to that of C3-C1. The reason why the signal is present in C3-C2 and not in C3-C1 is structural. C3 is fixed to 4, while C2 is on the extreme of Deck 2, which is constrained to the middle pier and free to move on 4, as described in Section 3.3. On the contrary, C1 is tied to 4. The origin of the period of the signal in \(X\) (longitudinal) of the base C3-C2 is essentially related to the seasonal oscillation of the temperature of the structure, as we prove later. Temperatures have been measured by a thermometer embedded in the structure. This signal is clearly more important in the longitudinal direction, which is also the direction of the base, than in the transversal direction. Thus, the oscillation of \(X\) (longitudinal) in C3-C2 represents the thermal linear elongation of the structure of the bridge over a length of about 32 m.  
To corroborate this statement, we can compare the plot of \(X\) of C3-C2 with that of the mean daily temperature of the bridge. Apart from the visual agreement, we find a correlation coefficient of \(\rho=0.95\), which is quite significant. Furthermore an elementary linear regression of \(\Delta X_{i}=x_{i}-\overline{x}\) over \(\Delta T_{i}=T_{i}-\overline{T}\) gives a linear regression coefficient  
\[a=\frac{E(\Delta X_{i}\Delta_{i})}{E(\Delta T_{i}^{2})}=0.000473\;\text{mK}^{-1}\]  
which, divided by the length of Deck 2, provides the mean thermal expansion coefficient  
\[\alpha=1.48\times 10^{-5}\;\text{K}^{-1}\]  
Finally, in Figures 7 and 8, we present the hourly solutions of the two bases, C3-C1 and C3-C2, over a period of 1 month. The first remark is that the series are quite noisier than those of daily solutions, as can be verified from Table 3.  
Figure 6: Daily solutions of C3–C2 baseline. Figure 8: Hourly positions for C3–C2 baseline. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Baseline** & **Length** & \(\sigma\)**X (Longitudinal)** & \(\sigma\)**Y (Transversal)** & \(\sigma\)**Z (Up)** \\ \hline C3–C1 & \
\begin{tabular}{c} horiz:: 69.1 m, \\ vertic:: \(-0.4\) m \\ \end{tabular} & 0.7 mm & 0.8 mm & 1.4 mm \\ C3–C2 &
\begin{tabular}{c} horiz:: 67.4 m, \\ vertic:: \(-0.4\) m \\ \end{tabular} & 0.9 mm & 0.7 mm & 1.1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Root mean squares (RMSs) of \(XYZ\) with respect to the mean motion (splines every 14 days) for the 2 baselines over the 1-month period. Figure 7: Hourly positions of C3–C1 baseline.