## 2 MDD/GNSS Installations and Measurement Methods  
### GNSS Configuration and Principle  
In this study, a Multi-Depth Deflectometer (MDD) and the Global Navigation Satellite System (GNSS) were employed as field instrumentation to measure the vertical deformation induced at different layers for periodic time monitoring (Figure 3). The MDD/GNSS system measures the deflection in track foundation soil at various depths in a vertical direction caused by the passing wheel load of railway cars. By analyzing the measured deflections at various depths, a vertical deflection profile can be obtained. The MDD can also measure the relative elastic deflection due to wheel load or the total permanent deformation of each layer in the slab track structure. To improve the MDD instrument, a laser sensor equipped with MDD was designed and used in this study. Four MDD laser sensing modules (KEYENCE(r) laser sensor IL-600 model, Keyence Corporation, Osaka, Japan) were installed in a single 15 cm diameter hole casing at a field site located between a tunnel entrance and the embankment. This place is a typical transition zone with many occurrences of track settlement reported before repair. The laser sensing modules were clamped against the sides of the hole casing at the designed depths. The head of the MDD casing was anchored at the top Track Concrete Layer (TCL) so that the MDD modules could measure net vertical deflections at various depths by assuming that the anchored fixing head and the casing move together with rail track TCL. This means that the net measured deflection at each depth of the installed module provides soil displacement only at the location itself. Figure 3: The concept of obtaining track deformation based on a GNNS baseline analysis method with carrier phase measurements. There are two ways to make the GNSS measurement: (1) point positioning for navigation, which obtains the three-dimensional absolute coordinates of latitude, longitude, and height of a measurement point with an accuracy of a few meters to ten meters or more, and (2) relative positioning using carrier phase measurements, which provides the three-dimensional relative coordinates between two points with an accuracy of millimeters to centimeters. The latter method was adapted in this study and has been used by many researchers to monitor displacement in different areas [18]. Referring to Figure 3, the carrier phase of the signals transmitted from satellite \(k\) at measurement point mi is obtained as follows [18]:  
\[\phi_{mi}^{k}=\frac{r_{mi}^{k}+I_{\phi}+T_{\phi}}{\lambda}+\frac{c\left(\delta t _{mi}+\delta t^{k}\right)}{\lambda}+N_{mi}^{k}+\epsilon_{\phi_{mi}}^{k} \tag{1}\]  
where \(r_{mi}^{k}\) is the distance between measurement point \(mi\) and satellite \(k\) as follows:  
\[r_{mi}^{k}=\sqrt{\left(x_{mi}-X_{k}\right)^{2}+\left(y_{mi}-Y_{k}\right)^{2}+ \left(z_{mi}-Z_{k}\right)^{2}} \tag{2}\]  
\(x_{mi},y_{mi},z_{mi}\) and \(X_{k},Y_{k},Z_{k}\) are the measurement coordinates at point \(m_{i}\) and satellite \(k\), respectively, \(I_{\phi}\) is the ionospheric delay, \(T_{\phi}\) is the tropospheric delay, and \(\lambda\) is the wavelength of the signal; \(\delta t_{mi}\) and \(\delta t^{k}\) are the biases of the receiver clock and the satellite clock, respectively, \(c\) is the velocity of the light, \(N_{mi}^{k}\) is the unknown integer ambiguity of the carrier phase, and \(\epsilon_{\phi_{mi}}^{k}\) is an observation error. On the other hand, the carrier phase of the signals transmitted from satellite \(k\) at reference point (base station) \(m_{b}\) is also obtained similarly. Then, the single-phase \(\left(\phi_{mi}^{k}-\phi_{mb}^{k}\right)\) difference between measurement point \(m_{i}\) and reference point \(m_{b}\) for satellite \(k\) is obtained. In the same process, another single-phase \(\left(\phi_{mi}^{l}-\phi_{mb}^{l}\right)\) between measurement point \(m_{i}\) and reference point \(m_{b}\) for satellite \(l\) is also taken.