## 4 Analysis of Settlement Data and Discussion
### The MDD/GNSS Settlement Prediction Model  
Statistical Analysis for Calculation of Model Coefficients  
To develop the settlement prediction model for MDD laser sensor measurements, the settlement of the MDD data obtained for each MDD position was analyzed by OriginPro(r) 2018, which is a program used for data analysis developed by OriginLab. It provides a range of features for analyzing and presenting data, including curve fitting, etc. [24]. In this study, we utilized a curve fitting method to construct the most effective model for predicting settlement graphs measured by MDD with the GNSS. The exponential model, a non-linear statistical function represented by Equation (8), was chosen for this purpose due to its suitability for analyzing MDD/GNSS data. OriginPro(r) 2018 software was utilized to perform the curve fitting and regression analysis, as its efficacy for this purpose has been extensively documented in the literature by [PERSON] [24]. \[y=y_{0}+Ae^{R_{0}x}, \tag{8}\]  
where \(y\) is the predicted vertical settlement, \(y_{0}\) = offset, \(A\) = initial value, \(R_{0}\) = rate, \(x\) = MDD measuring time and \(y_{0}\), \(A\), and \(R_{0}\) are the model coefficients that were obtained by curve fitting analysis. The analysis was performed first to determine the coefficients of the exponential model for variables \(y_{0}\), \(A\), and \(R_{0}\). The results of the analysis are presented in Table 2. The goodness of fit of the model was assessed using the R\({}^{2}\) value, which measures the degree of correlation between comparison and predicted values. The R\({}^{2}\) value was found to be close to one, indicating a strong relationship between comparison and predicted values. This suggests that the exponential model fits the data well and has a high level of confidence. Thus, through nonlinear statistical function analysis, the following model equations were derived for the prediction of MDD total settlement for each position. For MDD position #1:  
\[y=8.895-10.12e^{-0.00897x} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MDD** & **Parameter** & **Value** & **Standard Error** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{Position \#1} & y\({}_{0}\) & 8.895 & 0.00645 & \multirow{3}{*}{0.98868} \\  & A & \(-10.12\) & 0.03786 & \\  & R\({}_{0}\) & \(-0.00897\) & \(5.25641\times 10^{-5}\) & \\ \hline \multirow{3}{*}{Position \#2} & y\({}_{0}\) & 7.4166 & 0.000 & \multirow{3}{*}{0.99999} \\  & A & \(-8.37913\) & 0.000 & \\  & R\({}_{0}\) & \(-0.0075\) & 0.000 & \\ \hline \multirow{3}{*}{Position \#3} & y\({}_{0}\) & 7.34331 & 0.00642 & \multirow{3}{*}{0.98797} \\  & A & \(-8.16207\) & 0.03057 & \\ \cline{1-1}  & R\({}_{0}\) & \(-0.00692\) & \(4.25569\times 10^{-5}\) & \\ \hline \multirow{3}{*}{Position \#4} & y\({}_{0}\) & 7.04242 & 0.00668 & \multirow{3}{*}{0.9808} \\  & A & \(-8.27055\) & 0.04067 & \\ \cline{1-1}  & R\({}_{0}\) & \(-0.00945\) & \(7.22496\times 10^{-5}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of a nonlinear statistical function analysis. For MDD position #2:  
\[y=7.42-8.38e^{-0.0075x} \tag{10}\]  
For MDD position #3:  
\[y=7.34-8.162e^{-0.007x} \tag{11}\]  
For MDD position #4:  
\[y=7.04-8.27e^{-0.00945x} \tag{12}\]  
Furthermore, we assessed the applicability of the prediction model for MDD positions by comparing the total settlement results obtained from calibrated MDD data measurements to those predicted by the model. This comparison was carried out by plotting the results on a graph, as shown in Figures 20-23. The results revealed that the pattern of vertical displacement was highly consistent between MDD total settlement and the predicted results calculated using Equations (9)-(12). This indicates that the prediction model accurately predicted the total settlement behavior at MDD positions. As described in the previous chapter, it was carefully predicted that there could be a reduced size of the void space remaining below the track slab even after restoration.