## 2 Methodology
### Longitudinal Level Deterioration Model  
The power-time-transformed Wiener process (PTT-WP) method is a random method widely used in modeling research on facility performance deterioration and useful life prediction. The PTT-WP method is quite suitable for describing the long-time deterioration process with the uncertainty of continuous condition variables [38; 39; 40; 41]. Moreover, it can capture the characteristics of the deterioration process in which the rate of deterioration accelerates over time. In this study, the PTT-WP method was used to develop a deterioration prediction model for the longitudinal level of a 200 m track segment in the tamping maintenance cycle. According to the general form of PTT-WP, for track segment \(k\), the formula for the deterioration prediction model of the longitudinal level in the tamping maintenance cycle is as follows:  
\[X_{k}(t)=X_{0,k}+\beta_{k}\Lambda(t;\theta_{k})+\sigma_{k}B(\Lambda(t;\theta _{k})), \tag{1}\]  
Figure 2: The main models of heavy tamping machines in China and the main phases of a tamping process: (**a**) Hexic DCL-32 tamping machine [33], (**b**) tamping machine DWL-48 that integrates tamping and stabilization [33], and (**c**) the main phases of a tamping process [35]. where \(X_{k}(t)\) is the predicted value of the standard deviation of the longitudinal level (SDLL) at time \(t\) during the tamping maintenance cycle for track segment \(k\). \(X_{0,k}\) is the initial value of the SDLL during the tamping maintenance cycle. \(\beta_{k}\) is the drift coefficient. \(\sigma_{k}\) is the diffusion coefficient. The Brownian motion, denoted as \(B(\cdot)\), is employed to model the uncertainty inherent in the deterioration process, particularly capturing the improvements in the SDLL attributable to local maintenance activities. \(B(\cdot)\) is characterized by increments that follow a normal distribution. Specifically, in this study, the increment \(B(\Lambda(t;\theta_{k}))-B(\Lambda(s;\theta_{k}))\) follows a normal distribution with a mean of 0 and a variance of \(\Lambda(t;\theta_{k})-\Lambda(s;\theta_{k})\). Combined with the calculation formula for \(X_{k}(t)\) as shown in Equation (1), it can be further deduced that \(\Delta X_{k}=X_{k}(t)-X_{k}(s)\), representing the increment of SDLL at any two time points, is normally distributed with mean \(\beta_{k}(\Lambda(t;\theta_{k})-\Lambda(s;\theta_{k}))\) and variance \(\sigma_{k}^{2}\)-\((\Lambda(t;\theta_{k})-\Lambda(s;\theta_{k}))\). \(\Lambda(t;\theta_{k})\) is a timescale transformation function, for which the form \(\Lambda(t;\theta_{k})=t^{\theta_{k}}\) was adopted in this study, where \(\theta_{k}\) is a parameter to be estimated. Values of \(\theta_{k}>1\) indicate that the trend curve of the deterioration process is concave, characterized by an increasing deterioration rate over time. When \(\theta_{k}=1\), the trend curve of the deterioration process is linear, and in this case, the model corresponds to the traditional linear Wiener deterioration process. Values of \(\theta_{k}<1\) indicate that the trend curve of the deterioration process is convex, meaning that the condition gradually deteriorates to a saturation point or an asymptotic level. Consequently, the advantage of using the timescale transformation function in the form of \(t^{\theta_{k}}\) is that it does not necessitate the assumption that the deterioration process is linear or nonlinear. Instead, it estimates the value of \(\theta_{k}\) using actual condition inspection data of segment \(k\) itself. Based on the estimated result for \(\theta_{k}\), the characteristics of the deterioration process can be further analyzed for segment \(k\). This approach is more adaptable and flexible compared to the traditional modeling methods based on the linear Wiener process (i.e., in the case of \(\Lambda(t;\theta_{k})=t\)). It is important to note that a substantial body of relevant literature indicates that the track geometry deterioration between two consecutive tamping operations follows a linear or exponential trend [37].