## 2 Methodology  
### Problem Description  
In this study, the track tamping maintenance cycle period refers to the time interval between two successive tamping maintenance operations, and its length depends on the deterioration pattern of the track geometry during the tamping cycle. Because of the influences of various factors such as the track structure, transportation organization, and maintenance history, the deterioration rules of track geometry conditions at different spatial locations are inconsistent, exhibiting heterogeneity; at the same time, the deterioration process of the track geometry condition at a specific spatial location is also characterized by uncertainty. This study used a 200 m track segment as the basic research object, and based on historical inspection data of the longitudinal level of each track segment, the stochastic process method of the power-time-transformed Wiener process was used to develop the longitudinal-level deterioration prediction model between successive tamping maintenances. For each track segment, the deterioration model parameters were estimated individually using the historical track geometry inspection data of the track segment itself, with simultaneous consideration of the heterogeneity and uncertainty of the deterioration process. Based on the track geometry condition deterioration model with heterogeneous deterioration parameters across different segments, the time interval for the track geometry condition indicator of each segment to reach the tamping maintenance threshold, that is, the tamping maintenance cycle of each track segment, was predicted. We propose the following premises and hypotheses for the prediction model developed in this study.  
(1) Railway infrastructure managers in most countries (such as the United Kingdom, France, and Sweden) adopt the longitudinal level as the main track geometry parameter to trigger preventive tamping maintenance operations [32]. As illustrated in Figure 1, the longitudinal level represents the vertical plane irregularities of each rail [33]. The measurement is typically obtained using a track geometry car equipped with an inertial reference system. Some researchers have noted that the longitudinal-level deterioration trend over time between two successive tamping maintenance events is more evident and regular [34]. Therefore, in this study, the standard deviation of the longitudinal level over a 200 m track segment was used as the track geometry condition indicator triggering tamping maintenance.  
Figure 1: (a) Schematic diagram of the principle of track longitudinal level (red lines illustrating the deviation from the ideal geometry). (b) Schematic diagram depicting the principle of the inertial reference method.  
(2) Assumptions regarding the initial value of the longitudinal-level indicator in the deterioration process were as follows. As shown in Figure 2, in the tamping operation process, operators employ heavy tamping machines to lift, level, and align the track based on measurement results. This operation ensures that the track geometry parameters meet the requirements specified in track design standards or maintenance codes [33; 35]. Because a track dynamic stabilization car is used to stabilize a track quickly after a tamping operation [36], in this scenario, the bedding-in process (that is, the process by which the track ballast gradually stabilizes and track geometry gradually improves after the tamping operation) mentioned by [PERSON] and [PERSON] [37] could be ignored. Therefore, this study assumed that the initial value of the longitudinal-level indicator during the tamping maintenance cycle was approximately equal to the first value detected after the last tamping maintenance operation.