# Optimized Detection Algorithm for Vertical Irregularities in Vertical Curve Segments
## 1 Introduction  
With the continuous expansion of urban rail transit mileage and the significant increase in train speeds, the operational safety of locomotives and the comfort of passengers have garnered considerable attention [1]. Deviations between the geometric configuration of the track and design standards, referred to as track irregularities [2], have become a primary factor contributing to vibrations in railway vehicles and track structures [3]. The presence of track irregularities exacerbates wheel-rail wear, leads to the loosening of track structures, and diminishes both the safety and efficiency of train operations [4; 5; 6; 7]. Therefore, the rapid and accurate detection of track irregularities is essential for enhancing track quality, ensuring train safety, and facilitating effective track maintenance operations [8]. Based on existing dynamic inspection results and relevant literature, it is evident that the wavelength of irregularities affecting train ride comfort increases with operating speed [9]. However, detecting long-wavelength irregularities near vertical curves inthe track's longitudinal profile often leads to significant detection errors and frequent exceedances in measurement data [10]. Therefore, it is crucial to conduct in-depth research on the anomalies in detection data for vertical curve segments in the longitudinal profile. Previous studies have predominantly focused on the parameter settings for railway vertical curves and the analysis of vehicle dynamics, with some investigations exploring the relationship between vertical curve parameters and track dynamic irregularities. For instance, in [11], the authors analyzed the relationship between the maximum vertical vibration acceleration of a train and the vertical curve radius, gradient, and length of the intervening straight section from the perspective of dynamic responses as the train traverses a vertical curve. They also provided recommended values for vertical curve parameters in high-speed railways operating at 400 km/h. In [12], the authors conducted an in-depth analysis of wheel-rail dynamics, focusing on the track profile parameters and ride comfort. They identified that the primary vertical vibration frequency induced by vertical curves is associated with operating speed and gradient length, noting that a concise gradient length can lead to overlapping vibrations at the beginning and end of the vertical curve. In [13], the authors proposed design recommendations for the track profile parameters by comprehensively considering the dynamic responses of both the vehicle and track systems, determining the dynamic indicators sensitive to variation. In [14], using theoretical analysis, simulation, and data comparison, the authors studied the impact of vertical curve parameters on various dynamic performance metrics. They provided recommended values for vertical curve radii at 400 km/h, considering safety, comfort, and maintenance requirements. The authors of [15] provided recommended values for the minimum radius and length of high-speed railway vertical curves, considering both operational safety and passenger comfort. In [16], the authors conducted theoretical analyses and simulations to investigate the adaptability of existing high-speed railways operating at 250 km/h to increased operational speeds, optimizing related technical standards, including the minimum vertical curve radius. The authors of [17] referenced metro design specifications to calculate relevant technical requirements for metro lines operating at 140 km/h. Additionally, the authors of [18] analyzed measured data from the Shanghai Railway Bureau on the Hangzhou-Changsha Passenger Dedicated Line, revealing correlations between core vertical curve parameters, locomotive operating speeds, and long-wavelength irregularity exceedances in the time domain.