## 1 Introduction  
An improved and more astute comprehension of railway track conditions stands as a paramount concern within the industry, impacting operations, safety, passenger comfort, resource utilization, and beyond. Notably, a significant share of operational and maintenance costs, up to 60%, is attributed to ballast layer upkeep, necessitating the implementation of optimized strategies and resource allocation [1, 2, 3]. On the other hand, the most important factor ensuring the safe operation of rail vehicles lies in track geometry, which helps the efficiency and performance of railway tracks [4, 5]. Investigating the correlation between the geometrical conditions of the track and the performance and quality of the components of ballasted tracks can lead us to a reasonable understanding of the condition of railway tracks. Railway managers pay particular attention to the geometrical conditions of tracks, recognizing the challenge inherent in assessing their operational status [6, 7]. In modern railway maintenance practices such as track recording cars, mechanized machines play a pivotal role in recording, monitoring, and inspecting track geometrical conditions. While moving at the specified speed on the track, they can capture various parameters such as geometrical irregularities and track stiffness. Geometric parameters of railway tracks including longitudinal levels (LLs), horizontal alignments (ALs), track gauge (GAU), cross-level (XLV), and twist (TWI) play a crucial role in ensuring the safe operation of rail vehicles and improving the efficiency and performance of railway tracks. On the other hand, one of the other main factors that plays a key role in the maintenance of railway tracks is ballast stiffness. Even though track stiffness measurement is not commonly used in the inspection of railway tracks, there are a large amount of railway track specifications that are directly or indirectly related to track stiffness, which helps managers and railway engineers make maintenance decisions by discovering knowledge from these data [8; 9; 10]. So, the interpretation of track stiffness can help decision-makers and senior managers determine a strategy and plan for the maintenance and repair of railway tracks. In recent years, significant strides have been made in the development of continuous track measurement methods [11; 12; 13; 14; 15; 16; 17; 18]. Track stiffness plays a crucial role in influencing the dynamic behavior of railway tracks and their interaction with wheels. In addition, the geometric quality of tracks is directly correlated with track maintenance, and both are directly related to track stiffness. A high stiffness of the track enhances its load-carrying capacity and reduces the displacement of the track. On the other hand, high track stiffness leads to an increase in dynamic forces, especially in the contact area between the wheel and the rail, leading to increased damage to track components, including sleepers and the ballast [19]. Changes in track stiffness along the route lead to changes in track/wheel interaction forces, resulting in varying settlements and geometric deterioration along the track. Numerous studies have been conducted on track stiffness in recent years, illuminating its pivotal role in railway infrastructure [20; 21; 22; 23]. Generally, limited research has been performed to investigate the mutual effects of track stiffness, which represents the characteristics of the track substructure and track geometry, indicative of the track superstructure. [PERSON] et al. conducted a critical review of studies and their future trends regarding track stiffness irregularity (TSI) and presented a new concept of critical values of track stiffness irregularity for the integrated management of track geometry [24]. [PERSON] et al. delved into determining the optimal track stiffness concerning its effect on track geometric quality, considering it dependent on the individual stiffness of all superstructure and substructure elements, as well as their mutual compatibility [25].