## 3 Application of Drive-by Methodology on the Railway Track Subsystem
### Track Irregularities Assessment
A comparison between time domain and frequency domain-based formulations for assessing alignment and cross-level irregularities was conducted by [PERSON] et al. [34]. Two formulations based on state space models (time domain) were investigated, one grounded in the use of Kalman filters and the other on the Unknown Input Observer (UIO) technique. Additionally, one frequency domain approach based on the inversion of FRFs was investigated. The numerical studies performed demonstrated that all methods have comparable performance, except for the estimation of cross-level, where the UIO had worse performance. Additionally, it was demonstrated that both the frequency domain and the Kalman filter-based technique performed very well under high levels of measurement noise. Still concerning model-based methods, [PERSON] et al. [35] proposed the use of the cross-entropy optimization algorithm to determine the vertical irregularities profile, which minimizes an objective function composed of the squared sum of the differences between measured and simulated accelerations. The method was tested for a very simplified 2 DOF railway vehicle quarter-car model running over an infinitely rigid track, for which very accurate results were obtained. The formulation was later enhanced by [PERSON] et al. [36] considering a 10 DOF 2D vehicle model including the car body, two bogies, and four wheelsets, as well as a three-layer flexible track model. It has been numerically demonstrated that, in the absence of sensor noise, the irregularity profile is estimated with great accuracy. However, the presence of noise is responsible for generating drifts on the estimated profiles and convergence issues on better-quality tracks. Furthermore, the presented methods, which are based on numerical models to estimate irregularities, are very dependent on their quality. Obtaining a good quality model requires precise knowledge of the physical properties of the track and the vehicles, which is always a challenge when it comes to practical applications. A field validation of the method presented by [PERSON] et al. [36] was conducted by the same authors [37] during a one-month experimental campaign performed on an Irish Rail intercity train operating regularly between Dublin and Belfast. A triaxial accelerometer and a gyro-meter were installed in the bogie of the wagon (Figure 3), seeking the position as close as possible to the center of mass. From data collected by these sensors, the longitudinal level of a 200 m stretch of track was inferred by the methodology based on the cross-entropy optimization algorithm [36] and results were compared with those derived from conventional measurement techniques. Generally, it was possible to verify a good agreement between the shape of the inferred profiles and the reference values; however, the magnitude was severely overestimated. The main reasons pointed out for this error were: (i) errors in the gyro-meter calibration; (ii) the use of a 2D model, as well as calibration inaccuracies of this model, and (iii) the fact that the track was not loaded during the measurement of the profile used as reference. More recently, with the increasing use of deep learning techniques, some authors have chosen to train this type of model to infer track irregularities based on vehicle accelerations. [PERSON] et al. [38] trained an extended auto-encoder (EAE) model to predict track longitudinal level from car body vertical acceleration using both simulated and real data. The architecture of the model is presented in Figure 4a. During the training phase, the encoder is used to compress the irregularity information and create a latent representation that can be later reconstructed by the decoder. The estimator, in this case, a neural network, uses car body acceleration to estimate this latent representation learned by the encoder. As demonstrated in Figure 4b,c the model can accurately estimate the longitudinal level for most wavelengths even for real-world data.