## 2 Characterization of Rail Unevenness Profile
### Experimental Characterization  
For a more efficient operation of the track, maintenance planning, and assessment of traffic safety conditions, railway administrations systematically monitor and assess the geometric quality of railway lines. As an example, Figure 1a shows the EM120 inspection vehicle used by the Portuguese railway administration (IP) for geometric characterization of the line. This vehicle allows, among others, the registration of the irregularity profile along the rail. Figure 1b shows, as an example, the longitudinal level of both rails on a stretch of the Portuguese railway network with the recording of information every 25 cm. From the point of view of numerical modeling, it is usual to treat the longitudinal-level values of the track in the wavenumber domain, through power spectral density (PSD) functions [7]. To determine the PSD, it is firstly necessary to transform the signal from the  
Figure 1: Geometric characterization of the rail: (**a**) EM120 inspection vehicle; (**b**) spatial representation of unevenness profile on a stretch of the Portuguese railway network (both rails). space domain to the wavenumber domain (also called spatial frequency) by means of a Fourier transform. This process is identical to obtain a frequency spectrum of a temporal signal, only with the particularity that it is a transform in relationship to space in place of a transform regarding time, resulting in a spectrum of the signal as a function of the wavenumber. In a generic way, and according to the procedure described in Newland [8], for a given spatial signal \(f(x)\), its transformation to the wavenumber domain can be defined by the following expression:  
\[F(k_{1})=\int_{-\infty}^{+\infty}f(x)e^{-ik_{1}x}dx \tag{1}\]  
where \(k_{1}\) represents the longitudinal wavenumber and \(x\) is the longitudinal distance. As can be understood, the previous transform is only applied to analog signals. To deal with digital signals, it is necessary to adapt the previous equation to account for the discrete nature of the latter. In this sense, a discrete Fourier transform can be stated as follows:  
\[F_{m}=\sum_{n=1}^{N}f_{n}e^{-i2\pi(m-1)(n-1)/N} \tag{2}\]  
where \(N\) corresponds to the number of sampling points. The discrete finite Fourier transform can be calculated by means of a very efficient mathematical algorithm, called FFT (Fast Fourier Transform) [9]. The commercial software MatLab R2023b has this algorithm implemented, which allows its application in a direct way for a large set of measurement points. The final goal being the evaluation of the power spectral density function \(S\), and keeping in mind that the longitudinal-level data correspond to a continuous random signal, Equation (1), corresponding to the Fourier transform, is adapted in order to meet this typology of signals. To contain the growth of \(F(k_{1})\) to infinity, the limit given in the following expression is applied:  
\[F(k_{1})=\underset{X\rightarrow\infty}{lim}\frac{1}{X}\int_{-\infty}^{+\infty }f(x)e^{-ik_{1}x}dx \tag{3}\]  
To estimate the PSD function for the case of a digital signal, the initial spatial recording is subject to an additional treatment. In this case, the signal is divided into a set of blocks of period \(X_{b}\) and number of samples \(N_{b}\) in order to obtain a smoother configuration of the function. The overlapping of these blocks can also be considered, and it is usual to consider an overlapping close to half of the block length (overlapping of 50%). For each block, it is then possible to estimate the PSD function from the response spectrum in the wavenumber domain.