# Optimized Detection Algorithm for Vertical Irregularities in Vertical Curve Segments
## 2 Virtual Track Inspection Technology
### Validation of the Virtual Track Inspection Method  
To validate the accuracy of the virtual track inspection technology based on the acceleration integration method, excitation irregularities were introduced into the track, enabling the train to operate under vertical excitation. The virtual vertical irregularities computed by the software were then compared with the excitation irregularities in both the time and frequency domains. For a time-domain comparison, a 2.5 m step-size sliding window was used to calculate the standard deviation of the error between the virtual and excitation irregularities, capturing amplitude variation information. For a frequency-domain comparison, [PERSON]'s power spectral density (PSD) estimation method [27] was employed to calculate the power spectral density of the data sequences, providing an insight into wavelength components.  
The excitation irregularities adopted in this study were generated from the spectrum inversion of ballastless tracks on Chinese high-speed railways. It is described by a piecewise fitting approach using power functions, with each segment of the track spectrum represented by a unified expression [28], as shown in Equation (4):  
\[S(\phi)=\frac{A}{\phi^{n}} \tag{4}\]  
where \(S(\phi)\) represents the power spectral density function; \(\phi\) denotes the spatial frequency of the track irregularity, \(A\) and \(n\) are fitting coefficients, which can be obtained from Reference [29].  
The excitation irregularities applied in the simulation model can be converted from spatial spectra into spatial irregularity sample sequences through numerical simulation methods. Given the focus on detection accuracy for long-wavelength vertical irregularities, the wavelength range of the excitation irregularities was set to 1-100 m, with a spatial sampling interval of 0.25 m. Simulations were conducted to obtain virtual vertical irregularitiesat different inspection speeds of 80, 120, and 160 km/h. Figure 2 presents a comparison of the virtual and excitation irregularities at an inspection speed of 80 km/h.  
As shown in Figure 3a, although the calculation of virtual irregularities involves various interferences, including the train dynamics model, double integration, and high-pass filtering, the standard deviation of the error between the virtual and excitation irregularities remains generally within 0.05 mm. This indicates that the error is within an acceptable range.  
Figure 3b demonstrates that the PSD curves of the virtual and excitation irregularities are largely consistent in both amplitude and trend, indicating that the wavelength components of the virtual irregularities obtained through the acceleration integration method align with those of the excitation irregularities across different detection speeds. Therefore, the virtual irregularities calculated via the acceleration integration method accurately reflect the amplitude and wavelength composition of the excitation irregularities, thereby validating the effectiveness of this approach.  
Figure 3: Comparison of time and frequency domain indexes of virtual and incentive irregularities at different speeds (**a**) Error standard deviation comparison (**b**) Power spectral density comparison.  
Figure 2: Comparison of virtual and incentive irregularity waveforms.