# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 3 Detecting Time-Varying Dynamic Properties of Ballast Using GPR  
Since railway ballast properties are highly varied along the rail network, vehicle-track system modeling, evaluation, and validations are required to be implemented and verified at the network level instead of locally at the section level. Efficient measures to characterize track and ballast time/space-varying dynamic properties using GPR along railways have been widely applied in the community [57, 58, 59, 60, 69].  
As a very efficient method to obtain physical and geometric information for the railway ballast, the GPR technique is based on the transmission/reception of electromagnetic waves into the ground in a given frequency band. The main advantage of this approach is that drive-by high-speed sampling and minimum or non-destructive sampling are required to calibrate the system. The inhomogeneity of materials with different electromagnetic properties and the interface between different structured layers can lead to changes in the position and amplitude of the signal peaks, from which information on layer thickness and material properties can be inferred.  
Figure 5 illustrates a cross-sectional schematic of a railway substructure and the associated trajectories (A-scans) from a single GPR measurement, which include a signal generator (transmitter), transmitting and receiving antennas, and a recording device (receiver). The transmitter generates a pulse to detect objects and transmits it to the transmitting antenna. When an electromagnetic wave strikes an object whose electrical properties are different  
Figure 4: Force versus harmonic displacement.  
from other materials, part of the wave's energy is reflected to the receiving antenna. This energy is then sent to the receiver for storage and display.  
Because the frequency spectrum of the GPR return reveals the characteristics of the materials on the electromagnetic wave path, the frequency features are used to categorize ballast conditions automatically. Generally, the inference, identification, and classification of the GPR system comprise three steps: pre-processing, extracting features, and classification, as illustrated in Figure 6.  
The conventional approach to GPR detection and analysis is based on a linear framework in which linearity similarity is used to correlate mechanical dynamics with electromagnetic system/electric circuit systems characterized by discrete, specific natural frequencies. STFT has been used widely in GPR signal processing for frequency analysis. The method obtains data in the time and frequency domains by tracking changes in the frequency spectrum over time (or depth) and ballast properties (e.g., the correlation between the ballast's electromagnetic and mechanical properties).  
Conventionally, in the test analysis using GPR, the ballast system is assumed to be linear and time-invariant. The output of GPR is assumed to be a convolution of the incident signal or using the superposition principle. However, the ballast system is an inhomogeneous and non-isotropic nonlinear system, and its property is a time-variant nonlinear system in which the convolution or superposition principles cannot be used for high accuracy. As such, the existing approaches to extract features by using deconvolution, such as STFT, have limitations.