# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 1 Introduction
Stiffness is one of the basic performance parameters for railway track load-carrying capacity. Efficient and accurate stiffness measurement has been considered the foundation for further development of railway engineering and, therefore, has excellent theoretical and practical significance. Conventional methods for dynamic property characterization have been used for specific locations at the level of specific track sections. They are hard to use for quick and continuous measurements at the level of the railway network. GPR has been widely used for railway ballast inspection at high speeds of standard trains, by which the ballast dynamics properties can be inferred. Since the ballast layer is the most dominating component of the track stiffness, the use of GPR makes the indirect estimation and monitoring of track and ballast stiffness possible, and the application of GPR to assess the ballast condition and stiffness has been widely developed [7; 10; 57; 58; 59; 60; 61]. Currently, for surveying and analyzing long sections of transport infrastructure, the mainstream detection method is GPR detection [57; 58; 59; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79]. Even though GPR detection/inspection for ballast layers has matured, some challenges still need to be address, such as combining GPR results with track stiffness. Previous studies have revealed that the overall electrical properties of the base material can be used to infer its strength characteristics, assuming that information on the layered properties of the track substructure can be used to indirectly infer the modulus and stiffness of the track and ballast [58; 59]. These correlations can be used to predict the deformation properties of ballast and its dynamic behavior under heavy axle load in different conditions by using a GPR survey. Based on this technique and similar tests on different types of ballast under different conditions, a criterion can be developed to detect the railroad high-risk areas and predict the engineering properties of ballast through GPR surveys. The time-varying property of ballast GPR signals was investigated in [60]. The study investigated the frequency spectrum of GPR in the time domain using the short-time Fourier transform (STFT) method, in which the authors demonstrated the variation trends of the frequency energy when the ballast was subject to different fouling conditions. Ref. [61] introduced an algorithm to extract the magnitude spectra at salient frequencies and then to classify railway ballast conditions with the support of vector machines. Ref. [65] introduced a theoretical framework to bridge the working principles of electromagnetic and mechanical systems for GPR application on ballast. Ref. [66] investigated a method to assess railway track conditions according to GPR data in the frequency domain. The above methodologies successfully showed the reliability and effectiveness of GPR in assessing ballast stiffness and time-varying dynamic properties. [PERSON] et al. [67] used time-frequency and Wavelet techniques to process ballast GPR data to evaluate ballast degradation due to fouling detection. [PERSON] et al. [77] used STFT and Wavelet techniques to process ballast GPR data for efficient railroad inspection. Conventional time-frequency methods for ballast GPR signal, such as STFT or spectrogram, which is the squared magnitude of STFT, have the inherent weaknesses of cross-term errors and inaccuracy for nonlinear signals. On the other hand, Wavelet transform has worse accuracy for spectrum values compared with STFT performance due to its disadvantages of shift sensitivity, poor directionality, and lack of phase information. To further illustrate the time/space-varying property of the developed stiffness model of ballast, a detected ballast GPR signal is characterized using adaptive optimal kernel time-frequency representation (AOKTFR) [80; 81; 82]. In principle, the AOKTFR method has higher time-frequency resolution than conventional time-frequency methods and is more suitable for nonlinear time-variant systems.