## 2 Test Method
### Complex Signal Processing  
Complex signal processing decomposes the reflected wave information from electromagnetic recordings into the instantaneous amplitude, phase, and frequency. This technique primarily uses the Hilbert transform to convert real electromagnetic signals into complex ones and decompose them into the above parameters. These parameters are then used to analyze the response characteristics of anomalous bodies. According to the Hilbert transform, a signal's frequency domain and time domain exhibit an equivalence relationship after undergoing a Fourier transform, which is closely related to the symmetry of the Fourier transform.  
The reflected wave signal data obtained through the GprMax forward simulation are real signals (Equation (4)). By applying the Hilbert transform to the real signal \(x(t)\), the imaginary component of its complex signal can be obtained (Equation (5)). The real signal \(x(t)\) and its corresponding complex signal \(\hat{x}(t)\) can then be combined to form the complex GPR signal (Equation (6)) [23].  
\[x(t)=A(t)\cos[\omega_{0}t+\varphi(t)] \tag{4}\]  
\[\hat{x}(t)=x(t)\frac{1}{\pi t}=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{x(t) }{t-\pi}dt=A(t)\sin[\omega_{0}t+\varphi(t)] \tag{5}\]  
\[\begin{array}{l}f(t)=x(t)+i\hat{x}(t)=A(t)\cos[\omega_{0}t+\varphi(t)]+iA(t) \sin[\omega_{0}t+\varphi(t)]\\ =A(t)e^{i[\omega_{0}t+\varphi(t)]}\end{array} \tag{6}\]  
In the above equations, \(\varphi(t)\) is the phase function of the real signal; \(A(t)\) is the amplitude function; \(\omega_{0}\) is the center frequency; \(t\) is the time variable; and \(f(t)\) is the complex signal of the GPR.  
As shown in Equation (7), \(A(t)\) represents the amplitude of \(f(t)\), which reflects the strength of the reflected signal. The instantaneous amplitude is proportional to the square root of the energy of the georadar signal at that moment. When there is an abnormality within the roadbed, the magnitude of the electromagnetic wave signal's instantaneous amplitude exhibits a significant variation. This anomaly is evident in the instantaneous amplitude profile of the roadbed, where distinct changes occur at the locations of the roadbed's structural interfaces and internal anomalies.  
As shown in Equation (8), \(\theta(t)\) denotes the instantaneous phase of \(f(t)\). The instantaneous phase is primarily used to describe the continuity of the phase axis in the georadar profile. In the instantaneous phase profile of the roadbed, both strong and weak reflected waves are represented by their phases. Even weak-amplitude effective waves can be clearly displayed in the instantaneous phase profile. When an anomaly is present within the roadbed structure, the instantaneous phase of the electromagnetic wave will experience irregularities, resulting in discontinuities in the phase axis.  
As shown in Equation (9), the instantaneous frequency can be obtained by taking the derivative of \(\theta(t)\) with respect to \(t\). The instantaneous frequency is the rate of change in the instantaneous phase over time; it primarily reflects the variations in the physical properties of the subsurface layers. When electromagnetic waves propagate through different media, the frequency of the reflected signals will vary [24].  
\[A(t)=\sqrt{x^{2}(t)+\hat{x}(t)} \tag{7}\]  
\[\theta(t)=\omega_{0}t+\varphi(t)=\tan^{-1}\left[\frac{\hat{x}(t)}{x(t)}\right] \tag{8}\]  
\[S(t)=\frac{d\theta(t)}{dt}=\frac{d}{dt}\tan^{-1}\left[\frac{\hat{x}(t)}{x(t)}\right] \tag{9}\]