## 2 Materials and Methods
### Monitoring of Glacier Termini Movement Speed from GB Radar
#### 2.2.2 Principle of Radar Time-Series Interferometry Measurement  
1. Signal model  
The concept of SFCW was first proposed in 1972 for detecting buried objects. However, extensive research on SFCW radar sensors only began in the early 1990s [42]. The pulse width of the single frequency sub signal of SFCW is \(\tau\), the speed of light is \(c\), the starting frequency of the carrier frequency is \(f_{0}\), the center frequency is \(f_{c}\), the frequency step is \(\Delta f\), and the number of frequency steps is \(N\). The frequency of the SFCW signal is shown in Figure 6.  
\(B\) is the bandwidth of the transmitted signal, as follows:  
\[B=N\cdot\Delta f \tag{1}\]  
\(\delta_{r}\) is the resolution of the system, as follows:  
\[\delta_{r}=\frac{0.886\cdot c}{2B} \tag{2}\]  
The signal time is \(t\), \(n\) is an integer from 0 to \(N-1\), and the transmitted signal \(S_{t}(t)\) can be expressed as  
\[S_{t}(t)=\exp[2\pi(f_{0}+n\cdot\Delta f)t] \tag{3}\]  
Figure 5: User interface for system control. The echo signal \(S_{r}(f)\) generated by radar illumination from a target at a distance of \(R\) from the antenna is  
\[S_{r}(f)=\exp[2\pi(f_{0}+n\cdot\Delta f)(t-2R/c)] \tag{4}\]  
After quadrature demod, the normalized baseband signal \(S_{IF}(f)\) is  
\[S_{IF}(f)=\exp(-j4\pi f_{0}R/c)\cdot\exp(-j4\pi n\Delta f\cdot R/c) \tag{5}\]  
In the above equation, the first term is a constant, and the second term can be regarded as a frequency domain discrete signal with a linear change in frequency. Directly performing Inverse Fast Fourier Transform (IFFT) on the signal can complete distance compression and obtain a high-resolution one-dimensional range profile of the target [43], as follows:  
\[H(l)=\frac{1}{N}\exp\left(\frac{-j4\pi f_{c}R}{c}\right)\cdot\exp\left[\frac{j (N-1)\pi}{N}\left(l-\frac{2 NR\Delta f}{c}\right)\right]\cdot\frac{\sin[\pi(l- 2 NR\Delta f/c)]}{\sin[\pi/N(l-2 NR\Delta f/c)]} \tag{6}\]  
By taking the modulus of Equation (6), the following is obtained:  
\[|H(l)|=\left|\frac{\sin[\pi(l-2 NR\Delta f/c)]}{N\sin[\pi/N(l-2 NR\Delta f/c)]}\right| \tag{7}\]  
The distance sampling of the target is \(l\), at which point the amplitude of the signal is a sinc function, and the peak position \(l_{0}\) is the following:  
\[l_{0}=2 NR\cdot\Delta f/c \tag{8}\]  
The above equation indicates the correspondence between the peak position of the sampling point and the distance to the target. After distance compression, the position of the target to be measured is determined through the sampling point. If the scattering intensity of the target is high, it appears as a large amplitude sinc pulse in the time domain. Assuming that the distance between a target and the radar is 100 m, and the bandwidth of the transmitted signal is 300 MHz, simulate the raw data of the target and implement distance compression through IFFT. After interpolation, the signal amplitude is displayed in logarithmic form, and the simulation results of the one-dimensional distance profile are as follows \[B=N\cdot\Delta f \tag{1}\]0  
\[B=N\cdot\Delta f \tag{1}\]1 Schematic diagram of SFCW signal frequency. After IFFT, the signal exhibits a sinc function at 100 m with a 3 dB width of 0.448 m, which is consistent with the theoretical value shown in Equation (2). The SFCW signal processing technology has been verified through simulation. 2. Time-series interferometry measurement  
After signal processing, the phase containing deformation information can be extracted. Interference refers to obtaining surface change information from the phase of two radar images obtained in the same region at a relatively close time, specifically through phase subtraction or complex conjugate multiplication. When observing the target in practice, the radar will emit signals to the target at a certain PRF.