## 2 Materials and Methods
### Monitoring of Glacier Termini Movement Speed from GB Radar
#### 2.2.2 Principle of Radar Time-Series Interferometry Measurement
Set \(\eta\) as the sampling time; therefore, the original radar echo signal is a set of two-dimensional time-series signals, as shown in \[B=N\cdot\Delta f \tag{1}\]2  
The horizontal axis represents the frequency sampling and the vertical axis represents signal sampling, forming a two-dimensional signal matrix. The expression is as follows:  
\[B=N\cdot\Delta f \tag{1}\]3  
The difference from Equation (6) is that the instantaneous slant distance becomes a function of \(\eta\). After compression in the distance direction, the following is obtained:  
\[ \begin{array}{l}H(l,\eta)=\frac{1}{N}\exp\Bigl{(}\frac{-j4\pi f_{c}R(\eta)}{ c}\Bigr{)}\cdot\exp\Bigl{[}\frac{(N-1)\pi}{N}\Bigl{(}l-\frac{2 NR(\eta)\Delta f}{c} \Bigr{)}\Bigr{]}\cdot\\ \frac{\sin[\pi(l-2 NR(\eta)\Delta f/c)]}{\sin[\pi/N(l-2 NR(\eta)\Delta f/c)]} \end{array}  \tag{10}\]  
Figure 7: One-dimensional range profile of SFCW radar. \begin{table}
\begin{tabular}{c c c c c} \hline
**Pulse No.** & \(f\) & \(f_{1}\) & \(f_{2}\) & \(\cdots\) & \(f_{N}\) \\ \hline
1 & \(S_{11}\) & \(S_{12}\) & \(\cdots\) & \(S_{1N}\) \\
2 & \(S_{21}\) & \(S_{22}\) & \(\cdots\) & \(S_{2N}\) \\
3 & \(S_{31}\) & \(S_{32}\) & \(\cdots\) & \(S_{3N}\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ M & \(S_{M1}\) & \(S_{M2}\) & \(\cdots\) & \(S_{MN}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Two-dimensional raw signal matrix. At present, the horizontal axis is transformed from the frequency domain to the time domain, representing time or distance. Extracting the phase from a certain distance gate in the above equation yields the following:  
\[\phi(\eta)=\frac{-4\pi f_{c}R(\eta)}{c}+\frac{(N-1)\pi}{N}\bigg{(}l-\frac{2 NR( \eta)\Delta f}{c}\bigg{)} \tag{11}\]  
If it is within a distance gate, the phase will not be entangled, that is, there will be no phase jump. If it is greater than a distance gate, phase unwrapping is required to obtain the phase truth. The phase extracted from the first sampled signal is defined as the initial phase \(\phi_{0}\), which interferes with \(\phi(\eta)\) to obtain the following:  
\[\Delta\phi(\eta)=\phi(\eta)-\phi_{0} \tag{12}\]  
The position change in the target can be obtained based on the phase difference between two target echoes, as follows:  
\[\Delta R(\eta)=-\frac{c\cdot\Delta\phi(\eta)}{4\pi f_{c}} \tag{13}\]  
Take the phase of the timing echo signal of the distance gate from the target to be tested and subtract it from the initial phase. By inverting with Equation (13) and accumulating multiple times, the temporal micro deformation of the target can be obtained. The motion speed of the target can be calculated based on the deformation and acquisition time. 3. Linear least squares fitting  
Assuming that the variable changes uniformly over time, in an ideal situation, the cumulative phase is linearly related to time. However, due to factors such as clutter noise and the internal structure of ground targets in signal reception, the two are approximately linear. Therefore, better measurement results can be achieved by linearly fitting the deformation results. Define \(y_{t}\) as the cumulative linear fitting phase at time \(t\), and the relationship between the two is as follows:  
\[y_{t}=at+b \tag{14}\]  
Among them, \(a\) and \(b\) are constants, and the solution satisfies the minimum sum of squared errors [44], as follows:  
\[I=\sum\limits_{t=0}(y_{t}-\phi_{t})^{2}=\text{min} \tag{15}\]  
The goodness of fit refers to the degree to which the regression curve fits the observed values, and the determination coefficient (\(R^{2}\)) can measure the goodness of fit. The maximum value of \(R^{2}\) is 1, and the closer the value is to 1, the better the fit of the regression curve to the observed values. Calculate the mean as \(\overline{y}\) and the fitted data as \(\hat{y}\). The calculation is as follows:  
\[R^{2}=1-\frac{\sum\limits_{t=1}^{n}{(y_{t}-y_{t})^{2}}}{\sum\limits_{t=1}^{n}{ (y_{t}-\overline{y})^{2}}} \tag{16}\]