## 4 The Application of Tikhonov Regularization in THz SAR  
### Wavenumber Spectrum Reconstruction Model of THz SAR Azimuth Non-Uniform Sampling Signals  
Assuming that the radar platform operates in the stripmap mode with a side-looking configuration, where the platform's motion trajectory is parallel to the x-axis of the coordinate system, the radar center frequency is \(f_{c}\), and the antenna beamwidth is \(\theta_{bw}\). In the fast-time domain, the number of samples for each azimuthal sampling point is \(N_{k}\). Figure 6 shows the imaging geometry under non-uniform sampling conditions in the azimuth direction for the radar platform. Due to the deviation of sampling positions from the ideal uniformly sampled positions, the received echo data exhibits certain deviations. Let \(x_{1},x_{2},x_{3},\cdots\cdots x_{N_{m}}\) denote the non-uniform sampling positions in thr azimuth direction of the radar platform, which can be obtained through accurate inertial navigation systems or by using the raw data of SAR echo signals received by the radar platform. The position coordinates of point target \(B\) are denoted as \([x_{B},y_{B},0]\), and the closest slant range \(AB\) from the radar track to point target \(B\) is denoted as \(R_{B}\). The slant range from the \(i\)-th sampling position \(x_{i}\) to point target \(B\) is denoted as \(R(x_{i})\). It can be expressed as  
\[R(x_{i})=\sqrt{\left(x_{i}-x_{B}\right)^{2}+R_{B}^{2}} \tag{19}\]  
The SAR signal processing is based on the \"stop and go\" assumption. Under the condition that this assumption holds, the received SAR echo data can be considered as identical regardless of the platform's velocity, as long as the positions of the azimuth sampling points are fixed. Therefore, when processing the echo data in wavenumber domain, there is no need to consider the specific velocity of the radar platform [30]. In the case of SAR azimuth non-uniform sampling, the processing of SAR echo data in Doppler domain becomes more complicated due to uneven platform velocity. Therefore, we adopt a more convenient wavenumber domain processing method. Taking the commonly used LFM signal as an example, assuming the frequency modulation rate of radar transmission signal is \(K_{r}\) and the duration is \(T_{p}\), neglecting the range and azimuth  
Figure 6: Azimuth non-uniform sampling imaging geometry of THz SAR. envelopes of echo data and considering only the phase term, the point target model of the demodulated SAR echo signal can be expressed as  
\[s_{0}(x,t_{k})=\exp\{-j4\pi f_{c}R(x)/c\}\exp\{j\pi K_{r}(t_{k}-2R(x)/c)^{2}\} \tag{20}\]  
where \(c\) is the speed of light, \(x\) is the azimuth position, \(R(x)\) represents the distance from \(x\) to point target \(B\), and \(t_{k}\) represents the fast time in range direction. The ranges of \(t_{k}\) and \(x\) are  
\[t_{k}\in\left[2R(x)/c-\frac{T_{p}}{2},2R(x)/c+\frac{T_{p}}{2}\right] \tag{21}\]  
\[x\in\left[-\frac{\theta_{bw}}{2}\leq\arctan(\frac{|x-x_{B}|}{R_{B}})\leq\frac {\theta_{bw}}{2}\right] \tag{22}\]  
The range Fourier transform result can be obtained as  
\[S_{1}(x,f_{k})=\exp\{-j\frac{4\pi(f_{c}+f_{k})R(x)}{c}\}\exp\{-j\frac{\pi f_{k }^{2}}{K_{r}}\} \tag{23}\]  
where \(f_{k}\in\left[-K_{r}T_{p}/2,K_{r}T_{p}/2\right]\), combining Equations (19) and (23), the azimuth Fourier transform is obtained as  
\[S_{2}(K_{x},f_{k})=\exp\{-j2\pi\sqrt{\frac{4{(f_{c}+f_{k})}^{2}}{c^{2}}-K_{x}^ {2}}-j2\pi K_{x}x_{B}\}\exp\{-j\frac{\pi f_{k}^{2}}{K_{r}}\} \tag{24}\]  
The relationship between the azimuth wavenumber spectrum \(K_{x}\) and \(x\) is given by  
\[K_{x}=\frac{2(f_{c}+f_{k})}{c}\left[\frac{\left(x_{B}-x\right)}{\sqrt{R_{B}^{ 2}+\left(x_{B}-x\right)^{2}}}\right] \tag{25}\]  
Let  
\[\sin\theta=\frac{\left(x_{B}-x\right)}{\sqrt{R_{B}^{2}+\left(x_{B}-x\right)^{ 2}}} \tag{26}\]  
Based on the geometric relationship shown in Figure 6, \(\theta\) represents the angle between the line connecting the phase center of antenna to the point target and the closest slant range \(AB\).