## 3 Processing Method
### The Proposed Method
#### 3.2.2 TDCM Processing  
After data sorting, the rearranged data are processed using the TDCM. Using reference channel data at \(\eta\) as the starting point, a group of \(M+1\) adjacent data is formed for the time-domain cross-correlation operation. Since the adjacent data after rearrangement are  
Figure 4: Distributed SAR data sorting after rearrangement. Figure 3: Schematic diagrams of the coupling of the CTB measurement error and the motion error: (**a**) measurement error of the CTB, (**b**) motion error of the single channel, and (**c**) actual spatial arrangement of sampling points. not sampled at the same slow time, the phase difference obtained by the TDCM includes both the channel phase error and the motion error. From Equation (8), the phase difference between \(M\) adjacent data of a set of data can be expressed as  
\[\zeta_{m}(\eta)=\left\{ \begin{array}{ll}=\phi_{m}^{\prime}-\phi_{m-1}^{\prime} \\ -4\pi\frac{\Delta R_{muc}\left(\eta-mT_{prt}\right)-\Delta R_{muc}\left(\eta-(m- 1)T_{prt}\right)}{\lambda}\\ -2\pi f_{dc}\left(\frac{d_{m}-d_{m-1}}{\zeta_{\rm p}}-T_{prt}\right)\\ =-\phi_{M-1}^{\prime}\\ -4\pi\frac{\Delta R_{muc}\left(\eta+T_{prt}\right)-\Delta R_{muc}\left(\eta-( M-1)T_{prt}\right)}{\lambda}\\ -2\pi f_{dc}\left(MT_{prt}-\frac{d_{M-1}}{\zeta_{\rm p}}\right)\\ \end{array} \right.,m=M \tag{9}\]  
where \(T_{prt}\) represents pulse repetition time, \(f_{dc}\) represents the Doppler center of the signal, and \(\zeta_{m}(\eta)\) includes channel error and motion error. The sequence of phase errors by sampling point is shown in Figure 5. The phase error is a quadratic curve with slow time on single-channel data, while there are also phase jumps between channels generated by channel errors. Through the TDCM between adjacent data, the estimation of Equation (9) is  
\[\hat{\xi}_{m}(\eta)=\left\{ \begin{array}{ll}\angle\big{(}s_{m}(\eta-mT_{prt})s _{m-1}{}^{*}(\eta-(m-1)T_{prt})\big{)},m=1,2,\ldots,M-1\\ \angle\big{(}s_{0}(\eta+T_{prt})s_{M-1}{}^{*}(\eta-(M-1)T_{prt})\big{)},m=M\\ \end{array} \right. \tag{10}\]  
where \(\angle\big{(}\ )\) represents the phase and \(*\) is the complex conjugate sign. Summing \(\zeta_{m}(\eta)\) cumulatively, the phase difference between the channel \(m\) and the reference channel is obtained:  
\[ \begin{array}{ll}\omega_{m}(\eta)=\sum\limits_{k=1}^{m}\zeta_{k}(\eta)=\phi _{m}^{\prime}+2\pi\Big{(}\frac{d_{m}}{\zeta_{\rm q}}-mT_{prt}\Big{)}f_{dc}\\ \hskip 14.226378 pt+\frac{4\pi}{\lambda}mT_{prt}v_{c}+\frac{4\pi}{\lambda}\Big{(} m\eta T_{prt}-\frac{1}{2}m^{2}T_{prt}{}^{2}\Big{)}a_{c}\,m=1,2,\ldots,M-1\\ \end{array}  \tag{11}\]  
Figure 5: Phase error sequence after rearrangement and local enlargement. The estimation of \(\omega_{m}(\eta)\) is  
\[\hat{\omega}_{m}(\eta)=\sum_{k=1}^{m}\xi_{k}(\eta)\ \ \,m=1,2,\ldots,M-1 \tag{12}\]  
Except for the phase term associated with \(f_{dc}\), the remaining terms are the signal errors to be compensated before signal reconstruction. The phase difference between the adjacent slow times of the reference channels is  
\[\omega_{0}(\eta)=\sum_{k=1}^{M}\zeta_{k}(\eta)=2\pi T_{prt}f_{dc}-\frac{4\pi} {\lambda}T_{prt}v_{c}-\frac{4\pi}{\lambda}\bigg{(}\eta T_{prt}+\frac{1}{2}T_{ prt}{}^{2}\bigg{)}a_{c} \tag{13}\]  
The estimation of \(\omega_{0}(\eta)\) is  
\[\hat{\omega}_{0}(\eta)=\sum_{k=1}^{M}\xi_{k}(\eta) \tag{14}\]  
Based on Equations (11) and (13),  
\[ \begin{array}{l}\varphi_{m}(\eta)=\omega_{m}(\eta)-\frac{\hat{\omega}_{0}( \eta)}{T_{prt}}\bigg{(}\frac{d_{m}}{v_{a}}-mT_{prt}\bigg{)}\\ \hskip 14.226378 pt=\phi_{m}^{\prime}+\frac{4\pi}{\lambda}\frac{d_{m}}{v_{a}}v_ {c}\\ \hskip 14.226378 pt-\frac{4\pi}{\lambda}\bigg{(}\frac{1}{2}m(m+1)T_{prt}{}^{2}- \frac{1}{2}\frac{d_{m}}{v_{a}}T_{prt}\bigg{)}a_{c}+\frac{4\pi}{\lambda}\frac{d _{m}}{v_{a}}a_{c}\eta\ \,m=1,2,\ldots,M-1\end{array}