## 3 Theoretical Analysis
### Range Characteristic between MBS and EGT
Moreover, the synthetic aperture length of MBS is also longer than GEO-SAR's, thus increasing the EGT's exposure time. The Moon is locked by Earth's tides, and almost only one side faces the earth, which leads to a basically fixed line of sight for the Earth's observation. Therefore, it is necessary and beneficial to develop the MBS for Earth observation, although range discrepancy does exist. Finally, we will describe the necessity of using separate calculations for dual-delay time. Another processing is the fitting of actual trajectories deduced from ephemeris, such as those related to precession, nutation, apsidal motion, etc., in a specific scenario for better imaging. Through Equation (15), the fitting speed of MBS and maximum error in slant range (MESR), i.e., the absolute value of difference between fitting range and the range calculated in Section 3.1, around apogee, perigee, ascending, and descending nodes are displayed in Tables 4 and 5. Generally speaking, the fitting speed is close to the speed of the platform, but the actual fitting speed is significantly different from the speed of the MBS \begin{table}
\begin{tabular}{c c c c c c} \hline
**Location of MBS** & **Look Angle** & \begin{tabular}{c} **Difference at** \\ **Aspoee in 80 s** \\ \end{tabular} & \begin{tabular}{c} **Difference at** \\ **Perigee in 80 s** \\ \end{tabular} & \
\begin{tabular}{c} **Difference at** \\ **Aspoee in 150 s** \\ \end{tabular} &
\begin{tabular}{c} **Difference at** \\ **Descending Node** \\ **in 150 s** \\ \end{tabular} \\ \hline (30\({}^{\circ}\) W, 30\({}^{\circ}\) S) & 0.1\({}^{\circ}\) & 24.0298 & 22.1642 & 40.6371 & 42.2243 \\ (0\({}^{\circ}\), 0\({}^{\circ}\)) & 0.1\({}^{\circ}\) & 19.6996 & 16.5473 & 44.3090 & 47.1823 \\ (30\({}^{\circ}\) E, 30\({}^{\circ}\) N) & 0.1\({}^{\circ}\) & 14.2782 & 15.4220 & 49.0154 & 52.7617 \\ (30\({}^{\circ}\) W, 30\({}^{\circ}\) S) & 0.3\({}^{\circ}\) & 23.6288 & 21.1031 & 39.5041 & 39.4661 \\ (0\({}^{\circ}\), 0\({}^{\circ}\)) & 0.3\({}^{\circ}\) & 19.1191 & 16.0099 & 42.9002 & 46.0779 \\ (30\({}^{\circ}\) E, 30\({}^{\circ}\) N) & 0.3\({}^{\circ}\) & 13.8890 & 14.4862 & 48.2349 & 49.3998 \\ \hline \end{tabular}
\end{table} 
Table 2: The maximum difference in the delay range (m) (i.e., the maximum value between the downlink path and the uplink path) for various MBSs at one approximate apogee (i.e., 19:01:01 24 January 2001) and one approximate perigee (i.e., 09:01:01 10 January 2001), with the synthetic aperture time being about 80 s and 150, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Location of** & **Look Angle** & \begin{tabular}{c} **Difference at** \\ **Aspoee in 80 s** \\ \end{tabular} & \begin{tabular}{c} **Difference at** \\ **Descending Node** \\ **in 80 s** \\ \end{tabular} & \
\begin{tabular}{c} **Difference at** \\ **Ascending Node** \\ **in 150 s** \\ \end{tabular} &
\begin{tabular}{c} **Difference at** \\ **Descending Node** \\ **in 150 s** \\ \end{tabular} \\ \hline (30\({}^{\circ}\) W, 30\({}^{\circ}\) S) & 0.1\({}^{\circ}\) & 24.0298 & 22.1642 & 40.6371 & 42.2243 \\ (0\({}^{\circ}\), 0\({}^{\circ}\)) & 0.1\({}^{\circ}\) & 19.6996 & 16.5473 & 44.3090 & 47.1823 \\ (30\({}^{\circ}\) E, 30\({}^{\circ}\) N) & 0.1\({}^{\circ}\) & 14.2782 & 15.4220 & 49.0154 & 52.7617 \\ (30\({}^{\circ}\) W, 30\({}^{\circ}\) S) & 0.3\({}^{\circ}\) & 23.6288 & 21.1031 & 39.5041 & 39.4661 \\ (0\({}^{\circ}\), 0\({}^{\circ}\)) & 0.3\({}^{\circ}\) & 19.1191 & 16.0099 & 42.9002 & 46.0779 \\ (30\({}^{\circ}\) E, 30\({}^{\circ}\) N) & 0.3\({}^{\circ}\) & 13.8890 & 14.4862 & 48.2349 & 49.3998 \\ \hline \end{tabular}
\end{table} 
Table 3: The maximum difference in the delay range (m) (i.e., the maximum value between the downlink path and the uplink path) for various MBS at one approximate ascending node (i.e., 11:01:01 15 January 2001) and one approximate descending node (i.e., 22:01:01 2 January 2001), with the synthetic aperture times being about 80 s and 150, respectively. or EGT.