## 2 Algorithm for MBS
### Range History Calculation  
Since the dual-delay time of MBS can exceed 2.3 s, the transmitted and received positions of MBS are not the same; the stop-and-go model applied to a low orbit platform will no longer be applicable for MBS distributed around the near side of Moon. In an actual scenario, the long average distance from MBS to its achievable EGT, the rotation of Earth, the rotation of Moon, and the revolution of Moon, together with DEM of MBS and EGT will have different effects on their slant range. The near side of Moon is distributed with uneven craters, hills, uplifts, and potholes that have unequal DEM, and so does the EGT; the differences in locations of MBS and EGT bring diversity to the ground range and slant range, which have been considered in imaging processing; because of earth tide lock, the revolution and rotation of the Moon's near side periodically faces the Earth. The Earth's unequal rotation in latitude direction makes the exposure time, related to synthetic time and length, of EGT change unevenly. Without loss of generality and to simplify the analysis process, the diverse DEM of MBS and its corresponding EGT are omitted, and the two planets, i.e., the Earth and the Moon, are considered as follows: the Earth is considered to be a standard ellipsoid body, with an equator radius of about 6377 km, and a polar radius of about 6356 km, and the Moon is considered to be a standard sphere, with average radius of about 1737 km, respectively [18]. Then, consider the relative motion between MBS, e.g., its location is about 0\({}^{\circ}\), 0\({}^{\circ}\) on the Moon's surface, and its corresponding look angle is about 0.1\({}^{\circ}\), as shown in Figure 4. The distance from MBS to EGT with Earth's rotation (an oscillating smooth curve displayed in the black solid line) shows much different when compared to the results without Earth's rotation (a sine smooth curve displayed in the green solid line). The curve occlusion of the sphere is also ignored to provide a simplified description. Although their overall trends are similar, the actual distance shows that oscillation changes daily, so the Earth's rotation cannot be ignored for actual Earth-Moon SAR imaging, especially when determining MBS and accessible EGT. For a more realistic scenario, the ephemeris are adopted among calculations in this paper. While adopting JPL DE430, or other ephemerides at specific moments for researching, the dual-path method is used, i.e., uplink and downlink components, are calculated separately, with Earth's rotation been considered in every pulse [19,20]. Here, its antenna boresight lies in the corresponding zero-Doppler plane after using at least the two-dimensional attitude steering method, and the plane is assumed to remain unchanged during synthetic aperture time. The dual-path method from MBS to achievable EGT at the \(n\) pulse is shown as Figure 5, where \(\tau_{1}\) is delay time of the downlink path from MBS to EGT; \(\tau_{2}\) is delay time of the uplink from EGT to MBS; \(T\) is pulse repeat time (PRT) of MBS; \(n\) is number of pulses; \(nT\), \(nT+\tau_{1}\), \(nT+\tau_{1}+\tau_{1}+\tau_{1}+\tau_{2}\) are three adjacent discrete time variables, indicating the wave propagation of a single pulse; \(\mathbf{\tau}_{i}^{nT}\), \(\mathbf{\tau}_{i}^{nT+\tau_{1}}\), \(\mathbf{\tau}_{i}^{nT+\tau_{1}+\tau_{2}}\) are the corresponding position vectors of EGT; \(\mathbf{\Omega}_{m}^{nT}\), \(\mathbf{\Omega}_{m}^{nT+\tau_{1}}\), \(\mathbf{\Omega}_{m}^{nT+\tau_{1}+\tau_{2}}\) are the corresponding position vectors of the Moon's centroid; \(\mathbf{\tau}_{m}^{nT}\), \(\mathbf{\tau}_{m}^{nT+\tau_{1}}\), \(\mathbf{\tau}_{m}^{nT+\tau_{1}+\tau_{2}}\) are the corresponding position vectors of MBS; \(\mathbf{\
u}_{l}^{nT}\) is the velocity vector of EGT at \(nT\) pulse as related to the Earth's rotation; and \(\mathbf{\
u}_{l}^{nT}\) is the velocity vector of MBS at the \(nT\) pulse as related torotation and revolution of Moon.