## 2 Algorithm for MBS
### Division Mode in FBP  
According to the imaging process of BP algorithm, the dual-delay time from SAR platform to each point in imaging area is calculated; then compensation phase is added to the echo corresponding to each point in time domain regarding the range compressing process, and the final imaging is received by coherently accumulating the projection data consistent with different azimuth moments [16,17]. After the range matching and filtering, the echo of EGT is expressed in Equation (1), as shown below:  
\[\mathrm{s}(\eta,\tau)=A\cdot exp\left(\frac{-j\cdot 4\pi\cdot R(\eta)}{\lambda} \right)\cdot sinc\left(\tau-\frac{2\cdot R(\eta)}{c}\right) \tag{1}\]  
where \(\lambda\) is the wavelength of SAR platform; \(\mathrm{c}\) is light speed in vacuum; \(\tau\) is the fast time at range direction; \(R(\eta)\) is the slant range between MBS and its corresponding EGT in slow time \(\eta\) at azimuth direction; and \(A\) is the signal amplitude, which indicates the reflective properties of EGT and is often deemed as a constant of one for simplicity. Unlike the slant range calculated in a low orbit platform, its trajectory can be replaced with second-order hyperbola under a constant speed with little change, the curve motion and the locations' arbitrariness of MBS no longer defines trajectory and speed as platforms in a low orbit; a detail description will be provided in Section 2.2. After the echo has been received, each point in imaging area is projected to the echo domain, forming the migration trajectory of EGT; then, the complex data of EGT is obtained after the echo been superimposed. The complex data of each pixel (\(i\),\(k\)) in the imaging area can be expressed as shown below:  
\[f(i,k)=\int_{-0.5t}^{0.5t}s_{ik}(\eta,\tau(\eta))\cdot exp\left(\frac{j\cdot 4 \pi\cdot R(\eta)}{\lambda}\right)dt \tag{2}\]  
where \(\tau(\eta)=\frac{2\cdot R(\eta)}{c}\) indicates the dual-delay time at the azimuth time \(\eta\), \(t\) is the synthetic aperture time of MBS. Since each point in image area has been projected on echo domain in the BP algorithm, if the sampling numbers in azimuth direction is N\({}_{a}\), the pixel numbers along azimuth and range directions in imaging areas are M \(\times\) N, the computational cost of the projection calculation in imaging area is N\({}_{a}\)\(\times\) M \(\times\) N; the cost of phase compensation is also N\({}_{a}\)\(\times\) M \(\times\) N, and the cost of coherent superposition is \((\mathrm{N}_{a}-1)\)\(\times\) M \(\times\) N. Therefore, the total computational cost is the sum of above three computational costs, i.e., \((3\mathrm{N}_{a}-1)\)\(\times\) M \(\times\) N. Once  
Figure 2: The flow chart of FBP algorithm. the number of sampling points becomes large enough, the computational cost will exhibit undesired exponential growth. However, for MBS located in the near side of Moon, its synthetic aperture time can be hundreds of seconds longer than that on the space-born platform; moreover, the imaging area around the Earth's surface can be larger, e.g., exceeding millions of square kilometers, under a small gaze angle. Therefore, the computational cost of the BP algorithm in the MBS scenario might be intolerable large. Therefore, to bring down the cost, the traditional BP algorithm is modified into the FBP algorithm, shown as follows, based on sub-aperture and sub-imaging area division. In the proposed FBP algorithm, the synthetic aperture length and corresponding imaging area for MBS will be divided into sub-apertures and sub-imaging areas, as shown in Figure 3a. Here, the lengths in Figure 3a are not shown in the same ratio as the actual scenario, in order to create a clear display. The synthetic aperture length \(L_{s}\) is divided into \(\alpha_{1}\) small non-overlapping sub-apertures; therefore, the actual length of every sub-aperture is \(d=\frac{L_{s}}{\alpha_{1}}\).