## 2 Algorithm for MBS
### Range History Calculation
Figure 5 indicates that when MBS transmits radar wave at time \(nT\), the EGT is in \(\mathbf{\tau}_{i}^{nT}\); after a period of time \(\tau_{1}\), the EGT receives the wave in \(\mathbf{\tau}_{i}^{nT+\tau_{1}}\) and reflects it immediately, and the MBS has moved in \(\mathbf{\tau}_{m}^{nT+\tau_{1}}\). Again, after a period of time \(\tau_{2}\), the MBS gets the echo signal in \(\mathbf{\tau}_{m}^{nT+\tau_{1}+\tau_{2}}\), and the EGT has moved in \(\mathbf{\tau}_{i}^{nT+\tau_{1}+\tau_{2}}\); this process can be called an entire delay cycle for a single pulse; here the two velocities \(\mathbf{\
u}_{i}^{nT}\) and \(\mathbf{\
u}_{l}^{nT}\) are deemed to be unchanged in an entire delay cycle. The coordinate system for the calculation of the dual path involved with the Earth's central rotational coordinate system (ECR), the Earth's  
Figure 4: The approximate distance between MBS and its corresponding EGT with Earth rotation (the black solid line) and without Earth rotation (the green solid line) as recorded in January, 2001. The curve occlusion of the spheres between MBS and EGT are not considered. Figure 5: The signal transmission path between MBS and EGT at \(n\). central inertial coordinate system (ECI), the Moon's central Moon fixed coordinate system (MCMF), etc. In this study, the position and velocity of MBS and EGT should be represented in the same coordinate system, for example, in the ECR. From the cosine theorem of trigonometric functions, we can obtain the equations for the signal transmission path of downlink in Equation (10) and uplink in Equation (11), respectively:  
\[\langle\textbf{r}_{m}^{nT}-\textbf{r}_{t}^{nT},\textbf{r}_{m}^{nT}-\textbf{r} _{t}^{nT}\rangle+\langle\textbf{v}_{t}^{nT},\left(\textbf{v}_{t}^{nT}\right)^ {T}\rangle\cdot\textbf{r}_{1}^{2}+2\cdot\langle\textbf{r}_{m}^{nT}-\textbf{r} _{t}^{nT},\textbf{v}_{t}^{nT}\rangle\cdot\textbf{r}_{1}-\left(c\cdot\textbf{ r}_{1}\right)^{2}=0 \tag{10}\]  
\[\langle\textbf{r}_{m}^{nT+\eta}-\textbf{r}_{t}^{nT+\eta},\textbf{r}_{m}^{nT+ \eta}-\textbf{r}_{t}^{nT+\eta}\rangle+\langle\textbf{v}_{t}^{nT+\eta},\left( \textbf{v}_{t}^{nT+\eta}\right)^{T}\rangle\cdot\textbf{r}_{2}^{2}+2\cdot \langle\textbf{r}_{m}^{nT+\eta}-\textbf{r}_{t}^{nT+\eta},\textbf{v}_{t}^{nT+ \eta}\rangle\cdot\textbf{r}_{2}-\left(c\cdot\textbf{r}_{2}\right)^{2}=0 \tag{11}\]  
where the top right subscript \"\(T\)r \" means matrix transpose, and the symbol \"\(<\), \" signifies inner product operator. From Equations (11) and (12), the two delay time variables can be expressed in Equations (12) and (13) as shown below:  
\[\tau_{1}=\frac{\sqrt{\langle\textbf{r}_{m}^{nT}-\textbf{r}_{t}^{nT}\textbf{ v}_{t}^{nT}\rangle+\left(c^{2}-\textbf{v}_{t}^{nT},\left(\textbf{v}_{t}^{ nT}\right)^{T}\right)\cdot\langle\textbf{r}_{m}^{nT}-\textbf{r}_{t}^{nT}, \textbf{r}_{m}^{nT}-\textbf{r}_{t}^{nT}\rangle}}{c^{2}-\textbf{v}_{t}^{nT}, \left(\textbf{v}_{t}^{nT}\right)^{T}}-\frac{\langle\textbf{r}_{m}^{nT}- \textbf{r}_{t}^{nT},\textbf{v}_{t}^{nT}\rangle}{c^{2}-\textbf{v}_{t}^{nT}, \left(\textbf{v}_{t}^{nT}\right)^{T}} \tag{12}\]  
\[\tau_{2}=\frac{\langle\textbf{r}_{m}^{nT+\eta}-\textbf{r}_{t}^{nT+\eta}, \textbf{v}_{t}^{nT+\eta}\rangle}{c^{2}-\textbf{v}_{t}^{nT},\left(\textbf{v}_ {t}^{nT}\right)^{T}\rangle}+\frac{\sqrt{\left(c^{2}-\textbf{v}_{t}^{nT}, \left(\textbf{v}_{t}^{nT}\right)^{T}\right)\cdot\langle\textbf{r}_{m}^{nT+ \eta}-\textbf{r}_{t}^{nT+\eta},\textbf{r}_{m}^{nT+\eta}-\textbf{r}_{t}^{nT+ \eta}\rangle}}}{c^{2}-\textbf{v}_{t}^{nT},\left(\textbf{v}_{t}^{nT}\right)^{T }} \tag{13}\]  
Then consequently, the total range history at pulse \(nT\) that consists of both downlink propagation and uplink propagation, can be expressed as below:  
\[R^{nT}=R_{down}^{nT}+R_{up}^{nT}=c\cdot\textbf{\tau}_{1}+c\cdot\textbf{\tau }_{2} \tag{14}\]  
If we convert the pulse time \(nT\) into azimuth slow time \(\eta\), then Equation (14) can be transformed to a high-order polynomial series, as shown in Equation (15), to fit the range history and coherent superposition for imaging.