## 2 Algorithm for MBS
### Division Mode in FBP
Because the range error \(\Delta R(\theta)\) is far less than any range from MBS, to achieve EGT, the range error \(\Delta R\) can be approximately quantified by the equation shown below:  
\[\Delta R(\theta)=\frac{[R(\theta+\Delta\theta)+R(\theta)]\cdot[R(\theta+ \Delta\theta)-R(\theta)]}{R(\theta+\Delta\theta)+R(\theta)}\approx\frac{R( \theta+\Delta\theta)^{2}-R(\theta)^{2}}{2\cdot R(\theta)} \tag{4}\]  
According to the cosines law of triangles, the slant range \(R\) can be re-expressed as follows:  
\[R(\theta)=\sqrt{r^{2}+u^{2}-2r\cdot u\cdot cos\theta},\ u\leq 0.5d \tag{5}\]  
where \(u\) is the length from position of the beam to its initial position, in this paper, the initial position is reckoned as the center of sub-aperture, \(r\) is the length of centerline, and \(d\) is the length of sub-aperture. Therefore, Equation (4) can be reverted as follow. \[\Delta R(\theta)=\frac{cos\theta-cos(\theta+\Delta\theta)}{\sqrt{1-\frac{2u}{ r}\cdot cos\theta+\left(\frac{u}{r}\right)^{2}}}\cdot u \tag{6}\]  
In actual scenario, \(u\) is much smaller than \(r\). Therefore, considering the extreme situation, the maximum value of the range error in Equation (6) can be expressed as shown below:  
\[\left|\Delta R\right|_{max}=\frac{d\cdot\Delta\theta}{2}\approx\frac{d\cdot D} {4r},\ d<2r \tag{7}\]  
where \(D\) is the length of sub-image area in azimuth direction. Equation (7) indicates that the range error will fall within a limit, so long as the product, namely the sub-aperture's length and the sub-image area's length in azimuth direction, remains unchanged. It is generally assumed that the phase error caused by the range error should not be greater than \(\frac{\pi}{4}\); this is to say, the maximum value of the range error should be subject to Equation (8). \[\frac{4\pi}{\lambda}\left|\Delta R\right|_{max}\leq\frac{\pi}{4} \tag{8}\]The maximum range error in Equation (8) will often be less than \(\frac{\lambda}{16}\), and the denominator constant can be substituted with a control factor \(\delta\). With a combination of Equations (7) and (8), the sub-image area's length can be expressed as shown below:  
\[f(i,k)=\int_{-0.5t}^{0.5t}s_{ik}(\eta,\tau(\eta))\cdot exp\left(\frac{j\cdot 4 \pi\cdot R(\eta)}{\lambda}\right)dt \tag{2}\]0  
In Equation (9), the product of the sub-image area's length and sub-aperture's length has a negative correlation with the control factor \(\delta\). Once \(\delta\) becomes too large, the sub-aperture's length will be closer to a pulse, and thus cannot achieve the effect of aperture division and computational cost reduction, so a suitable \(\delta\) should make \(d\) equal to the length of the exponential power of 2, i.e., the sub-aperture's length contains 16, 32, or even 64 pulses. With suitable parameters before being set, the true position showed in Figure 3b still might not project on an existing sampling point of the centerline, shown as a solid red dot; this situation will cause an error expressed as \(dr\). In order to reduce such an error, a step of different multiples interpolation, shown in Box 2 of Figure 2, will be conducted during the imaging process. Theoretically, \(dr\) can be reduced by increasing the interpolation multiple; the higher the interpolation multiple, the smaller the error. However, \(dr\) will accumulate with the superposition of sub-apertures; the more sub-apertures, the greater the error accumulated, which may affect the imaging quality. After the FBP algorithm has been introduced, the next step is to derive the range or phase history of the echo; both are influenced by Earth-Moon motion and the actual location of MBS. For a better description of the processing of echo, the subsequent section introduces a detailed method of range history estimation. In this method, the locations of the MBS during periodic Earth-Moon motion are all taken into consideration.