## 2 Data and Methods
### Reference Systems and Transformations  
To analyze the observational geometry and Earth observation coverage in this study, it is necessary to describe the relative position and attitude of the Moon-based platform, satellite platform, and Earth, which are defined in different reference systems. As a result, reference system transformations are crucial. The reference system comprises both coordinate and time systems. Therefore, in studying the geometric characteristics and coverage of MS-BiSAR, it is necessary to bring the Moon-based observatory, satellites, and Earth under a unified time system (UT1) and coordinate system (ITRS). Due to the abundance of reference materials available for time system transformations [23], this paper will not delve into them in detail; instead, this paper primarily focuses on coordinate system transformation. To simplify the calculations and incorporate existing research on potential Moon base sites [18; 24], this paper makes three assumptions, as follows:  
1. The Earth is a sphere with a radius of \(a_{E}=6378\) km;
2. The Moon is a sphere with a radius of \(a_{M}=1737\) km;
3. The Moon-based observatory is located at the center of the lunar near side (\(0^{\circ},0^{\circ}\)). Three types of reference systems are introduced in this paper. The first type is the geodetic reference system, used to describe the positions of Moon-based observatories on the Moon and targets on the Earth. The geographic reference system (GRS) and the selenographic reference system (SRS) are examples of such reference systems in this paper. The second type is a reference system dealing with the parameters related to the celestial body surface, its origin is defined at the center of the celestial body, and its X-axis lies at the intersection of latitude \(0^{\circ}\) and longitude \(0^{\circ}\) degrees on the celestial surface, with its basic plane in the celestial equatorial plane forming a right-handed coordinate system. In this paper, such reference systems include the mean Earth/polar axis reference system (MERS), the principal axis lunar reference system (PALRS), and the ITRS. The third type is the celestial reference system (CRS), with an origin at the center of the celestial body, and its fundamental plane being the mean equator plane containing the celestial's center of epoch J2000.0. Its principal X-axis is pointing to the mean vernal equinox of epoch J2000.0. In this paper, the GCRS with an origin at the center of the Earth and the selenocentric celestial reference system (SCRS) with an origin at the center of the Moon are involved. While some papers describe the methods of reference system transformation in detail, this paper briefly introduces them [25; 26]. Figure 4 shows the key reference system transformation processes involved in this paper. The reference systems transformations start from the Moon-based observatory's coordinate, which is given by the latitude \(\phi_{SRS}\) and longitude \(\phi_{SRS}\) in SRS. It can be expressed as a Cartesian coordinate in the MERS. Figure 4: Flowchart of reference systems transformation. \[ \begin{bmatrix}X_{M}\\ Y_{M}\\ Z_{M}\end{bmatrix} _{MER}= \begin{bmatrix}a_{M}\cos\phi_{SRS}\cos\psi_{SRS}\\ a_{M}\sin\phi_{SRS}\cos\psi_{SRS}\\ a_{M}\sin\psi_{SRS}\end{bmatrix}  \tag{1}\]  
Then, with the following transformation equation, it can be converted into the GCRS:  
\[ \begin{bmatrix}X_{M}\\ Y_{M}\\ Z_{M}\end{bmatrix} _{GCRS}=R_{ICRS}\cdot R_{PALRS} \begin{bmatrix}X_{M}\\ Y_{M}\\ Z_{M}\end{bmatrix} _{MER}+ \begin{bmatrix}X_{Moon}\\ Y_{Moon}\\ Z_{Moon}\end{bmatrix} _{ICRS}- \begin{bmatrix}X_{Earth}\\ Y_{Earth}\\ Z_{Earth}\end{bmatrix} _{ICRS} \tag{2}\]  
where \(\left(X_{Moon},Y_{Moon},Z_{Moon}\right)_{ICRS}\) and \(\left(X_{Earth},Y_{Earth},Z_{Earth}\right)_{ICRS}\) are the ICRS coordinates of the Moon and Earth, respectively. \(R_{ICRS}\) and \(R_{PALRS}\) are the rotation matrices for the coordinate transformation from MERS to PALRS and PALRS to ICRS, respectively.