## 2 Data and Methodology
### Transformation of Coordinate Systems  
In order to compare the geometric characteristics of Moon-based platform and a halo orbiter, it is important to obtain the relative motion of the orbiter to the ground, which requires the positions of the sensor and the ground target in the same coordinate system. There are several reference systems related to the processing of DE data of the Earth and Moon, such as the Heliocentric Celestial Reference System (HCRS), the Selenocentric Celestial Reference System (SCRS), and the Geocentric Reference System (GCRS) which is an Earth-Centered Inertial Coordinate System (ECI). The processing of JPL DE430 and coordinate transformations between the above reference systems have been studied in numerous studies and have not been detailed enough to avoid repetition [26; 33]. This study concentrates more on the coordinate transformation related to the halo orbiter and the newly introduced reference system worthy of specification is the L1-centered Syzygy Coordinate System (LSCS). LSCS is an L1-centered rotational right-handed intermediate reference system used for describing the position of the halo orbiter, with the origin located at the L1, the \(x\)-axis directed toward the Moon's barycenter, the \(z\)-axis along the angular momentum direction of L1 around Earth, and the \(y\)-axis following the right-hand rule. Figure 1 shows the sketch of the main reference systems involved in this study and the approximate relative location of L1 and the halo orbit, where the RH, RH, and RS are respectively the position vectors of the ground target point P and the halo orbiter PH. For the sake of clarity, this diagram is not to scale.  
Figure 2 presents the general flowchart of processing procedures and the transformations of relevant reference systems in this study. The remainder of this section is a brief introduction to the required transformations in the program.  
The transformation from LSCS to GCRS:  
\[\mathbf{R}_{H}^{GCRS}=\mathbf{R}_{L}^{GCRS}+\mathbf{R}_{z}^{LSCS}(\theta_{RAAN})\cdot\mathbf{R}_{ x}^{LSCS}(\theta_{m})\cdot\mathbf{R}_{z}^{LSCS}(u)\cdot\mathbf{R}_{H}^{LSCS}, \tag{3}\]  
where \(\theta_{m}\), \(\theta_{RAAN}\) and \(u\) are the lunar orbital inclination, right ascension at the ascending node, and latitude amplitude, respectively. \(\mathbf{R}_{z}^{LSCS}(\theta_{s})\), \(\mathbf{R}_{y}^{LSCS}(-\theta_{m})\), \(\mathbf{R}_{Z}^{LSCS}(u)\) are the rotation matrix for transforming \(R_{SAR}^{LSCS}\) from LSCS to a local GCRS (an intermediate coordinate system with the origin of GCRS translated to L1). However, as mentioned above, the actual position and velocity of the Moon slightly deviate from the case of a uniform circular motion; thus, the angular parameters (\(\theta_{m}\), \(\theta_{RAAN}\), \(u\)) cannot describe the pose of LSCS at every moment like most previous studies. With the use of JPL DE430 to locate the L1, this paper uses the following instantaneous transformation matrix \(\mathbf{R}_{T}\), which varies with the actual location and velocity as well as the irregular deviation to transform the coordinates of the halo orbiter from LSCS to GCRS:  
\[\mathbf{R}_{T}=[\mathbf{I}_{X},\mathbf{I}_{Y},\mathbf{I}_{Z}], \tag{4}\]  
\[\mathbf{I}_{X}=\frac{\mathbf{R}_{L}^{GCRS}}{|\mathbf{R}_{L}^{GCRS}|} \tag{5}\]  
where \(\mathbf{I}_{X},\mathbf{I}_{Y},\mathbf{I}_{Z}\) are column vectors that make up the transformation matrix and \(\mathbf{V}_{L}^{GCRS}\) is the velocity of L1 in GCRS and \(\mathbf{R}_{T}\) is the substitution of the product of the three rotation matrices above.  
Figure 1: The scheme of the geometric model between Earth and halo orbit.  
Figure 2: Flowchart of data processing.