## 2 Materials and Methods
### SLR Validation Method  
SLR stations record the turnaround time \(t\), which represents the interval of a signal emitted from the station and received after being reflected at LRAs of the target satellite. The time of flight multiplied by the speed of light in vacuum \(c\) can be transformed into a range value. Throughout this paper, the SLR residual is defined as the difference \(\rho_{0}\) between the observed range and the computed range \(\rho_{r}^{R}\) obtained from Equations (1) and (2). \[\rho_{0}=\frac{1}{2}\cdot ct-\rho_{r}^{R} \tag{1}\]  
The range value \(\rho_{r}^{R}\) denotes the computed distance between the SLR station and the LRAs of the target satellite. Although SLR measurements are relatively free from bias, some effects of signal propagation need to be accurately modeled to correct the range value. Thus, the corrected SLR range value measurement is given by Equation (2). \[\rho_{r}^{R}=\frac{1}{2}\cdot\left(\rho_{up}(t)+\rho_{down}(t)\right)+\Delta \rho_{rel}+\Delta\rho_{trop}+\varepsilon \tag{2}\]  
In Equation (2), the \(\rho_{up}\) is the distance from the station to the satellite, while the \(\rho_{down}\) is the distance from the satellite to the station. \(\Delta\rho_{rel}\) denotes the time delay related to the relativistic effect [25]. \(\Delta\rho_{trop}\) is the tropospheric delay due to the atmospheric friction. \(\varepsilon\) consists of observation noises and unmodeled errors. Considering the relative motion between the station and the satellite caused by the Earth's rotation and the satellite revolution, the value for the distance of uplink and downlink in an inertial reference frame can be obtained by Equation (3). \[ \begin{array}{c}\rho_{up}(t)=\left\|\mathbf{R}(t-\tau_{down}-\tau_{up}\right)- \mathbf{r}(t-\tau_{down})\right\|\\ \rho_{down}(t)=\left\|\mathbf{R}(t)-\mathbf{r}(t-\tau_{down})\right\|\end{array}  \tag{3}\]  
Figure 2: Monthly statistics of SLR observations. ILRS has been tracking BDS-2 satellites since before 2017 (C08, C10, C11, and C13), while the SLR observations of BDS-3 MEO satellites began in August 2018 (C20 C21, C29, and C30). where \(\mathbf{r}\) and \(\mathbf{R}\) denote the time-dependent position vectors of the SLR station and the satellite, respectively. In this case, \(\rho_{up}\) is not strictly equal to \(\rho_{down}\) and the difference is less than a millimeter [26]. Therefore, we used to assume that the time of uplink \(\tau_{up}\) is approximately equal to that of downlink \(\tau_{down}\), which are half of the time measurement value \(\tau\). Thus, the simplified expressions of Equation (2) are obtained by Equation (4). \[ \begin{array}{l}\rho_{up}=\left\|\mathbf{R}(t-\tau)-\mathbf{r}(t-\
icefrac{{\tau}}{ {2}})\right\|\\ \rho_{down}=\left\|\mathbf{R}(t)-\mathbf{r}(t-\
icefrac{{\tau}}{{2}})\right\|\end{array}  \tag{4}\]  
where the coordinates and the eccentricity correction of the SLR station are determined as part of the SLR contribution (SLRF2014 is used in this paper) to the International Terrestrial Reference Frame (ITRF) [27] and the position of the target satellite is derived from the precise orbit product in the International Celestial Reference Frame (ICRF) with the added satellite-independent LRA center-of-mass correction in a spacecraft frame (SCF). The alternative equations are described by Equation (5). \[ \begin{array}{l}\rho_{up}(t)=\left\|\mathbf{P}_{\text{ITRF}}^{\text{ICRF}}(+ \Delta cECC)\mathbf{R}^{\text{ITRF}}(t-\tau)-\mathbf{P}_{\text{SCF}}^{\text{ICRF}}(+ \Delta cLRA)\mathbf{r}\big{(}t-\frac{\tau}{2}\big{)}\right\|\\ \rho_{down}(t)=\left\|\mathbf{P}_{\text{ITRF}}^{\text{ICRF}}(+\Delta cECC)\mathbf{R} ^{\text{ITRF}}(t)-\mathbf{P}_{\text{SCF}}^{\text{ICRF}}(+\Delta cLRA)\mathbf{r} \big{(}t-\frac{\tau}{2}\big{)}\right\|\end{array}  \tag{5}\]  
where \(\mathbf{P}_{\text{SCF}}^{\text{ICRF}}\) and \(\mathbf{P}_{\text{ITRF}}^{\text{ICRF}}\) are the time-dependent rotation matrices from SCF and ITRF to ICRF, respectively.