## 2 Materials and Methods
Referring to the existing research [9], we express the transformed clock-free ISL observation equation at time \(\bar{t}\) as  
\[\mathbf{L}_{isl}=\mathbf{F}_{isl}(\delta\mathbf{x}_{o},\delta\mathbf{p}^{s},\delta\mathbf{x}_{ \text{pco\_isl}},\delta\mathbf{x}_{\text{bias\_isl}},\bar{t})+\varepsilon_{isl}, \tag{2}\]  
where \(\mathbf{L}_{isl}\) expresses the transformed clock-free ISL observation, and \(\mathbf{F}_{isl}()\) denotes the functional relationship between the observation and the parameters to be estimated. \(\delta\mathbf{x}_{o}\) is the state parameter related to the orbit of satellite, while \(\delta\mathbf{p}^{s}\) represents the perturbation parameter. \(\delta\mathbf{x}_{\text{pco\_isl}}\) and \(\delta\mathbf{x}_{\text{bias\_isl}}\) are the PCO and hardware delay of the ISL. \(\varepsilon_{isl}\) denotes the noise of the transformed clock-free ISL observations. The principle of SLR technology involves using the ground-based satellite laser ranging system to transmit laser pulses to a target observation satellite. After reflection by the target satellite, these pulses are received by the ground system. According to the round-trip time interval \(\tau\) of the measured laser pulse between the ground station and the target satellite, combined with the speed of light c in vacuum, the distance between the station and the satellite can be calculated as  
\[\rho_{0}=\frac{c\cdot\tau}{2}. \tag{3}\]  
In fact, in SLR observation, there are various error terms in addition to the geometric distance between the satellite and the ground station. Taking into account various errors, the observation equation for SLR can be expressed in detail as follows [26]:  
\[\rho_{0}=\rho_{slr}-\big{(}\Delta\rho_{occ}+\Delta\rho_{\text{tidal }}+\Delta\rho_{\text{bias }}+\Delta\rho_{t}+\Delta\rho_{b}+\Delta\rho_{\text{atm}}+\Delta\rho_{\text{ rel}}+\Delta\rho_{\text{ecc}}^{\text{sat}}+\varepsilon\big{)}, \tag{4}\]  
where \(\rho_{\text{slr}}\) signifies the actual geometric distance between the ground station and the satellite. \(\Delta\rho_{ecc}\) is the eccentricity correction term of the station, adjusting for the ground station's non-central position within the earth's gravitational field. The \(\Delta\rho_{\text{tidal}}\) is the tidal displacement correction, which corrects for the deformation of the earth's shape due to tidal forces exerted by other celestial bodies. In addition, \(\Delta\rho_{\text{bias}}\) and \(\Delta\rho_{t}\) are the corrections addressing systematic biases in distance measurements and the synchronization errors related to the timing at the ground station, respectively. \(\Delta\rho_{b}\) accounts for delays in the ground system's signal transmission time, crucial for calibrating the time delay in sending and receiving laser pulses. \(\Delta\rho_{atm}\) and \(\Delta\rho_{rel}\) express the atmospheric refraction correction and the general relativity correction, respectively. \(\Delta\rho_{\text{ecc}}^{\text{sat}}\) is the satellite eccentricity correction, compensating for the satellite orbital deviations from a perfect circle. Finally, \(\varepsilon\) represents the systematic error and random noise of the ground station. Most of the errors can be corrected with the current model, which will be introduced in the following sections. Then, briefly, the SLR observation equation can be expressed as  
\[\mathbf{L}_{slr}^{bds}=\mathbf{F}_{slr}^{bds}(\delta\mathbf{x}_{o},\delta\mathbf{x}_{slr},\bar {t})+\varepsilon_{slr}, \tag{5}\]  
where \(\mathbf{L}_{slr}^{bds}\) expresses the SLR observation, and \(\mathbf{F}_{slr}^{bds}()\) expresses the functional relationship between the SLR observation and the parameters to be estimated. \(\delta\mathbf{x}_{slr}\) is the parameter related to the SLR station. \(\varepsilon_{slr}\) denotes the noise of the SLR observations. The other parameters have the same meaning as those in Equation (2). The precise orbit determination for BDS-3 satellites can be conducted by combining the ISL and SLR observation equations above.