## 2 Materials and Methods  
For the ISL measurements, a wireless link for communication and ranging between satellites, is described by a dual one-way ranging model that follows a time division multiple access (TDMA) scheme. The SLR measurements refer to the ranging between the SLR station on the ground and the satellites of BDS-3. The orbit determination for BDS-3 satellites can be achieved by integrating the SLR measurement with the ISL measurement. Taking three BDS-3 satellites and two SLR ground stations as an example, Figure 1 illustrates the schematic diagram of the orbit determination for BDS-3 satellites based on SLR and ISL observations. As mentioned above, the ISL measurements employ a dual one-way ranging measurement approach based on a time division multiple access scheme. The time difference between the forward and backward observations is less than 3 s. As a consequence, it is essential to convert the dual one-way measurements taken at different times to a common epoch. By synchronizing the dual one-way measurements to a common epoch, clock-free and geometry-free measurements can be derived through subtraction and addition of the synchronized dual one-way measurement, respectively. The former, known as the clock-free measurement, is usually used in the AOD due to its properties of being unaffected by clock error and only including the geometric distance [27]. Herein, we present the ISL observation equation between satellite A and B after transformation to the common time \(\bar{t}\) without derivation as Equation (1) [9]:  
\[\left\{ \begin{array}{l}P_{\text{AB}}(\bar{t})=P_{\text{AB}}(t_{1})+\Delta P_ {\text{AB}}\\ =\rho_{\text{AB}}(\bar{t},\bar{t})+\text{c}\cdot\left[dt_{\text{B}}(\bar{t})-dt _{A}(\bar{t})\right]+c\left(\delta^{\text{rec}}_{B}+\delta^{\text{send}}_{A} \right)+PCO_{\text{AB}}+\varepsilon_{1}\\ P_{\text{BA}}(\bar{t})=P_{\text{BA}}(t_{2})+\Delta P_{\text{BA}}\\ =\rho_{\text{AB}}(\bar{t},\bar{t})+\text{c}\cdot\left[dt_{A}(\bar{t})-dt_{B}( \bar{t})\right]+c\left(\delta^{\text{rec}}_{A}+\delta^{\text{send}}_{B} \right)+PCO_{\text{BA}}+\varepsilon_{2}\end{array} \right., \tag{1}\]  
where \(P_{\text{AB}}(\bar{t})\) and \(P_{\text{BA}}(\bar{t})\) denote the transformed measurements from satellite A to satellite B and from satellite B to satellite A at time \(\bar{t}\), respectively. \(P_{AB}(t_{1})\) and \(P_{BA}(t_{2})\) are the forward and backward ISL measurements at the receiver time. \(\Delta P_{AB}\) and \(\Delta P_{BA}\) represent the correction values of the satellite distance and clock biases between the observed epoch and the target epoch \(\bar{t}\). \(\rho_{AB}(\bar{t},\bar{t})\) is the instantaneous distance between the two satellites. c is the speed of light. Moreover, \(dt_{\text{A}}(\bar{t})\) and \(dt_{\text{B}}(\bar{t})\) are the clock biases of satellite A and B at the same time \(\bar{t}\). \(\delta^{\text{rec}}_{A}\) and \(\delta^{\text{send}}_{A}\) are the hardware delay at the receiver and transmitter of satellite A, respectively. Analogously, \(\delta^{\text{rec}}_{B}\) and \(\delta^{\text{send}}_{B}\) are the hardware delay at the receiver and transmitter of satellite B, respectively. It should be noted that the hardware delay mentioned here refers to the time-lag phenomenon caused by factors such as the characteristics of physical components or signal processing procedures during the  
Figure 1: Schematic diagram of POD based on ISL and SLR observation. transmission and processing of signals in hardware devices. \(PCO_{AB}\) and \(PCO_{BA}\) are the phase center offsets (PCO) of the receiving and transmitting antennas of satellite A and B, respectively. \(\varepsilon_{1}\) and \(\varepsilon_{2}\) represent the corresponding measurement errors. Subtracting two original expressions in the above equation yields a combined observation that only includes the geometric distance and is independent of clock offsets, usually used for orbit determination.