## 2 Real-Time BDS-3/BDS-2 PPP Model  
The ionospheric-free (IF) combination is adopted in the real-time BDS-3/BDS-2 PPP model, for the purpose of eliminating first-order effects of ionospheric delay. The overlap signals only include B3 and B1 for BDS-2 and BDS-3 satellites. The B1/B3 dual-frequency IF combined phase and code observables are exhibited as follows:  
\[\left\{ \begin{array}{c}P_{IF}=a_{13,1}\times P_{1}+a_{13,3}\times P_{3}=\rho+ c\mathrm{d}t_{r}-c\mathrm{d}t+T+d_{r,IF}+d_{IF}+\varepsilon p_{IF}\\ \Phi_{IF}=a_{13,1}\times\Phi_{1}+a_{13,3}\times\Phi_{3}=\rho+c\mathrm{d}t_{r}- c\mathrm{d}t+T+N_{IF}+b_{r,IF}+b_{IF}+\varepsilon\Phi_{IF}\end{array} \right. \tag{1}\]  
with  
\[\left\{ \begin{array}{c}N_{IF}=a_{13,1}\times N_{1}+a_{13,3}\times N_{3}\\ d_{r,IF}=a_{13,1}\times d_{r,1}+a_{13,3}\times d_{r,3}\\ d_{IF}=a_{13,1}\times d_{1}+a_{13,3}\times d_{3}\\ b_{r,IF}=a_{13,1}\times b_{r,1}+a_{13,3}\times b_{r,3}\\ b_{IF}=a_{13,1}\times b_{1}+a_{13,3}\times b_{3}\end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{c}a_{13,1}=\frac{f_{1}^{2}}{f_{1}^{2}-f_{3}^{2}}\\ a_{13,3}=-\frac{f_{3}^{2}}{f_{1}^{2}-f_{3}^{2}}\end{array} \right. \tag{3}\]  
where \(P_{i}\) (\(i=1,\ 3\)) denotes the code observations on B1 or B3 frequency, \(\Phi_{i}\) is the corresponding phase observations, \(a_{13,i}\) is the dual-frequency IF combination coefficient, \(f_{i}\) denotes the carrier frequency, \(d_{i}\) and \(d_{r,i}\) are the code hardware delays at the satellite and receiver, respectively, \(N_{i}\) is the phase ambiguity, \(b_{i}\) and \(b_{r,i}\) denote the satellite- and receiver-specific phase hardware delays, respectively, \(P_{IF}\), \(\Phi_{IF}\), \(N_{IF}\), \(d_{r,IF}\), \(d_{IF}\), \(b_{IF}\) and \(b_{IF}\) are the corresponding terms for the dual-frequency IF combination, \(\varepsilon p_{IF}\) and \(\varepsilon\Phi_{IF}\) are the code and phase observation noises, respectively, \(T\) is the slant tropospheric delay, \(c\mathrm{d}t_{r}\) and \(c\mathrm{d}t\) are the physical clock errors of the receiver and the satellite, respectively, and \(\rho\) denotes the geometric distances. For a period of time, all the hardware delay terms are considered to be stable [32]. The corrections of satellite clocks in the real-time scenario, which are contained in the real-time stream CLK93 provided by CNES, are computed with the use of B1/B3 dual-frequency IF combination. Since the hardware delay terms are ignored when generating the real-time stream CLK93, the derived corrections of satellite clocks in the real-time situation include a B1/B3 dual-frequency IF combination of code hardware delays at the satellite, besides the physical clock errors. With the use of real-time precise products, the satellite clocks and orbits are fixed. We estimate the wet component of tropospheric delay from the observations, and employ an a priori model to correct the dry component. After grouping the parameters with similar characteristics, the linearized observation model of real-time BDS-3/BDS-2 PPP reads:  
\[\left\{ \begin{array}{c}\cdot\\ p_{\text{IF}}=\mu\times X+c\dd{t_{r}}+m\times Z+\varepsilon_{p_{IF}}\\ \cdot\\ \phi_{\text{IF}}=\mu\times X+c\dd{t_{r}}+m\times Z+N_{IF}+\varepsilon_{\phi_{IF}} \end{array} \right. \tag{4}\]  
with  
\[\left\{ \begin{array}{c}\cdot\\ c\dd{t_{r}}=c\dd{t_{r}}+d_{r,\text{IF}}\\ \cdot\\ N_{IF}=N_{IF}+b_{r,\text{IF}}+b_{\text{IF}}-d_{r,\text{IF}}-d_{IF}\end{array} \right.