# Assessment of BeiDou-3 and Multi-GNSS Precise Point Positioning Performance
## 2 Multi-GNSS Ionospheric-Free PPP Model  
The ionospheric-free (IF) model is generally utilized in PPP to mitigate the first order ionospheric delay with dual-frequency pseudorange and carrier-phase observations and the higher-order ionospheric delay is absorbed by observation noises, which can be expressed as [12; 13]  
\[\left\{ \begin{array}{l}\mathrm{P_{IF}=R_{r}^{s}+\left(c\!\cdot\!\delta t_{r}+l_ {bp}\right)-\left(c\!\cdot\!\delta t^{s}+l_{B_{P}}\right)+T_{r}^{s}+\varepsilon _{P}}\\ \mathrm{\Phi_{IF}=R_{r}^{s}+\left(c\!\cdot\!\delta t_{r}+l_{bv}\right)-\left(c \!\cdot\!\delta t^{s}+l_{B_{P}}\right)+T_{r}^{s}+\alpha_{1,2}\!\cdot\!\lambda_{1 }\!\cdot\!\mathrm{N_{1}}+\beta_{1,2}\!\cdot\!\lambda_{2}\!\cdot\!\mathrm{N_{2} }+\varepsilon_{\Phi}}\end{array} \right. \tag{1}\]  
with  
\[\left\{ \begin{array}{l}\alpha_{1,2}=\frac{\ell_{1}^{2}}{\ell_{1}^{2}-\ell_{2 }^{2}},\qquad\quad\beta_{1,2}=\frac{-\ell_{2}^{2}}{\ell_{1}^{2}-\ell_{2}^{2}} \\ \mathrm{l_{B_{P}}=\alpha_{1,2}\!\cdot\!B_{P_{1}}+\beta_{1,2}\!\cdot\!B_{P_{2}},l_{bp}=\alpha_{1,2}\!\cdot\!b_{P_{1}}+\beta_{1,2}\!\cdot\!b_{P_{2}}}\\ \mathrm{l_{B_{P}}=\alpha_{1,2}\!\cdot\!B_{\varphi_{1}}+\beta_{1,2}\!\cdot\!b_{ \varphi_{2}},l_{b_{p}}=\alpha_{1,2}\!\cdot\!b_{\varphi_{1}}+\beta_{1,2}\!\cdot \!b_{\varphi_{2}}}\end{array} \right. \tag{2}\]  
where \(\mathrm{P_{IF}}\) is pseudorange IF observation; \(\mathrm{\Phi_{IF}}\) is the IF carrier-phase observation; \(\mathrm{R_{r}^{s}}\) denotes the computed geometrical range; \(\mathrm{T_{r}^{s}}\) is the tropospheric delay;\(\mathrm{B_{P_{1}}}\) and \(\mathrm{b_{P_{1}}}\) are the frequency-dependent satellite and receiver uncalibrated code delays (UCDs); \(\mathrm{B_{\varphi_{1}}}\) and \(\mathrm{b_{\varphi_{1}}}\) are the frequency-dependent satellite and receiver uncalibrated phase delays (UPDs); \(\mathrm{N_{1}}\) and \(\mathrm{N_{2}}\) are the integer phase ambiguity on the frequency \(\mathrm{f_{1}}\) and \(\mathrm{f_{2}}\); \(\lambda_{1}\) and \(\lambda_{2}\) are the wavelengths of the different carrier phase; \(\varepsilon_{P}\) and \(\varepsilon_{\Phi}\) are the pseudorange and carrier phase observation noises and multipath error for pseudorange and carrier observation; and, \(\alpha_{1,2}\) and \(\beta_{1,2}\) are the frequency-dependent factors, which are independent of the satellite number. By convention, the IGS precise satellite clock products are generated using IF observables and the satellite clock offsets absorb the IF combination of satellite UCDs [14], which can be expressed, as follows  
\[c\!\cdot\!\delta t_{IF}^{s}=c\!\cdot\!\delta t^{s}+\alpha_{1,2}\!\cdot\!B_{P_{ 1}}+\beta_{1,2}\!\cdot\!B_{P_{2}} \tag{3}\]  
The influence of the satellite UCDs can be eliminated if we use the observations of the same frequency as the satellite clock product for PPP. Furthermore, we use the observation whose frequency is different from satellite clock offset, the differential code bias (DCB) products could be used to correct the satellite UCDs [12; 15], which can be expressed as  
\[\left\{\begin{array}{l}\mathrm{P_{IF}=R_{r}^{s}+\left(c\!\cdot\!\delta t_{r}+ l_{bp}\right)-c\!\cdot\!\delta t_{IF}^{s}+T_{r}^{s}+\varepsilon_{P}}\\ \mathrm{\Phi_{IF}=R_{r}^{s}+\left(c\!\cdot\!\delta t_{r}+l_{bp}\right)-c\!\cdot \!\delta t_{IF}^{s}+T_{r}^{s}+\lambda_{IF}\!\cdot\!\mathrm{N_{IF}}+\varepsilon_ {\Phi}}\  
In order to investigate the ability of the combined PPP global positioning, the equations of multi-GNSS PPP can be established, as follows:  
\[\left\{