# Precise Point Positioning Using Triple GNSS Constellations in Various Modes
## 2 Un-Differenced Post-Processing PPP Models
In addition, BeiDou satellite clock correction includes the effect of the ionosphere-free linear combination of the satellite hardware delays of B1/B2 code [11]. By applying the precise clock products for GPS, Galileo, and BeiDou observations, Equations (1)-(6) will take the following form:  
\[P_{G_{IF}}=\rho_{G}+c[dt_{rG}-dt^{s}_{prec}]+c[adp_{1}-\beta d_{P2}]_{r}+T_{G}+ \varepsilon_{PG_{IF}} \tag{10}\]  
\[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-dt^{s}_{prec}]+c[ad_{E1}-\beta d_{E5a}]_{r}+T_{E}+ \varepsilon_{E_{IF}} \tag{11}\]  
\[P_{B_{IF}}=\rho_{B}+c[dt_{rG}-dt^{s}_{prec}]+c[ad_{B1}-\beta d_{B2}]_{r}+T_{B}+ \varepsilon_{B_{IF}} \tag{12}\]  
\[\Phi_{G_{IF}}=\rho_{G}+\delta t_{rG}-c[dt_{prec}^{s}+[ad_{B1}-\beta d_{E2}]_{r} +c[ad_{B1}-\beta d_{E2}]_{r}-[ac_{B1}-\beta d_{E2}]_{r}+T_{G}+\psi_{G_{G_{IF}}}+ \psi_{G_{G_{IF}}}+\varepsilon c_{G_{IF}} \tag{13}\]  
\[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]0  
\[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]1  
For simplicity, the receiver and satellite hardware delays are written as:  
\[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]2 \[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]3 \[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]4 \[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]5 \[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]6 \[P_{E_{IF}}=\rho_{E}+c[dt_{rG}-GGTO-dt^{s}]+c[\alpha d_{E1}-\beta d_{E5a}]_{r}+c[ \alpha d_{E1}-\beta d_{E5a2}]^{s}+T_{E}+\varepsilon_{E_{IF}} \tag{2}\]7  
In the combined GPS, Galileo and BeiDou un-differenced post-processed PPP solution, the GPS receiver clock error is lumped to the GPS receiver differential code biases. To maintain consistency in the estimation of a common receiver clock offset, this convention is used when combining the ionosphere-free linear combination of GPS L1/L2, Galileo E1/E5a, and BeiDou B1/B2 observations in the post-processed PPP solution. This, however, introduces an additional bias in the Galileo ionosphere-free PPP mathematical model, which represents the difference in the receiver differential code biases of both systems. Such an additional bias is commonly known as the inter-system bias, which is referred to as \(ISB\) in this paper. In our PPP model, the Hopfield tropospheric correction model along with the Vienna mapping function are used to account for the hydrostatic component of the tropospheric delay [27,31]. Other corrections are also applied, including the effect of ocean loading [32,33], Earth tide [28], carrier-phase windup [8,34], Sagnac [35], relativity [7], and satellite and receiver antenna phase-center variations [36]. The noise terms are modeled stochastically using an exponential model, as described in [37].