# Precise Point Positioning Using Triple GNSS Constellations in Various Modes
## 3 BSSD Post-Processing PPP Models
If a GPS satellite is selected as a reference for all GNSS observables, using Equations (16)-(21), BSSD mathematical models can be written as:  
\[\rho_{G,G}^{ij}+m_{f_{G,G}}^{ij}zpd_{w}+\widetilde{\varepsilon}_{PG_{IF}}^{ij} -\widetilde{P}_{G_{IF}}^{ij}=0 \tag{25}\]  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]  
\[\rho_{G,G}^{ij}+m_{f_{G,G}}^{ij}zpd_{w}+\widetilde{N}_{G_{IF}}^{ij}+\widetilde {\varepsilon}_{\Phi G_{IF}}^{ij}-\widetilde{\Phi}_{G_{IF}}^{ij}=0 \tag{28}\]  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{N}_{EG_{IF}}^{ik}+ \widetilde{\varepsilon}_{\Phi E_{IF}}^{ik}-\widetilde{\Phi}_{EG_{IF}}^{ik}=0 \tag{29}\]  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{N}_{BG_{IF}}^{ih}+ \widetilde{\varepsilon}_{\Phi B_{IF}}^{ih}-\widetilde{\Phi}_{BG_{IF}}^{ih}=0 \tag{30}\]  
where \(\widetilde{N}_{G_{IF}}^{ij}\), \(\widetilde{N}_{EG_{IF}}^{ik}\) and \(\widetilde{N}_{BG_{IF}}^{ih}\) are given by:  
\[\widetilde{N}_{G_{IF}}^{ij}=N_{G_{IF}}^{i}-N_{G_{IF}}^{j}+b_{G\Phi}^{ij}-b_{P}^{ij} \tag{31}\]  
\[\widetilde{N}_{EG_{IF}}^{ik}=N_{G_{IF}}^{i}-N_{E_{IF}}^{k}+b_{r_{\Phi\Phi}}-b_{ r_{\Phi}}+b_{E\Phi}^{k}-b_{\Phi}^{i}+b_{P}^{j}-b_{E}^{k} \tag{32}\]  
\[\widetilde{N}_{BG_{IF}}^{ih}=N_{G_{IF}}^{i}-N_{B_{IF}}^{h}+b_{r_{\Phi\Phi}}-b_{ r_{\Phi}}+b_{B\Phi}^{h}-b_{\Phi}^{i}+b_{P}^{j}-b_{B}^{h} \tag{33}\]  
Similarly, when a Galileo satellite is selected as a reference, using Equations (16)-(21), BSSD mathematical models can be written as:  
\[\rho_{G,E}^{ij}+m_{f_{G,E}}^{ij}zpd_{w}-ISB_{GE}+\widetilde{\varepsilon}_{PG_{ IF}}^{ij}-\widetilde{P}_{GE_{IF}}^{ij}=0 \tag{34}\]  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]0  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]1\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]2  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]3  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]4  
where, \(\widetilde{N}^{lj}_{GE_{IF}}\), \(\widetilde{N}^{lk}_{E_{IF}}\), and \(\widetilde{N}^{lh}_{BE_{IF}}\) are the BSSD non-integer ambiguity parameters lumped to the receiver and satellite hardware delays, which are given by:  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]5  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]6  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]7  
When selecting a BeiDou satellite as a reference, using Equations (16)-(21), BSSD mathematical models can be written as:  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]8  
\[\rho_{E,G}^{ik}+m_{f_{E,G}}^{ik}zpd_{w}+ISB_{GE}+\widetilde{e}_{PE_{IF}}^{ik}- \widetilde{P}_{EG_{IF}}^{ik}=0 \tag{26}\]9  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]0  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]1  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]2  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]3  
where, \(\widehat{N}^{hj}_{GB_{IF}}\), \(\widehat{N}^{hk}_{EB_{IF}}\), and \(\widetilde{N}^{hu}_{B_{IF}}\) are the BSSD non-integer ambiguity parameters lumped to the receiver and satellite hardware delays, which are given by:  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]4  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]5  
\[\rho_{B,G}^{ih}+m_{f_{B,G}}^{ih}zpd_{w}+ISB_{GB}+\widetilde{e}_{PB_{IF}}^{ih}- \widetilde{P}_{BG_{IF}}^{ik}=0 \tag{27}\]6  
Under the assumption that the observations are uncorrelated and the errors are normally distributed with zero mean, the covariance matrix of the un-differenced observations takes the form of a diagonal matrix.