## 2 Methods
### BDS Ionospheric-Free SPP Model  
The IF pseudorange observation model can be expressed as [9; 27]:  
\[\text{P}_{\text{r,IF}}^{\text{S}}=\text{u}_{\text{r}}^{\text{S}}\cdot\text{x} +\text{c}\cdot\left(\delta\text{t}_{\text{r}}-\delta\text{t}^{\text{S}}\right) +\text{T}_{\text{trop}}+\text{c}\cdot\left(\text{IF}_{\text{bp}}-\text{IF}_{ \text{Bp}}\right)+\epsilon_{\text{r,IF}}^{\text{S}} \tag{8}\]  
where \(\text{P}_{\text{r,IF}}^{\text{S}}\) denotes the IF pseudorange observation; \(\text{u}_{\text{r}}^{\text{S}}\) is the unit vector form receiver r to satellite s; x refers to the vector of the receiver position increments in three-dimensional (3D) directions; \(\delta\text{t}_{\text{r}}\) and \(\delta\text{t}^{\text{S}}\) are receiver and satellite clock offsets, respectively. \(\text{T}_{\text{trop}}\) denotes tropospheric delay; \(\text{IF}_{\text{bp}}\) and \(\text{IF}_{\text{Bp}}\) denote the receiver and satellite IF UCDs, respectively. \(\epsilon_{\text{r,IF}}^{\text{S}}\) is the IF pseudorange observation noises containing multipath and higher-order ionospheric delay. Combing Equations (1), (2) and (8), the BDS IF SPP model can be expressed as:  
\[\text{P}_{\text{r,IF}}^{\text{S}}=\text{u}_{\text{r}}^{\text{S}}\cdot\text{x} +\text{c}\cdot\left(\delta\text{t}_{\text{r}}+\text{IF}_{\text{bp}}+\text{dD} \right)-\text{c}\cdot\delta\text{t}_{\text{IF}}^{\text{S}}+\text{T}_{\text{ trop}}+\epsilon_{\text{r,IF}}^{\text{S}} \tag{9}\]  
Now turn to BDS-2 and BDS-3 combined SPP model, the receiver clock offsets of BDS-2 and BDS-3 absorb the receiver IF UCDs and the datum of satellite clock offsets of each system. \[\left\{ \begin{array}{l}\text{P}_{\text{r,IF}}^{\text{BDS-2}}=\text{u}_{ \text{r}}^{\text{BDS-2}}\cdot\text{x}+\text{c}\cdot\left(\delta\text{t}_{ \text{r}}+\text{IF}_{\text{bp}}^{\text{BDS-2}}+\text{dD}^{\text{BDS-2}}\right) -\text{c}\cdot\delta\text{t}_{\text{IF}}^{\text{BDS-2}}+\text{T}_{\text{ trop}}+\epsilon_{\text{r,IF}}^{\text{BDS-2}}\\ \text{P}_{\text{r,IF}}^{\text{BDS-3}}=\text{u}_{\text{r}}^{\text{BDS-3}}\cdot \text{x}+\text{c}\cdot\left(\delta\text{t}_{\text{r}}+\text{IF}_{\text{bp}}^{ \text{BDS-3}}+\text{dD}^{\text{BDS-3}}\right)-\text{c}\cdot\delta\text{t}_{ \text{IF}}^{\text{BDS-3}}+\text{T}_{\text{ trop}}+\epsilon_{\text{r,IF}}^{\text{BDS-3}}\end{array} \right. \tag{10}\]  
If no differences exist between BDS-2 and BDS-3 for the receiver IF UCDs and the datum of satellite clock offsets, the BDS-2 and BDS-3 combined IF SPP model can be expressed as:  
\[\left\{ \begin{array}{l}\text{P}_{\text{r,IF}}^{\text{BDS-2}}=\text{u}_{ \text{r}}^{\text{BDS-2}}\cdot\text{x}+\text{c}\cdot\left(\delta\text{t}_{ \text{r}}+\text{IF}_{\text{bp}}+\text{dD}\right)-\text{c}\cdot\delta\text{t}_{ \text{IF}}^{\text{BDS-2}}+\text{T}_{\text{ trop}}+\epsilon_{\text{r,IF}}^{\text{BDS-2}}\\ \text{P}_{\text{r,IF}}^{\text{BDS-3}}=\text{u}_{\text{r}}^{\text{BDS-3}}\cdot \text{x}+\text{c}\cdot\left(\delta\text{t}_{\text{r}}+\text{IF}_{\text{bp}}+ \text{dD}\right)-\text{c}\cdot\delta\text{t}_{\text{IF}}^{\text{BDS-3}}+\text{T}_{ \text{ trop}}+\epsilon_{\text{r,IF}}^{\text{BDS-3}}\end{array} \right. \tag{11}\]  
The estimated vector E1 in the IF SPP model includes receiver 3D position increments and the receiver clock offsets can be expressed as:  
\[\text{E1}=\left[ \begin{array}{cc}\text{x}&\delta\text{t}_{\text{IF}}^{\text{ r}}\end{array} \right] \tag{12}\]  
where \(\delta\text{t}_{\text{IF}}^{\text{r}}=\delta\text{t}_{\text{r}}+\text{IF}_{ \text{bp}}+\text{dD}\). If the differences exist between BDS-2 and BDS-3 in the receiver IF UCDs and the datum of satellite clock offsets, the BDS-2 and BDS-3 receiver clock offsets are inconsistent. The receiver clock offsets absorbed the BDS-2 receiver IF UCDs and the datum of BDS-2 satellite clock offsets are considered as the datum of the receiver's clock estimation. The BDS-2 and BDS-3 combined IF SPP model can be expressed as:\[\left\{