# Estimating BDS-3 Satellite Differential Code Biases with the Single-Frequency Uncombined PPP Model
## 2 Methods
### Extraction of Ionospheric Observables Using the SF Uncombined PPP Model
\begin{array}{l}\overline{dt}_{r}(i)=dt_{r}(i)+b_{r,1}\\ \overline{ION}^{s}_{r,1}(i)=ION^{s}_{r,1}(i)+\frac{f_{2}^{s}}{f_{1}^{s}-f_{2} ^{s}}\left(b^{s}_{1}-b^{s}_{2}\right)\\ \overline{N}^{s}_{r,1}=\lambda_{1}N^{s}_{r,1}+\delta_{r,1}-\delta^{s}_{r,1}-b_ {r,1}+\frac{f_{2}^{s}+f_{2}^{s}}{f_{1}^{s}-f_{2}^{s}}b^{s}_{r,1}-\frac{2f_{2}^ {s}}{f_{1}^{s}-f_{2}^{s}}b^{s}_{r,2}\end{array} \right. \tag{5}\]  
In Equation (4), there is still a rank deficiency of size one between the re-parametrized \(\overline{dt}_{r}\), \(\overline{ION}^{s}_{r,1}\) and \(\overline{N}^{s}_{r,1}\), which means that they cannot be estimated independently. Through recombining the biased receiver clock offset parameters into the variation amount of the receiver clock offset of subsequent epochs relative to the first epoch receiver clock offset, and recombining the parameters in Equation (4), the resulting equation is a full-rank SF uncombined PPP model equation:  
\[\left\{ \begin{array}{l}\Delta P^{s}_{r,1}(1)=m^{s}_{r}(1)Z_{r}(1)+\overline{ ION}^{s}_{r,1}(1)+\varepsilon_{p}(1)\\ \Delta\Phi^{s}_{r,1}(1)=m^{s}_{r}(1)Z_{r}(1)-\overline{\overline{ION}}_{r,1} (1)+\overline{N}^{s}_{r,1}+\varepsilon_{\Phi}(1)\\ \Delta P^{s}_{r,1}(i)=m^{s}_{r}(i)Z_{r}(i)+\overline{dt}_{r}(i)+\overline{ ION}^{s}_{r,1}(i)+\varepsilon_{p}(i)\;(i\geq 2)\\ \Delta\Phi^{s}_{r,1}(i)=m^{s}_{r}(i)Z_{r}(i)+\overline{\overline{dt}}_{r}(i)- \overline{\overline{ION}}^{s}_{r,1}(i)+\overline{N}^{s}_{r,1}+\varepsilon_{ \Phi}(i)\;(i\geq 2)\end{array} \right. \tag{6}\]  
with  
\[\left\{ \begin{array}{l}\overline{\overline{dt}_{r}}(i)=\overline{dt}_{r}(i) -\overline{dt}_{r}(1)\\ \overline{ION}^{s}_{r,1}(i)=\overline{ION}^{s}_{r,1}(i)+\overline{dt}_{r}(1)= ION^{s}_{r,1}(i)-\frac{f_{2}^{s}}{f_{1}^{s}-f_{2}^{s}}(b^{s}_{2}-b^{s}_{r,1})+ \overline{dt}_{r}(1)\\ \overline{N}^{s}_{r,1}=\overline{N}^{s}_{r,1}+2\overline{dt}_{r}(1)\end{array} \right. \tag{7}\]  
In Equation (7), \(DCB^{S}_{2,1}=b^{s}_{2}-b^{s}_{,1}\), the estimable parameter \(\overline{\overline{ION}}^{s}_{r,1}\) can be represented as:  
\[\overline{\overline{ION}}^{s}_{r,1}(i)=ION^{s}_{r,1}(i)-\frac{f_{2}^{s}}{f_{1}^ {s}-f_{2}^{s}}DCB^{S}_{2,1}+\overline{dt}_{r}(1) \tag{8}\]  
The estimable ionospheric parameter \(\overline{\overline{ION}}^{s}_{r,1}\), known as the ionospheric observables, encompasses the slant ionospheric delay, satellite DCB and \(\overline{dt}_{r}(1)\). Note that \(\overline{dt}_{r}(1)\) represents the combined value of the first epoch receiver clock offset and the receiver code hardware delay at the first frequency. Due to the expected stability of the receiver code hardware delay under normal environmental conditions [8; 18], \(\overline{dt}_{r}(1)\) is treated as a constant during the daily processing. However, it is essential to be aware that if the data collection is interrupted for any reason, a gap will appear between subsequent and preceding results. This gap occurs because a new receiver clock offset is introduced as the first epoch receiver clock offset in such cases.