# 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  
The equations representing the raw code and carrier phase observations for the first frequency BDS measurements are as follows [21; 22]:  
\[\left\{ \begin{array}{l}P_{r,1}^{s}(i)=\rho_{r}^{s}(i)+m_{r}^{s}(i)Z_{r}(i)+ dt_{r}(i)-dt^{s}(i)+ION_{r,1}^{s}(i)+b_{r,1}-b_{1}^{s}+\varepsilon_{P}(i)\\ \Phi_{r,1}^{s}(i)=\rho_{r}^{s}(i)+m_{r}^{s}(i)Z_{r}(i)+dt_{r}(i)-dt^{s}(i)-ION_{ r,1}^{s}(i)+\lambda_{1}N_{r,1}^{s}+\delta_{r,1}-\delta_{r,1}^{s}+\varepsilon_{ \Phi}(i)\end{array} \right. \tag{1}\]  
In Equation (1), \(P_{r,1}^{s}(i)\) and \(\Phi_{r,1}^{s}(i)\) respectively denote the code and carrier phase observations from BDS satellite \(s\) to receiver \(r\) on the first frequency at epoch \(i\); \(\rho_{r}^{s}\) is the geometric distance of the transmitted signal from the satellite to the receiver; \(m_{r}^{s}\) is the corresponding troposphere mapping function; \(Z_{r}\) is the zenith tropospheric delay at receiver \(r\); \(dt_{r}\) and \(dt^{s}\) represent the receiver clock offsets and the satellite clock offsets, respectively; \(ION_{r,1}^{s}\) represents the slant ionospheric delay from the receiver to the satellite at the first frequency. \(b_{r,1}\) and \(b_{1}^{s}\) respectively represent the code hardware delays of the receiver and satellite at the first frequency. \(\lambda_{1}\) denotes the carrier phase wavelength of the first frequency. \(N_{r,1}^{s}\) is the carrier phase integer ambiguity at the first frequency. \(\delta_{r,1}\) and \(\delta_{r,1}^{s}\) denote the carrier phase hardware delays for the receiver and the satellite. \(\varepsilon_{p}\) and \(\varepsilon_{\Phi}\) respectively represent the combination of observational noise and errors such as multipath effects in the code and carrier phase observations. Given the availability of high-precision priori known station coordinates and precise ephemeris data released from IGS analysis centers, it becomes feasible to compute the geometric distance at each epoch. The precise satellite clock products \(\hat{dt}^{s}\) provided by IGS analysis centers are derived through the ionosphere-free (IF) combination of code and carrier-phase observations. This means that the precise satellite clock products incorporate the IF combination of the satellite code hardware delays. It can be expressed as:  
\[\hat{dt}^{s}(i)=dt^{s}(i)+\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{,1}^{s}- \frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{2}^{s} \tag{2}\]  
With the priori known precise clock products, Equation (1) can be expressed as:  
\[\left\{ \begin{array}{l}P_{r,1}^{s}(i)-\rho_{r}^{s}(i)+\hat{dt}^{s}(i)=m_{r}^ {s}(i)Z_{r}(i)+dt_{r}(i)+b_{r,1}+ION_{r,1}^{s}(i)+\frac{f_{2}^{2}}{f_{1}^{2}-f _{2}^{2}}\left(b_{,1}^{s}-b_{2}^{s}\right)+\varepsilon_{p}(i)\\ \Phi_{r,1}^{s}(i)-\rho_{r}^{s}(i)+\hat{dt}^{s}(i)=m_{r}^{s}(i)Z_{r}(i)+dt_{r}(i )-ION_{r,1}^{s}(i)+\lambda_{1}N_{r,1}^{s}+\delta_{r,1}-\delta_{r,1}^{s}\\ \hskip 85.358268 pt+\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{2}}b_{,1}^{s}-\frac{f_{2 }^{2}}{f_{1}^{2}-f_{2}^{2}}b_{,2}^{s}+\varepsilon_{\Phi}(i)\end{array} \right. \tag{3}\]  
Equation (3) exhibits a rank deficiency function among the ionosphere, receiver clock offset, phase ambiguity parameters, and code hardware delay. This rank deficiency leads to non-unique solutions of Equation (3) [23; 24]. To simplify the expression, let us assume that \(\Delta P^{s}_{r,1}=P^{s}_{r,1}-\rho^{s}_{r}+\hat{dt}^{s}\) and \(\Delta\Phi^{s}_{r,1}=\Phi^{s}_{r,1}-\rho^{s}_{r}+\hat{dt}^{s}\). After reparameterization and recombination, the resulting equation can be derived as follows:  
\[\left\{ \begin{array}{l}\Delta P^{s}_{r,1}(i)=m^{s}_{r}(i)Z_{r}(i)+\overline{ dt}_{r}(i)+\overline{ION}^{s}_{r,1}(i)+\varepsilon_{p}(i)\\ \Delta\Phi^{s}_{r,1}(i)=m^{s}_{r}(i)Z_{r}(i)+\overline{dt}_{r}(i)-\overline{ION }_{r,1}(i)+\overline{N}^{s}_{r,1}+\varepsilon_{\Phi}(i)\end{array} \right. \tag{4}\]  
with  
\[\left\{