## 2 Methods
### General Observation Models and OSB Concept  
The GNSS original observation equations with the time-variant parts of pseudorange and phase hardware delays can be written as follows [31]:  
\[\left\{ \begin{array}{l}p_{r,j}^{s}(t)=\rho_{r}^{s}(t)+dt_{r}(t)-dt^{s}(t)+b_{ r,j}-b_{j}^{s}+b_{r,j}(t)-b_{j}^{s}(t)+T_{r}^{s}(t)+I_{r,j}^{s}(t)+\varepsilon_{p_{ j}}(t)\\ \phi_{r,j}^{s}(t)=\rho_{r}^{s}(t)+dt_{r}(t)-dt^{s}(t)+B_{r,j}-B_{j}^{s}+B_{r,j}(t )-B_{j}^{s}(t)+T_{r}^{s}(t)-I_{r,j}^{s}(t)+N_{r,j}^{s}+\varepsilon_{\phi_{j}}(t )\\ \end{array} \right. \tag{1}\]  
where \(p_{r,j}^{s}(t)\) and \(\phi_{r,j}^{s}(t)\) are the pseudorange and phase observations on frequency \(j\) in units of meter, respectively; \(\rho_{r}^{s}(t)\) is the geometrical range from satellite \(s\) to receiver \(r\) at epoch \(t\); \(dt_{r}(t)\) and \(dt^{s}(t)\) denote the original receiver and satellite clock offsets; and \(b_{r,j}\) and \(b_{j}^{s}\) represent the receiver and satellite time-invariant uncalibrated code delays (UCDs), while \(b_{r,j}(t)\) and \(b_{j}^{s}(t)\) represent the corresponding time-variant parts; similarly, \(B_{r,j}\) and \(B_{j}^{s}\) depict the time-invariant uncalibrated phase delays (UPDs) and \(B_{r,j}^{s}(t)\) are the time-dependent parts; \(T_{r}^{s}(t)\) depicts the slant tropospheric delay; \(I_{r,j}^{s}(t)\) is the slant ionospheric delay; \(N_{r,j}^{s}\) denotes the integer ambiguity; and \(\varepsilon_{p_{j}}(t)\) and \(\varepsilon_{\phi_{j}}(t)\) are the pseudorange and phase observation noises containing multipath and unmodeled errors. The IGS and iGMAS ACs typically employ dual-frequency undifferenced IF combination observations to generate satellite clock offsets [31; 32]:  
\[\left\{\begin{array}{l}p_{r,I\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{ \bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{ \bar{r}_{\bar{r}_{\bar{r}_{\bar{r}_{\bar{r}}_{\bar{r}}_{\bar{r}}  
The satellite clock offsets absorbed with the hardware delays make the measurement inconsistent at the specific frequency. The satellite clock offsets for the specific frequency can be expressed as follows:  
\[\left\{ \begin{array}{l}d\overline{t}_{i}^{s}(t)=dt_{IF_{ij}}^{s}(t)+\delta_{p_{ i}}+\delta_{\phi_{i}}(t)+dt_{ref}=dt_{IF_{ij}}^{s}(t)+\delta_{M_{i}}^{s}(t)+dt_{ref} \\ d\overline{t}_{j}^{s}(t)=dt_{IF_{ij}}^{s}(t)+\delta_{p_{j}}+\delta_{\phi_{j}}(t) +dt_{ref}=dt_{IF_{ij}}^{s}(t)+\delta_{M_{j}}^{s}(t)+dt_{ref}\\ d\overline{t}_{k}^{s}(t)=dt_{IF_{ij}}^{s}(t)+\delta_{p_{k}}+\delta_{\phi_{k}}(t) +dt_{ref}=dt_{IF_{ij}}^{s}(t)+\delta_{M_{k}}^{s}(t)+dt_{ref}\\ \end{array} \right. \tag{5}\]  
where \(d\overline{t}_{i}^{s}(t)\), \(d\overline{t}_{j}^{s}(t)\), and \(d\overline{t}_{k}^{s}(t)\) denote the satellite clock offsets for the basic \(i\)th, \(j\)th, and the third \(k\)th frequency signals. \(\delta_{p_{i}}\) and \(\delta_{p_{j}}\) denote the pseudorange OSB for the baisc frequency signals; \(\delta_{p_{k}}\) denotes the pseudorange OSB for the third frequency signal; and, in analogy, \(\delta_{\phi_{i}}(t)\), \(\delta_{\phi_{j}}(t)\), and \(\delta_{\phi_{k}}(t)\) are carrier-phase time-variant OSB for the basic and the third frequency. Normally, the definition of satellite IFCB is the bias between the satellite clock offsets for the different frequency, which usually contains CIFCB (also known as DCB or pseudorange OSB) and PIFCB (also known as the phase time-variant OSB). The \(\delta_{p_{k}}\) and \(\delta_{\phi_{k}}(t)\) can be combined into the mixed IFCB with OSB type \(\delta_{M_{k}}^{s}(t)\). Generally, the pseudorange OSB is usually determined by the CCL method. However, the phase time-variant OSB for the basic frequency cannot be calculated by the methods of obtaining pseudorange OSB. As shown in (2), it will be absorbed into the other parameters such as clock offsets and residuals. As for the dual-frequency PPP, the phase time-variant OSB for the basic frequency will not have any negative impact.