## 2 Methods
### Traditional GFIF Phase Time-Variant OSB Model  
The phase time-variant OSB is commonly generated by the ED GFIF model. The GFIF combination between the third and the basic frequency can be written as follows [33]:  
\[\phi_{r,GFIF_{ijk}}^{s}(t)=\phi_{r,IF_{ik}}^{s}(t)-\phi_{r,IF_{ij}}^{s}(t)= \delta_{r,\phi_{IF_{ik}}}(t)-\delta_{\phi_{IF_{ik}}}^{s}(t)+\overline{N}_{r,GFIF _{ijk}}^{s} \tag{7}\]  
where \(\overline{N}_{r,GFIF_{ijk}}^{s}\) denotes the differential IF-phase ambiguity. If there is no cycle slip, and the epoch difference strategy can be used to eliminate ambiguity, which can be written as follows:  
\[\Delta\delta_{r,\phi_{IF_{ik}}}(t,t-1)-\Delta\delta_{\phi_{IF_{ik}}}^{s}(t,t-1 )=\phi_{r,GFIF_{ijk}}^{s}(t)-\phi_{r,GFIF_{ijk}}^{s}(t-1) \tag{8}\]  
where \(\Delta\delta_{r,\phi_{IF_{ik}}}(t,t-1)\) and \(\Delta\delta_{\phi_{IF_{ik}}}^{s}(t,t-1)\) denote the differential IF PIFCB.  
Because the PIFCB of the same satellite at different stations has good consistency, the receiver PIFCB can be combined into satellite PIFCB [33], which reads as follows:  
\[\Delta\delta_{r,\phi_{IF_{ik}}}^{s}(t,t-1)=\phi_{r,GFIF_{ijk}}^{s}(t)-\phi_{r,GFIF _{ijk}}^{s}(t-1) \tag{9}\]  
where \(\Delta\delta_{r,\phi_{IF_{ik}}}^{s}(t,t-1)\) denotes the corresponding IFCB for the paired receiver and satellite.  
Assuming that satellite \(s\) can be tracked by \(m\) common viewing stations in the observational network at epoch \(t\) and \(t-1\), the epoch-differenced PIFCB is expressed as follows:  
\[\left\{\begin{array}{l}\Delta\delta^{s}_{\phi_{IF_{R}}}(t,t-1)=\frac{\sum \limits_{i=1}^{n}\left[\Delta\delta^{s}_{\phi_{IF_{R}}}(t-1)\cdot\omega^{s}_{ \tau}(t,t-1)\right]}{\sum\limits_{i=1}^{n}\left[\omega^{s}_{\tau}(t,t-1) \right]}\\ \omega^{s}_{\tau}(t,t-1)=\left\{\begin{array}{l}\sin(El\epsilon^{s}_{\tau}( t,t-1)),El\epsilon^{s}_{\tau}(t,t-1)<40^{\circ}\\ 1,El\epsilon^{s}_{\tau}(t,t-1)\geq 40^{\circ}\end{array}\right.\end{array}\right. \tag{10}\]  
where \(El\epsilon^{s}_{\tau}(t,t-1)\) represents the average elevation at epoch \(t\) and \(t-1\). The PIFCB can be accumulated through epoch-differenced values, as shown below:  
\[\delta^{s}_{\phi_{IF_{R}}}(t)=\delta^{s}_{\phi_{IF_{R}}}(t)+\sum\limits_{t=t_{ 0}+1}^{t}\Delta\delta^{s}_{\phi_{IF_{R}}}(t,t-1) \tag{11}\]  
with  
\[\delta^{s}_{\phi_{IF_{R}}}(t)=B^{s}_{IF_{R}}(t)-B^{s}_{IF_{R}}(t) \tag{12}\]  
where \(t_{0}\) is the initial epoch time and the initial undifferenced PIFCB, \(\delta^{s}_{\phi_{IF_{R}}}(t_{0})\), is usually set to zero. \(\Delta\delta^{s}_{\phi_{IF_{R}}}(t,t-1)\) is the ED IFCB between epoch \(t\) and \(t-1\). \(\delta^{s}_{\phi_{IF_{R}}}(t)\) can be accumulated by the \(\Delta\delta^{s}_{\phi_{IF_{R}}}(t,t-1)\) from the initial epoch time. The \(\delta^{s}_{\phi_{IF_{R}}}(t)\) is the IF-combined PIFCB, which is difficult to be corrected in the complex PPP models such as non-basic frequency PPP, undifferenced PPP, MFPPP, and so on. According to the SINEX_BIAS Version 1.0 [34], the bias in the form of OSB type can better adapt to multi-frequency observations and the complex PPP algorithms. Combining (6) and (12), the IF-combined PIFCB can be transformed into the phase time-variant OSB, which can be expressed as follows.  
\[\delta^{s}_{\phi_{R}}(t)=\beta^{-1}_{ik}\cdot\delta^{s}_{\phi_{IF_{R}}}(t) \tag{13}\]  
The phase time-variant OSB for basic and third frequency can generated by using Equations (6) and (13).