## 2 Methods
### Undifferenced Mixed OSB Model Coupled with Satellite Clock Offsets  
Through the comprehensive analysis of the above calculation process, it can be found that the ED GFIF model has three major flaws. The first is that it neglects the receiver PIFCB. It assumes that the PIFCB of the same satellite at different receivers has good consistency and the receiver PIFCB can be combined into satellite PIFCB. The second is that it neglects the relationship between two adjacent epochs because it obtains PIFCB by averaging and accumulating. The third and most important is the MFPPP need to correct both pseudorange OSB and PIFCB, which makes the client algorithms more complex. For traditional methods, two bias corrections are required to achieve MFPPP, which makes the client algorithms more complex. To avoid the above defects, the undifferenced pseudorange and phase time-variant mixed OSB model coupled with satellite clock offsets can be expressed as follows:  
\[\left\{ \begin{array}{l}p^{s}_{r,IF_{R}}(t)=\rho^{s}_{r}(t)+dt_{r,IF_{R}}(t) -dt^{s}_{IF_{R}}(t)-\delta^{s}_{IF_{R}}(t)+T^{s}_{r}(t)+\bar{\tau}_{p_{IF_{R}} }(t)\\ \phi^{s}_{r,IF_{R}}(t)=\rho^{s}_{r}(t)+dt_{r,IF_{R}}(t)-dt^{s}_{IF_{R}}(t)- \delta^{s}_{IF_{R}}(t)+T^{s}_{r}(t)+\bar{N}^{s}_{r,IF_{R}}+\epsilon_{\phi_{IF _{R}}}(t)\end{array} \right. \tag{14}\]  
where \(\delta^{s}_{IF_{R}}(t)\) is the satellite pseudorange and phase time-variant mixed IFCB. The IGS legacy clocks are fixed throughout the undifferenced mixed OSB model; hence, we further rewrite (14) by introducing (4) as follows:  
\[\left\{ \begin{array}{l}p^{s}_{r,IF_{R}}(t)+d\bar{t}^{s}_{IF_{R}}(t)=\rho^{s} _{r}(t)+d\bar{t}_{r,IF_{R}}(t)-\delta^{s}_{IF_{R}}(t)+T^{s}_{r}(t)+\bar{\tau}_ {p_{IF_{R}}}(t)\\ \phi^{s}_{r,IF_{R}}(t)+d\bar{t}^{s}_{IF_{R}}(t)=\rho^{s}_{r}(t)+d\bar{t}_{r,IF _{R}}(t)-\delta^{s}_{IF_{R}}(t)+T^{s}_{r}(t)+\bar{N}^{s}_{r,IF_{R}}+\epsilon_{ \phi_{IF_{R}}}(t)\end{array} \right. \tag{15}\]where \(d\overline{t}_{I\bar{F}_{ij}}^{s}(t)\) is IGS legacy clock offsets and \(d\overline{t}_{r,I\bar{F}_{ik}}(t)\) is the recombined receiver clock offsets, which absorb the reference clock \(dt_{ref}\). Obviously, there is a rank-deficient problem between \(d\overline{t}_{r,I\bar{F}_{ik}}(t)\) and \(\delta_{I\bar{F}_{ik}}^{s}(t)\). To avoid the negative impact of receiver hardware delay on the satellite IFCB estimation, the zero-mean condition (ZMC) for a constellation is used to separate the receiver and satellite IFCB. Naturally, there are two technical routes to obtain satellite IFCB according to whether it is with or without pseudorange OSB correction for basic frequency. If the basic frequency observations are corrected by pseudorange OSB, the third satellite IFCB will not contain satellite pseudorange OSB, which can be expressed as follows. \[\delta_{M_{I\bar{F}_{ik}}}^{s}(t)=\beta_{ik}\cdot(b_{k}^{s}-b_{l}^{s})+B_{I \bar{F}_{ik}}^{s}(t)-B_{I\bar{F}_{ij}}^{s}(t)=\delta_{p_{I\bar{F}_{ik}}}^{s}+ \delta_{\bar{p}_{I\bar{F}_{ik}}}^{s}(t) \tag{16}\]  
The IF-combined mixed IFCB \(\delta_{M_{I\bar{F}_{ik}}}^{s}(t)\) can be further converted into OSB type, which is expressed as follows. \[\left\{ \begin{array}{c}\delta_{M_{I\bar{F}_{ik}}}^{s}(t)=\delta_{p_{i}}= \beta_{ij}\cdot\left(b_{i}^{s}-b_{j}^{s}\right)\\ \delta_{M_{i}}^{s}(t)=\delta_{p_{j}}=-\alpha_{ij}\cdot\left(b_{i}^{s}-b_{j}^{ s}\right)\\ \delta_{M_{k}}^{s}(t)=b_{k}^{s}-b_{i}^{s}\\ +\alpha_{ij}\cdot\left(1-\mu_{k}/\mu_{i}\right)\cdot B_{i}^{s}(t)-\beta_{ij} \cdot\left(\mu_{k}-1\right)\cdot B_{j}^{s}(t)-B_{k}^{s}(t)\\ =\delta_{p_{k}}+\delta_{\phi_{k}}(t)=\beta_{ik}^{-1}\cdot\delta_{I\bar{F}_{ik} }^{s}(t)\end{array} \right\}\text{basic frequency pair} \tag{17}\]  
If the basic frequency observations are not corrected by pseudorange OSB, the estimated mixed IFCB will absorb satellite pseudorange OSB. Hence, the mixed IFCB for the third frequency can be given as follows.