## 2 The Unified Forms of Code Biases
### Inter Frequency Clock Bias  
If we compare with double-frequency, the triple-frequency model add a new carrier signal to the observations. The triple-frequency observation equations are illustrated as follows.  
\[P_{1}^{sys}= \Delta\rho^{sys}+\delta I^{sys}+\varepsilon_{P_{1}^{sys}} \tag{11}\] \[P_{2}^{sys}= \Delta\rho^{sys}+\gamma_{2}^{sys}\delta I^{sys}+\varepsilon_{P_{ 2}^{sys}}\] (12) \[P_{3}^{sys}= \Delta\rho^{sys}+\gamma_{3}^{sys}\delta I^{sys}-c(\beta^{sys}( \gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{sys}}^{r})\] \[+c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{ sys}}^{s})+\varepsilon_{P_{3}^{sys}}\] (13) \[L_{1}^{sys}= \Delta\rho^{sys}-\delta I^{sys}-\lambda_{1}^{sys}\delta N_{1}^{ sys}+\varepsilon_{L_{1}^{sys}}\] (14) \[L_{2}^{sys}= \Delta\rho^{sys}-\gamma_{2}^{sys}\delta I^{sys}-\lambda_{2}^{sys }\delta N_{2}^{sys}+\varepsilon_{L_{2}^{sys}}\] (15) \[L_{3}^{sys}= \Delta\rho^{sys}-\gamma_{3}^{sys}\delta I^{sys}-c(\beta^{sys}( \gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{sys}}^{r})\] \[+c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{ sys}}^{s})-\lambda_{3}^{sys}\delta N_{3}^{sys}+\varepsilon_{L_{3}^{sys}} \tag{16}\]  
where \(\Delta\rho^{sys}=\rho^{sys}+cdt_{P_{ClF2}^{sys}}-cdT_{P_{ClF2}^{sys}}+T_{w}\), \(\delta I^{sys}=I_{1}^{sys}+c\beta^{sys}DCB_{P1P2^{sys}}^{r}-c\beta^{sys}DCB_{ P1P2^{sys}}^{s}\) and \(\delta N_{i}^{sys}=N_{i}^{sys}-f_{i,sys}bias_{i}^{s}\).  
The DCBs in the first and second frequency are eliminated through merging transformation. When the same operation happens in the third frequency, there are residual DCBs in third frequency pseudorang \(P_{3}\) and carrier phase \(L_{3}\). All residual DCBs in third frequency pseudorang \(P_{3}\) can be estimated as a parameter called inter frequency clock bias (IFB), which can be expressed as a function of the DCBs (\(DCB_{P_{1}-P_{2}}\) and \(DCB_{P_{1}-P_{3}}\)). To keep physical significances of both the estimated ionospheric delay and fractional cycle bias (FCB) in the ambiguities consistent in different frequencies, IFB is also introduced into third frequency carrier phase \(L_{3}\).  
\[IFB_{sys}=c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{s}+DCB_{P1P3^{sys} }^{s})-c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{sys}}^{ r}) \tag{17}\]  
According to the type of the terminals (receiver, satellite), IFB can be divided into satellite IFB \(IFB_{sys}^{s}\) and receiver IFB \(IFB_{sys}^{r}\) as follows.  
\[IFB_{sys}^{s}=c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{s}+DCB_{P1P3^{ sys}}^{s}) \tag{18}\]  
\[IFB_{sys}^{r}=c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{P1P2^{sys}}^{r}+DCB_{P1P3^{ sys}}^{r}) \tag{19}\]When IFB is regarded as variable, it can be estimated according to the references [11; 14; 15; 16; 17]. The root mean square (RMS) values of IFBs in GPS and BDS are less than 0.3 mm [18; 32]. They are far less than the RMS value of noise in carrier phase (3 mm) [10]. We ignore time-variant component of IFB and regard IFB as a constant [19]. The satellite IFB needs to be corrected using the DCB files published by IGS organization, and the receiver IFB needs to be estimated together with other unknown parameters. Namely, the estimated IFB in the actual PPP is the receiver component of IFB.  
\[IFB_{sys}^{estimated}=-IFB_{sys}^{r}=-c(\beta^{sys}(\gamma_{3}^{sys}-1)DCB_{ P1P2^{sys}}^{r}+DCB_{P1P3^{sys}}^{r}) \tag{20}\]