# Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning
## 2 Materials and Methods
### Triple-Frequency Precise Point Positioning Model  
Actually, the IFCB estimate obtained from the above IFCB estimation process is a linear combination of the phase biases on B1, B2 and B3 frequencies, and thus it is termed phase-specific IFCB. In addition to the phase-specific IFCB, the whole IFCB estimate should also contain the code-specific IFCB, which is a linear combination of the constant code hardware biases on B1, B2 and B3 frequencies. The code-specific IFCB can be computed through the triple-frequency code observations, and it can also be expressed as a function of differential code biases (DCBs) [21]. A new type of elevation- and frequency-dependent systematic bias, termed satellite-induced code bias (SICB), was identified to exist in the code observations from BDS-2 satellites [28]. The SICB is ignored during the generation process of DCB products. Therefore, the SICB will affect the estimation of code-specific IFCB when using either triple-frequency code observations or DCBs. Although some literatures, such as [21], claimed that the code-specific IFCB is stable, its stability needs further investigation as the phase-specificIFCB shows variations of several centimeters during a day. The stability of code-specific IFCB will also affect its estimation. For completeness, the results of triple-frequency PPP using \(L_{1}/L_{2}\) and \(L_{1}/L_{3}\) phase observations as well as \(P_{1}/P_{2}\) and \(P_{1}/P_{3}\) code observations are also presented in the next section. The code-specific IFCB is simply calculated using the DCB products. Compared to the results of triple-frequency PPP ignoring \(P_{1}/P_{3}\) code observations, the convergence time is reduced by 1.1\(-\)3.6 min, but their positioning accuracy is found to be at the same level. It is indicated that the positive role that the code observations could play in the improvement for PPP on positioning accuracy and convergence time is limited at present. Despite this, greater benefits from \(P_{1}/P_{3}\) code observations for triple-frequency PPP can be expected after carefully handling the above issues on code-specific IFCB. In the future, we will thoroughly analyze the code-specific IFCB, including the mitigation of SICB, the effect of SICB on its estimation, the stability, the effect of stability on its estimation, and its effect on triple-frequency PPP. After this, the \(P_{1}/P_{3}\) code observations and code-specific IFCB corrections will be included in the triple-frequency PPP model. In this paper, we mainly focus on the phase-specific IFCB. In the triple-frequency PPP model developed here, only the ionospheric-free linear combination of \(P_{1}/P_{2}\) code observations is applied and the \(P_{1}/P_{3}\) ionospheric-free combined code observations are not used. \(P_{k}\) (\(k=1\), 2, 3) is the measured pseudorange on B1, B2 and B3 frequencies. The first-order ionospheric delay errors are removed by utilizing ionospheric-free combined observables. The triple-frequency PPP observation model with consideration of IFCB may be given as follows:  
\[IF(P_{1},P_{2})=\rho+c\overline{\mathrm{d}t}_{r}-c\overline{\mathrm{d}t}_{12}+T \tag{9}\]  
\[IF(L_{1},L_{2})=\rho+c\overline{\mathrm{d}t}_{r}-c\overline{\mathrm{d}t}_{12} +T+\overline{N}_{12} \tag{10}\]  
\[IF(L_{1},L_{3})=\rho+c\overline{\mathrm{d}t}_{r}-c\overline{\mathrm{d}t}_{13} +T+\overline{N}_{13} \tag{11}\]  
\[c\overline{\mathrm{d}t}_{13}=c\overline{\mathrm{d}t}_{12}+\delta \tag{12}\]  
where \(c\overline{\mathrm{d}t}_{12}\) is the precise satellite clock corrections generated and released by Multi-GNSS Experiment (MGEX), \(c\overline{\mathrm{d}t}_{r}\) is the receiver clock offset grouped with code hardware bias at the receiver, and \(\overline{N}\) is the phase ambiguity term that has been grouped with the receiver-dependent uncalibrated phase delay (UPD) and code hardware bias as well as the corresponding satellite-specific ones.