# Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning
## 2 Materials and Methods
### Triple-Frequency Precise Point Positioning Model
The unknown parameters including three-dimensional (3D) coordinates, tropospheric zenith wet delay (ZWD), receiver clock offset and real-valued B1/B2 and B1/B3 phase ambiguity parameters need to be estimated. The number of ambiguity parameters is twice as large as the BDS satellite number. Appropriate stochastic models should be provided for the measurements in the Kalman filter Kalman (1954) applied to BDS triple-frequency PPP. For either code or carrier phase, the ionospheric-free combined observations are a linear combination of original measurements on two different frequencies. Assuming there is no correlation between the measurements on different frequencies, of different types or from different satellites, the covariance matrix of observations for a BDS satellite is obtained by the law of random error propagation as given below:  
\[Q=\left[ \begin{array}{ccc}(a_{12,1}^{2}+a_{12,2}^{2})\cdot\sigma_{P}^{2}&0& 0\\ 0&(a_{12,1}^{2}+a_{12,2}^{2})\cdot\sigma_{L}^{2}&(a_{12,1}\cdot a_{13,1})\cdot \sigma_{L}^{2}\\ 0&(a_{12,1}\cdot a_{13,1})\cdot\sigma_{L}^{2}&(a_{13,1}^{2}+a_{13,2}^{2})\cdot \sigma_{L}^{2}\\ \end{array} \right] \tag{13}\]  
with \(a_{12,1}=f_{1}^{2}/\left(f_{1}^{2}-f_{2}^{2}\right)\), \(a_{12,2}=f_{2}^{2}/\left(f_{1}^{2}-f_{2}^{2}\right)\), \(a_{13,1}=f_{1}^{2}/\left(f_{1}^{2}-f_{3}^{2}\right)\), and \(a_{13,2}=f_{3}^{2}/\left(f_{1}^{2}-f_{3}^{2}\right)\) where \(\sigma_{P}^{2}\) is the variance of code observations, and \(\sigma_{L}^{2}\) is the variance of carrier phase observations. Regarding the actual variance, it is defined as a function of the satellite elevation angle and initial variance ([PERSON], 1966). The file \"igs08_wwww.atx\" provided by International GNSS Service (IGS) is adopted to correct the phase center offsets (PCOs) of satellite antenna for BDS on B1, B2 and B3 frequencies. Currently, the BDS satellite phase center variations (PCVs) on the three frequencies are still unavailable. In order to keep consistency with the strategies adopted in the BDS precise satellite orbit determination and clock estimation, the PCO and PCV corrections of receiver antenna on GPS L1 and L2 frequencies are simply used for BDS B1 and B2 signals, respectively [8]. In view that there is only a small difference between GPS L2 and BDS B3 frequencies, we simply use the PCO and PCV corrections of receiver antenna on L2 frequency for B3 frequency. The data processing models and strategies for BDS triple-frequency PPP are summarized in Table 1. In addition, Table 1 also details the specific implementation of dynamic models for state vector and stochastic models for measurements [9]. The code and phase observation precision for GPS are usually set to 0.3 and 0.002 m, respectively [31; 32]. Compared with GPS, BDS satellite orbit and clock are at a relatively lower accuracy and BDS measurements exhibit relatively higher noise level [9; 17]. Therefore, the code and phase observation precision for BDS are empirically set to 0.6 and 0.004 m, respectively. [PERSON] et al. [9] adopted same values for the precision of BDS observations. The stochastic model of measurements for BDS triple-frequency PPP needs further investigation.