# Considering Inter-Frequency Clock Bias for BDS Triple-Frequency Precise Point Positioning
## 2 Materials and Methods  
### Estimation of Inter-Frequency Clock Bias  
The carrier phase measurements \(L_{k}\) (\(k\) = 1, 2, 3) collected on the B1, B2 and B3 frequencies may be described as follows:  
\[L_{k}=\rho+c\mathrm{d}t_{r}+(-c\mathrm{d}t+b_{k})+T-I_{1}\cdot f_{1}^{2}/f_{k} ^{2}+N_{k} \tag{1}\]  
where \(\rho\) is the geometric range, \(c\mathrm{d}t_{r}\) is the receiver clock offset, \(c\mathrm{d}t\) is the satellite clock offset included in a parenthesis, \(b_{k}\) is the satellite-, signal- and time-dependent phase bias and the parenthesis shows a combination of \(b_{k}\) and \(c\mathrm{d}t\), \(T\) is the tropospheric delay, \(f\) is the carrier phase frequency, \(I_{1}\) is the ionospheric delay on B1 frequency, and \(N\) is the phase ambiguity term grouped with the stable receiver-dependent phase hardware delay as well as the constant portion of the satellite phase hardware delay. According to [PERSON] et al. (2017), the time-varying portion of phase hardware delay at the satellite may result in the phase bias \(b_{k}\). In the BDS satellite orbit and clock determination, the B1/B2 ionospheric-free combined carrier phase observations are used. The satellite clock offset estimate can be shown below:  
\[c\overline{\mathrm{d}t}_{12}=c\mathrm{d}t-(\frac{f_{1}^{2}}{f_{1}^{2}-f_{2}^{ 2}}\cdot b_{1}-\frac{f_{2}^{2}}{f_{1}^{2}-f_{2}^{2}}\cdot b_{2}) \tag{2}\]  
where \(c\overline{\mathrm{d}t}_{12}\) is the satellite clock offset estimate using B1/B2 observations. As can be seen from Equation (2), the satellite clock offset estimate is different from the corresponding true value and there is an extra linear combination term of phase biases on B1 and B2 frequencies. The satellite clock offset estimated with B1/B3 observations may be different from the estimate using legacy B1/B2 measurements because of the presence of frequency-dependent phase bias \(b_{k}\). From Equations (1) and (2), we can recognize that the difference between two ionospheric-free combinations of B1/B2 and B1/B3 carrier phase observations equals the sum of the difference between B1/B3 and B1/B2 satellite clock estimates and an additional ambiguity term. The ambiguity term is constant over each continuous ambiguity block. The differenced ionospheric-free (\(DIF\)) measurements, namely the triple-frequency carrier phase combination, can be expressed as:  
\[DIF(L_{1},L_{2},L_{3})=IF(L_{1},L_{2})-IF(L_{1},L_{3})=c\overline{\mathrm{d}t} _{13}-c\overline{\mathrm{d}t}_{12}+N_{DIF} \tag{3}\]  
where \(N_{DIF}\) is the phase ambiguity of DIF measurement. Rewriting Equation (3), the following equation can be obtained:  
\[\delta=c\overline{\mathrm{d}t}_{13}-c\overline{\mathrm{d}t}_{12}=DIF(L_{1},L_ {2},L_{3})-N_{DIF} \tag{4}\]  
where \(\delta\) is the inter-frequency clock bias. Based on Equation (4), the IFCB is directly observable from DIF measurements. It is assumed that the similar phase errors, which are frequency dependent and change over time, do not exist in the receiver. For the purpose of fast IFCB estimation, we adopt the epoch-differenced (ED) strategy in this study (K  
where \(E_{r}^{s}(t,t-1)\) is the average satellite elevation angle over the epochs \(t\) and \(t-1\) at station \(r\) for satellite \(s\), and \(w\) is the weight, which is elevation dependent and decomposed into a sinusoidal function and a constant function over different elevation ranges. In general, the measurement noises on a single frequency increase as the satellite elevation angles decrease [17]. Considering the linear combination, similar discussions on measurement noises can be found in the literatures [22; 23; 24]. Therefore, the ED IFCB solutions at the station with lower elevation angles are down-weighted. After comprehensively considering the accuracy and reliability of ED IFCB estimates, an empirical determination of \(40^{\circ}\) for elevation threshold is adopted. Multipath is a significant and largely unmodeled source for satellite positioning errors.