# Real-Time Phase Bias Estimation for BeiDou Satellites Based on Consideration of Orbit Errors
## 2 Methods
### Effects of Orbit Error on FCB Estimation  
To generate the satellite FCB in PPP analysis, the satellite orbit and clock are fixed, which means that the orbit error is not separable from the IF ambiguity, and the second equation from Equation (1) will become:  
\[L_{IF,i}^{k}=\rho_{i}^{k}+cdt_{i}-cdt^{k}+\widetilde{N}_{IF,i}^{k}+\varepsilon _{IF,i}^{k} \tag{2}\]  
where \(\widetilde{N}_{IF,i}^{k}=N_{IF,i}^{k}+orb_{i}^{k}\). The corresponding IF ambiguity can be expressed using a combination of the wide-lane (WL) and narrow-lane (NL) ambiguities for fixing:  
\[\widetilde{N}_{IF,i}^{k}=\frac{f_{2}^{k}}{f_{1}^{k}+f_{2}^{k}}\lambda_{w}^{k} N_{w,i}^{k}+\lambda_{n}^{k}(N_{n,i}^{k}+\phi_{n,i}-\phi_{n}^{k}+orb_{i}^{k}/ \lambda_{n}^{k}) \tag{3}\]where \(N_{w,i}^{k}\) and \(N_{n,i}^{k}\) are the WL and NL ambiguities, respectively, which have corresponding wavelengths of \(\lambda_{w}^{k}\) and \(\lambda_{n}^{k}\). \(\phi_{n,i}\) and \(\phi_{n}^{k}\) are the receiver and satellite NL UPDs, respectively, and \(orb_{i}^{k}/\lambda_{n}^{k}\) is the NL orbit error. In PPP, the WL ambiguity is usually resolved using the Hatch-Melbourne-Wubbena (HMW) combination [34; 35; 36] of the carrier phase and code observations as follows:  
\[ \begin{array}{ll}\widetilde{N}_{w,i}^{k}&=\left(\frac{L_{1,i}}{\lambda_{1}} -\frac{L_{2,i}}{\lambda_{2}}-\frac{\int_{1}^{k}P_{1,i}^{k}+\int_{2}^{k}P_{2,i} ^{k}}{(\int_{1}^{k}+\int_{2}^{k})\lambda_{n}^{k}}\right)\\ &=N_{w,i}^{k}+\phi_{w,i}-\phi_{w}^{k}\end{array}  \tag{4}\]  
where \(N_{w,i}^{k}\) is the integer WL ambiguity, and \(\phi_{w,i}\) and \(\phi_{w}^{k}\) are the receiver and satellite WL UPDs, respectively. The advantage of this approach is that it is not affected by atmospheric (ionosphere and troposphere) bias or any geometry-related bias. Therefore, it will not be affected by the orbit errors. In high-precision satellite clock estimation, part of the orbital error can be compensated by the satellite clock, while the residual part will appear as code residuals or can be absorbed into the float IF ambiguity in PPP, which then goes into the NL ambiguity, as shown in Equation (3). According to [PERSON] et al. [37], the proportion corresponding to the residual orbital error is governed by the term \(2\cdot|\sin(\theta/2)|\), which is called the scale factor. Here, \(\theta\) represents the maximum value of the included angle between a vector from the satellite to one receiver and another vector from the same satellite to another receiver. The tracking network used in China covers an area with a radius of approximately 1500 km, and we can thus calculate the scale factor; then, based on the 3D orbit error given in Table 1, we can also calculate the effects of the residual orbital error on the NL ambiguity. The resulting values are summarized in Table 2. The results show that the orbit errors of the GPS satellites can be neglected, whereas the orbit errors of all three types of BDS satellites must be estimated along with the NL FCB parameters. After the WL ambiguities are fixed, the float NL ambiguities can then be obtained using Equation (3):  
\[ \begin{array}{ll}\widetilde{N}_{n,i}^{k}&=(\widetilde{N}_{IF,i}^{k}-\frac{ \int_{2}^{k}}{\int_{1}^{k}+\int_{2}^{k}}\lambda_{w}^{k}N_{w,i}^{k})/\lambda_{n }^{k}\\ &=N_{n,i}^{k}+\phi_{n,i}-\phi_{n}^{k}+orb_{i}^{k}/\lambda_{n}^{k}\end{array}  \tag{5}\]  
The orbit error is expressed as:  
\[orb_{i}^{k}=l_{i}^{k}\cdot\Delta x^{k}+m_{i}^{k}\cdot\Delta y^{k}+n_{i}^{k} \cdot\Delta z^{k} \tag{6}\]  
where \(\Delta x^{k}\), \(\Delta y^{k}\), and \(\Delta z^{k}\) are the satellite errors expressed in an Earth-centered and Earth-fixed coordinate system; and \(l_{r}^{k}\), \(m_{r}^{k}\) and \(n_{r}^{k}\) are the corresponding direct cosines of the satellite. To generate the GPS + BDS FCBs in the PPP analysis, the estimated parameters required include two receiver clock parameters, one zenith tropospheric delay (ZTD) parameter, and the undifferenced IF ambiguities.