## 2 Elevation-Dependent Modeling of Satellite-Induced Code Bias
### Improved SICB Modeling  
Actually, we do not need to compute the float phase ambiguities. In this study, the between-elevation single-difference (BESD) is used to remove the invariable integer phase ambiguities and constant hardware delay terms so that the SICB modeling is free of the effects of MP combination inconsistency caused by the cycle slips and different tracking of a satellite at different stations. Elevation-dependent piecewise linear modeling is employed for SICB [5]. To describe more precise SICB variations, a denser elevation node separation of 1\({}^{\circ}\), rather than the 5\({}^{\circ}\) or 10\({}^{\circ}\) adopted by previous studies, is used. The elevation node separation of 1\({}^{\circ}\) is empirically determined. If the separation  
Figure 4: MP combination time series of B1 signal at 800 consecutive epochs for BDS-2 MEO satellite C14 at stations ASCG and NKLG on 19 August 2016. Figure 5: Dependence of MP combination time series of B1 signal on elevations for BDS-2 MEO satellite C14 at stations ASCG and NKLG on 19 August 2016.  
is oversized, the accuracy of the developed model will degrade, because a lot of details about SICB variations are missing. If the separation is undersized, the errors caused by multipath effects and measurement noises will be enlarged, since the BESD processing is conducted. In view of the fact that the MP values at integer elevation angles cannot usually be obtained, the epochs with float elevation angles nearest to the respective integers are first singled out. The difference between the float elevation angle and its nearest integer should be smaller than 0.1\({}^{\circ}\) in this contribution. Assuming that the SICB linearly changes over elevation angles for a very small elevation range, the elevation-differenced (ED) SICB on the \(i\)th frequency for satellite \(s\) between two consecutive integer elevation angles \(E_{k+1}\) and \(E_{k}\) at station \(r\) can be computed as:  
\[\Delta\beta_{i}(E_{k+1},E_{k})_{r}^{s}=\left[\mathrm{MP}_{i}(t_{k+1},E_{f,k+1} )_{r}^{s}-\mathrm{MP}_{i}(t_{k},E_{f,k})_{r}^{s}\right]/(E_{f,k+1}-E_{f,k}) \tag{5}\]  
where \(\Delta\) stands for the BESD processing, and \(E_{f,k}\) and \(E_{f,k+1}\) are the corresponding float elevation angles at epochs \(t_{k}\) and \(t_{k+1}\), respectively. It is assumed that no cycle slips occur between epochs \(t_{k}\) and \(t_{k+1}\). Assuming there are a total of \(n\) sets of ED SICB estimates on the \(i\)th frequency for satellite \(s\) between integer elevation angles \(E_{k+1}\) and \(E_{k}\) for all the selected stations and days, the average processing can be carried out to mitigate the effects of multipath errors and measurement noises, that is:  
\[\Delta\beta_{i}(E_{k+1},E_{k})^{s}=\left(\sum\limits_{r=1}^{n}\Delta\beta_{i} (E_{k+1},E_{k})_{r}^{s}\right)/n \tag{6}\]  
According to Equation (6), we can only obtain a series of precise ED SICB estimates. Assuming that the SICB at the elevation angle 45\({}^{\circ}\) (\(E_{k0}=45^{\circ}\)) on the \(i\)th frequency for satellite \(s\) is \(d\), the SICB estimates at an integer elevation angle \(E_{k}\) can be expressed as:  
\[\left\{ \begin{array}{rl}\beta_{i}(E_{k})^{s}=d_{i}^{s}+\sum\limits_{p=k_{0} +1}^{k}\Delta\beta_{i}(E_{p},E_{p-1})^{s}&E_{k}>45^{\circ}\\ \beta_{i}(E_{k})^{s}=d_{i}^{s}&E_{k}=45^{\circ}\\ \beta_{i}(E_{k})^{s}=d_{i}^{s}-\sum\limits_{p=k+1}^{k_{0}}\Delta\beta_{i}(E_{ p},E_{p-1})^{s}&E_{k}<45^{\circ}\end{array} \right. \tag{7}\]  
For the purpose of fixing the absolute level of the SICB model, a zero-mean restriction is introduced for the model values of all used MP combinations [5]. Assuming that there is a total of \(n\) sets of MP combinations for all the selected stations and days on the \(i\)th frequency from satellite \(s\), the following condition is introduced:  
\[\sum\limits_{p=1}^{n}\beta_{i}(E_{k,p})^{s}=0 \tag{8}\]  
With Equation (8), the numerical value of the unknown parameter \(d\) can be solved. The absolute SICB model can be constructed with the solved parameter \(d\).