# BeiDou Geostationary Satellite Code Bias Modeling Using Fengyun-3C Onboard Measurements
## 3 Results
### Satellite- and Receiver-Induced Code Biases and Modeling
Afterward, the sky-view grid maps of the corrected IGSO and MEO MP variations for B1 and B2 frequencies were derived, which were believed to be receiver-induced MP effects; these results are shown in Figure 9. In the IGSO and MEO panels of Figure 9, a decreasing trend of receiver-induced MP effects from the signal asc  
Figure 6: Average \(2^{\circ}\times 2^{\circ}\) grid map of each BDS satellite type for the BDS B1 (**top**) and B2 (**bottom**) frequencies (unit: m). Figure 7: Average \(2^{\circ}\times 2^{\circ}\) MP grid maps of GPS for L1 (**left**) and L2 (**right**) frequencies (unit: m). signal descending direction can be observed and is very similar to GPS in Figure 7; it varies by more than 1 m. After subtracting between IGSO and MEO, the trend is significantly reduced. However, some small trend components are still visible. This may be caused by the relatively larger local MP noise or the inaccuracy of the BDS code bias corrections. Based on the analysis, it is quite certain that the static near-field MP pattern exhibits only a frequency-dependent characteristic and is independent on the BDS satellite type. Therefore, when computing the precise values of elevation-dependent BDS-satellite-induced biases using Fengyun-3C onboard measurements, removal of local MP effects must be considered. Supposing that each of the grid elements consists of 4 node points (A, B, C, and D) and the elevation  
Figure 8: The subtraction between different BDS satellite type grids of B1 (**top**) and B2 (**bottom**) frequencies (unit: m). Figure 9: the \(2^{\circ}\times 2^{\circ}\) resolution grid of the static receiver-induced MP pattern in the Fengyun-3C environment for BDS B1 (**top**) and B2 (**bottom**) frequencies when correcting satellite-induced biases using Wanninger and Beer’s values (unit: m)  
and azimuth angles of these points are \((e_{1},a_{1})\), \((e_{1},a_{1}+2^{\circ})\), \((e_{1}+2^{\circ},a_{1}+2^{\circ})\), and \((e_{1}+2^{\circ},a_{1})\) respectively, the MP observables are expressed as  
\[ \begin{array}{l}M\mathrm{P}_{i}^{s}(e,a)=(1-\alpha)(1-\beta)P_{i,A}+\alpha(1- \beta)P_{i,B}\\ \quad+\alpha\beta P_{i,C}+(1-\alpha)\beta P_{i,D}-b_{i}^{s}(e)+\varepsilon_{i} \\ \alpha=\frac{(a-a_{1})}{2};\;\beta=\frac{(e-e_{1})}{2}\\ a_{1}=2[\frac{a}{2}];\;e_{1}=2[\frac{a}{2}]\end{array}  \tag{2}\]  
with  
\[b_{i}^{s}(e)=b_{i,k}^{s}+\frac{e-e_{k}}{e_{k+1}-e_{k}}(b_{i,k+1}^{s}-b_{i,k}^{s}) \tag{3}\]  
where subscript \(i\) and superscript \(s\) are used to denote the different frequencies and groups of BDS satellites respectively, \(P\) is the receiver-induced MP value of each grid point, \(\varepsilon\) is the noise, \([\bullet]\) stands for the down rounding operation, \(b_{i}^{s}(e)\) is biases in BDS satellites and can be expressed with the linear piece-wise model, and \(b_{i,k}^{s}\), \(b_{i,k+1}^{s}\) are the bias values of two adjacent nodes. Hence, using Equations (2) and (3) as the observation equation in a least-squares sense, we again estimated 7 groups of elevation-dependent linear piece-wise node values \(b_{i,k}^{s}\), including GEO C01 to C04 B1, GEO C05 B1, GEO B2, IGSO B1 and B2, and MEO B1 and B2, together with 2 groups of the receiver-induced MP grid values \(P\) for B1 and B2 based on the whole data set. Because of a different behavior observed in C05 MP1 variations as above, a new group was taken into account in this process. Figure 10 shows the final elevation-dependent linear piece-wise models which can be expressed as Equation (3) using derived node values \(b_{i,k}^{s}\) from \(0^{\circ}\) to \(90^{\circ}\) for each group, and Table 1 gives the linear piece-wise model corrections for BDS-satellite-induced biases. Because the distribution of the derived receiver-induced MP grid values \(P\) is similar to that in Figure 9, we do not display it in figures. In contrast to Figure 5, clear elevation-dependent variations can be observed in both BDS GEO B1 and B2 frequencies.