## 3 Results
### Impact of CNCORS on BDS/GPS Joint POD
#### 3.1.4. Orbit Dilution of Precision Analysis  
Based on the positioning dilution of precision (PDOP) theory [32], the orbit dilution of precision (ODOP) is used to quantitatively analyze the distribution of ground stations to verify the influence of the CNCORS on the BeiDou POD [12]. Various error corrections are ignored to illustrate the ODOP computation process; thus, the following is the observation equation [33]:  
\[\rho_{k}=\left[\left(x_{k}{}^{j}-X_{ik}\right)^{2}+\left(y_{k}{}^{j}-Y_{ik} \right)^{2}+\left(z_{k}{}^{j}-Z_{ik}\right)^{2}\right]^{1/2}+\varepsilon_{k} \tag{1}\]  
In the equation, \(\rho_{k}\) is the distance measurement; \(t_{k}\) is the observation epoch; \((X_{ik},Y_{ik},Z_{ik})\), \((x_{k}{}^{j},y_{k}{}^{j},z_{k}{}^{j})\) are the position vectors of satellite j and observation station \(i\) in the epoch \(k\), respectively; and \(\varepsilon_{k}\) is the measurement error. The above equation is rewritten as the error equation and linearized as [34; 35; 36]:  
\[\Delta\rho_{k}=H_{k}\Delta X+\varepsilon_{k} \tag{2}\]  
where:  
\[H_{k}=\left[ \begin{array}{cc}\frac{x_{k}{}^{j}-X_{ik}}{R_{k}{}^{j}}&\frac{y_ {k}{}^{j}-Y_{ik}}{R_{k}{}^{j}}&\frac{z_{k}{}^{j}-Z_{ik}}{R_{k}{}^{j}}\\ \end{array} \right]=\frac{1}{R_{ik}{}^{j}}\left[ \begin{array}{cc}\Delta x_{ik }{}^{j}&\Delta y_{ik}{}^{j}&\Delta z_{ik}{}^{j}\\ \end{array} \right]\]  
According to Equation (2), the correction of the satellite position can be solved by using the least squares adjustment: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**PRN**} & \multirow{2}{*}{**Result**} & \multirow{2}{*}{**Normal**} & \multicolumn{2}{c}{**MEAN**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**RMS**} \\ \cline{3-10}  & & & **S1** & **S2** & **S1** & **S2** & **S1** & **S2** \\ \hline \multirow{3}{*}{C01} & C01 & 69 & \(-\)63.7 & \(-\)63.1 & 17.0 & 17.7 & 65.9 & 65.6 \\  & C08 & 20 & 1.0 & 0.4 & 1.4 & 2.4 & 1.7 & 2.4 \\ \cline{1-1}  & C10 & 33 & \(-\)0.3 & \(-\)3.1 & 5.3 & 5.4 & 5.3 & 6.2 \\ \cline{1-1}  & C11 & 65 & \(-\)1.8 & \(-\)1.4 & 2.8 & 3.2 & 3.3 & 3.5 \\ \hline \end{tabular}
\end{table} 
Table 6: SLR assessment between strategies 1 and 2 (unit: cm). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Satellite Type**} & \multirow{2}{*}{**PRN**} & \multicolumn{2}{c}{**Along-Track**} & \multicolumn{2}{c}{**Cross-Track**} & \multicolumn{2}{c}{**Radial**} & \multicolumn{2}{c}{**3D**} \\ \cline{3-10}  & & **S1** & **S2** & **S1** & **S2** & **S1** & **S2** & **S1** & **S2** \\ \hline \multirow{3}{*}{IGSO} & C15 & 16.6 & 12.3 & 16.3 & 15.2 & 5.1 & 5.5 & 24.0 & 23.1 \\  & **mean** & **14.0** & **13.6** & **19.7** & **17.8** & **7.4** & **8.4** & **25.7** & **25.0** \\ \hline \multirow{3}{*}{MEO} & C11 & 10.2 & 8.1 & 5.9 & 5.5 & 3.9 & 3.1 & 12.4 & 9.9 \\  & C12 & 10.8 & 8.5 & 6.2 & 5.9 & 4.2 & 3.7 & 13.2 & 11.2 \\ \cline{1-1}  & C14 & 8.3 & 8.5 & 9.9 & 9.2 & 3.8 & 4.4 & 13.7 & 11.8 \\ \cline{1-1}  & **mean** & **9.8** & **8.4** & **7.3** & **6.9** & **4.0** & **3.7** & **13.1** & **11.0** \\ \hline \end{tabular}
\end{table} 
Table 5: _Cont._\[\Delta X=\left(H^{\text{T}}H\right)^{-1}H^{\text{T}}\Delta\rho_{k} \tag{3}\]  
The following is the orbit determination error of Equation (3) [32]:  
\[\sigma_{r}=\left(\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}\right)^{ \frac{1}{2}}=\left[tr\left(H^{\text{T}}H\right)^{-1}\right]^{\frac{1}{2}}\cdot \partial_{0} \tag{4}\]  
\[ODOP=\left[tr\left(H^{\text{T}}H\right)^{-1}\right]^{\frac{1}{2}} \tag{5}\]  
In Equation (4), \(\partial_{0}\) is the standard error of unit weight, and \(tr\) is the trace of the matrix. ODOP is closely related to the spatial distribution of the ground stations. Usually, a smaller ODOP value and standard error of the unit weight yields a higher orbit determination accuracy. Given that the actual observations cannot reach the theoretical observations, using the theoretical satellite visible arc of the tracking station to compute the ODOP value is improper.