# New Satellite Selection Approach for GPS/BDS/GLONASS Kinematic Precise Point Positioning
## 3 Establishment of the NSS Model in Kinematic PPP
### The Inheritance of Ambiguity  
With the satellite selection algorithm, the possible problems should also be considered in its application in kinematic PPP. Due to satellite movement and the application of the satellite selection algorithm, it is common that satellites selected in different epochs are not all the same. One satellite selected in the previous epoch may not be selected in the next epoch. This will lead to a circumstance that the positioning process is based on discontinuous observations although the satellite is continuously observed. At this time, if we apply the satellite selection in SPP, there are no problems in the data process because SPP only uses pseudorange observations for positioning. However, when it comes to the kinematic PPP solution, the positioning will be affected by the constantly re-estimated ambiguity because of the application of phase observations. The satellite selection algorithm will cause a poor accuracy and reliability in kinematic PPP. Thus, a method that useful ambiguity information is inherited in the data process is proposed. In the process of this method, only the ambiguity of each satellite at the previous and the next epoch are kept. It is a simple storage operation and its computational costs can be ignored when the reduced computation in the satellite selection algorithm are compared. The main contents are as follows.  
In the process of parameter estimation, the estimated values and variances of float ambiguity of all satellites are saved in a list which is continuously kept. Suppose the related information of satellite \(s\) at epoch \(a-1\) has been saved. From epoch \(a\) to the epoch \(m-1\), satellite \(s\) is not selected but continuously observed. At epoch \(m\), satellite \(s\) is selected again. At this time, we add the previously estimated parameter and variance of ambiguity to the optimal estimation of the previous epoch. The method is not rigorous, but we can use it in actual situations. Taking the undifferenced and uncombined dual-frequency PPP model as an example. Satellite \(s\) is selected at epoch \(a-1\) and epoch \(m\). In addition, it is continuously observed from epoch \(a-1\) to epoch \(m\) while the total number of visible satellites is \(n\) at epoch \(m-1\), then the transformation of the last best estimation (State matrix and Variance matrix) at epoch \(m\) can be shown as follows. Equation (10) is the transformation of the best estimation for the state matrix. Equation (11) is the transformation of the best estimation for the variance matrix.  
\[\begin{split}&\mathbf{X}_{m-1}=[\mathbf{u},cd\bar{t}_{r},\text{ZWD}_{r}, \overbrace{\widehat{t}_{r,1}^{1},\ldots,\widehat{t}_{r,1}^{m-1}}^{\text{Ionospheric parameter}},\underbrace{\mathbf{N}_{r,1}^{1},\ldots,\overline{N}_{r,1}^{m-1}}_{ \text{Arbiguity on $f_{1}$}},\overbrace{\widehat{N}_{r,2}^{1},\ldots,\widehat{N}_{r,2}^{m-1}} ^{\text{Ambiguity on $f_{2}$}}]_{\text{$\chi(3n+2)$}}^{\text{T}}\\ &\rightarrow[\mathbf{u},cd\bar{t}_{r},\text{ZWD}_{r},\widehat{t}_{r, 1}^{1},\ldots,\widehat{t}_{r,1}^{m-1},\overline{N}_{r,1}^{1},\ldots,\widehat{ N}_{r,1}^{m-1},\overline{N}_{r,2}^{1},\ldots,\overline{N}_{r,2}^{m-1}, \overline{N}_{r,1}^{1},\overline{N}_{r,2}^{1}]_{\text{$\chi(3n+4)$}}^{\text{T}} \end{split} \tag{10}\]  
Figure 3: Flowchart of the NSS model.  
\[Q_{m-1}=\left[\begin{array}{ccccc}(\alpha_{\rm X}^{2})_{3\times 3}&&\\ &\ddots&\\ &&&(\alpha_{\rm N}^{2})_{2n\times 2n}\end{array}\right]\rightarrow\left[\begin{array}{ccccc}( \alpha_{\rm X}^{2})_{3\times 3}&&\\ &\ddots&&\\ &&(\alpha_{\rm N}^{2})_{2n\times 2n}&&\\ &&&\alpha_{N_{s,1}}^{2}\\ &&&\alpha_{N_{s,2}}^{2}\end{array}\right], \tag{11}\]  
where \(\mathbf{X}_{m-1}\) is the best estimation for the state vector at epoch \(m-1\); \(Q_{m-1}\) is the best estimation for the variance vector at epoch \(m-1\). \(\sigma\) is the standard deviation of the estimated parameter.