# Research on Satellite Selection Strategy for Receiver Autonomous Integrity Monitoring Applications
## 2 Problem Statement
### Impacts of Subset Size on Accuracy and Integrity Performances
#### 2.2.1 Accuracy Performance Improved by Subset Size Growth  
When the accuracy of the pseudo-range measurements is known to differ, it is uneasy to find a common configuration of the satellite geometry that can reach the smallest 2 drms of the least-squares solution. In this situation, assuming the measurement errors are independent and identically distributed is an effective way to investigate the highest accuracy in each subset size. Thus, two simulation scenes are considered to illustrate the accuracy performance improved by subset size growth. The first one brute force searches all the possible ideal satellite geometries above the mask angle of 5 degrees for each subset size under independent and identical measurement error assumption. Then it finds the subset with the highest accuracy, i.e., the smallest 2 drms value in each given subset size. In fact, the high accuracy geometries for different size subsets have similar distributions that the satellites equally spaced on the low elevation bottom of the visible area. However, for the benefit of vertical accuracy performance, several satellites should be at the zenith of the sky [31]. Take the single constellation navigation system, for example, an observation matrix for the ideal subset with \(n\) satellites can be summarized as:  
\[G=\left[ \begin{array}{cccc}cos(\frac{2\pi}{n-a})cos\gamma&sin(\frac{2\pi}{n -a})cos\gamma&sin\gamma&1\\ \vdots&\vdots&\vdots&\vdots\\ cos(2\pi)cos\gamma&sin(2\pi)cos\gamma&sin\gamma&1\\ cos(\alpha_{1})cos\beta_{1}&sin(\alpha_{1})cos\beta_{1}&sin\beta_{1}&1\\ \vdots&\vdots&\vdots&\vdots\\ cos(\alpha_{a})cos\beta_{a}&sin(\alpha_{a})cos\beta_{a}&sin\beta_{a}&1\\ \end{array} \right]n-a \tag{11}\]  
where \(a\) is the satellite number at the zenith and \(\gamma\) is the mask angle. \(\beta_{1}\) to \(\beta_{a}\) are the elevation angles of satellites at the zenith and \(\alpha_{1}\) to \(\alpha_{a}\) are the azimuth angles of the zenith satellites. The minimal 2 drms values for different size subsets can be found in Figure 1a when \(\sigma_{0}=10.5\) m and the \(a\) is set to 1 in this study. It is no doubt that the minimal 2 drms value is decreased when more satellites are considered in the subset, as a better geometry can be structured. However, the improvement becomes less obvious when more satellites have been considered in the former. For example, if we need to reach a 2 drms value of about 30% smaller than the 18.85 m in the 6 satellites subset, we should at least choose 12 satellites. However, if we need to reach a 2 drms value of about 30% better than the 12 satellites subset, at least 24 satellites should be in the subset. In another word, if sufficient satellites have been used to achieve an acceptable accuracy, it is inefficient to improve the navigation accuracy by simply considering more satellites in the subset. The curve of 2 drms reduction to the former subset in Figure 1a also supports this conclusion. The second scene turns to a practical satellite distribution which includes 23 visible satellites from the GPS and the BeiDou satellite navigation system (BDS-2). The differences between pseudo-range measurement errors are taken into consideration and the accuracy varying range for each given size subset is shown in Figure 1b in the form of upper and lower bounds. As illustrated in the figure, the 2 drms value of a given size subset can vary  
Figure 1: (**a**) Minimal 2 drms value and 2 drms reduction to the former for each size subset in the ideal scene; (**b**) Accuracy variation range for each size subset in the practical scene. in a wide range, especially when only a small part of the all-in-view satellites is chosen. Additionally, we can notice that the 6 satellites subset can reach a 2 drms value of 11.92 m if properly chosen, while the randomly chosen subset can maintain a better performance only if the subset size increases to 19 or larger. This means, the properly constructed subset can achieve a significantly better accuracy compared with the randomly chosen subset, even when the later owns more satellites.