## 4 Simulation and Analysis  
To investigate the impact of the improved second group of parameters (\(P_{\text{sat}}\) and _URA_ on the RAIM integrity risk) the global integrity risk simulation experiment was conducted based on the proposed worst-case integrity risk method. The impact of satellite geometry and the third group of parameters on the integrity loss caused by the improved second group of parameters was discussed. By comparing the global worst-case integrity risk simulation result and integrity risk requirement, the acceptable second group of parameters was provided. We mainly focused on the vertical direction because that is where the geometric diversity of the satellite constellation was the poorest [23], causing a vertical integrity risk larger than for the horizontal. The following error model was used to describe the STD of the \(i\)th satellite observation error [24]:  
\[\sigma_{i}=\sqrt{URA_{i}^{2}+\sigma_{\text{trop},i}^{2}+\sigma_{\text{user},i}^{ 2}} \tag{12}\]where, \(\sigma_{\text{trop},i}\) and \(\sigma_{\text{user},i}\) are the STD tropospheric delay and user related errors, which are defined in [25]. _URA_ is currently recommended to range from 0.5 to 2.5 m for GPS and BDS, respectively [9; 10; 11; 24; 25; 26]. Considering practical application, we mainly discussed the influence of _URA_ from 0.5 to 3.5 m on integrity risk performance. Moreover, the _URA_ of 1.0, 1.5 and 2.0 m were specially chosen for discussing the integrity loss caused by improved _URA_. The \(P_{\text{sat}}\) for GPS satellites ranged from \(10^{-5}\) to \(10^{-4}\) according to [7; 8; 24; 26; 27] and for BDS it was usually from \(10^{-3}\) to \(10^{-4}\)[10; 11; 26; 28]. Three different orders of \(P_{\text{sat}}\) (\(10^{-5}\), \(10^{-4}\) and \(10^{-3}\)) were selected to indicate the GPS and BDS constellation performance. We mainly focused on the integrity risk requirement of \(10^{-7}\) for LPV-200. According to the minimum operational performance standards (MOPS) promulgated by RTCA [29], the third group of parameters--\(P_{\text{fa}}\), \(P_{\text{md}}\) and VAL--were chosen as \(10^{-5}\), \(10^{-3}\) and 35 m, respectively. The vertical integrity risk requirement of LPV-200 was allocated as 9 \(\times\)\(10^{-8}\)[30]. A stricter requirement of vertical integrity risk should have been considered in this contribution because we only accounted for the single-satellite fault mode. We assume that the single-satellite fault mode is allocated half of the vertical integrity risk requirement of LPV-200: \(4.5\times 10^{-8}\). Based on the open-source software MAAST of Stanford University ([[https://gps.stanford.edu/resources/software-tools/maast](https://gps.stanford.edu/resources/software-tools/maast)]([https://gps.stanford.edu/resources/software-tools/maast](https://gps.stanford.edu/resources/software-tools/maast)), accessed on 4 June 2023), the GPS and BDS dual constellations were used for the global simulation experiment. The GPS almanac contained in MAAST was used, including 24 MEO satellites. The BDS almanac was provided by the Test and Assessment Research Center of China Satellite Navigation Office ([[http://www.csno-tarc.cn/system/almanac](http://www.csno-tarc.cn/system/almanac)]([http://www.csno-tarc.cn/system/almanac](http://www.csno-tarc.cn/system/almanac)), accessed on 4 June 2023), including 24 MEO and 3 IGSO satellites. The global simulation experiment parameters are shown in Table 1. The global simulation experiment was based on a 5 \(\times\)\(5^{\circ}\) user grid, for 10 days with a sampling interval of 10 min, which generated 1440 worst-case integrity samples per user grid. The cut-off elevation angle was set to \(10^{\circ}\) to balance observation accuracy and the visible satellite. At each user grid, the 99.5 percentile of the worst-case integrity risk over the simulation duration was selected to show the worst-case integrity risk performance.