# Impact Analysis of Satellite Geometry Variation on ARAIM Integrity Risk over Exposure Interval
## 4 Experiment and Analysis
### Tightness Analysis of Integrity Risk Evaluate Results  
In this section, we will analyze the bounding tightness of the proposed method for the reference. Bounding tightness is a critical characteristic for assessing the performance of an integrity risk-evaluation method. Firstly, it necessitates that the evaluation results are more conservative than the reference results. Secondly, the closer the evaluation results are to the reference, the better the bounding tightness. The purpose of this analysis is to prove the proposed method's effectiveness in verifying the performance of ARAIM. If the evaluation results of the proposed method are consistently lower than the reference, the subsequent performance verification will be too optimistic. Conversely, if they are significantly higher than the reference, it would lead to an excessively conservative performance verification. We chose the MAAST for ARAIM Version 0.4 software (from Stanford University in Stanford, CA, USA) as the simulation platform, and took the Beijing site as an example to show the relationship between the evaluated results obtained from the proposed method and the reference. It should be noted that the reference was obtained by MC simulation based on (1). Satellite observation error in MC simulation follows the first-order Gaussian Markov model, and the specific model setting can be referred to in [15]. Unfortunately, a full numerical evaluation based on MC simulation would be too computationally taxing for an \(P_{\text{HMI,REQ}}\) of \(10^{-7}\) over exposure interval. For instance, considering the RNP 0.1 phase, this would require the simulation of a Gaussian Markov process for at least \(10^{7}\) h, which is more than a thousand years. This limitation is the reason that MC simulation is not directly employed to analyze the performance of the ARAIM. In the tightness analysis simulations, we implemented two settings to mitigate the time-consuming simulations: Firstly, we only tested the Galileo constellation fault situation, as it is the most crucial fault mode impacting the integrity performance of the ARAIM. Secondly, we artificially set VAL to 15 m and HAL to 20 m. This was performed to avoid a considerable number of invalid MC simulations caused by extremely low integrity risk. Our tests mainly include two situations: LPV-200 for V-ARAIM and RNP 0.1 flight phase for H-ARAIM. The total simulation duration for both situations was 10 days, including the LPV-200 approach for 5760 times and the RNP 0.1 phase simulation for 960 times. It should be noted that in the RNP 0.1 phase simulation, a sliding window of 3600 s with the step of 4 s was utilized to increase the test samples. The number of MC simulations was \(10^{5}\), which is significantly higher than the required number of MC simulations converted from \(P_{\text{HMI,REQ}}/10^{-4}\). It should be noted that \(10^{-4}\) represents the prior probability of the Galileo constellation fault, and the calculation is based on the R\({}_{\text{const}}\) and MFD. We have collected four types of integrity risk results: the integrity risk reference derived from MC simulation, the minimum and maximum integrity risks within the exposure interval obtained from NES-based ARAIM, and the actual integrity risk obtained from the proposed method. Figures 3 and 4 illustrate the evaluation tightness of the proposed method. \(\beta\) = 10 and \(\beta\) = 1000 were utilized to represent the cases with weak and strong time correlations of positioning errors, respectively. In cases of the low error correlation, as indicated in Figure 3,