## 2 Problem Statement  
In the milestone reports of ARAIM TSG, the roadmap of ARAIM development is divide into two steps: Horizontal ARAIM and Vertical ARAIM. The Horizontal ARAIM can be considered as the solution of transitional stage from traditional RAIM to ARAIM, which aims at providing horizontal navigation for en route and terminal approach. The Vertical ARAIM aims at providing LPV-200 for worldwide aircraft landing navigation. The LPV-200 is the last period in the non-precision approach and after that is the precision approach Category I. As the Vertical ARAIM can be considered as the ultimate goal of ARAIM application, we will only consider Vertical ARAIM in this paper (hereinafter referred to as ARAIM) and the ARAIM availability criteria is seen as LPV-200.  
The baseline ARAIM user algorithm is based on snapshot multiple hypothesis solution separation (MHSS). Multiple hypothesis determines the fault modes that need to be monitored to meet the integrity risk with the assist of integrity support message (ISM). Solution separation tests the consistency between the present all-in-view position solution and the fault-tolerant position corresponding to the fault modes determined in multiple hypothesis and test the differences to prompt whether a fault exists. Finally, the LPV-200 criteria, including horizontal protection level (HPL), vertical protection level (VPL), EMT, and vertical accuracy, will be used to evaluate the ARAIM performance.  
The number of subsets and corresponding fault modes probabilities are calculated within the satellite prior fault probability and constellation prior fault probability which is provided by the ISM. However, ISM does not specify which fault modes need to be monitored. This determination must be made by the receiver based on the contents of ISM and the positioning resolution. Different from the traditional RAIM, which can only monitor a single satellite fault, the constellation faults and multiple simultaneous satellite fault can be detected with the assist of ISM. In ARAIM, the constellation faults and satellite faults are independent events in fault modes determination as described in:  
\[\begin{array}{l}P_{event,i}=P_{sat,i}\\ P_{event,Nsat+j}=P_{const,j}\end{array} \tag{1}\]  
where \(P_{sat,i}\) and \(P_{const,j}\) represent the \(i\)th satellite and \(j\)th constellation in ARAIM, respectively, and \(N_{sat}\) is the number of satellites in the positioning calculation.  
A satellite fault is a signal in space fault state which may be caused by erroneous satellite navigation data or anomalous satellite payload events. The constellation fault is a common cause which may originate at the constellation service provides (CSPs) ground segment. They are independent events in terms of prior fault probability in the content of ISM. A fault mode is a combination of events. With the prior fault probability received from the ISM, the probability of different fault modes \(p_{faultmode}\) can be calculated as:  
\[p_{fault}=\prod_{i\in idx}(P_{event,i})\prod_{j\
otin idx}\left(1-P_{event,j}\right) \tag{2}\]  
where \(idx\) refers to the set of fault events.  
Not all fault modes need to be monitored. ARAIM introduces \(P_{THRES}\) as the threshold for the integrity risk coming from unmonitored faults. One fault mode corresponds to one fault subset. The ARAIM can work into the step of fault-tolerant positioning after the probability of unmonitored fault modes is less than \(P_{THRES}\).