# BDS-3 Integrity Risk Modeling and Probability Evaluation
## 3 BDS-3 Integrity Failure Model and Effect Analysis Algorithm
### BDS-3 Integrity Risk Tree Modeling
#### 3.4.2 Integrity Risk Probability Evaluation Method  
The design value of the single-satellite integrity risk probability of the BDS-3 system is 1e-5/h, which means the simultaneous occurrence probability of an abnormal satellite event and an abnormal monitoring miss event. However, since both the system anomaly and the monitoring miss are small probability events and BDS-3 has a short service time, there are fewer historical samples for statistics. If only the ratio of the historical statistics number of small samples to the statistical time is used as the system anomaly occurrence probability or the monitoring miss, it easily leads to evaluation deviation, that is, the calculated event occurrence probability is greatly affected by the system's state in the statistical historical period (for example, during the statistical historical period, the space segment is relatively stable, and the number of system anomalies is less, then the anomaly probability is low). Conversely, if, during the statistical historical time, the space segment is more active, the number of system anomalies is greater, and the probability of anomalies is higher. \begin{table}
\begin{tabular}{c c c c c} \hline
**Section** & **Anomaly Models** & **Impact on SISRE** & **Spatial Correlation** & **Receiver Detectability** \\ \hline \multirow{8}{*}{Space segment} & Satellite clock & \multirow{2}{*}{Step/tilt} & No & No \\  & anomaly & & & No \\  & Satellite attitude & & & No \\  & anomaly & & & \\  & Signal measurement & & & No \\  & anomaly & & & No \\  & Abnormal signal & & No & No \\  & property parameter & & & No \\  & broadcast error & & & No \\  & Satellite autonomous & & & No \\  & integrity not monitored & & & No \\ \hline \multirow{8}{*}{Control segment} & Monitoring network & \multirow{2}{*}{Step/tilt/sine curve} & Certain extent & No \\  & data anomalies & & & No \\  & Injection anomaly at the & & Inclination & Certain extent & No \\  & upper injection station & & & No \\  & Calculation anomaly of & & & No \\  & orbit/clock difference & & & No \\  & Abnormal calculation of & & Multiplicity & Certain extent & Most conditions \\  & ephemeris fitting parameters & & & No \\  & Track and time synchronization & & & No \\  & processing equipment anomalies & & & No \\  & Ground integrity & & & No \\  & parameter broadcast error & & & No \\  & Ground integrity & & No & Yes \\  & not monitored & & & No \\ \hline \end{tabular}
\end{table} 
Table 3: GNSS anomaly models and their impacts. To solve the problem of accurate and reasonable event probability estimation based on small sample historical events, this paper proposes an integrity risk probability assessment method based on the Bayesian estimation theory. The so-called integrity risk probability, as shown in Figure 3, refers to the system occurring at the same time as system anomalies and monitoring missing alarm events and can be expressed as:  
\[IR=P_{fail}\times P_{md} \tag{1}\]  
where _IR_ represents the integrity risk probability, \(P_{fail}\) is the system anomaly occurrence probability, and \(P_{md}\) is the abnormal monitoring missing alarm rate. The system anomaly probability can be shown as follows:  
\[P_{fail}=\sum_{m}P(m) \tag{2}\]  
where \(m\) is the bottom event of the system abnormal branch in Figure 3, and _P(m)_ represents the estimated probability of event \(m\). The monitoring miss probability can be expressed as:  
\[P_{md}=\sum_{n}P(n) \tag{3}\]  
where \(n\) is the bottom event of the anomaly monitoring miss branch in Figure 3, and _P(n)_ represents the estimated probability of event n.  
Therefore, the key to evaluating the system's integrity risk probability is to objectively and reasonably estimate the risk tree bottom events occurrence probability (hereinafter referred to as abnormal events for the convenience of elaboration). The abnormal events probability can usually be counted according to the statistical number and evaluation time of the abnormal event, as follows:  
\[P(i)=\frac{k}{T}\times MTTN \tag{4}\]  
Figure 3: Model of BDS-3 integrity risk tree.