## 4 Abnormal Measurement Exclusion Using RAIM Multiple Fault Detection and Exclusion  
The idea of RAIM FDE is to exclude the satellite with larger pseudorange residual. As mentioned earlier, it is capable of excluding abnormal signals. An illustration of the devastated effect of the particle filter is depicted in Figure 3. This paper therefore proposes the idea of RAIM to overcome the drawback. Figure 4 shows the flowchart of the use of the proposed RAIM FDE in the 3D map method. The red-frame of Figure 4 is the newly proposed method in this paper. As shown in Figure 4, the RAIM FDE will be used after calculating the simulated pseudorange of a candidate. It is interesting to note the idea of the pseudorange difference of each candidate in the proposed 3D map positioning method is similar to the pseudorange residual of the RAIM algorithm. In the proposed method, the pseudorange difference is defined as the pseudorange measurement minus the simulated pseudorange. The simulated pseudorange is calculated as:  
\[\hat{\rho}_{n}^{(i)}=R_{n}^{(i)}+c\big{(}\delta t^{\tau(i)}-\delta t_{n}^{ \text{sv}}\big{)}+I_{n}+T_{n}+\varepsilon_{n}^{\text{refl}(i)} \tag{1}\]  
where \(n\) denotes the index of the satellite, \(i\) denotes the index of a position candidate (particle), \(R_{n}^{(i)}\) denotes geometric distance between the satellite \(n\) and the candidate \(i\), \(c\) denotes the speed of light, the satellite clock and orbit offset \(\delta_{n}^{\tau^{m}}\) are corrected using the satellite broadcast model. The ionospheric delay \(I\) and the tropospheric delay \(T\) are obtained from the Klobuchar and Saastamoinen models, respectively. The reflection delay is estimated by the ray tracing and city building models [23; 24; 25]. The receiver clock offset, \(\delta t^{\tau^{(i)}}\), for the candidate is modified to minimize the difference between the simulated set and the measured set. If an abnormal signal is used in the calculation of the receiver clock bias minimization, the optimized receiver clock bias will be inaccurate. The flowchart of the RAIM multiple fault exclusion in the proposed method is shown in Figure 5.  
Figure 4: Flowchart of the particle filter using 3D city building models and receiver autonomous integrity monitoring (RAIM) multiple fault detection and exclusion (FDE). Firstly, all valid satellites are selected into a subset. The valid satellite means a satellite that is classified to same signal type by both the C/N\({}_{0}\) and ray-tracing classifications. Afterwards, the pseudorange difference (\(\varepsilon_{n}^{pr}\)) is calculated as:  
\[\varepsilon_{n}^{pr}=\rho_{n}-\hat{\rho}_{n}^{(I)}\Big{(}\partial t_{optimized}^{(I)}\Big{)} \tag{2}\]  
where \(\partial t_{optimized}^{(I)}\) is the receiver clock bias for a candidate based on the previous developed optimization method [23]. The root of the sum of the squared pseudorange difference, which defined as a test statistic (TS) in this paper, is as:  
\[TS=\sqrt{\sum\
olimits_{n=1}^{N^{sim}}\big{(}\varepsilon_{n}^{pr}\big{)}^{2}} \tag{3}\]  
where \(N^{sim}\) denotes the number of simulated pseudoranges. This paper assumes the pseudorange difference is normally distributed zero mean random variable. Thus the pseudorange difference can be tested by the chi-square test according to appropriate degree of freedom (DOF) and probability of false alarm. The DOF is calculated using Equation (4) because it requires at least one satellite from the same GNSS constellation to calculate the optimized receiver clock bias. \[DOF=N^{sim}-1 \tag{4}\]  
Theoretically speaking, two satellites from the same constellation mean the \(DOF\) equals 1, which can be checked by the consistency of receiver clock bias. However, this check could be difficult in the practical implementation. For example, in the case of two satellites are received, one is a clean LOS signal and the other one is abnormal NLOS signal. In this case, the \(DOF\) is 1 theoretically, and it should be checked by the constraint of receiver clocks.