## 3 First Step of Identification Algorithm
As depicted in Figure 3a, when received signals are all LOS signal, the distance between MS and common chord intersection point is very short, and \(\Delta r_{i}\) (\(i=1,\ 2,\ 3\)) are all small. However, when the received signal of any BS in BS combination is due to NLOS transmission, i.e., \(e_{i}\
eq 0\) (\(i=1\) or \(2\) or \(3\)), for instance BS1 in Figure 3b, corresponding NLOS error will result in the distance between MS and common chord intersection point increases greatly, accordingly, \(\Delta\theta\) increases greatly, too. According to Equation (17), the estimate of \(\Delta R\) will increase greatly, and the estimate of \(\Delta r_{j}\) (\(j\
eq i\)) will increase greatly, too. Thus, the probability of \(P(\Delta r_{1}<\sigma\) & \(\Delta r_{2}<\sigma\) & \(\Delta r_{3}<\sigma)\) can be approximated with \(P(\Delta r_{i}<\sigma\) ) (\(i=1\) or \(2\) or \(3\)). For Gaussian distribution, 68.2% of the measurement noise errors lie within 1 standard deviations of the mean and 95.5% lie within 2 standard deviations. Let 1 standard deviations \(\sigma\) as the identification threshold value of \(\Delta r_{i}\) satisfies Gaussian distribution; in other words, if received signals at the three BSs are all due to LOS transmission, they should satisfy the following inequality,  
\[\Delta r_{1}<\sigma\ \&\ \Delta r_{2}<\sigma\ \&\ \Delta r_{3}<\sigma \tag{20}\]  
Then, when only measurement noise error exists, the probability of a LOS BS combination is misjudged as a NLOS BS combination, denoted by \(P_{mis}\), is estimated as following  
\[P_{mis}=1-P(\Delta r_{1}<\sigma\ \&\ \Delta r_{2}<\sigma\ \&\ \Delta r_{3}<\sigma) \cong 1-P(\Delta r_{i}<\sigma\ )\cong 31.8\% \tag{21}\]  
If there are four BSs are LOS BS in all BSs, for any LOS BS of the four LOS BS, it belongs to \(C_{3}^{2}=3\) LOS BS combinations, thus, the probability of this LOS BS is misjudged as NLOS BS is  
\[P_{mis}{{}^{3}}\cong 3.2\ \% \tag{22}\]As shown in Equations (21) and (22), when only measurement noise error exists, for the case of four BSs are LOS BS in all BSs, if we choose one standard deviations \(\sigma\) as the identification threshold value of \(\Delta r_{i}\) only resulting from measurement noise errors, although the probability of a LOS BS combination is misjudged as a NLOS BS combination is up to 31.8%, the probability of LOS BS is misjudged as NLOS BS is small enough to be neglected. Therefore, we can choose 1 standard deviations \(\sigma\) as the identification threshold value. As the number of LOS BS continues to increase, the probability of LOS BS is misjudged will decrease dramatically. Nevertheless, if there are only three BSs are LOS BS in all BSs, for any BS of the three LOS BSs, it only belongs to one BS combination, the probability of LOS BS is misjudged, i.e., the probability of the sole LOS BS combination is misjudged as NLOS BS combination, is up to 31.8%. Therefore, when only measurement noise error exists, for the case of only three BSs are LOS BS in all BSs, identification threshold value of \(\Delta r_{i}\) only resulting from measurement noise errors, should be amplified from 1 standard deviations to 2\(\sigma\), corresponding identification inequality is  
\[\Delta r_{1}<2\sigma\ \&\ \Delta r_{2}<2\sigma\ \&\ \Delta r_{3}<2\sigma \tag{23}\]  
Figure 3: Demonstration of the variation of distance between MS and common chord intersection point (**a**) in absence of NLOS error (**b**) in presence of NLOS error \(e_{1}\
eq 0\).