## 3 First Step of Identification Algorithm  
For LOS BS combination, there is only measurement noise error, the distance error corresponding to \(r_{i}\), i.e., \(\Delta r_{i}\), should satisfy Gaussian distribution. Hence, if we can estimate \(\Delta r_{i}\) on the basis of measurements of the distance between BS and MS, i.e., \(r_{i}{}^{\prime}\)\((i=1,\ 2,\ 3)\), we can identify the LOS BS combination based on whether estimate of \(\Delta r_{i}\) satisfies Gaussian distribution. Without loss of generality, suppose that \(\Delta r_{i}\) satisfies Gaussian distribution with zero mean and standard deviation \(\sigma\) when there is only measurement noise error. If there is only measurement noise error, we can obtain  
\[r_{i}\gg\Delta r_{i}\ (i=1,\ 2,\ 3) \tag{14}\]  
so, we can approximate \(\Delta\delta\) as  
\[\Delta\delta\cong \begin{bmatrix}2\Delta r_{1}r_{1}-2\Delta r_{2}r_{2}\\ 2\Delta r_{1}r_{1}-2\Delta r_{3}r_{3}\\ 2\Delta r_{2}r_{2}-2\Delta r_{3}r_{3}\end{bmatrix} = \begin{bmatrix}2r_{1}&-2r_ {2}&0\\ 2r_{1}&0&-2r_{3}\\ 0&2r_{2}&-2r_{3}\end{bmatrix} \cdot \begin{bmatrix}\Delta r_{1}\\ \Delta r_{2}\\ \Delta r_{3}\end{bmatrix} =R\cdot\Delta R \tag{15}\]  
where  
\[R= \begin{bmatrix}2r_{1}&-2r_{2}&0\\ 2r_{1}&0&-2r_{3}\\ 0&2r_{2}&-2r_{3}\end{bmatrix} ,\ \Delta R= \begin{bmatrix}\Delta r_{1}\\ \Delta r_{2}\\ \Delta r_{3}\end{bmatrix}  \tag{16}\]  
Substitute (15) and (16) into (13) yields  
\[R\cdot\Delta R=A\cdot\Delta\theta \tag{17}\]  
According to Equation (17), if we can obtain the coordinate deviation of common chord intersection point from MS, i.e., \(\Delta\theta\), and \(R\), we can obtain an estimate of \(\Delta R\), i.e., estimate of \(\Delta r_{1}\), \(\Delta r_{2}\) and \(\Delta r_{3}\), and then identify LOS BS combination based on whether the estimate of \(\Delta r_{i}\) (referred as \(\Delta r_{i}\) for simplicity in the following) follows Gaussian distribution, because the estimate of \(\Delta r_{i}\) should follow the Gaussian distribution if only measurement noise error exists. Note \(R\), i.e., \(r_{i}\), is the actual distance between \(i\)th BS and MS, which is unknown. Nevertheless, because of measurement noise error satisfies Gaussian distribution with zero mean, \(r_{i}\) is an unbiased estimate of the measurement of the distance between \(i\)th BS and MS, i.e., \(r_{i}{{}^{\prime}}\). Thus, we approximate \(r_{i}\) in \(R\) with \(r_{i}{{}^{\prime}}\) in (17) for the case of only measurement noise error exists, that is  
\[R\cong R^{\prime}= \begin{bmatrix}2r_{1}{{}^{\prime}}&-2r_{2}{{}^{\prime}}&0\\ 2r_{1}{{}^{\prime}}&0&-2r_{3}{{}^{\prime}}\\ 0&2r_{2}{{}^{\prime}}&-2r_{3}{{}^{\prime}}\\ \end{bmatrix}  \tag{18}\]  
On the other hand, the coordinate deviation of common chord intersection point from MS, i.e., \(\Delta\theta\) is unknown, too. Similar to reference [39], for the case of only measurement noise error exists, we can adopt the mean value of the coordinates of three intersection points of three localization circles to approximate the actual coordinate of MS. Meanwhile, the coordinate of common chord intersection point can be solved according to Equation (2). On the basis of approximation of MS coordinate and common chord intersection point coordinate, corresponding coordinate deviation \(\Delta\theta\) can be solved. Substitute the approximation of \(\Delta\theta\) and \(R\) into (17), and with least squares method, the estimate of \(\Delta R\), denoted by \(\Delta R_{est}\), can be obtained as  
\[\Delta R_{est}=argmin\left\|R^{\prime}\cdot\Delta R-A\cdot\Delta\theta\right\| ^{2} \tag{19}\]  
It should be noted that, though it is \(\Delta R\) cause \(\Delta\theta\) instead of \(\Delta\theta\) cause \(\Delta R\), we can inversely deduce the estimate of \(\Delta R\) according to \(\Delta\theta\), and then identify whether the estimate of \(\Delta R\) satisfies Gaussian distribution. Without loss of generality, we can suppose that \(e_{i}\gg n_{i}\).