## 2 System Model
On the basis of LOS BS combination identification, all BSs of LOS BS combination are identified as LOS BSs  
According to this idea, all BSs receiving signal from the same MS are divided into multiple BS combinations with every three BSs, for instance, \(n\) BSs can be divided into  
Figure 2: Demonstration of MS and common chord intersection point (P) in presence of distance error. BS combinations, one BS can belong to multiple BS combinations when the number of BS is more than three. Each BS belonging to LOS BS combination is identified as LOS BS and a BS can be identified as LOS BS because it belongs to any LOS BS combination. The signals received at LOS BSs are identified as LOS signals. Let the coordinate of the common chord intersection point is \((x^{\prime},y^{\prime})\), the coordinates of three BSs are \((x_{1},y_{1}),\ (x_{2},y_{2})\) and \((x_{3},y_{3})\), respectively. Thus, for the case of measurement noise error or NLOS error exist, i.e., \(\Delta r_{i}\
eq 0\ (i=1,\ 2,\ 3)\), the linear equations of the three common chords can be expressed as Equations (2):  
\[ \begin{cases}k_{1}-k_{2}+2x^{\prime}\cdot(x_{2}-x_{1})+2y^{\prime}\cdot(y_{2}- y_{1})={r_{1}}^{\prime 2}-{r_{2}}^{\prime 2}\\ k_{1}-k_{3}+2x^{\prime}\cdot(x_{3}-x_{1})+2y^{\prime}\cdot(y_{3}-y_{1})={r_{1}}^ {\prime 2}-{r_{3}}^{\prime 2}\\ k_{2}-k_{3}+2x^{\prime}\cdot(x_{3}-x_{2})+2y^{\prime}\cdot(y_{3}-y_{2})={r_{2}} ^{\prime 2}-{r_{3}}^{\prime 2}\end{cases}  \tag{2}\]  
where \(k_{i}={x_{i}}^{2}+{y_{i}}^{2}\), \(i=1,\ 2,\ 3\). Equations (2) can be rewritten as the form of matrix,  
\[\delta^{\prime}=A^{\prime}\cdot\theta^{\prime} \tag{3}\]  
where  
\[\delta^{\prime}= \begin{bmatrix}{r_{1}}^{\prime 2}-{r_{2}}^{\prime 2}-k_{1}+k_{2 }\\ {r_{1}}^{\prime 2}-{r_{3}}^{\prime 2}-k_{1}+k_{3}\\ {r_{2}}^{\prime 2}-{r_{3}}^{\prime 2}-k_{2}+k_{3}\end{bmatrix}  \tag{4}\]  
\[A^{\prime}= \begin{bmatrix}2(x_{2}-x_{1})\ 2(y_{2}-y_{1})\\ 2(x_{3}-x_{1})\ 2(y_{3}-y_{1})\\ 2(x_{3}-x_{2})\ 2(y_{3}-y_{2})\end{bmatrix}  \tag{5}\]  
\[\theta^{\prime}= \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}  \tag{6}\]  
Similar to Equation (3), for the case of without measurement noise error and without NLOS error, i.e., \(\Delta r_{i}=0\ (i=1,\ 2,\ 3)\), we can obtain  
\[\delta=A\cdot\theta \tag{7}\]  
where \(\delta\), \(A\) and \(\theta\) correspond one-to-one to \(\delta^{\prime}\), \(A^{\prime}\) and \(\theta^{\prime}\) for the case of measurement noise error or if the NLOS error exists. Suppose that coordinate of MS is \((x,y)\), and the coordinate deviation of common chord intersection point from MS is \((\Delta x,\ \Delta y)\), i.e., \(x^{\prime}=x+\Delta x\) and \(y^{\prime}=y+\Delta y\), then  
\[\theta^{\prime}= \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix} = \begin{bmatrix}x\\ y\end{bmatrix} + \begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix} =\theta+\Delta\theta \tag{8}\]  
where \(\theta= \begin{bmatrix}x\\ y\end{bmatrix} \) and \(\Delta\theta= \begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix} \). Obviously, \(\theta\) is independent of the measurement noise error and NLOS error, but the two categories error all have impact on \(\Delta\theta\). Similarly, dividing the other two variables of Equation (3) into two parts: error-related and error-independent, Equation (3) can be written as  
\[(\delta+\Delta\delta)=(A+\Delta A)\cdot(\theta+\Delta\theta) \tag{9}\]where \(\delta^{\prime}=\delta+\Delta\delta\), \(A^{\prime}=A+\Delta A\), and  
\[\delta^{\prime}= \begin{bmatrix}(r_{1}+\Delta r_{1})^{2}-(r_{2}+ \Delta r_{2})^{2}-k_{1}+k_{2}\\ (r_{1}+\Delta r_{1})^{2}-(r_{3}+\Delta r_{3})^{2}-k_{1}+k_{3}\\ (r_{2}+\Delta r_{2})^{2}-(r_{3}+\Delta r_{3})^{2}-k_{2}+k_{3}\end{bmatrix}  \tag{10}\] \[=