## 3 Localization Method
### Gauss-Newton Iteration  
Taking the Taylor-series expansion of \(\mathbf{\alpha}(\mathbf{\beta})\) at \(\mathbf{\beta}^{(0)}\) and retaining the first-order terms give a quadratic approximation of \(J(\mathbf{\beta})\)  
\[J(\mathbf{\beta})\approx\bigg{[}\mathbf{\alpha}-\bar{\mathbf{\alpha}}(\mathbf{\beta}^{(0)})- \left.\frac{\partial\mathbf{\alpha}}{\partial\mathbf{\beta}}\right|_{\mathbf{\beta}=\mathbf{ \beta}^{(0)}}(\mathbf{\beta}-\mathbf{\beta}^{(0)})\bigg{]}^{T}\mathbf{Q}_{\mathbf{\alpha}} ^{-1}\bigg{[}\mathbf{\alpha}-\bar{\mathbf{\alpha}}(\mathbf{\beta}^{(0)})-\left.\frac{ \partial\mathbf{\alpha}}{\partial\mathbf{\beta}}\right|_{\mathbf{\beta}=\mathbf{\beta}^{(0)} }(\mathbf{\beta}-\mathbf{\beta}^{(0)})\bigg{]}. \tag{16}\]  
Differentiating it with respect to \(\mathbf{\beta}\) and setting the result to zero yield the GN iterative equation for asymptotically minimizing \(J(\mathbf{\beta})\), after replacing the superscript,  
\[\mathbf{\beta}^{(g+1)}=\mathbf{\beta}^{(g)}+\Big{(}\mathbf{G}^{(g)T}\mathbf{W}\mathbf{ G}^{(g)}\Big{)}^{-1}\mathbf{G}^{(g)T}\mathbf{W}\Big{(}\mathbf{\alpha}-\mathbf{\alpha}^{(g)} \Big{)},\ g=0,1,\cdots,L. \tag{17}\]  
where \(g\) is the iteration count and \(L\) is the number of iterations. \(\mathbf{\beta}^{(0)}\) is the initial solution guess whose choice will be elaborated later, which should not be confused with \(\mathbf{\beta}^{\circ}\) that represents the true value. In (17), \(\mathbf{W}=\mathbf{Q}_{\mathbf{\alpha}}^{-1}\) and \(\mathbf{\alpha}^{(g)}\) are the re-generated measurement vectors with \(\mathbf{\beta}^{(g)}\) replacing \(\mathbf{\beta}^{\circ}\) in \(\mathbf{\alpha}^{\circ}\). \(\mathbf{G}^{(g)}\) is the Jacobian matrix given by  
\[\mathbf{G}^{(g)}=\left.\frac{\partial\mathbf{\alpha}}{\partial\mathbf{\beta}^{T}} \right|_{\mathbf{\beta}=\mathbf{\beta}^{(g)}}= \begin{bmatrix}\frac{\partial\mathbf{\tau} }{\partial\mathbf{\mathfrak{u}}(\mathbf{\mathfrak{s}})^{T}}&\frac{\partial\mathbf{\mathfrak{ u}}}{\partial\mathbf{\mathfrak{u}}(\mathbf{\mathfrak{s}})^{T}}\\ \frac{\partial\mathbf{\mathfrak{t}}}{\partial\mathbf{\mathfrak{u}}(\mathbf{\mathfrak{s}})^ {T}}&\frac{\partial\mathbf{\mathfrak{t}}}{\partial\mathbf{\mathfrak{u}}(\mathbf{\mathfrak{ s}})^{T}}\end{bmatrix} = \begin{bmatrix}\mathbf{R}^{(g)}&\mathbf{B}^{(g)}\\ \bar{\mathbf{R}}^{(g)}&\bar{\mathbf{B}}^{(g)}\end{bmatrix} . \tag{18}\]  
The \(((k-1)(M-1)+(i-1))\)-th row of \(\mathbf{R}^{(g)}\) is  
\[\mathbf{R}^{(g)}((k-1)(M-1)+(i-1),:)=\frac{\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{ s}_{i,k}\Big{)}^{T}}{d_{i,k}^{(g)}}-\frac{\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{s}_{1,k }\Big{)}^{T}}{d_{1,k}^{(g)}}, \tag{19}\]  
\[k=1,\cdots,K,\quad i=2,\cdots,M,\]  
where \(d_{i,k}^{(g)}\) and \(d_{1,k}^{(g)}\) are from (4) with the true value \(\mathbf{u}^{\circ}\) replaced by \(\mathbf{u}^{(g)}\), and similarly the \(((k-1)(M-1)+(i-1))\)-th row of \(\bar{\mathbf{R}}^{(g)}\) is  
\[ \begin{split}&\bar{\mathbf{R}}^{(g)}((k-1)(M-1)+(i-1),:)\\ &=\frac{\bar{\mathbf{u}}^{(g)T}-\mathbf{s}_{i}^{T}}{d_{i,k}^{(g)} }-\frac{d_{i,k}^{(g)}}{d_{i,k}^{(g)}}\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{s}_{ i,k}\Big{)}^{T}-\frac{\bar{\mathbf{u}}^{(g)T}-\mathbf{s}_{1}^{T}}{d_{1,k}^{(g)}}+ \frac{d_{1,k}^{(g)}}{d_{1,k}^{(g)}}\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{s}_{1,k }\Big{)}^{T},\end{split}  \tag{20}\]  
The \(((k-1)(M-1)+(i-1))\)-th row of \(\mathbf{B}^{(g)}\) is  
\[ \begin{split}\mathbf{B}^{(g)}((k-1)(M-1)+(i-1),:)& =\tau(k-1)\mathbf{R}^{(g)}((k-1)(M-1)+(i-1),:)\\ &=\tau(k-1)\left(\frac{\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{s}_{ i,k}\Big{)}^{T}}{d_{i,k}^{(g)}}-\frac{\Big{(}\mathbf{u}_{k}^{(g)}-\mathbf{s}_{ i,k}\Big{)}^{T}}{d_{1,k}^{(g)}}\right),\end{split}  \tag{21}\]  
and the \(((k-1)(M-1)+(i-1))\)-th row of \(\dot{\mathbf{B}}^{(g)}\) is  
\[