## 4 Proposed Localization Method
### Algorithm Development
#### 4.1.1 Calibration Stage
To resolve this contradiction, we preliminarily let \(\mathbf{W}_{1}=\mathbf{I}_{MN\times MN}\) and use (67) to compute a least squares estimate of \(\theta^{\mathrm{o}}\), and then use the estimated \(\theta^{\mathrm{o}}\) to update \(\mathbf{W}_{1}\) for another repetition. Based on the WLS theorem, it can be deduced that the estimate \(\theta\) is approximately unbiased and the corresponding covariance matrix can be obtained, given sufficiently small BR measurement noise and transmitter/receiver position error, as  
\[\mathrm{cov}(\theta)=\left(\mathbf{G}_{1}^{\mathrm{T}}\mathbf{W}_{1}\mathbf{G }_{1}\right)^{-1} \tag{69}\]  
Next, the functional relation between the target position \(\mathbf{u}^{\mathrm{o}}\) and the introduced nuisance parameters \(R_{\mathrm{t},1}^{\mathrm{o}},R_{\mathrm{t},2}^{\mathrm{o}},\ldots,R_{\mathrm{ t},M}^{\mathrm{o}}\), is explored to compute the final estimate of target position. To this end, reorganize the functional relation in (4) as  
\[2(\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}})^{\mathrm{T}}\mathbf{u}^{\mathrm{o}}= (\mathbf{u}^{\mathrm{o}})^{\mathrm{T}}\mathbf{u}^{\mathrm{o}}+(\mathbf{s}_{ \mathrm{t},m}^{\mathrm{o}})^{\mathrm{T}}\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}} -(R_{\mathrm{t},m}^{\mathrm{o}})^{2} \tag{70}\]  
Denoting the estimation error of \(\theta\) by \(\Delta\theta\), mathematically we arrive at  
\[\mathbf{u}^{\mathrm{o}} =\theta(1:3)-\Delta\theta(1:3) \tag{71}\] \[R_{\mathrm{t},m}^{\mathrm{o}} =\theta(3+m)-\Delta\theta(3+m) \tag{72}\]  
Putting (71), (72) into the right side of (70) and \(\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}=\hat{\mathbf{s}}_{\mathrm{t},m}-\left( \mathbf{\Delta s}_{\mathrm{t},m}-\mathbf{\Delta\hat{s}}_{\mathrm{t},m}\right)\) into the both sides, we have after ignoring second-order error terms,  
\[ \begin{split} 2\hat{\mathbf{s}}_{\mathrm{t},m}^{\mathrm{T}} \mathbf{u}^{\mathrm{o}}&=\theta(1:3)^{\mathrm{T}}\theta(1:3)- \theta(m+3)^{2}+\hat{\mathbf{s}}_{\mathrm{t},m}^{\mathrm{T}}\hat{\mathbf{s}}_ {\mathrm{t},m}\\ &-2\theta(1:3)\Delta\theta(1:3)+2\theta(m+3)\Delta\theta(m+3)+2( \mathbf{u}^{\mathrm{o}}-\hat{\mathbf{s}}_{\mathrm{t},m})^{\mathrm{T}}( \mathbf{\Delta s}_{\mathrm{t},m}-\mathbf{\Delta\hat{s}}_{\mathrm{t},m})\end{split}  \tag{73}\]  
The final estimate of target position should satisfy (73) and meanwhile retain as close as possible to the estimated values of target position in \(\theta\). In line with this principle, one has the following set of equations  
\[\mathbf{G}_{2}\mathbf{u}^{\mathrm{o}}=\mathbf{h}_{2}+\mathbf{\Delta h}_{2} \tag{74}\]where  
\[\mathbf{G}_{2}=\left[ \begin{array}{c}\mathbf{I}_{3\times 3}\\ 2\mathbf{s}_{t,1}^{\mathrm{T}}\\ 2\mathbf{s}_{t,2}^{\mathrm{T}}\\ \vdots\\ 2\mathbf{s}_{t,M}^{\mathrm{T}}\end{array} \right] \tag{75}\]  
\[\mathbf{h}_{2}=\left[ \begin{array}{c}\theta(1:3)\\ \theta(1:3)^{\mathrm{T}}\theta(1:3)-\theta(1+3)^{2}+\hat{\mathbf{s}}_{t,1}^{ \mathrm{T}}\hat{\mathbf{s}}_{t,1}\\ \theta(1:3)^{\mathrm{T}}\theta(1:3)-\theta(2+3)^{2}+\hat{\mathbf{s}}_{t,2}^{ \mathrm{T}}\hat{\mathbf{s}}_{t,2}\\ \vdots\\ \theta(1:3)^{\mathrm{T}}\theta(1:3)-\theta(M+3)^{2}+\hat{\mathbf{s}}_{t,M}^{ \mathrm{T}}\hat{\mathbf{s}}_{t,M}\end{array} \right] \tag{76}\]  
\[\mathbf{\Delta h}_{2}=\mathbf{B}_{2}\Delta\theta+\mathbf{D}_{2}(\Delta \mathbf{s}-\Delta\hat{\mathbf{s}}) \tag{77}\]  
\[\mathbf{B}_{2}=\left[ \begin{array}{cccc}-\mathbf{I}_{3\times 3}&0_{3\times 1}&0_{3 \times 1}&\cdots&0_{3\times 1}\\ -2\theta(1:3)&2\theta(1+3)&0&\cdots&0\\ -2\theta(1:3)&0&2\theta(2+3)&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -2\theta(1:3)&0&0&\cdots&2\theta(M+3)\end{array} \right] \tag{78}\]  
\[\mathbf{D}_{2}=\left[ \begin{array}{cccc}\mathbf{O}_{3\times 3M}& \mathbf{O}_{3\times 3N}\\ 2\mathrm{diag}\big{[}(\mathbf{u}^{\mathrm{o}}-\hat{\mathbf{s}}_{t,1})^{ \mathrm{T}},(\mathbf{u}^{\mathrm{o}}-\hat{\mathbf{s}}_{t,2})^{\mathrm{T}}, \ldots,(\mathbf{u}^{\mathrm{o}}-\hat{\mathbf{s}}_{t,M})^{\mathrm{T}}\big{]}& \mathbf{O}_{3M\times 3N}\end{array}