# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 3 Pseudo-Linear Equation  
### DOA Pseudo-Linear Equation for Land-Based Station  
The linear equation of azimuth is first established. Based on (4), we have  
\[\sin(\theta_{i}^{o})\mathbf{b}_{i,2}^{\mathrm{T}}(\mathbf{u}-\mathbf{s}_{i}^{o})=\cos( \theta_{i}^{o})\mathbf{b}_{i,1}^{\mathrm{T}}(\mathbf{u}-\mathbf{s}_{i}^{o}),1\leq i\leq N _{1} \tag{14}\]  
Simplifying (14) gives  
\[\left(\sin(\theta_{i}^{o})\mathbf{b}_{i,2}-\cos(\theta_{i}^{o})\mathbf{b}_{i,1}\right) ^{\mathrm{T}}\mathbf{u}=\left(\sin(\theta_{i}^{o})\mathbf{b}_{i,2}-\cos(\theta_{i}^{o} )\mathbf{b}_{i,1}\right)^{\mathrm{T}}\!\mathbf{s}_{i}^{o},1\leq i\leq N_{1} \tag{15}\]  
Combining (15) of all land-based stations obtains  
\[\mathbf{G}_{\mathbf{\theta}}^{o}\mathbf{u}=\mathbf{y}_{\mathbf{\theta}}^{o} \tag{16}\]  
where  
\[\mathbf{G}_{\mathbf{\theta}}^{o}= \begin{bmatrix}\left(\sin(\theta_{1}^{o})\mathbf{b}_{1, 2}-\cos(\theta_{1}^{o})\mathbf{b}_{1,1}\right)^{\mathrm{T}}\\ \left(\sin(\theta_{2}^{o})\mathbf{b}_{2,2}-\cos(\theta_{2}^{o})\mathbf{b}_{2,1}\right)^ {\mathrm{T}}\\ \vdots\\ \left(\sin(\theta_{N_{1}}^{o})\mathbf{b}_{N_{1},2}-\cos(\theta_{N_{1}}^{o})\mathbf{b} _{N_{1},1}\right)^{\mathrm{T}}\end{bmatrix} \in\mathbb{R}^{N_{1}\times 3} \tag{17}\]  
Converting (7) gives  
\[\left(R_{o}+h_{i}^{o}\right)\sin(\beta_{i})\sin(\varphi_{i}^{o})=\left(\left(R _{o}+h_{i}^{o}\right)\cos(\beta_{i})-R_{o}\right)\cos(\varphi_{i}^{o}) \tag{19}\]Substituting \(\beta_{i}\) into (19) yields  
\[(R_{o}+h_{i}^{o})\sin(\varphi_{i}^{o})||\mathbf{u}-\mathbf{s}_{i}^{o}||_{2}+2R_{o}^{2}\cos (\varphi_{i}^{o})=(R_{o}+h_{i}^{o})\cos(\varphi_{i}^{o})\sqrt{4R_{o}^{2}-||\mathbf{u }-\mathbf{s}_{i}^{o}||_{2}^{2}} \tag{20}\]  
Squaring and simplifying both sides of (20) simultaneously yields  
\[c_{i,1}||\mathbf{u}-\mathbf{s}_{i}^{o}||_{2}^{2}+c_{i,2}||\mathbf{u}-\mathbf{s}_{i}^{o}||_{2}+c_ {i,3}=0 \tag{21}\]  
where \(c_{i,1}=\big{(}R_{o}+h_{i}^{o}\big{)}^{2}\), \(c_{i,2}=2R_{o}^{2}\big{(}R_{o}+h_{i}^{o}\big{)}\sin\big{(}2\varphi_{i}^{o}\big{)}\), and \(c_{i,3}=-4R_{o}^{2}\Big{(}\big{(}h_{i}^{o}\big{)}^{2}+2R_{o}h_{i}^{o}\Big{)} \big{(}\cos\big{(}\varphi_{i}^{o}\big{)}\big{)}^{2}\). (21) is a quadratic equation about \(||\mathbf{u}-\mathbf{s}_{i}^{o}||_{2}\). Appendix A proves that \(||\mathbf{u}-\mathbf{s}_{i}^{o}||_{2}\) is the only positive root of the equation shown in (21). Therefore, we have  
\[c_{i,4}=\frac{\sqrt{c_{i,2}^{2}-4c_{i,1}c_{i,3}}-c_{i,2}}{2c_{i,1}}=||\mathbf{u}- \mathbf{s}_{i}^{o}||_{2} \tag{22}\]  
Squaring both sides of (22) simultaneously yields  
\[2\mathbf{s}_{i}^{o\mathsf{T}}\mathbf{u}-||\mathbf{u}||_{2}^{2}=||\mathbf{s}_{i}^{o}||_{2}^{2} -c_{i,\mathcal{A}}^{2},1\leq i\leq N_{1} \tag{23}\]  
The pseudo-linear equation for elevation is obtained as follows:  
\[\mathbf{G}_{\mathbf{\varphi}}^{o}\mathbf{\eta}_{a}^{o}=\mathbf{y}_{\mathbf{\varphi}}^{o} \tag{24}\]  
where  
\[\mathbf{G}_{\mathbf{\varphi}}^{o}= \begin{bmatrix}2\big{(}\mathbf{s}_{1}^{o}\big{)}^{ \mathsf{T}}&-1\\ 2\big{(}\mathbf{s}_{2}^{o}\big{)}^{\mathsf{T}}&-1\\ \vdots&\vdots\\ 2\Big{(}\mathbf{s}_{N_{1}}^{o}\big{)}^{\mathsf{T}}&-1\end{bmatrix} \in\mathbb{R}^{ N_{1}\times 4},\mathbf{y}_{\mathbf{\varphi}}^{o}= \begin{bmatrix}||\mathbf{s}_{1}^{o}||_{2}^{2 }-c_{1,\mathcal{A}}^{2}\\ ||\mathbf{s}_{2}^{o}||_{2}^{2}-c_{2,\mathcal{A}}^{2}\\ \vdots\\ ||\mathbf{s}_{N_{1}}^{o}||_{2}^{2}-c_{N_{1}}^{2}\end{bmatrix} \in\mathbb{R}^{N_{1} \times 1} \tag{25}\]  
\[\mathbf{\eta}_{a}^{o}= \begin{bmatrix}\mathbf{u}\\ ||\mathbf{u}||_{2}^{2}\end{bmatrix} \in\mathbb{R}^{4\times 1} \tag{26}\]  
Combining (16) and (24) yields the DOA pseudo-linear equation as  
\[\mathbf{G}_{a}^{o}\mathbf{\eta}_{a}^{o}=\mathbf{y}_{a}^{o} \tag{27}\]  
where  
\[\mathbf{G}_{a}^{o}= \begin{bmatrix}\mathbf{G}_{\mathbf{\theta}}^{o}&\mathbf{0}_{N_{1}\times 1} \\ \mathbf{G}_{\mathbf{\varphi}}^{o}\end{bmatrix} \in\mathbb{R}^{2N_{1}\times 4},\mathbf{y}_{a}^{o}= \begin{bmatrix}\mathbf{y}_{\mathbf{\varphi}}^{o}\\ \mathbf{y}_{\mathbf{\varphi}}^{o}\end{bmatrix} \in\mathbb{R}^{2N_{1}\times 1},\mathbf{\eta}_{a}^{o}=