# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 5 Proposed Method  
### Proposed Method for Land-Based Station  
#### 5.1.1 Localized Method for Land-Based Station  
Since in the practical process, only the measurement with errors is available, the following error equation can be established:  
\[\mathbf{\varepsilon}_{a}=\mathbf{y}_{a}-\mathbf{G}_{a}\mathbf{\eta}_{a} \tag{51}\]  
Substituting the measurements with errors into (25) gives \(\mathbf{y}_{a}\) and \(\mathbf{G}_{a}\). Performing first-order Taylor expansion for \(\mathbf{y}_{a}\) and \(\mathbf{G}_{a}\) yields  
\[\left\{ \begin{array}{l}\mathbf{y}_{a}\approx\mathbf{y}_{a}^{o}+\mathbf{A}_{1}\mathbf{n}_{a}+ \mathbf{A}_{2}\mathbf{n}_{h}\\ \mathbf{G}_{a}\approx\mathbf{G}_{a}^{o}+\sum_{d_{1}=1}^{2N_{1}}<\mathbf{z}>_{d_{1}}\mathbf{B}_ {1,d_{1}}+\sum_{d_{2}=1}^{N_{1}}<\mathbf{h}>_{d_{2}}\mathbf{B}_{2,d_{2}}\end{array} \right. Therefore, the DOA localization problem for land-based stations can be modeled as  
\[\left\{ \begin{array}{l}\underset{\eta_{a}}{\min}J_{a}(\eta_{a})=(\mathbf{y}_{a}-G_{a} \mathbf{\eta}_{a})^{\mathsf{T}}\mathbf{W}_{a}(\mathbf{y}_{a}-\mathbf{G}_{a}\mathbf{\eta}_{a})\\ \text{s.t. }\mathbf{u}^{\mathsf{T}}\mathbf{u}-||\mathbf{u}||_{2}^{2}=0\\ \mathbf{u}^{\mathsf{T}}\mathbf{\Lambda}_{1}\mathbf{u}=R_{e}^{2}\end{array} \right. \tag{55}\]  
\(\mathbf{W}_{a}\) is the weighted matrix, which is defined as  
\[\mathbf{W}_{a}=\left(\mathbb{E}[\epsilon_{a}\epsilon_{a}^{\mathsf{T}}]\right)^{-1}= \left(\mathbf{C}_{1}\mathbf{Q}_{a}\mathbf{C}_{1}^{\mathsf{T}}+\mathbf{C}_{2}\mathbf{Q}_{h}\mathbf{C}_{2 }^{\mathsf{T}}\right)^{-1} \tag{56}\]  
As can be seen from (55), the cost function is convex and the equation constraints are nonconvex. The equation constraints are written in matrix form as  
\[\mathbf{D}_{a}\mathbf{\eta}_{a}=\mathbf{d}_{a} \tag{57}\]  
where  
\[\mathbf{D}_{a}= \begin{bmatrix}\mathbf{u}^{\mathsf{T}}&-1\\ \mathbf{u}^{\mathsf{T}}\mathbf{\Lambda}_{1}&0\end{bmatrix} ,\mathbf{d}_{a}= \begin{bmatrix} 0\\ R_{e}^{2}\end{bmatrix}  \tag{58}\]  
\(\mathbf{D}_{a}\) is related to \(\mathbf{u}\), so the estimate of \(\mathbf{u}\) can be substituted into \(\mathbf{D}_{a}\) to obtain \(\mathbf{D}_{a}^{k}\) during the iteration process. \(\mathrm{J}_{a}(\mathbf{\eta}_{a})\) in (55) can be expressed as  
\[\mathrm{J}_{a}(\mathbf{\eta}_{a})=\mathbf{y}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{y}_{a}-2\bm {y}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{G}_{a}\mathbf{\eta}_{a}+\mathbf{\eta}_{a}^{\mathsf{T} }\mathbf{C}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{G}_{a}\mathbf{\eta}_{a} \tag{59}\]  
\(\mathbf{y}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{y}_{a}\) is a constant with respect to \(\mathbf{\eta}_{a}\). Therefore, substituting the results of the \(k\)th iteration into (57), (55) can be converted to  
\[\left\{ \begin{array}{l}\underset{\eta_{a}}{\min}J_{a}(\mathbf{\eta}_{a})=\mathbf{ \eta}_{a}^{\mathsf{T}}\mathbf{G}_{a}\mathbf{\eta}_{a}-2\mathbf{y}_{a}^{\mathsf{T}}\mathbf{\eta} _{a}\\ \text{s.t. }\mathbf{D}_{a}^{k}\mathbf{\eta}_{a}=\mathbf{d}_{a}^{k}\end{array} \right. \tag{60}\]  
where \(\mathbf{G}_{a}=\mathbf{G}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{G}_{a}\) and \(\mathbf{\tilde{y}}_{a}=\mathbf{G}_{a}^{\mathsf{T}}\mathbf{W}_{a}\mathbf{y}_{a}\). \(k\) is the iteration number. \(\mathbf{D}_{a}^{k}\) and \(\mathbf{d}_{a}^{k}\) can be obtained by substituting the \(k\)th iteration result \(\mathbf{u}_{a}^{k}\) into (58). As can be seen in (60), the cost function is still convex. However, by using the results of the previous iteration, the nonconvex quadratic equation constraints can be converted to linear constraints, which can ensure that the algorithm converges to the globally optimal solution.