# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 2 Measurement Models
### DOA Measurement Model for Land-Based Station  
Figure 1 is the IVH model, which shows the short-wave signal sent by the source reaching the land-based station after being reflected by the ionosphere. \(\theta_{i}^{o}\) represents the true value of the azimuth and \(\varphi_{i}^{o}\) is the true value of elevation. \(\theta_{i}^{o}\) is defined as the angle between the projection of the incidence direction on the surface plane and the local north direction of the station. \(\varphi_{i}^{o}\) is defined as the angle between the incidence direction and the surface plane of the station. \(h_{i}^{o}\) is the ionosphere reflection height and \(R_{o}=6371.393\) km is the average radius of the Earth. Combining all position vectors of land-based stations yields: \(s_{a}^{o}=\big{[}\big{(}s_{1}^{o}\big{)}^{\mathrm{T}},(s_{2}^{o})^{\mathrm{T} },\ldots,\big{(}s_{N_{1}}^{o}\big{)}^{\mathrm{T}}\big{]}^{\mathrm{T}}\). The ionosphere reflection height is often difficult to measure accurately and is \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Explanations** \\ \hline \(\lfloor\cdot\rceil^{\mathrm{T}}\) & Matrix or vector transpose \\ \(\lfloor\cdot\rfloor^{\sharp}\) & Generalized inverses \\ \(\lfloor\cdot\rfloor^{-1}\) & The inverse of the matrix \\ \(trace(\cdot)\) & Traces of the matrix \\ E\(\lfloor\cdot\rfloor\) & Mathematical expectation \\ \(diag\{\cdot\}\) & Diagonal matrix \\ \(blkdiag\{\cdot\}\) & Chunked diagonal matrix \\ \(||\cdot||_{2}\) & Euclidean norm \\ \(<\cdot>_{i}\) & The \(i\)th element of the vector \\ \(\prod^{\perp}[A]\) & Orthogonal projection matrix onto the \\ \(\mathbb{R}^{m\times n}\) & Real matrices of size \(m\times n\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical symbols and explanations. therefore usually modeled as measurements with errors: \(\mathbf{h}_{i}=\mathbf{h}_{i}^{o}+\delta\mathbf{h}_{i},1\leq i\leq N_{1}\). The reflection height measurement vector is thus given as \(\mathbf{h}=\left[h_{1},h_{2},\dots,h_{N_{1}}\right]^{\mathrm{T}}=\mathbf{h}^{o}+\mathbf{n}_ {h}\), and \(\mathbf{h}^{o}=\left[h_{1}^{o},h_{2}^{o},\dots,h_{N_{1}}^{o}\right]^{\mathrm{T}}\) denotes the true value of the ionosphere reflection height. \(\mathbf{n}_{h}=\left[\delta h_{1},\delta h_{2},\dots,\delta h_{N_{1}}\right]^{ \mathrm{T}}\) denotes the measurement error, and its covariance matrix is denoted as \(\mathbf{Q}_{h}\). In the local coordinate system of the station, the target coordinate vector \(\mathbf{u}\) can be transformed:  
\[\mathbf{u}_{i}^{{}^{\prime}}= \begin{bmatrix}-\sin(\omega_{i,1})&\cos(\omega_{i,1} )&0\\ -\cos(\omega_{i,1})\sin(\omega_{i,2})&-\sin(\omega_{i,1})\sin(\omega_{i,2})& \cos(\omega_{i,2})\\ \cos(\omega_{i,1})\cos(\omega_{i,2})&\sin(\omega_{i,1})\cos(\omega_{i,2})& \sin(\omega_{i,2})\end{bmatrix} ^{\mathrm{T}}(\mathbf{u}-\mathbf{s}_{i}^{o}) \tag{3}\]  
The origin of the local coordinate system is the station. Therefore, \(\theta_{i}^{o}\) can be given as  
\[\theta_{i}^{o}=arctan\left(\frac{\mathbf{b}_{i,1}^{\mathrm{T}}\left(\mathbf{u}-\mathbf{s} _{i}^{o}\right)}{\mathbf{b}_{i,2}^{\mathrm{T}}\left(\mathbf{u}-\mathbf{s}_{i}^{o}\right)}\right) \tag{4}\]  
where \(\mathbf{b}_{i,1}=\left[-sin(\omega_{i,1}),cos(\omega_{i,1}),0\right]^{\mathrm{T}}\), \(\mathbf{b}_{i,2}=\left[-cos(\omega_{i,1})sin(\omega_{i,2}),-sin(\omega_{i,1})sin( \omega_{i,2}),cos(\omega_{i,2})\right]^{\mathrm{T}}\). According to (4), the true value vector of azimuth can be given as \(\mathbf{\theta}^{o}=\left[\theta_{1}^{o},\theta_{2}^{o},\dots,\theta_{N_{1}}^{o}\right] ^{\mathrm{T}}\).