# A GNSS-Cellular Network Hybridization Strategy for Robust Positioning
## 2 Materials and Methods
### Measurement-Level Hybridization Method
#### 2.2.1 Pseudoranges Between GNSS Satellites and the UE  
Standard point positioning (SPP) in GNSS typically relies on pseudorange code observations to determine the user's position, offering meter-level accuracy. However, this accuracy is influenced by several factors, including hardware delays, ionospheric delays, and multipath errors. Even though modeled terms, such as tropospheric and ionospheric delays and relativistic effects, are corrected, residual errors still remain [32]. The GNSS pseudorange observation equation \(R_{r}^{s}\) can be expressed as follows [33]:  
\[R_{r}^{s}=\rho_{r}^{s}+d\rho_{r}^{s}+\text{c}(dt_{r}-dt^{s})+I_{R_{r}^{s}}+T_ {R_{r}^{s}}+dm_{R_{r}^{s}}+E_{rd}+r_{tn} \tag{14}\]  
where  
* \(\rho_{r}^{s}\) is the observed geometric distance between the receiver r and the satellite s;
* \(d\rho_{r}^{s}\) is the linearization of the satellite-receiver geometric range;
* \(c\) is the speed of light;
* \(dt_{r}\) is the receiver clock error;
* \(dt^{s}\) is the satellite clock error;
* \(I_{R_{r}^{s}}\) is the ionospheric delay on the GNSS signals;
* \(T_{R_{r}^{s}}\) is the tropospheric delay on the GNSS signals;
* \(dm_{R_{r}^{s}}\) is the multipath error for the pseudorange;\(E_{rel}\) is the relativistic error;
* \(r_{tn}\) represents residual errors. Equation (14) can be rewritten as follows:  
\[R_{r}^{s}-D_{r}^{s}-\rho_{r}^{s}=d\rho_{r}^{s}+cdt_{r} \tag{15}\]  
where  
\[D_{r}^{s_{1}}=-cdt^{s}+I_{R_{r}^{s}}+T_{R_{r}^{s}}+dm_{R_{r}^{s}}+E_{rel}+r_{tn} \tag{16}\]  
The right-hand side of Equation (15) contains the unknown parameters: the receiver's position correction \(d\rho_{r}^{s}\) and the receiver clock offset \(dt_{r}\). Next, linearizing the satellite-receiver geometric range gives the following:  
\[d\rho_{r}^{s}=\left[\frac{x^{s}-x_{r}}{\rho_{r}^{s}}\right]dx+\left[\frac{y^{ s}-y_{r}}{\rho_{r}^{s}}\right]dy+\left[\frac{z^{s}-z_{r}}{\rho_{r}^{s}} \right]dz \tag{17}\]  
where \((x^{s},y^{s},z^{s})\) and \((x_{r},y_{r},z_{r})\) represent the geocentric coordinates of the satellite and receiver, respectively. In matrix form, assuming that \(m\) satellites are observed, the \(A\) matrix is as follows:  
\[A=\left[ \begin{array}{cccc}l_{x}^{s_{1}}&l_{y}^{s_{1}}&l_{z}^{s_{1}}&1\\ l_{x}^{s_{2}}&l_{y}^{s_{2}}&l_{z}^{s_{2}}&1\\ &\vdots\\ l_{x}^{s_{m}}&l_{y}^{s_{m}}&l_{z}^{s_{m}}&1\end{array} \right]_{m\
eq 4} \tag{18}\]  
where the coefficients \(l_{x}^{s}\), \(l_{y}^{s}\), and \(l_{z}^{s}\) are derived from Equation (17):  
\[l_{x}^{s}=\frac{x^{s}-x_{r}}{\rho_{r}^{s}};\;l_{y}^{s}=\frac{y^{s}-y_{r}}{\rho _{r}^{s}};\;l_{z}^{s}=\frac{z^{s}-z_{r}}{\rho_{r}^{s}} \tag{19}\]  
The unknowns in vector \(x\) (Equation (1)) are the position corrections and the receiver clock offset:  
\[x=\left[ \begin{array}{c}dx\\ dy\\ dz\\ cdt_{GNSS}\end{array} \right]_{4x1} \tag{20}\]  
The vector \(b\) of the GNSS measurements is as follows:  
\[b_{GNSS}=\left[ \begin{array}{c}R_{r}^{s_{1}}-D_{r}^{s_{1}}-\rho_{r}^{s_{1} }\\ R_{r}^{s_{2}}-D_{r}^{s_{2}}-\rho_{r}^{s_{2}}\\.\\.\\ R_{r}^{s_{m}}-D_{r}^{s_{m}}-\rho_{r}^{s_{m}}\end{array} \right]_{m\
eq 3} \tag{21}\]The weight matrix \(P_{GNSS}\) is as follows:  
\[P_{GNSS}=\left[ \begin{array}{ccc}\frac{1}{\sigma_{\text{ICSNS}}^{2}}&0&0\\ 0&\frac{1}{\sigma_{\text{ICSNS}}^{2}}&0\\.&.&.\\.&.&.\\.&.&.\\ 0&0&\frac{1}{\sigma_{\text{ICSNS}}^{2}}\end{array} \right]_{m\mathbf{x}m} \tag{22}\]  
and the diagonal elements of matrix \(P_{GNSS}\) are a function of the \(m\) satellite elevation angle [34]:  
\[\sigma_{\text{{}_{\text{SCNSS}}}}^{2}=\frac{\sigma_{\text{{}_{\text{GSNS}}}}^{ 2}}{\left(m\text{ {sat }s\text{ elevation angle}}\right)^{2}} \tag{23}\]  
where \(\sigma_{\text{{}_{\text{GSNS}}}}^{2}\) is a constant variance value. In Equation (23), the scalar value \(\sigma_{\text{{}_{\text{GSNS}}}}^{2}\) corresponds to a reference satellite elevation angle of 45\({}^{\circ}\), selected as a representative intermediate value between 0\({}^{\circ}\) and 90\({}^{\circ}\) to balance the signal quality and availability in the weighting scheme.