# Evaluation of Ionospheric Delay Effects on Multi-GNSS Positioning Performance
## Materials and Methods
### Multi-GNSS Positioning Models
#### 2.1.2 Single-Frequency PPP  
One crucial problem in single-frequency PPP, is that the ionospheric delay cannot be mitigated by the combination of observations on different frequencies. In the GRAPHIC approach, the arithmetic mean of the pseudo-range and carrier phase is considered as the observation where the ionospheric delays are canceled out since the pseudo-range and carrier phase delays have the same values but opposite signs. Thus, the observations with \(m\) observed satellites can be expressed as \[\left[\begin{array}{c}(p_{r,1}^{s,1}+\lambda_{1}\varphi_{r,1}^{s,1})/2-\overline{ \rho}_{r}^{s}\\ \vdots\\ (p_{r,1}^{s,m}+\lambda_{1}\varphi_{r,1}^{s,m})/2-\overline{\rho}_{r}^{s}\\ \end{array}\right]=\left[\begin{array}{cc}-\mathbf{u}_{r}^{s}&\mathbf{1}&\mathbf{M}_{\mathbf{ W}}&\mathbf{R}_{1}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ \begin{array}{c}d\overline{t}_{r}\\ Z_{w}\\ \overline{N}_{r,1}^{s}\end{array}\right]+\left[\varepsilon_{p,\varphi}\right],\mathbf{Q}_{L} \tag{6}\]  
\[d\overline{t}_{r}=dt_{r}+d_{r,1}/2+\lambda_{1}\cdot b_{r,1}/2 \tag{7}\]  
\[\overline{N}_{r,1}^{s}=(\lambda_{1}(N_{r,1}^{s}-b_{1}^{s})+d_{IF}^{s})/\lambda _{1}/2 \tag{8}\]  
where \(\mathbf{M}_{\mathbf{W}}\) is the wet mapping function of tropospheric delay; \(\mathbf{R}_{1}\) denotes the corresponding wavelength of carrier phase to the ambiguity parameters on the first frequency; \(Z_{w}\) is the tropospheric zenith wet delay (ZWD); and \(\overline{N}_{r,1}^{s}\) denotes the estimated float ambiguity values on the first frequency.  
For the single-frequency UC PPP, the pseudo-range and carrier phase observations with \(m\) observed satellites for a single epoch can be written as  
\[\left[\begin{array}{c}p_{r,1}^{s,1}-\overline{\rho}_{r}^{s}\\ \varphi_{r,1}^{s,1}-\overline{\rho}_{r}^{s}\\ \vdots\\ p_{r,1}^{s,m}-\overline{\rho}_{r}^{s}\\ \end{array}\right]=\left[\begin{array}{cc}-\mathbf{u}_{r}^{s}&\mathbf{1}&\mathbf{M}_{\mathbf{ W}}&\mathbf{K}&\mathbf{R}_{1}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ d\overline{t}_{r}\\ Z_{w}\\ \frac{l_{r,1}^{s}}{N_{r,1}^{s}}\end{array}\right]+\left[\begin{array}{c} \varepsilon_{p}\\ \varepsilon_{\varphi}\end{array}\right],\mathbf{Q}_{L} \tag{9}\]  
\[d\overline{t}_{r}=dt_{r}+d_{r,1} \tag{10}\]  
\[\overline{N}_{r,1}^{s}=N_{r,1}^{s}+b_{r,1}-b_{1}^{s}-d_{r,1}/\lambda_{1}+d_{ IF}^{s}/\lambda_{1} \tag{11}\]  
where the matrix K denotes the corresponding coefficient of slant ionospheric delay, the element for pseudo-range observation is \(1\) while the element for carrier phase is \(-1\).  
Compared to single-frequency UC PPP, the ionosphere-constrained single-frequency PPP adds virtual ionospheric observations from the external model with their corresponding constraints, which can be expressed as  
\[\left[\begin{array}{c}p_{r,1}^{s,1}-\overline{\rho}_{r}^{s}\\ \varphi_{r,1}^{s,1}-\overline{\rho}_{r}^{s}\\ \vdots\\ p_{r,1}^{s,m}-\overline{\rho}_{r}^{s}\\ \varphi_{r,1}^{s,m}-\overline{\rho}_{r}^{s}\\ \overline{\gamma}_{r,1}^{s,1}\\ \vdots\\ \overline{\gamma}_{r,1}^{s,m}\end{array}\right]=\left[\begin{array}{cccc}-\mathbf{u }_{r}^{s}&\mathbf{1}&\mathbf{M}_{\mathbf{W}}&\mathbf{K}&\mathbf{R}_{1}\\ \mathbf{O}&\mathbf{O}&\mathbf{O}&\mathbf{I}&\mathbf{O}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ d\overline{t}_{r}\\ Z_{w}\\ \frac{l_{r,1}^{s}}{N_{r,1}}\end{array}\right]+\left[\begin{array}{c} \varepsilon_{p}\\ \varepsilon_{\varphi}\\ \varepsilon_{ion}\end{array}\right],\mathbf{Q}_{L},\mathbf{Q}_{I} \tag{12}\]  
where \(\widetilde{l}_{r,1}^{s}\) is the virtual ionospheric observation derived from external ionospheric models; \(\mathbf{O}\) denotes zero matrix; \(\mathbf{I}\) is the identity matrix; and \(\mathbf{Q}_{\mathbf{I}}\) denotes the stochastic model of virtual ionospheric observations.