# Evaluation of Ionospheric Delay Effects on Multi-GNSS Positioning Performance
## Materials and Methods
### Multi-GNSS Positioning Models
#### Dual-Frequency PPP  
The IF combination is often utilized for dual-frequency PPP to remove the first order ionospheric delay, which can be expressed as \[\left[\begin{array}{c}\alpha_{1,2}^{s}\cdot p_{r,1}^{s,1}+\beta_{1,2}^{s}\cdot p _{r,2}^{s,1}-\overline{p}_{r}^{s}\\ \alpha_{1,2}^{s}\cdot\varphi_{r,1}^{s,1}+\beta_{1,2}^{s}\cdot\varphi_{r,2}^{s,1} -\overline{p}_{r}^{s}\\ \vdots\\ \alpha_{1,2}^{s}\cdot p_{r,1}^{s,m}+\beta_{1,2}^{s}\cdot p_{r,2}^{s,m}- \overline{p}_{r}^{s}\\ \alpha_{1,2}^{s}\cdot\varphi_{r,1}^{s,m}+\beta_{1,2}^{s}\cdot\varphi_{r,2}^{s, m}-\overline{p}_{r}^{s}\end{array}\right]=\left[\begin{array}{cc}-\mathbf{u}_{r}^{s}&\mathbf{1}&\mathbf{M}_{ \mathbf{W}}&\mathbf{R}_{1}\end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ d\overline{t}_{r}\\ Z_{w}\\ \overline{N}_{r,1F}^{s}\end{array}\right]+\left[\begin{array}{c}\varepsilon _{p}\\ \varepsilon_{\varphi}\end{array}\right],\mathbf{Q}_{L} \tag{13}\]  
\[d\overline{t}_{r}=dt_{r}+d_{r,IF} \tag{14}\]  
\[\overline{N}_{r,IF}^{s}=(\alpha_{1,2}^{s}\cdot\lambda_{1}\cdot N_{r,1}^{s}+ \beta_{1,2}^{s}\cdot\lambda_{2}\cdot N_{r,2}^{s}+b_{r,IF}-b_{IF}^{s}-d_{r,IF} +d_{IF}^{s})/\lambda_{1} \tag{15}\]  
where \(\overline{N}_{r,IF}^{s}\) denotes the estimated float ambiguity values.  
For the dual-frequency UC PPP, the receiver UCD will be absorbed by the receiver clock offset and slant ionospheric delay and the model can be expressed as  
\[\left[\begin{array}{c}\sigma_{r,1}^{s,1}-\overline{p}_{r}^{s}\\ \sigma_{r,1}^{s,1}-\overline{p}_{r}^{s}\\ \vdots\\ \sigma_{r,2}^{s,m}-\overline{p}_{r}^{s}\end{array}\right]=\left[\begin{array}[ ]{cc}-\mathbf{u}_{r}^{s}&\mathbf{1}&\mathbf{M}_{\mathbf{W}}&\mathbf{K}&\mathbf{R}_{1}&\mathbf{R}_{2}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ d\overline{t}_{r}\\ \overline{T}_{r,1}^{s}\\ \overline{N}_{r,1}^{s}\\ \overline{N}_{r,2}^{s}\end{array}\right]+\left[\begin{array}{c}\varepsilon _{p}\\ \varepsilon_{\varphi}\end{array}\right],\mathbf{Q}_{L} \tag{16}\]  
\[\left\{\begin{array}{c}d\overline{t}_{r}=dt_{r}+d_{r,IF}\\ \overline{T}_{r,1}^{s}=\beta_{1,2}^{s}(\text{DCB}_{1,2}^{s}-\text{DCB}_{r,1,2} )\\ \lambda_{1}\overline{N}_{r,1}^{s}=\lambda_{1}(N_{r,1}^{s}+b_{r,1}-b_{1}^{s})+d _{IF}^{s}-d_{r,IF}+\beta_{1,2}^{s}(\text{DCB}_{1,2}^{s}-\text{DCB}_{r,1,2})\\ \lambda_{2}\overline{N}_{r,2}^{s}=\lambda_{2}(N_{r,2}^{s}+b_{r,2}-b_{2}^{s})+d _{IF}^{s}-d_{r,IF}+(f_{1}^{s}/f_{2}^{s})^{2}\beta_{1,2}^{s}(\text{DCB}_{1,2}^{ s}-\text{DCB}_{r,1,2})\end{array}\right. \tag{17}\]  
For the ionosphere-constrained dual-frequency PPP, virtual ionospheric observations with their constraints are added and the model can be expressed as  
\[\left[\begin{array}{c}p_{r,1}^{s,1}-\overline{p}_{r}^{s}\\ \varphi_{r,1}^{s,1}-\overline{p}_{r}^{s}\\ \vdots\\ p_{r,2}^{s,m}-\overline{p}_{r}^{s}\\ \overline{l}_{r,1}^{s,1}\\ \vdots\\ \widehat{l}_{r,1}^{s,m}\end{array}\right]=\left[\begin{array}{cccccc}-\mathbf{u}_{r }^{s}&\mathbf{1}&\mathbf{I}&\mathbf{M}_{\mathbf{W}}&\mathbf{K}&\mathbf{R}_{1}&\mathbf{R}_{2}\\ \end{array}\right]\left[\begin{array}{c}\mathbf{x}\\ d\overline{t}_{r}\\ \text{DCB}_{r,1,2}\\ Z_{w}\\ \overline{l}_{r,1}^{s}\\ \overline{N}_{r,1}^{s}\\ \overline{N}_{r,2}^{s}\end{array}\right]+\left[\begin{array}{c}\varepsilon _{p}\\ \varepsilon_{\varphi}\\ \varepsilon_{\varphi}\end{array}\right],\mathbf{Q}_{L^{\prime}}\mathbf{Q}_{I} \tag{18}\]  
\[\left\{\begin{array}{c}\lambda_{1}\overline{N}_{r,1}^{s}=\lambda_{1}(N_{r,1 }^{s}+b_{r,1}-b_{1}^{s})+d_{IF}^{s}-d_{r,IF}\\ \lambda_{2}\overline{N}_{r,2}^{s}=\lambda_{2}(N_{r,2}^{s}+b_{r,2}-b_{2}^{s})+d _{IF}^{s}-d_{r,IF}\\ \end{array}\right. \tag{19}\]  
In matrix \(\mathbf{J}\), the element for \(p_{r,1}^{s}-\overline{p}_{r}^{s}\) is set to \(\beta_{1,2}^{s}\), and the element for \(p_{r,2}^{s}-\overline{p}_{r}^{s}\) is set to \(-\alpha_{1,2}^{s}\), to acquire the corresponding receiver DCB.