# Assessment of Affordable Real-Time PPP Solutions for Transportation Applications
## 2 Materials and Methods
### Mathematical Model of Multi-GNSS Kinematic RT-PPP  
The Basic GNSS observation formulas for both the code range and carrier phase can be expressed mathematically [50]:  
\[P_{i}^{I}=\rho_{r}^{I}+cdt_{r,J}-cdt^{I}+\frac{f_{i}^{2}}{f_{0}^{2}}I_{i}^{I}+T_ {r}^{I}+\varepsilon_{p,i}^{I} \tag{1}\]  
\[\Phi_{i}^{I}=\rho_{r}^{I}+cdt_{r,J}-cdt^{J}-\frac{f_{i}^{2}}{f_{0}^{2}}I_{i}^{I} +T_{r}^{I}+\lambda_{i}N_{i}+\varepsilon_{\Phi,i}^{I} \tag{2}\]  
where \(i\), \(J\), and \(r\) denote the signal frequency, GNSS satellite, and receiver, respectively; \(P\) and \(\Phi\) are the code range and carrier phase measurements, respectively; \(\rho\) defines the satellite-receiver geometric range; \(c\) represents the speed of light in a vacuum; \(dt_{r,J}\) is the receiver clock error; \(dt^{I}\) represents the satellite clock errors; \(f_{i}\) represents the carrier phase frequency; \(f_{0}\) is the basic GNSS frequency (\(f_{0}=10.23\) MHz); \(I_{i}^{I}\) represents the ionospheric delay; \(T\) is the tropospheric delay; \(\lambda_{i}\) is the carrier phase's wavelength; \(N_{i}\) is the real-value ambiguity term; \(\varepsilon_{p,i}\) and \(\varepsilon_{\Phi,i}\) are the noise in the code and carrier. It is well known that ionospheric delay is the dominant error source for GNSS measurements; therefore, ionosphere-free linear combinations with both the code and the carrier are used to remove the ionospheric delay. Furthermore, a precise point positioning processing model is applied depending on the ionosphere-free linear combination, in addition to other corrections, such as using accurate satellite orbit and clock parameters. The fundamental multi-GNSS ionosphere-free dual-frequency real-time precise point positioning mathematical model can be defined as follows [50]:  
\[P_{3}^{G}=\rho_{r}^{G}+c\Big{(}dt_{r,G}-dt^{G}\Big{)}+c\Big{(}b_{r,G,3}-b_{3}^ {G}\Big{)}+T_{r}^{G}+\varepsilon_{p,3}^{G} \tag{3}\]  
\[\Phi_{3}^{G}=\rho_{r}^{G}+c\Big{(}dt_{r,G}-dt^{G}\Big{)}+c\Big{(}\delta_{r,G,3} -\delta_{3}^{G}\Big{)}+T_{r}^{G}+\lambda N_{3}^{G}+\varepsilon_{\Phi,3}^{G} \tag{4}\]  
\[P_{3}^{I}=\rho_{r}^{I}+c\Big{(}dt_{r,G}-dt^{I}\Big{)}+c\Big{(}b_{r,G,3}-b_{3}^ {I}\Big{)}+T_{r}^{I}+ISB^{I}+\varepsilon_{p,3}^{I} \tag{5}\]  
\[\Phi_{3}^{J}=\rho_{r}^{I}+c\Big{(}dt_{r,G}-dt^{I}\Big{)}+c\Big{(}\delta_{r,G,3 }-\delta_{3}^{I}\Big{)}+T_{r}^{I}+ISB^{I}+\lambda N_{3}^{J}+\varepsilon_{\Phi,3}^{I} \tag{6}\]  
where \(P_{3}\) and \(\Phi_{3}\) indicate the ionosphere-free pseudorange and phase linear combinations, respectively; \(G\) is the GPS satellite system; \(J\) refers to other navigation systems (i.e., GLONASS or Galileo); \(b_{r,G,3}\) denotes the ionosphere-free receiver differential code bias (DCB); \(\delta_{r,G,3}\) represents the ionosphere-free receiver differential phase bias (DPB); \(b_{3}^{G}\) and \(b_{3}^{J}\) are the ionosphere-free satellite differential code biases for GPS and other satellite systems, respectively; \(\delta_{3}^{G}\) and \(\delta_{3}^{I}\) refer to the ionosphere-free satellite differential phase biases for GPS and other navigation systems, respectively; \(ISB^{I}\) corresponds to the inter-system bias (ISB), which includes the difference in receiver DCB between the GPS and other satellite systems; \(\lambda N_{3}^{G}\) and \(\lambda N_{3}^{I}\) represent the ionosphere-free real-value ambiguity terms for GPS and GNSS, respectively. Regarding the RT-PPP processing model, the satellite and receiver DCBs are grouped into corresponding clock parameters; in addition, the satellite clock parameter is corrected using real-time satellite clock products.