# Estimation and Evaluation of Zenith Tropospheric Delay from Single and Multiple GNSS Observations
## 2 Observation Model and Processing Strategy  
A combination of dual-frequency carrier-phase and pseudo-range (LC, PC) is commonly used to eliminate the first-order ionospheric delay in PPP processing. The observation equation is given as follows [31],  
\[L_{r,j}^{s}=\rho_{r\,g}^{s}-t^{s}+\lambda_{j}\Big{(}b_{r,j}-b_{j}^{s}\Big{)}+ \lambda_{j}N_{r,j}^{s}-I_{r,j}^{s}+T_{r}^{s}+\varepsilon_{r,j}^{s} \tag{1}\]\[P_{r,j}^{s}=\rho_{r}^{s}\,g-t^{s}+t_{r}+c\left(d_{r,j}-d_{j}^{s}\right)+I_{r,j}^{s} +T_{r}^{s}+e_{r,j}^{s} \tag{2}\]  
where \(s\), \(r\), and \(j\) denote GNSS satellite, receiver, and frequency, respectively; \(\rho_{g}\) is the geometric distance; \(t^{s}\) and \(t_{r}\) refer to satellite and receiver clock offset; \(\lambda_{j}\) is the wavelength; \(b_{r,j}\) and \(b_{r}^{s}\) are the uncalibrated phase delays; \(T_{r}^{s}\) is the tropospheric delay; at different frequencies, the ionospheric delays \(I_{r,j}^{s}\) can be expressed as \(I_{r,j}^{s}=\kappa_{j}\cdot I_{r,1}^{s}\), \(\kappa_{j}=\lambda_{j}^{2}/\lambda_{1}^{2}\); and \(N_{r,j}^{s}\) is the inter ambiguity; \(e_{r,j}^{s}\) and \(e_{r,j}^{s}\) are the pseudo-range and carrier phase observations noises, which cannot be modeled; and \(d_{r,j}\) and \(d_{j}^{s}\) are the code biases. The offsets of antenna and phase wind-up can be rectified [32]. By interpolating the tidal constituents at each station according to Finite Element Solution 2004 and [PERSON] et al. 's models [33;34], oceanic and atmospheric tidal loading can be corrected. The GPS + BDS + Galileo + GLONASS observation model is given as follows in a multi-GNSS environment,  
\[\left\{ \begin{array}{l}L_{r,j}^{G}=\rho_{r}^{G}-t^{G}+t_{r}+\lambda_{jG} \left(b_{rG,j}-b_{j}^{G}\right)+\lambda_{jG}N_{r,j}^{G}-\kappa_{jG}\cdot I_{r, 1}^{G}+T_{r}^{s}+\epsilon_{r,j}^{G}\\ L_{r,j}^{C}=\rho_{r}^{C}g-t^{C}+t_{r}+\lambda_{jC}\left(b_{rC,j}-b_{j}^{C} \right)+\lambda_{jC}N_{r,j}^{C}-\kappa_{jC}\cdot I_{r,1}^{C}+T_{r}^{C}+ \epsilon_{r,j}^{C}\\ L_{r,j}^{E}=\rho_{r}^{F}g-t^{E}+t_{r}+\lambda_{jE}\left(b_{rE,j}-b_{j}^{F} \right)+\lambda_{jE}N_{r,j}^{E}-\kappa_{jE}\cdot I_{r,1}^{E}+T_{r}^{F}+ \epsilon_{r,j}^{F}\\ L_{r,j}^{R_{k}}=\rho_{r}^{R}g-t^{R}+t_{r}+\lambda_{jR_{k}}\left(b_{rR_{k},j}-b_ {j}^{R}\right)+\lambda_{jR_{k}}N_{r,j}^{R}-\kappa_{jR_{k}}\cdot I_{r,1}^{R}+T _{r}^{R}+\epsilon_{r,j}^{R}\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}P_{r,j}^{G}=\rho_{r}^{G}g-t^{G}+t_{r}+c\left(d_{r,G,j }-d_{j}^{G}\right)+\kappa_{jG}\cdot I_{r,1}^{G}+T_{r}^{G}+\epsilon_{r,j}^{G} \\ P_{r,j}^{C}=\rho_{r}^{C}g-t^{C}+t_{r}+c\left(d_{r,G,j}-d_{j}^{C}\right)+\kappa _{jC}\cdot I_{r,1}^{C}+T_{r}^{C}+\epsilon_{r,j}^{C}\\ P_{r,j}^{E}=\rho_{r}^{E}g-t^{E}+t_{r}+c\left(d_{r,E,j}-d_{j}^{E}\right)+\kappa _{jE}\cdot I_{r,1}^{E}+T_{r}^{E}+\epsilon_{r,j}^{F}\\ P_{r,j}^{R_{k}}=\rho_{r}^{R}g-t^{R}+t_{r}+c\left(d_{rR,j}-d_{j}^{R}\right)+ \kappa_{jR_{k}}\cdot I_{r,1}^{R}+T_{r}^{F}+\epsilon_{r,j}^{R}\end{array} \right. \tag{4}\]  
where G, C, E, and R stand for the GPS, BDS, Galileo, and GLONASS satellites, respectively; \(R_{k}\) is the GLONASS satellite with frequency k; and the code biases \(d_{rG}\), \(d_{rC}\), \(d_{rE}\), and \(d_{rR_{k}}\) are different in the receiver. It is crucial to consider the intersystem biases (ISBs), which refer to differences between systems, and the interfrequency biases (IFB), which are different between the receiver code biases \(d_{rR_{k}}\) for GLONASS satellites with different frequencies when processing multi-GNSS combined observations. The GPS code bias is set to zero and the ISBs are in relation to the GPS satellites' biases. Firstly, the precise satellite orbits and clocks are determined using the observation data of 55 stations, which are shown in Figure 1. It is important to note that for each system and for each GLONASS frequency, zero mean conditions for the ISB/IFB parameters are involved in the determination of satellite orbits and clocks.