# Estimation and Evaluation of Zenith Tropospheric Delay from Single and Multiple GNSS Observations
## 2 Observation Model and Processing Strategy
Due to short-term fluctuations, station coordinates and fixed satellite orbits allow the calculation and updating of satellite clocks every epoch. In the observation equations, the correlative terms can be eliminated when satellite orbits, clock corrections, and station coordinates are fixed, and the observation model could be interpreted as:  
\[\left\{\begin{array}{l}I_{r,j}^{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}\\ I_{r,j}^{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}\\ I_{r,j}^{E}=t_{r}+\lambda_{jE}\left(b_{rE,j}-b_{j}^{E}\right)+\lambda_{jE}N_{r, j}^{E}-\kappa_{jE}\cdot I_{r,1}^{E}+T_{r}^{F}+\epsilon_{r,j}^{E}\\ I_{r,j}^{R_{k}}=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_  
where \(l_{r,j}^{s}\) is \"observed minus computed\" phase and \(p_{r,j}^{s}\) is the pseudo-range observable. The tropospheric delay \(T_{r}^{s}\) can be calculated from the hydrostatic components \(Zh_{r}\), wet components \(Zw_{r}\), gradients, and mapping functions:  
\[T_{r}^{s}=Mh_{r}^{s}.Zh_{r}+Mw_{r}^{s}[Zw_{r}+\cot(e)\cdot(G_{N}\cdot\cos(a)+G_ {E}\cdot\sin(a))] \tag{7}\]  
With the Saastamoinen model and meteorological data, hydrostatic components \(Zh_{r}\) can be calculated very accurately, while the wet components \(Zw_{r}\) and gradients must be estimated as unknown parameters due to their variation in the atmosphere [35]; \(Mh_{r}^{s}\) and \(Mw_{r}^{s}\) are the hydrostatic and wet coefficients of the mapping function; \(G_{N}\) and \(G_{E}\) are the gradients in north and east directions; \(e\) and \(a\) are the elevation and azimuth angle. By using an ionosphere-free linear combination, ionospheric delays are eliminated in time-consuming network solutions, greatly decreasing the number of calculated parameters. To enhance the PPP performance, we utilize the raw GNSS observation model with temporal-spatial ionospheric constraints [36]. The vector X for estimated parameters can be expressed as:  
\[X=\left(Zw_{r}G_{N}G_{E}t_{r}d_{rE}d_{rC}d_{rR_{k}}l_{r,1}^{s}\overline{N_{r} }^{s}\right)^{T} \tag{8}\]  
\[\overline{N_{r}^{s}}=N_{r}^{s}+b_{r}+b^{s} \tag{9}\]  
In PPP processing, a sequential least squares filter is employed to estimate unknown parameters. The clock bias of receiver \(t_{r}\) is calculated to be white noise. The ionospheric delays \(l_{r,1}^{s}\) are estimated through dual-frequency raw pseudo-range and phase observations for each satellite at each epoch. The ISB and IFB parameters are calculated as constant. The phase ambiguity parameters \(\overline{N_{r}^{s}}\) are calculated as constant for each continuous arc and can absorb the phase delays \(b_{r}\) and \(b^{s}\). The global mapping function (GMF) [37] is used to map tropospheric parameters. The tropospheric wet delay \(Zw_{r}\) is modeled as a random walk process with the noise intensity of 5-10 mm/\(\sqrt{hour}\)[17; 22]. The stochastic model adopts the elevation-dependent weighting strategy \(Q=1/sin^{2}(ele)\), where \(ele\) is the elevation of the satellite. The elevation cut-off angle is set as \(7^{\circ}\). Table 1 presents a summary of the multi-GNSS data processing strategy used for ZTD estimation. Figure 1: The distribution of GNSS stations from MGEX networks and radiosonde stations. The black triangle is the GNSS station, and the red circle is the radiosonde station.