## 2 Zenith and Slant troposphere Modelling Strategy  
Materials in PPP processing, precise satellite orbits, and clocks are usually corrected with external products, then the basic PPP processing model can be expressed as follows:  
\[\begin{array}{c}L_{r}^{s}=\rho_{r}^{s}+c(\delta t_{r}-\delta t^{s})+m_{r}^{s}T _{r}-I_{r}^{s}+\lambda(N_{r}^{s}+B_{r}-B^{s})+\ell_{L}\\ P_{r}^{s}=\rho_{r}^{s}+c(\delta t_{r}-\delta t^{s})+m_{r}^{s}T_{r}+I_{r}^{s}+c(b _{r}-b^{s})+\ell_{P}\end{array}, \tag{1}\]  
where \(L\) and \(P\) refer to un-differenced observations of the original carrier phase and pseudo-range, respectively; \(\rho\) represents the geometric distance from the satellite to the receiver; \(\delta t_{r}\) and \(\delta t^{s}\) represent the difference between the receiver and the satellite clock, respectively; \(T\) and \(m\) represent ZTD and the corresponding mapping function, respectively; \(I\) represents the ionospheric delay in the slant direction; \(N_{r}^{s}\) represents the ambiguity parameter of carrier phase observation, \(B_{r}\) and \(B^{s}\) represent carrier phase hardware delays for receiver and satellite, respectively; \(b_{r}\) and \(b^{s}\) denote code hardware delay of receiver and satellite, respectively; and \(\ell_{L}\) and \(\ell_{P}\) denote the unmodeled errors of carrier phase observations and pseudo-range observations respectively, and in which including the influence of multipath error, observation noise, and so on.  
Zenith tropospheric delay reflects the vertical changes of the atmosphere, whereas the gradient parameters reflect the horizontal changes of the atmosphere. On this basis, ZTD and the gradient parameters can be used to construct STD, which can reflect the 3D distribution of atmosphere. The equation can be written as:  
\[\delta T=m_{H}ZHD+m_{W}ZWD+m_{G}[G_{N}\cos(A)+G_{E}\sin(A)], \tag{2}\]  
where _ZHD_ and _ZWD_ represent the dry and wet components of the tropospheric delay in the zenith direction, respectively; \(G_{N}\) and \(G_{E}\) represent the tropospheric gradient in the direction of horizontal east and north, respectively [14]; \(m_{H}\), \(m_{W}\), and \(m_{G}\) represent the mapping function of the tropospheric dry delay, wet delay, and horizontal gradients, respectively.  
During the PPP processing, ZTDs and associated gradients are estimated with proper stochastic models, which means that the estimates are constrained by the prior variance and its propagation value. Owing to the high variability of the water vapor distribution, or the lower weight of the satellite in low elevations, the estimated tropospheric parameters cannot reflect the true variations of atmosphere. Therefore, parts of the tropospheric delays can be absorbed by the carrier phase residuals. Based on this, we propose to construct the STD considering the residuals, which can be expressed as follows:  
\[STD=\delta T+res, \tag{3}\]  
where _res_ represents the carrier phase observation residual.  
Carrier phase observation residuals are combinations of the unmodeled errors, multipath errors, and observations noise. Figure 1 illustrates the carrier phase residual from forward filtering (SMT00) and backward filtering (SMT24). The posterior residual changes with the satellite elevations, and the backward filter can reduce the residual without affecting its distribution. However, the residuals can be affected by the systematical error, which is clearly observed from BDS residuals.  
As the final satellite orbit and clock products are used for post-processing, the unmodeled errors in the residuals are mainly the unmodeled troposphere; satellite antenna phase center errors may also exist [15]. Therefore, the use of raw phase residuals from backward PPP filters to reconstruct the STD products can help improve the performance of post-processed PPP.