# Improved PPP Ambiguity Resolution with the Assistance of Multiple LEO Constellations and Signals
## 2 Data and Methods
### Multi-Frequency LEO-Augmented PPP AR Method  
To achieve the multi-frequency LEO-augmented PPP ambiguity resolution, the extra-wide-lane, wide-lane, and narrow-lane UPD corrections of GNSS and LEO satellites should be estimated first. The float extra-wide-lane and wide-lane ambiguities (\(\overline{N}_{r,ewl}^{s}\) and \(\overline{N}_{r,wl}^{s}\)) can be calculated by the Melbourne-Wubbena (MW, [34; 35]) combinations, and the float narrow-lane ambiguities (\(\overline{N}_{r,wl}^{s}\)) can be derived from the integer wide-lane ambiguities and ionosphere-free ambiguities. The calculation formulation of extra-wide-lane, wide-lane, and narrow-lane ambiguities can be expressed as follows:  
\[\overline{N}_{r,wl}^{s} =(\frac{L_{r,2}^{s}}{\lambda_{2}}-\frac{L_{r,3}^{s}}{\lambda_{3} }-\frac{f_{2}P_{r,2}^{s}+f_{3}P_{r,3}^{s}}{(f_{2}+f_{3})\lambda_{ext}}) \tag{3}\] \[=N_{r,ewl}^{s}+d_{r,ewl}-d_{ewl}^{s}\]  
\[\overline{N}_{r,wl}^{s} =(\frac{L_{r,1}^{s}}{\lambda_{1}}-\frac{L_{r,2}^{s}}{\lambda_{2} }-\frac{f_{1}P_{r,1}^{s}+f_{2}P_{r,2}^{s}}{(f_{1}+f_{2})\lambda_{ext}}) \tag{4}\] \[=N_{r,wl}^{s}+d_{r,wl}-d_{wl}^{s}\]\[ \begin{array}{ll}\lambda_{nl}\cdot\overline{N}_{r,nl}^{s}&=\lambda_{IF}\cdot \overline{N}_{r,IF}^{s}-\frac{c\frac{f_{2}}{f_{1}^{2}-f_{2}^{2}}\cdot N_{r,nl}^{ s}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad =N_{r,nl}^{s}+d_{r,nl}-d_{nl}^{s}\end{array}  \tag{5}\]  
where \(N_{r,ext}^{s}\), \(N_{r,ext}^{s}\) and \(N_{r,nl}^{s}\) are integer extra-wide-lane, wide-lane, and narrow-lane ambiguities, respectively; \(\lambda_{ext}\), \(\lambda_{wl}\) and \(\lambda_{nl}\) denote the wavelength of the extra-wide-lane, wide-lane, and narrow-lane ambiguities, which can be written as \(\lambda_{eval}=\frac{c}{f_{2}-f_{3}}\), \(\lambda_{wl}=\frac{c}{f_{1}-f_{2}}\) and \(\lambda_{nl}=\frac{c}{f_{1}+f_{2}}\); \(\overline{N}_{r,IF}^{s}\) denotes ionosphere-free ambiguity and \(\lambda_{IF}\) refers to the corresponding wavelength; and \(d_{r,ext}^{s}\), \(d_{r,wl}\) and \(d_{r,nl}\) denote extra-wide-lane, wide-lane, and narrow-lane UPDs at the receiver side, respectively, whereas \(d_{ext}^{s}\), \(d_{url}^{s}\) and \(d_{nl}^{s}\) denote the corresponding satellite UPDs. Based on the ambiguities from a global MGEX tracking network, the extra-wide-lane, wide-lane, and narrow-lane UPDs can be precisely estimated with the method of least squares estimation [7; 36]. We estimated the extra-wide-lane and wide-lane UPDs as constant during a whole day since the extra-wide-lane and wide-lane UPDs barely change over several months [37], while the narrow-lane UPD is estimated epoch by epoch considering its temporal characteristic [3]. Several specific issues of the UPD estimation for each system should be mentioned. To avoid the influence of the GLONASS inter-frequency biases on the UPD estimation, stations with homogeneous receivers should be employed to estimate the GLONASS UPD. Since the observation of LEO is designed to have the same frequencies of GPS, we can ignore the inter-system biases between these two systems in the UPD estimation. With the improvement of model precision of observations, the fractional parts of narrow-lane ambiguities are relatively stable and can be forecasted for a few hours. Thus, in the process of narrow-lane UPD estimation, if the satellite cannot be observed at this epoch at a station, the fractional parts of ambiguities at the last epoch will be used for narrow-lane UPD estimation to ensure that enough observations are made. With the UPD corrections, the multi-frequency LEO-augmented PPP AR was achieved by fixing extra-wide-lane, wide-lane, and narrow-lane ambiguities step by step. The satellite UPDs can be removed by the UPD products, while the receiver UPDs of extra-wide-lane and wide-lane ambiguities were obtained by averaging fractional parts of ambiguities with the satellite UPDs corrected. After the removal of the UPDs, the extra-wide-lane (\(N_{eval}^{s}\)) and wide-lane (\(N_{val}^{s}\)) ambiguities were fixed by a rounding strategy [38].