# Development of a Submillimetric GNSS-Based Distance Meter for Length Metrology +
## 3 GNSS Multipath Mitigation
Once the consecutive days are matched using azimuth and elevation, the median of the residuals is retained as multipath error. The result is a site-specific and time-specific multipath model for the combination used in the PPP solution. Interestingly, the model formed may contain some possible antenna calibration mismodelling since this error also repeats with azimuth and elevation. In the third step, the site-specific and time-specific models that were built up for each satellite are combined to form the required ionosphere-free DD residuals. This combination largely cancels those components of the residuals which originate in the satellites, such as orbits and clocks, those related to the receiver clock offsets and possible residual atmospheric errors. Finally, the L1/E1 DD multipath corrections are obtained for each day by using the ionosphere-free DD corrections and the corresponding DD residuals from an independent DD auxiliary solution. For that independent DD solution several frequencies such as L2, L5, E5a or E5b can be used as long as they were employed in the PPP solution. Obviously, the noise of the new multipath-corrected DD equation is higher then the original one, but that is not a problem for the case at hand as long as the multipath error is mitigated in such a way that the resulting distance is not biased. The observation-domain sidereal filtering for GPS Precise Point Positioning (PPP) developed at the SGNL-UCL [31] uses the ionosphere-free linear combination \(L_{3}\) with the following combination factors  
\[\begin{array}{llll}\alpha=\frac{77\lambda_{3}}{\lambda_{1}}&=\frac{f_{1}^{2 }}{f_{1}^{2}-f_{2}^{2}}&=2.5457277801631\\ \beta=-\frac{60\lambda_{3}}{\lambda_{2}}&=-\frac{f_{2}^{2}}{f_{1}^  
\[\lambda_{3}=\frac{\lambda_{1}\lambda_{2}}{77\lambda_{2}-60\lambda_{1}}\simeq 0.006 \ \text{m}\]  
and with standard noise values \(\sigma_{L_{1}}=\sigma_{L_{2}}=2\) mm, the resulting \(L_{3}\) noise is  
\[\sigma_{\text{L3}}=\sqrt{a^{2}\sigma_{L_{1}}^{2}+\beta^{2}\sigma_{L_{2}}^{2}}= 0.006\ \text{m}\]  
Nonetheless, modern receivers can measure the carrier phase with better precision and from our experience the values \(\sigma_{L_{1}}=\sigma_{L_{2}}=1\) mm are more realistic so that the resulting noise for the \(L_{3}\) combination can be considered around 3 mm. Since the noise of the ionosphere-free combination is \(\sigma_{L3}\approx 3\) mm, the obtained DD residuals are expected to have a theoretical noise \(\sigma_{DD_{L3}}\approx 6\) mm that could hamper the determination of a \(L_{1}\) site-specific multipath model by using a PPP approach. Surprisingly, if \(DD_{L1}\) residuals are obtained straight from the \(DD_{L3}\) residuals by substracting the corresponding \(DD_{L2}\) residuals, the resulting residuals have a noise \(\sigma_{DD_{L1}}\approx 2\) mm. The reason is that the coefficients are smaller than one. \[L_{1}=\frac{1}{\alpha}L_{3}-\frac{\beta}{\alpha}L_{2} \tag{10}\]  
and therefore,  
\[\sigma_{L_{1}}^{2}=\left(\frac{1}{\alpha}\right)^{2}\sigma_{L_{3}}^{2}+\left( \frac{\beta}{\alpha}\right)^{2}\sigma_{L_{2}}^{2}\approx 2\ \text{mm} \tag{11}\]  
Figure 5 illustrates the PPP ionosphere-free phase residuals for pillar 3 (PL3A) (see Figure 1) and satellite G12 after the matching process. As can be seen, the residuals from the four consecutive days of the experiment are consistent and show a clear sidereal pattern. Therefore, the initial assumption for the PPP residuals to be mostly multipath error is confirmed. When the residuals from the consecutive days are compared with each other, the differences are normally below 5 mm, thus confirming the correctness of the functional model used in the PPP implementation and its suitability for this application. Nevertheless, in some cases the PPP residuals differ by up to 1 cm, possibly due to a slightly different satellite trajectory, different humidity of the ground or technicalities of the particular PPP implementation.