## 2 GNSS Measurement Residuals
### Residual Computation
As the receivers share the same antenna, the effects of the atmosphere including ionospheric and tropospheric delays are cancelled out along with the multipath. \[\Delta\phi_{AB}^{P}=c(\delta t_{A}-\delta t_{B})+(N_{A}^{P}-N_{B}^{P})\lambda+ \epsilon_{AB}^{P} \tag{5}\]  
If we take the difference of the single differences between two satellites, \(P\) and \(Q\), we obtain the final ZBDD (\(\
abla\Delta\)) residual shown in (6). \[\
abla\Delta\phi_{AB}^{PQ}=\
abla\Delta N_{AB}^{PQ}\lambda+\epsilon_{AB}^{PQ} \tag{6}\]  
Although this method is effective in easily removing most of the non-noise terms, it has the disadvantage of underestimating the noise as well. All of the multipath and atmospheric noises between the satellites and the antenna splitter are common, so they are cancelled out [15]. Therefore, ZBDD will only be able to model the independent pure thermal noise terms present in the devices. SBDD is essentially the same as ZBDD, but the configuration does not share a single antenna. Therefore, the issues that stem from the use of a single antenna, which include underestimated noise and no information about the multipath and atmospheric noise,have been mitigated. However, the SBDD geometric range term and integer ambiguity resolution must be resolved as shown in (7). \[\
abla\Delta\phi_{AB}^{PQ}=\
abla\Delta R_{AB}^{PQ}+\
abla\Delta N_{AB}^{PQ} \lambda+\
abla\Delta M_{AB}^{PQ}+e_{AB}^{PQ} \tag{7}\]  
The geometric range term can be estimated if we know the baseline vector between the antennas corresponding to antennas \(A\) and \(B\). Assuming the baseline \(\Delta\overset{\rightarrow}{x}_{AB}\) is short enough to let the line-of-sight vectors \(\overset{\rightarrow}{G}_{A}\) and \(\overset{\rightarrow}{G}_{B}^{P}\) be the same as \(\overset{\rightarrow}{G}^{P}\) shown in (8), the relative geometry can be computed as (9). The geometry double difference (\(\
abla\Delta R_{AB}^{PQ}\)) can be calculated with the satellite-difference line-of-sight term (\(\
abla\overset{\rightarrow}{G}^{PQ}\)) and the baseline vector (\(\Delta\overset{\rightarrow}{x}_{AB}\)) as described in (10). \[\left. \begin{array}{l}\overset{\rightarrow}{G}=\left.(\overset{ \rightarrow}{x}^{P}-\overset{\rightarrow}{x}_{A})\right/\left|\overset{ \rightarrow}{x}^{P}-\overset{\rightarrow}{x}_{A}\right|\\ \overset{\rightarrow}{G}^{Q}=\left.(\overset{\rightarrow}{x}^{Q}-\overset{ \rightarrow}{x}_{A})\right/\left|\overset{\rightarrow}{x}^{Q}-\overset{ \rightarrow}{x}_{A}\right|\end{array} \right.\right\} \tag{8}\]  
where \(\overset{\rightarrow}{x}\) is the position vector of the receiver or the satellite. \[\Delta R_{AB}^{P}=\overset{\rightarrow}{G}\cdot\overset{\rightarrow}{( \overset{\rightarrow}{x}_{B}-\overset{\rightarrow}{x}_{A})}=\overset{ \rightarrow}{G}\cdot\Delta\overset{\rightarrow}{x}_{AB} \tag{9}\]  
\[\
abla\Delta R_{AB}^{PQ}=\Delta R_{AB}^{P}-\Delta R_{AB}^{Q}=\left(\overset{ \rightarrow}{G}^{P}-\overset{\rightarrow}{G}^{Q}\right)\cdot\Delta \overset{\rightarrow}{x}_{AB}=\
abla\overset{\rightarrow}{G}^{PQ}\cdot \Delta\overset{\rightarrow}{x}_{AB} \tag{10}\]  
For the integer ambiguities, there exist several methods to estimate and resolve the unknowns. These might include fixing the ambiguities, grid searching candidates, and variance reduction using the sequential quasi-Monte Carlo method [18; 22; 23]. For accurate noise modeling, it is important to exclude the outliers in carrier phase measurements, such as a cycle slip or clock jump, as they can introduce significant errors in the model. Fortunately, the detection of these outliers can be achieved with relative ease and accuracy through post-processing. The integers are fixed throughout the pass-over by dividing all the double-difference terms in the outlier-free session by the wavelength (\(\lambda\)), and they are rounded to the nearest integer as shown in (11).