# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Error Sources
#### 3.3.5 Multipath Error  
In ideal circumstances, each signal received from an overhead satellite arrives only along the least-time path. In practice, however, this so-called line-of-sight (LOS) component is accompanied by other components due to signal diffraction and single- or multiple-signal reflections off surrounding surfaces and obstacles (e.g., the glass facade of a nearby building, poles, trees, etc.). The complex baseband representation of the \(N\) signal components received from a particular satellite at a particular frequency and code is  
\[r(t)=\sum_{i=0}^{N-1}A_{i}(t)C[t-\tau_{i}(t)]\exp[j\theta_{i}(t)],\]  
where \(A_{i}\) is the amplitude of the \(i\)th component, \(C(t)\) is the GNSS code modulation, \(\tau_{i}(t)\) is the delay of the \(i\)th signal component relative to an unobstructed LOS signal, and \(\theta_{i}(t)\) is the beat carrier phase of the \(i\)th component. The combination of multiple components distorts the received signal and causes errors in the pseudorange and phase measurements. Unlike the study of ionospheric modeling errors, for application in urban mapping, multipath errors cannot be characterized with data from survey stations with a clear view of the sky. This section considers a simulation approach for scalable analysis of multipath tracking errors in an urban environment. The objective of this study was to inspect the presence of persistent biases caused by multipath due to the surrounding structure in the navigation solution averaged over multiple sessions  
Scenario Setup  
The present simulation study was based on the open-access Land Mobile Satellite Channel Model (LMSCM) [49], itself based on extensive experimentation with a wideband airborne transmitter at GNSS frequencies in urban and suburban environments. First, an urban corridor was simulated stochastically following the procedure described in [50]. The corridor was composed of buildings, trees, and poles. Some of the important parameters for the generation of the scene are summarized in Table 2, and a part of the scene realization is shown in Figure 3. Multi-GNSS satellite trajectories were generated at randomly-selected times based on GPS and Galileo satellite almanac data. An average of 25 satellites were available above an elevation mask of \(5^{\circ}\), consistent with modern multi-GNSS receivers. The satellites were assumed to be stationary over the simulation period of 60 s. Navigation solution errors were computed over 1000 60-s sessions. The vehicle trajectory was kept consistent across all 1000 driving sessions to avoid decorrelation of multipath error due to variable receiver motion. The trajectory was parametrized by its speed and heading, as described in [50]. The vehicle started at the zero coordinate on the along-track axis, and traveled in the positive direction, which was assumed to be aligned with the local north. The simulated trajectory was 60 s long and simulated a vehicle in stop-and-go traffic executing one 90\({}^{\circ}\) right turn, as shown in Figure 4. The vehicle traveled roughly 430 m and faced eastwards at the end of the trajectory. The three low-speed intervals, marked with red line segments in Figure 3, are expected to present severe multipath effects since multipath errors decorrelate slowly, and thus tend to reinforce one another within the navigation filter, when the vehicle moves slowly. \begin{table}
\begin{tabular}{l l l} \hline \hline Distance from road center to buildings & 24 m & Distance from road center to vehicle & 5 m \\ Mean distance between road center and trees & 20 m & Antenna height & 2 m \\ Mean building width & 30 m & Building width standard deviation & 25 m \\ Mean building height & 40 m & Building height standard deviation & 20 m \\ Probability of gap between buildings & 0.5 & Mean gap width & 30 m \\ Mean distance between trees & 60 m & Mean distance between poles & 25 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some urban scenario parameters. Figure 3: Initial segment of the simulated urban corridor.