# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Error Sources
#### 3.3.5 Multipath Error
Red lines across the road denote the positions where the vehicle is momentarily stopped. Multipath Simulation  
The LMSCM generates power, delay, and carrier phase for \(N\) LOS and echo signals. The interaction of the LOS with the simulated obstacles is governed by deterministic models for attenuation, diffraction, and delay. The LOS components of the combined signal, denoted \(r_{\text{LOS}}(t)\), may be composed of multiple components due to signal diffraction. These components are modeled as  
\[r_{\text{LOS}}(t)=\sum_{i=0}^{N_{\text{LOS}}-1}A_{i}(t)C[t-\tau_{i}(t)]\exp[j \theta_{i}(t)].\]  
In the special case of an unobstructed LOS signal, \(N_{\text{LOS}}=1\), \(A_{0}(t)=1\), \(\tau_{0}(t)=0\), and  
\[\theta_{0}(t)=\frac{\|r_{\text{R}}(t)-r_{\text{S}}(t)\|\cdot 2\pi}{\lambda}+ \gamma_{0},\]  
where \(\lambda\) denotes the wavelength and \(\gamma_{0}\) is a constant due to phase initialization in the satellite and receiver [51]. The LMSCM generates the \(N-N_{\text{LOS}}\) NLOS echoes stochastically based on satellite azimuth and elevation, receiver dynamics, and general characteristics of the scene (e.g., an _urban car_ scenario). This stochastic procedure might not be representative of multipath over multiple sessions through the same urban corridor, where certain echoes might persist over different sessions. To address this limitation, the LMSCM was augmented by the present authors to generate one- and two-bounce deterministic reflective NLOS echoes off the simulated buildings, and a one-bounce NLOS echo off the ground surface. These three additional reflective NLOS echoes, denoted \(r_{\text{DET}}(t)\), were added to \(r(t)\) and are modeled as  
\[r_{\text{DET}}(t)=\sum_{i=N}^{N+2}b_{i}(t)A_{i}(t)C[t-\tau_{i}(t)]\exp\bigl{[} j\bigl{(}\theta_{i}(t)+\theta_{i}^{\prime}(t)\bigr{)}\bigr{]},\]  
where \(b_{i}(t)\in\{0,1\}\) denotes whether the surrounding geometry supports the reflective echo. Since these reflections are expected to be the stronger than other diffracted and multiple-bounce NLOS echoes, the amplitudes \(A_{i}(t),i\in\{N,N+2\}\) for reflective echoes were drawn from the distribution of the strongest echo generated stochastically by the LMSCM at each epoch. By experiment, this distribution was found to be log-normal with \(20\log_{10}(A_{i})\sim\mathcal{N}(-22,5),i\in\{N,N+2\}\). The delays for the reflective echoes are given as  
\[\tau_{i}(t)=\frac{\|r_{\text{R}}^{\prime}(t)-r_{\text{S}}(t)\|-\|r_{\text{R}} (t)-r_{\text{S}}(t)\|}{c},\;i\in\{N,N+2\},\]  
Figure 4: Vehicle speed (solid line) and heading (dashed line) simulating stop-and-go motion with a 90\({}^{\circ}\) right turn. where \(\mathbf{r}_{\text{R}}^{\prime}(t)\) is the position of the imaginary _image_ antenna [52] about the reflecting plane (building or ground). Similarly, the carrier-phase of the reflective echoes is computed geometrically as  
\[\theta_{i}(t)=\frac{\|\mathbf{r}_{\text{R}}^{\prime}(t)-\mathbf{r}_{\text{S}}(t)\|\cdot 2 \pi}{\lambda}+\gamma_{0},\ i\in\{N,N+2\}.\]  
A random carrier-phase offset \(\theta_{i}^{\prime}(t)\in[0,2\pi)\) was added at the reflection point every time a new reflective echo was spawned to simulate the material-specific phase offset introduced by the reflection process. Receiver  
A receiver simulator was developed to account for the mediating effects that a receiver's tracking loops and navigation filter have on multipath-induced position errors in a receiver's reported position solution. The simulated receiver tracks the combination of all \(N_{\text{LOS}}\) line-of-sight signals and \(N+2-N_{\text{LOS}}\) multipath echoes for a given signal.