# A Post-Processing Multipath/NLoS Bias Estimation Method Based on DBSCAN
## 4 Experiments
### Dynamic Experiment
#### 4.2.2 Experimental Results of Multipath/NLoS Estimation  
Figure 12 illustrates all the \(L^{s}(t)\) values for the pseudorange measurements computed using the corrections and models outlined in Table 1. It is worth noting that, due to the Trimble receiver's relatively large clock bias and drift, the dominating factor in the \(L^{s}(t)\) values is the term \(c\cdot dt_{\text{rcv}}\). Consequently, the visual impact of \(L^{s}(t)\) values influenced by multipath/NLoS effects may not be immediately apparent in Figure 12. However, the DBSCAN algorithm is still able to recognize and segregate \(L^{s}(t)\) values affected by multipath/NLoS effects. This is achieved by identifying deviations from the general trend formed by \(L^{s}(t)\) values that are free from multipath/NLoS effects. This can be observed in the magnified section of Figure 12.  
To mitigate the impact of clock bias and drift on visualization, the DBSCAN algorithm is initially employed to identify \(L^{s}(t)\) values unaffected by multipath/NLoS effects within the largest cluster. Subsequently, the time series of the estimated \(c\cdot\hat{dt}_{\text{rcv}}\) is computed by averaging all the \(L^{s}(t)\) values within this cluster for each epoch. Figure 13 illustrates the time series of \(L^{s}(t)-c\cdot\hat{dt}_{\text{rcv}}\) within the largest cluster. As per (3), \(L^{s}(t)-c\cdot\hat{dt}_{\text{rcv}}\) should only be contributed by receiver noises for the cluster free from multipath/NLoS effects. The majority of the values in Figure 13 fall within the \(\pm 2\,\mathrm{m}\) range, which has a similar level to the receiver noise. This observation validates the analysis in Section 3.1, which further shows the effectiveness of the DBSCAN algorithm.  
Figure 12: Computed leftover terms \(L^{s}(t)\) for GPS satellites.  
Then, the \(\hat{F}^{s}(t)\) values can be separated from the leftover terms out of the largest cluster based on (7). Figure 14 presents \(\hat{F}^{s}(t)\) for each satellite. It is observed that PRN 5 produced significant multipath/NLoS biases, reaching up to 120 m. This is more likely caused by NLoS rather than multipath, considering that the maximum multipath bias for GPS L1 C/A signals is estimated to be around 70 m, as discussed in the theoretical analysis in [18].  
Like the static dataset, a new GNSS dataset is generated by applying these \(\hat{F}^{s}(t)\) to pseudorange measurements to evaluate the effectiveness of the estimated \(\hat{F}^{s}(t)\) values. Subsequently, new positioning results are obtained using these adjusted pseudoranges.  
Figure 15 shows a comparison of positioning trajectories with and without multipath/NLoS compensation. The positioning accuracy after the multipath/NLoS compensation proves that these computed \(\hat{F}^{s}(t)\) values are close to the real pseudorange biases caused by multipath/NLoS effects. Figure 16 illustrates the time series of positioning errors in both the horizontal and vertical directions before and after the multipath/NLoS compensation. The CDFs of these positioning errors are presented in Figure 17. The 3D RMSE shows a significant improvement, decreasing from 26.0103 m to 5.5094 m (reduced by 79%), using the new dataset that compensates for multipath/NLoS biases.  
Figure 14: Estimated multipath/NLoS bias on pseudorange.  
Figure 15: Positioning trajectory before and after pseudorange multipath/NLoS compensation in the horizontal plane.  
Table 3 provides the horizontal and vertical errors, specifically the 95 th percentile of the CDFs, obtained from the GNSS datasets before and after the compensation for multipath/NLoS effects.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Horizontal [m]** & **Vertical [m]** \\ \hline Before Compensation & 38.75 & 28.68 \\ After Compensation & 9.98 & 6.43 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Horizontal and vertical positioning errors at 95 th percentile of the CDFs.  
Figure 16: Positioning error time series before and after pseudorange multipath/NLoS compensation in East, North, and Up directions.