# A Post-Processing Multipath/NLoS Bias Estimation Method Based on DBSCAN
## 4 Experiments
### Static Experiment
#### 4.1.2 Experimental Results of Multipath/NLoS Estimation  
As an initial step, all the corrections and models listed in Table 1 were computed using the library provided by RTKLIB [24]. Subsequently, the pseudorange leftover terms \(L^{s}(t)\) for all epochs were calculated applying these corrections and models. Figure 5 depicts all the \(L^{s}(t)\) values, which are marked in different colors to distinguish the corresponding satellite pseudorandom noise. It can be seen that most \(L^{s}(t)\) values exhibit similar trends, indicating they are influenced solely by \(c\cdot dt_{\rm rev}\) and \(c_{t}^{s}(t)\). Conversely, certain \(L^{s}(t)\) values deviate from this trend, suggesting the presence of the large biases \(F^{s}(t)\) caused by multipath/NLoS effects. PRN 18 shows significantly different \(L^{s}(t)\) values compared to other PRNs, indicating a high probability of suffering from multipath/NLoS effects. Based on the skyplot in Figure 4 and the surroundings of the GNSS receiver in Figure 3, the reflected signal for PRN 18 may be attributed to the building on the west side of the road. Figure 4: Skyplot with corresponding signal \(C/N_{0}\). Figure 5: Computed leftover terms \(L^{s}(t)\) for GPS satellites. Then, Figure 6 shows all the \(\hat{L}^{s}(t)\) values in the largest cluster provided by DBSCAN, together with their mean value estimated as \(c\cdot\hat{d}_{\text{rcv}}\). Given the estimated \(c\cdot\hat{d}_{\text{rcv}}\), the \(\hat{F}^{s}(t)\) values can be obtained for the leftover terms not belonging to the largest cluster based on (7), and they are depicted in Figure 7. It can be seen that positive values of \(\hat{F}^{s}(t)\) are observed more frequently than negative values, consistent with the theoretical analysis in [2]. In fact, multipath can introduce both positive and negative biases to pseudoranges, while NLoS tends to produce only positive biases. Therefore, positive biases are more frequent. Figure 6: Leftover terms in the largest cluster determined by DBSCAN and \(c\cdot\hat{d}_{\text{rcv}}\). Figure 7: Estimated multipath/NLoS bias on pseudoranges. After obtaining the \(\hat{F}^{s}(t)\), it is necessary to prove the reliability of these estimates. Since there are no ground truth values for pseudorange measurement, in this work, we use a Generalized Least Squares (GLSs) positioning algorithm operating on pseudoranges corrected for the estimated values of \(\hat{F}^{s}(t)\). Even if, as explained in the experimental setup, a reliable ground truth for the position can be obtained, it is not straightforward to obtain the ground truth for each pseudorange measurement since, in particular, a true value for the receiver clock \(dt_{r}(t)\) in (1) is not known. Only the estimated value is available, and such estimates are affected by the presence of multipath/NLoS biases. For this reason, to validate the accuracy of the estimated multipath/NLoS biases, we prove the effectiveness of the method at the Position, Velocity, and Time level by comparing the accuracy of the solution obtained using pseudoranges with and without the application of the correction for the multipath/NLoS biases. Following this principle, new positioning results are computed using a new GNSS dataset, which is generated by compensating for \(\hat{F}^{s}(t)\) in the pseudorange measurements. This compensating action is conducted by subtracting the estimated \(\hat{F}^{s}(t)\) from the corresponding pseudorange measurements. Figure 8 compares positioning scatters with and without multipath/NLoS compensation. Given that \(\hat{F}^{s}(t)\) is estimated from the leftover term, which contains ground truth information, the scatter of positioning after multipath/NLoS compensation is concentrated around the true position with high accuracy. This indicates that the computed \(\hat{F}^{s}(t)\) values closely approximate the actual pseudorange biases caused by multipath/NLoS effects. Figure 9 shows the positioning error time series in both the horizontal and vertical directions before and after the multipath/NLoS compensation.