# A Post-Processing Multipath/NLoS Bias Estimation Method Based on DBSCAN
## 3 Multipath/NLoS Bias Estimation Using a Clustering Algorithm
### Statistical Characterization of the Leftover Terms  
Analyzing \(L^{s}(t)\) as in (3), different components with specific features can be recognized:  
1. The error contribution of the pseudorange measurement due to the multipath/NLoS interference \(F^{s}(t)\) is usually different from one satellite to another. Furthermore, \(F^{s}(t)\) is zero if a satellite is free from both multipath and NLoS interference. 2. The receiver noise \(e_{r}^{s}(t)\) is commonly characterized by a Gaussian distribution with a zero mean and constant variance \(\sigma^{2}\) under multipath/NLoS-free conditions [19]. However, when multipath/NLoS effects occur, the receiver noise still follows a Gaussian distribution with a zero mean and a different variance \(\sigma_{F}^{2}\)[20]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Term** & **Computation Method** \\ \hline \(p_{r}^{s}(t)\) & Measurements from the receiver \\ \(\mathbf{r}^{t}(t)\) & IGS final precise orbits products \\ \(\mathbf{r}_{r}(t)\) & RTK or RTK/INS positioning solutions \\ \(\delta_{r}^{s}(t)\) & Absolute IGS phase center corrections (igs14.atx) \\ \(I^{s}(t)\) & Final solution of IGS combined GIMs \\ \(T^{s}(t)\) & Saastamonem model \\ \(d^{s}\) & TGD provided by navigation messages \\ \(dt^{s}(t)\) & Clock biases of the satellites from navigation messages \\ \(\delta t^{s,\text{rel}}_{\text{stc}}(t)\) & See Appendix A \\ \(\delta t^{s,\text{rel}}_{\text{clk}}(t)\) & See Appendix B \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Corrections and models for pseudorange measurement. 3. The user clock bias term \(dt_{\text{rcv}}\) keeps the same value for every satellite for a certain epoch. In both conditions, with and without multipath/NLoS effects, \(e^{s}_{r}(t)\) is relatively small compared to \(F^{s}(t)\). Typically, in the absence of multipath/NLoS effects, the range error due to receiver noise is in the order of \(\pm 1\,\text{m}\) for a geodetic-quality receiver and antenna [21]. According to [20], the variance \(\sigma_{F}^{2}\) increases by approximately two times compared to \(\sigma^{2}\) when multipath/NLoS effects occur. However, the ranging bias due to multipath can reach up to about \(70\,\text{m}\) for GPS L1 C/A signals with one-chip early-to-late spacing [22]. Moreover, NLoS conditions may induce biases in pseudorange measurements spanning several kilometers [3]. Given that \(dt_{\text{rcv}}\) and \(F^{s}(t)\) are constant for a given epoch, the leftover term \(L^{s}(t)\) for a specific satellite can be statistically modeled as Gaussian random variables in both conditions, whether multipath/NLoS effects occur or not. \[L^{s}(t)\sim \begin{cases}\mathcal{N}\big{(}c\cdot dt_{\text{rcv}},\sigma^{2} \big{)}&\text{without multipath/NLoS}\\ \mathcal{N}\big{(}c\cdot dt_{\text{rcv}}+F^{s}(t),\sigma_{F}^{2}\big{)}&\text{ with multipath/NLoS}\end{cases}  \tag{4}\]  
The proposed method in this research leverages the consistency checking principle to detect and estimate the multipath/NLoS effects using leftover terms. This research assumes that at least two satellites are free from the impact of multipath/NLoS effects, which can cover most conditions according to the previous study [23]. Under this assumption, \(L^{s}(t)\) without multipath/NLoS effects is expected to follow the distribution \(\mathcal{N}\big{(}c\cdot dt_{\text{rcv}},\sigma^{2}\big{)}\), forming a cluster whose size is controlled by \(e^{s}_{r}(t)\). On the other hand, \(L^{s}(t)\) with multipath/NLoS effects should follow the distribution \(\mathcal{N}\big{(}c\cdot dt_{\text{rcv}}+F^{s}(t),\sigma_{F}^{2}\big{)}\), exhibiting a mean \(F^{s}(t)\). Given that \(F^{s}(t)\) is significantly larger than both \(\sigma\) and \(\sigma_{F}\), the leftover terms affected by multipath/NLoS will be separated from the cluster formed by leftover terms without multipath/NLoS effects. However, it is unlikely that the leftover terms impacted by multipath/NLoS can form a consistent cluster due to the distinct values of \(F^{s}(t)\) for each satellite resulting from different reflection paths.