# A Post-Processing Multipath/NLoS Bias Estimation Method Based on DBSCAN
## 2 Definition and Computation of the Leftover Term  
This section recalls the derivation of the pseudorange observation equation, as outlined in [18], in order to provide the definition of the leftover term containing the pseudorange biases due to multipath/NLoS effects, along with other receiver-related error components.  
The GNSS receiver generates pseudorange measurements by multiplying the signal travel time from the satellite to the receiver with the speed of light. It is noted that the superscript \(s\) is used to represent the pseudorandom noise (PRN) code of a specific GNSS satellite, and the subscript \(r\) is used to represent the GNSS receiver. The pseudorange measurement function is written as  
\[p_{r}^{s}(t)=c(d_{r}+dt_{r}(t))+\left\|\mathbf{r}^{s}(t)+\mathbf{r}_{r}( t)\right\|\\ +c\left(d^{s}-dt^{s}(t)+\delta t^{s,\text{rel}}_{\text{stc}}(t)- \delta t^{s,\text{rel}}_{\text{clk}}(t)\right)\\ +\xi^{s}_{r}(t)+\Gamma^{s}(t)+\Gamma^{s}(t)+\Gamma^{s}(t)+e^{s}_{ r}(t) \tag{1}\]  
where  
* \(p_{r}^{s}(t)\) is the pseudorange measurement obtained from the GNSS receiver for satellite \(s\) at time \(t\);
* \(c\) is the speed of light in vacuum;
* \(d_{r}\) is the signal instrumental delay of the receiver;
* \(dt_{r}(t)\) is the clock bias of the receiver at time \(t\);
* \(\mathbf{r}^{s}(t)\) is the mass center position of the satellite under Earth Centered Earth Fixed (ECEF) frames at time \(t\);
* \(\mathbf{r}_{r}(t)\) is the antenna reference point position of the receiver under ECEF frames at time \(t\);
* \(d^{s}\) is the signal instrumental delay of the satellite;
* \(dt^{s}(t)\) is clock bias of the satellite at time \(t\);
* \(\delta t^{s,\text{rel}}_{\text{clk}}(t)\) is the delay caused by space-time curvature of the relativistic effect at time \(t\);
* \(\delta t^{s,\text{rel}}_{\text{clk}}(t)\) is satellite clock bias caused by the relativistic effect at time \(t\);
* \(\xi^{s}_{r}(t)\) is antenna phase center corrections for both transmitting and receiving antennas at time \(t\);
* \(\Gamma^{s}(t)\) is the error contribution of the pseudorange measurement due to the ionospheric delay, expressed in meters at time \(t\);
* \(\Gamma^{s}(t)\) is the error contribution of the pseudorange measurement due to the tropospheric delay, expressed in meters at time \(t\);
* \(F^{s}(t)\) is the error contribution of the pseudorange measurement due to the multipath/NLoS interference at time \(t\), expressed in meters;
* \(e^{s}_{r}(t)\) is the error contribution of the pseudorange measurement due to the receiver noise at time \(t\), expressed in meters.  
The Sagnac effect caused by Earth's rotation should also be compensated for \(p_{r}^{s}(t)\) according to [18] (Chapter 19.1). By adjusting both sides of (1), we can obtain  
\[F^{s}(t)+e_{r}^{s}(t)+c(d_{r}+dt_{r}(t))\\ =p_{r}^{s}(t)-\|r^{s}(t)-r_{r}(t)\|-\delta_{r}^{s}(t)-I^{s}(t)-T^{ s}(t)\\ -c\left(d^{s}-dt^{s}(t)+\delta t^{s,\text{rel}}_{\text{stc}}(t)- \delta t^{s,\text{rel}}_{\text{clk}}(t)\right) \tag{2}\]  
Thanks to the advancements in the development of physical models and corrections offered by International GNSS Service (IGS), all the terms on the right-hand side of (2) can be computed in a post-processing manner, with relatively high accuracy, as outlined in Table 1. The left-hand side of (2) is designated as the leftover term \(L^{s}(t)\) of a pseudorange measurement:  
\[L^{s}(t) =F^{s}(t)+e_{r}^{s}(t)+c(d_{r}+dt_{r}(t)) \tag{3}\] \[=F^{s}(t)+e_{r}^{s}(t)+c\cdot dt_{\text{rev}}\]  
Since \(d_{r}\) and \(dt_{r}(t)\) are produced by the GNSS receiver itself, \(d_{r}\) remains constant under a specific epoch \(t\) for different satellites. Furthermore, the same \(dt_{r}(t)\) is also shared for every satellite under a certain epoch \(t\). Therefore, these two terms can be combined as a single term, which is denoted as \(dt_{\text{rev}}\).