## 3 Measurement Weighting and Exclusion (WE)
### Measurement Weighting  
The exclusion of measurements obtained under NLOS conditions is the most appropriate thing to do in current GNSS receivers when using 2 pol antennas. Nevertheless, when the LOS is present we can do better trying to reduce the effects of multipath into the obtained measure. To do so, three different concepts were proposed in Ref. [9], namely  
1. **Measurement weighting**: The effects of multipath into the PVT solution can be reduced by estimating the standard deviation of the range measurements due to multipath and pass it to the navigation processor, so that the measures can be weighted accordingly [10; 20]. 2. **Range correction**: The idea is to estimate the offset included by multipath into the range measurements and apply them directly into the provided measurements [21]. 3. **Tracking correction**: The idea here is to use a common tracking loop for both RHCP and LHCP components that is closed using together the outputs of both the RHCP and LHCP tracking outputs to mitigate multipath [6]. So, whenever NLOS signals are detected (exclusion area) these should be discarded from the navigation solution. The rest of signals should be used after applying some multipath countermeasure. In this paper, we will focus on the measurement weighting solution. Specifically, in order to estimate the standard deviation of the measurements we will use an experimental model based on the RHCP and LHCP components, namely  
\[\sigma=f\left(P_{\mathrm{L}},P_{\mathrm{R}},\mathrm{CN0_{R}};a,b\right), \tag{3}\]  
where \(P_{\mathrm{L}}\) and \(P_{\mathrm{R}}\) stand for the prompt correlator value of the LHCP and RHCP component, respectively, \(\mathrm{CN0_{R}}\) the \(\mathrm{CN0}\) of the RHCP component, and \(\{a,b\}\) are the model parameters. The form of the model (i.e., \(f(\cdot)\)) may be derived theoretically, but the parameters have to be determined empirically. In doing so, we will take into account several practical aspects such as receiver design and/or antenna artifacts. A proper model may be of the form  
\[f\left(P_{\mathrm{L}},P_{\mathrm{R}},\mathrm{CN0_{R}}\right)=\sigma_{0}( \mathrm{CN0_{R}})\left[1+c_{1}\frac{P_{\mathrm{L}}}{P_{\mathrm{R}}}\right]+c _{2}, \tag{4}\]  
where \(\{c_{1},c_{2}\}\) are two constants, and \(\sigma_{0}(\cdot)\) the traditional model for the standard deviation used for \(\mathrm{CN0}\)- or elevation-based weighting [20; 22]. The weighting function parameters are optimized by using PVT residuals statistics. Figure 8: (a) RLR in a sky plot for the urban environment. Elevation masks based on the known location of the buildings in the scenario (see (b)). The idea in Equation (4) is to add a correction factor to the traditional weighting models. This correction factor increases as long as the LHCP to RHCP ratio increases (i.e., the RLR decreases), thus increasing the modeled standard deviation as long as the multipath effects are stronger (with respect to the LOS signal). This concept is similar as the one adopted in Refs. [23; 24] in order to model NLOS and ionospheric errors, respectively. Specifically, we consider the following model:  
\[\sigma^{2}=a\frac{1}{\text{CN0}_{\text{R}}}\left[1+c\frac{P_{\text{L}}}{P_{ \text{R}}}\right]=a\frac{1}{\text{CN0}_{\text{R}}}+b\frac{P_{\text{L}}}{\text{ CN0}_{\text{R}}P_{\text{R}}}, \tag{5}\]  
with \(\sigma\) the standard deviation of the measurement error. From Equation (4), we have used \(c_{1}=c\) and \(c_{2}=0\) in Equation (5) and the traditional CN0 weighting model. As already stated, the model parameters \(\{a,b\}\) should be determined empirically. Specifically, in order to include a proper correction factor, a two-dimensional optimization will be considered. This is for making sure that the correction factor included in Equation (5) increases when the multipath effects are stronger and not because any other effect such as signal attenuation. This fact is illustrated in Figure 9 in which we see how for a given value of CN0 in the RHCP component we have different pseudorange errors depending on the scenario we are.