# The Local Integrity Approach for Urban Contexts:
## 3 Experimental Validation and Data Analysis
### Simplified URE Model for Demonstration Purposes  
It must be noticed that the described effective URE data base has applicability limited to a specific urban area and to a specific time slot. Aiming at developing a proof-of-concept demonstrator suitable to a wider urban area, a simplified URE model was extracted from the available GNSS measurements by means of a curve fitting procedure.  
The measurements obtained from the five consumer-grade receivers mentioned in Section 3.1 were comparatively assessed. The available pseudorange residuals were analysed _versus_ the satellite elevations and _versus_ the Carrier-to-Noise density ratio (C/N\({}_{0}\)) values, as reported in Figure 8a,b, respectively.  
Figure 8: Results of the linear fitting for the covariance of the residuals (linear scale on vertical axes):  
(**a**) _versus_ satellite elevations; (**b**) _versus_ C/N\({}_{0}\) measurements.  
Figure 7: Assessment of the conservativeness of LPL values in terms of Stanford plots: (**a**) Vertical direction; (**b**) Along-Track direction; (**c**) Cross-Track direction.  
Quite clear trends can be identified for all the receivers, showing better pseudorange measurement quality (_i.e._, lower covariance of the residuals) in case of high elevation satellites and high C/N\({}_{0}\) levels. Similar behaviours can be noticed in Figure 8 for almost all the tested receivers. The only exception is the uBlox-4T, which was the oldest device under test and showed almost flat covariance values _versus_ elevation and C/N\({}_{0}\). Thus, apart this specific device, it is possible to conclude that it is reasonable to build an effective UERE data base by combining measurements from different receivers from same category (consumer-grade), even if they belong to different manufactures. In any case possible calibration factors can be off-line estimated in order to finely tune possible differences between individual devices and/or to compensate for slightly different trends.  
Two piecewise linear fits (based on just five points each) were extracted from the trends, as reported by the thicker green lines in Figure 8. Aiming at better highlighting the trends of the covariance values _versus_ satellite elevations and C/N\({}_{0}\) measurements, previous results are plotted with a logarithmic scale for the vertical axis in Figure 9.  
In this case the measurements from the uBlox-4T receiver have been excluded, demonstrating a good agreement between the performances of other receivers under test. In addition, the plots in Figure 9 further confirm the validity of the two piecewise linear fits (thicker green lines).  
At this point, both the linear fits were used to build a simplified UERE model: starting from the instantaneous elevation and C/N\({}_{0}\) values for each \(j\)-th satellite used by a receiver, two separate covariance values can be obtained from the fits in Figure 8a,b, respectively. A weighted average of these two values can be computed and used as an estimate of the covariance of the residuals for each \(j\)-th satellite (_i.e._, \(\hat{\sigma}_{\mathbf{w}}^{2}\)). The obtained covariance values have been used as the diagonal elements of an approximated \(\overline{\boldsymbol{\Sigma}}_{xy}\) matrix, setting other off-diagonal elements to zero.  
This approach represents a simplified UERE model, which is intended for demonstration purposes only, because the underlying assumptions clearly degrade the reliability of resulting LPL values that must be computed with Equations (3)-(5). Nonetheless, it allows the estimation of effective UERE with a limited computational burden, avoiding the use of a huge data base for storing the covariance matrixes for each grid cell and for each time slot. For this reason, it has been adopted in the prototype implementation presented in next section.