# The Local Integrity Approach for Urban Contexts:
## 3 Experimental Validation and Data Analysis
### Computation of the Local Protection Levels  
After building an effective UERE data base as described above (_i.e._, a different matrix \(\overline{\Sigma}_{w}\) for each cell of the spatial grid), the \(\hat{\sigma}_{\text{UERE},eff}\) values were computed as from Equations (3) and (4). The actual number of satellites used by a specific receiver (\(N_{sat}\)) was considered during the computations in order to accordingly reshape the \(\overline{\Sigma}_{w}\) matrix (_i.e._, excluding rows and columns related to unused satellites). The next step for computing the Local Protection Levels, as from Equation (5), is the definition of proper \(k_{\text{AT}}\), \(k_{\text{CT}}\), and \(k_{\text{V}}\) multiplicative factors. It must be pointed out that their complete characterization in vehicular cases is an open issue (out of the scope of this work), because they depend on the specific application requirements and on the actual receiver performance. However, in order to validate the feasibility and to simply demonstrate the performance of the local integrity approach, empirical \(k_{\text{AT}}\), \(k_{\text{CT}}\), and \(k_{\text{V}}\) factors were set. Such choice aimed at ensuring a sufficient conservativeness of the ATPL, CTPL, and VPL with respect to actual Positioning Error (PE) values. These errors were computed in the Along-Track, Cross-Track and Vertical directions (ATPE, CTPE, VPE) as the difference between a reference track, obtained from a survey-grade receiver (shown as a green line in Figure 4), and the instantaneous positions estimated by the receiver under test (shown as a red line in Figure 4). An arbitrary success rate of 99% was targeted, intended as the percentage of positive cases where the local protection levels are conservative with respect to the absolute values of the Positioning Errors (in formulas, this condition corresponds to \(\text{ATPL}\geq\,|\,\text{ATPE}\,|\,\text{CTPL}\geq\,|\,\text{CTPE}\,|\,\text{ and VPL}\geq\,|\,\text{VPE}\,|\)). In this example, most of the errors had magnitude lower than 8 m: focusing on worst-case oriented statistics, the absolute errors assume maximum values equal to 6.8, 8.6 and 8.9 m for \(|\,\text{ATPE}\,|\,\text{,}\)\(|\,\text{CTPE}\,|\,\text{ and }\,|\,\text{VPE}\,|\,\text{,}\) whereas their 95% quantiles are 3.6, 5.2 and 5.7 m, respectively. In this case the minimum factors (\(k_{\text{AT}}\), \(k_{\text{CT}}\), or \(k_{\text{V}}\)) sufficient for ensuring at least the 99% of positive cases was 0.6, as demonstrated in Figure 5, with minor differences between the three orthogonal directions. At this point, the LPL values were computed choosing for simplicity \(k_{\text{AT}}=k_{\text{CT}}=k_{\text{V}}=0.6\) and the obtained results are reported in Figure 6, showing instantaneous absolute positioning errors, LPL values, and related statistics. In detail, the maximum LPL values in Figure 5a are equal to 80.3, 145.6 and 198.4 m for the ATPL, CTPL, and VPL, and 95% quantiles equal to 53.3, 84.1 and 109.7 m, respectively. It must be noticed that values larger than 100 m are obtained in three short time intervals only, corresponding to degraded geometrical conditions (_i.e._, reduced satellite visibility). The conservativeness of these values (especially VPL) is demonstrated in Figure 5b: in fact, non-integer events along the test (_i.e._, \(|\,\text{PE}\,|\,>\,\text{PL}\)) represent the 0.0%, 0.9%, and 0.5% of the cases for the Vertical, Along-Track, and Cross-Track directions, respectively. Aiming to clearly highlight the performance of the proposed approach, Stanford diagrams have also been computed from previous results (by means of MATLAB(r) scripts adapted from [19]). Figure 7 presents the obtained diagrams for the three orthogonal directions, where it can be noticed that most of the points are located in the upper-left areas of the plots (_i.e._, corresponding to conservative LPL values with respect to the positioning errors).