# The Local Integrity Approach for Urban Contexts:
## 2 Local GNSS Integrity Approach
### Effective UERE and Local Protection Levels  
The considerations in previous sections can be used in order to estimate the quality of the pseudoranges without resorting to generic analytical UERE models, as in the classic integrity framework. A possible solution is to adopt a novel _\"effective UERE\"_ parameter, which is defined in [10] as the ensemble average, at a certain position and time, of several \"instantaneous\" estimates of the covariance of the residuals. In synthesis, this parameter can be computed by means of the following operative steps:  
1. Collect \(M\) independent observations of the vector of pseudorange residuals \(\mathbf{w}_{n}\) (with \(n=1,\,\ldots,\,M\)) from a set of \(N_{sat}\) satellites used in PVT solution. The observations \(\mathbf{w}_{n}\) are obtained from different closely-spaced receivers in the VANET and/or combining measurements performed in different days, but with nearly the same geometrical conditions, velocity, and then similar signal degradations.
2. Compute the ensemble mean of the covariance matrix of the residuals as: \[\overline{\mathbf{\Sigma}}_{w}=\frac{1}{M}\sum_{n=1}^{M}\left(\mathbf{w}_{n} \,\mathbf{w}_{n}^{\mathrm{T}}\right)\] (2) where it must be noted that \(\left(\mathbf{w}_{n}\,\mathbf{w}_{n}^{\mathrm{T}}\right)\) is a square matrix \(\left(\mathbf{R}^{N_{sat}\times N_{sat}}\right)\) representing an instantaneous estimate of the covariance matrix at the \(n\)-th time instant.
3. Estimate the _effective UERE variance_\(\left(\hat{\sigma}_{\mathrm{UERE},eff}^{2}\right)\), starting from the Frobenius-norm of \(\overline{\mathbf{\Sigma}}_{w}\) (_i.e._, the square rooted trace of \(\overline{\mathbf{\Sigma}}_{w}^{T}\overline{\mathbf{\Sigma}}_{w}\)) and the number of satellites \(N_{sat}\). \[\hat{\sigma}_{\mathrm{UERE},eff}^{2}=\frac{\left|\,\overline{\mathbf{\Sigma}} _{w}\,\right|\,\mathrm{I}_{\mathrm{Fro}}}{\sqrt{N_{sat}-4}}=\sqrt{\frac{\sum \mathrm{diag}\left\{\overline{\mathbf{\Sigma}}_{w}^{T}\overline{\mathbf{ \Sigma}}_{w}\right\}}{N_{sat}-4}}\] (3)
4. Obtain the _effective UERE standard deviation_\(\left(\hat{\sigma}_{\mathrm{UERE},eff}\right)\) as: \[\hat{\sigma}_{\mathrm{UERE},eff}=\sqrt{\hat{\sigma}_{\mathrm{UERE},eff}^{2}}\] (4)  
Such an aggregated \(\left(\hat{\sigma}_{\mathrm{UERE},eff}\right)\) parameter represents a statistical characterization of the \"local\" environmental effects close to the receiver position, taking into account the satellite/receiver geometry and the measurement quality.  
As conceptually shown in Figure 1, this aggregated position- and time-dependent information can be stored in a centralized data base and then properly re-routed across the VANET in order to predict the GNSS signal quality in each position and at each hour in the GNSS time (orbital) frame.  
These operations represent the first step in order to estimate LPL values. In detail, LPLs can be defined adopting an Along-Track, Cross-Track, Up (ACU) reference frame, related to the orientation of a car in a road segment, as in Figure 2.  
Figure 2: Concept of Along-Track, Cross-Track, and Vertical Protection Levels.  
The Along-Track, Cross-Track, and Vertical Protection Levels (ATPL, CTPL, VPL) can be expressed as:  
\[<x>\text{PL}=k_{<x>}\cdot\hat{\sigma}_{\text{UEEE},eff}\sqrt{\left[\left(\mathbf{ H}_{\text{ACU}}^{T}\mathbf{H}_{\text{ACU}}\right)^{-1}\right]_{(x,jx)}} \tag{5}\]  
where \(<x>\) stands for either {AT, CT, V} and \(j_{x}\) = {1, 2, 3} respectively; furthermore, \(\mathbf{H}_{\text{ACU}}\) is the linear connection matrix computed in the ACU frame (depending on the satellite/user geometry) and \(k_{\text{AT}}\), \(k_{\text{CT}}\), and \(k_{\text{V}}\) are three multiplicative dimensionless factors which propagate the local position confidences to a level compatible with proper integrity requirements (these should be related to the intended application). Interested readers can find more details and the complete analytical derivation of previous expressions in [10].