# A Performance-Based Airspace Model for Unmanned Aircraft Systems Traffic Management
## 3 Urban Airspace Model
### CNS Risk Protection Volume Generation
#### 3.1.2 Surveillance Inflations  
As anticipated in Table 2, the uncertainty in the position of a detected intruder aircraft is dependent on two factors. First, the error in localizing the aircraft, and second, the latency between such detection and the position estimate in a downstream separation assurance system. A conventional surveillance scenario is illustrated in Figure 7. When Primary Surveillance Radar (PSR) is utilized, the estimated range, azimuth and elevation to the intruder aircraft is used to compute its position in a cartesian reference frame. Assuming the reference geometry depicted in Figure 7, target state vector information is measured relative to the radar site in a spherical coordinate system in range, elevation and azimuth (\(\tau_{\text{RDR}}\), \(\eta_{\text{RDR}}\), \(\epsilon_{\text{RDR}}\) respectively). The measurements in each of the elements are prone to an SNR dependent random range measurement error, which can be calculated as:  
\[\sigma_{\text{rRDR}}^{2}\ =\frac{\text{c}}{2\text{B}\sqrt{2(\text{SNR})}} \tag{5}\]  
where B is waveform bandwidth, c is the speed of light and signal to noise ratio (SNR). Radar angular measurements are commonly made using monopulse receive antennas that provide a difference pattern characterized by a deep null on boresight. The difference pattern formed by these beams may be used to measure the target angular position with a single signal transmission. The measurement accuracy in each angular coordinate is characterized by the RMS of the SNR dependent random angular measurement error, angular bias, and random measurement error. As with the range error, we assume angular error to be normally distributed:\[\sigma_{\text{eRDR}}^{2} = \sigma_{\text{AN}_{\text{e}}}^{2}\ +\ \sigma_{\text{AF}_{\text{e}}}^{2}\ +\ \sigma_{\text{AB}_{\text{e}}}^{2} \tag{6}\] \[\sigma_{\text{nRDR}}^{2} = \sigma_{\text{AN}_{\text{n}}}^{2}\ +\ \sigma_{\text{AF}_{\text{n}}}^{2}\ +\ \sigma_{\text{AB}_{\text{n}}}^{2} \tag{7}\]  
As with the range errors, the SNR dependent error dominates the radar angle error:  
\[\sigma_{\text{AN}}=\frac{v}{\text{k}_{\text{m}}\sqrt{2(\text{SNR})}} \tag{8}\]  
where: \(v\) is the radar beamwidth in the angular coordinates and \(\text{k}_{\text{m}}\) is the monopulse pattern difference slope. The tracking covariance is then:  
\[\text{Q}_{\text{TRK}}^{\text{SPH}}=\left[\begin{array}{ccc}\sigma_{\text{TRK }}^{2}&0&0\\ 0&\sigma_{\text{FRK}}^{2}&0\\ 0&0&\sigma_{\text{TRK}}^{2}\end{array}\right] \tag{9}\]  
Errors in these measurements propagate to the Cartesian position uncertainty. A more detailed treatment of these errors and the resulting ellipsoid is presented in [42]. On the other hand, when ADS-B is employed as the primary means of surveillance, the accuracy of localizing the intruder aircraft is closely related to GNSS performance. In addition, the tracking uncertainty ellipsoid must also account for the error due to latency. The total tracking uncertainty \(\sigma_{\text{tracking}}\) is then:  
\[\sigma_{\text{tracking}}^{2}=\sigma_{\text{NAV}}^{2}+\sigma_{\text{com}}^{2} \tag{10}\]  
where \(\sigma_{\text{NAV}}\) is the uncertainty in GNSS-estimated position, and \(\sigma_{\text{com}}\) is the uncertainty due to communication latency.  
The overall latency of a SA/CA procedure depends not only on the surveillance latency but also on the time required to assess a collision threat and to generate and execute a resolution. In conventional operations, these tasks would have been performed by the pilot and ATC operator. In the UTM context, however, the bulk of these tasks are assumed to be performed autonomously, with manual intervention required only in emergency conditions.  
The total deconfiction time (\(\text{t}_{\text{SA/CA}}\)) is the result of the sequential tasks illustrated in Figure 8, which includes the time taken to process the tracks of intruder aircraft, assess potential collision threats, generate avoidance trajectories and execute them.  
Figure 7: Conventional surveillance scenario.