# GNSS Performance Modelling and Augmentation for Urban Air Mobility +
## 3 GNSS Performance Characterization
### Error Modelling
#### 3.1.2 Atmospheric Errors  
Atmospheric errors include both ionospheric and tropospheric delays. As the satellite signal passes through the ionosphere, there are two sources of delay: Firstly, propagation through a non-vacuum material delays the Pseudo Random Noise (PRN) codes while advancing the carrier phase. Secondly, signal refraction induces further delay. Error due to refraction can be neglected for most applications if the elevation of the satellite in question is \(15^{\circ}\) or more. The magnitude of the delay depends on the number of electrons that the signal encounters along its propagation path. Therefore, it is influencedboth by the ionosphere characteristics, which vary depending on the time of day and the season, and the elevation angle of the satellite. An approximation of the ionospheric delay is given by [23]:  
\[\Delta\tau=40.31\frac{\text{TEC}}{\text{cf}^{2}} \tag{5}\]  
where TEC is the total electron content integrated along the LOS to the satellite in units of electrons/m2, f is the frequency in Hz and c is the speed of light. TEC varies with time and depends on the location of the ionosphere 'pierce-point', or the intersection point of the LOS vector with the ionospheric layer. At the L1 frequency (1575.42 MHz), (5) can be written for a satellite that is not at the zenith:  
\[\Delta\tau=0.162\frac{\text{F.TEC}_{\text{v}}}{\text{c}} \tag{6}\]  
where TEC\({}_{\text{v}}\) is the TEC value for a vertical column located at the pierce point, in units of 1016 electrons per m\({}^{2}\) and is the obliquity factor. The active region of the ionosphere is typically represented as a thin shell at an elevation of 350 km, the obliquity can be approximated as a function of the satellite elevation angle in degrees (E):  
\[\text{F}=1+2.74\times 10^{-6}(96-\text{E})^{3} \tag{7}\]  
Depending on the receiver design, different models can be employed to compute the correction terms to be applied to the pseudorange before estimating position. L1 code-range receivers use a sinusoidal model of the ionosphere (the Klobuchar model), which accounts for the temporal variations of the ionospheric layers (low over night, rapidly increase after dawn, slow increase during the afternoon and rapid decrease after sunset). The sinusoidal parameters (amplitude and period) are transmitted in the navigation message:  
\[\text{IDV}=\left\{ \begin{array}{ccc}\text{DC}+\text{Acos}\Big{[}\frac{2 \pi(\text{t}-\Phi)}{\text{P}}\Big{]}&,&\text{day}\\ \text{DC}&,&\text{night}\end{array} \right. \tag{8}\]  
where _IDV_ (Ionospheric Vertical Delay) is expressed in nsec, _DC_ is the constant night-day offset (5 nsec), \(A\) is the amplitude (whose value is between 10 and 100 nsec), \(\Phi\) is the constant phase offset (14.00 hours), t is the local time, and P is the period. The two factors A and P are transmitted as coefficients of a cubic equation representing a model of the ionosphere with varying latitude. As the delay also depends on obliquity of the path, elevation is included as an additional factor in the equation [10]:  
\[\text{TID}=\Big{[}1+16(0.53-\text{E}_{h})^{3}\Big{]}\text{IDV} \tag{9}\]  
where TID is the Total Ionospheric Delay (nsec) and _E_\({}_{h}\) is the elevation angle of the satellites over the local horizon. As the ionosphere is dispersive at radio frequencies, the magnitude of the delay for two signals at different frequencies will be different. Dual frequency GNSS receivers (e.g., GPS P-code receivers) can therefore measure the difference between the time of reception of L1 (1575.42 MHz) and L2 (1227.60 MHz), and evaluate the delay associated with both of them. For L1 the delay is given by:  
\[\Delta\tau_{\text{L1}}=\Delta\tau\Bigg{[}\Bigg{(}\frac{\text{f}_{\text{L1}}}{ \text{f}_{\text{L2}}}\Bigg{)}^{2}-1\Bigg{]}^{-1} \tag{10}\]  
where \(\Delta\text{T}=40.31\frac{\text{TEC}}{\text{cf}^{2}}\left(\frac{1}{\text{f}_{ \text{L2}}^{2}}-\frac{1}{\text{f}_{\text{L1}}^{2}}\right)\).