## 5 Simulation Method Uncertainties
### Angular Error
#### 5.2.2 Atmospheric Approximations  
An additional source of error in our calculations comes from atmospheric bending. This can be accounted for in a number of ways, including, in order of increasing complexity: neglecting it (as with the Cartesian and spherical earth approximations in the preceding section), using a four-third earth-radius model [34], using a standard-atmosphere model appropriate for the time and place, using a measured/modeled vertical structure for that time and place, or using a full 3D measured/modeled atmosphere for the area. The last of these is beyond the scope of this paper, but we can form an idea of the extent of the error introduced from the first four.  
For this, we make use of the temperature, pressure and humidity profiles for standard atmospheres given by the ITU in [36], calculating the resultant air refractive index [37]. We then use this to calculate the propagation of the signal through an assumed spherically stratified atmosphere (standard method, found in, e.g., Appendix A of [34]) to determine the propagation path through the atmosphere to the aircraft. The key parameters that we determine are the incidence and scattering elevation angles at the aircraft. We assume for this error analysis that the transmitter and receiver antenna gains are constant and that the small shift in range due to atmospheric effects is negligible. Were the last of these not the case, the same method could also be used to take this into account.  
In addition, MeteoSwiss provided us with 7 km resolution COSMO-7 [38] temperature, pressure and humidity reanalysis data at 10-min increments for the measurement time and area, interpolated to a 100 m vertical resolution. We averaged these data at each altitude and used the same method as for the standard atmospheres to calculate the resultant path through this atmosphere. We use this as a baseline against which to compare the other models.  
We show the results of this comparison along our flight track in Figure 16. Of the standard atmospheres, our latitude (46-47\({}^{\circ}\) N) puts us just into the 'upper latitude' model (threshold 45\({}^{\circ}\) N, and since the measurements were made in July, summer is the more appropriate version of this standard atmosphere to use, but we also show the US standard atmosphere and the mid-latitude summer atmospheres for comparison. In practice, in our specific case, the four-thirds earth model atmosphere produces angles very close to those from the COSMO model, most notably for the transmitter angle.  
From this we conclude that the error associated with the no-atmosphere approximations is significantly larger than the difference between any of the with-atmosphere results.  
Figure 16: The error in the elevation angles resulting from various methods of accounting for atmospheric bending, compared with a COSMO-7 reanalysis of the atmosphere at the time of the measurements: (**a**) transmitter; (**b**) receiver.  
The maximum variation of the apparent elevation angle of the transmitter at the location of the aircraft with different atmospheres is 0.034\({}^{\circ}\), and 0.008\({}^{\circ}\) for the closer receiver. The maximum difference between the four-thirds earth model and the most plausible high-latitude summer model is 0.009\({}^{\circ}\) for the transmitter and 0.002\({}^{\circ}\) for the receiver, i.e., far less than the variation between the different atmospheric models. In practice, for our case, we find that the four thirds earth model atmosphere is actually closest to our most sophisticated model, so we use this for our calculations. However, this is not a general result; if working under different conditions it would be necessary to revisit this assumption.