## 2 Materials and Methods
### Turbulence Clearance
#### 2.2.2 Methods  
After having defined the theoretical basis of the turbulence effect on aircrafts due to the presence of mountain waves, this section specifies the assumptions made and the relationships and equations developed to calculate the vertical displacement that an aircraft would experience under these conditions. In the PANS-OPS, the ICAO defines the different aircraft categories according to their indicated airspeeds (IAS) [50]. From these categories, an aircraft model was chosen as a reference to define the wing area (S) and the aircraft's weight (W), since these characteristics affect the turbulence perception and differ per category. Moreover, each category is associated with the maximum indicated airspeed (IAS) corresponding to the speed range according to the ICAO PANS-OPS categorization by approach speed (Table 4), since the maximum airspeed gives the largest vertical displacements. From the indicated airspeed and flight altitude, the true airspeed (TAS) is obtained using the following expression, provided by the PANS-OPS, for a non-interational system unit [46]:  
\[\text{TAS}=\text{IAS}\text{-}171233\text{-}\ \frac{\left(288+\text{VAR}\,-\,0.00198 \cdot\text{H}\right)^{0.5}}{\left(288\,-\,0.00198\cdot\text{H}\right)^{2.628}} \tag{17}\]  
where the following applies:  
* IAS is the indicated airspeed in knots [kt];
* VAR is the temperature variation in ISA in \({}^{\circ}\)C. By default, ISA+15 is used, and thus, VAR takes the value 15;
* H is the altitude in feet [ft]. Two tables are shown below; the first one corresponds to the table provided by the ICAO PANS-OPS, where the different aircraft categories are listed according to their indicated airspeeds (IASs) above the runway threshold (Vat) [50]. The second one shows the aircrafts that have been taken as a reference in each category in order to define the wing area and the weights assumed, which will be explained later. In Table 5, the maximum take-off weight (MTOW) and wing area (S) were obtained from the corresponding aircraft reference manual [51; 52; 53; 54]. \"W\" was defined as the approximate minimum weight at which the aircraft could operate in order to obtain the maximum vertical displacement achievable, since the lower the weight is, the greater the altitude loss from turbulence is. Therefore, the MTOW is not used, because the vertical displacements associated with this weight would be the smallest achieved. For categories A and B, the weight used in this methodology was calculated taking into account the Operational Empty Weight (OEW) and estimating a minimum payload and fuel load. This results in an excess altitude loss, ensuring the vertical protection of the aircraft in case of any turbulence. In contrast, for categories C and D, their weights (W) were obtained by analysing the landing weights of a sample of 110 and 155 real flights, respectively, which would be equivalent to the lowest weights with which the reference aircraft usually operates. Once \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Aircraft Category** & \(V_{at}\) & **Range of Speeds for Initial Approach** & **Range of Final Approach** & **Maximum Speeds for Visual Manouevring (Circling)** & **Maximum Speeds for Missed Approach** \\ \hline A & \textless{}91 & 90/150 & 70/100 & 100 & 110 \\ B & 91/120 & 120/180 & 85/130 & 135 & 130 & 150 \\ C & 121/140 & 160/240 & 115/160 & 180 & 160 & 240 \\ D & 141/165 & 185/250 & 130/185 & 205 & 185 & 265 \\ E & 166/210 & 185/250 & 155/230 & 240 & 230 & 275 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Aircraft approach categories (PANS-OPS, ICAO). Speed in knots. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Category** & **Reference Aircraft** & **S [m\({}^{2}\)]** & **MTOW [kg]** & **W [kg]** \\ \hline A & C172 & 16.17 & 1157 & 1040 \\ B & ATR 72–600 & 61 & 23,000 & 20,000 \\ C & A320 & 122.6 & 78,000 & 56,400 \\ D & A340-600 & 439.4 & 380,000 & 226,100 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wing area and weights according to aircraft category.