# Estimating Eddy Dissipation Rate with QAR Flight Big Data
## 2 Data and Methodology
### EDR Estimation
#### 2.3.3 Estimating EDR with QAR Data  
As shown in Figure 1, the key to estimate EDR with QAR data is to utilize the QAR records for all the parameter calculations. In this study, we designed a procedure for this calculation, and it is shown in Figure 1 as the following steps:  
1. Estimate the basic parameter vertical wind \(w_{g}\). 2. Build the aircraft response function. 3. Estimate the acceleration response energy. 4. Estimate EDR. (1) Estimate Basic Parameters  
In order to obtain EDR value, the first basic parameter, vertical wind \(w_{g}\), must be estimated. Vertical wind can be estimated from the above parameters, as shown in Equation (14):  
\[w_{g}=-TAS*(\cos\theta\sin AOA\cos\phi-\cos AOA\sin\theta)-IVV \tag{14}\]  
where \(\theta\) is the pitch angle PITCH of the QAR data and \(\varnothing\) is the roll angle ROLL of the QAR data. AOA (angle of attack, unit: Radians) refers to the angle between the velocity vector and the aircraft's  
Figure 1: Procedure of estimating EDR with QAR data. string, PITCH (unit: Radians) is the airplane's pitch angle, IVV (unit: m/s) is the instantaneous hoisting rate, and TAS (unit: m/s) is the true airspeed. Figure 2 exemplifies these attitude angles geometrically. As shown in Figure 2, the angles of attack recorded by QAR cannot be used directly. They need to be converted into an angle of attack based on the body axis before being used in the calculation of vertical wind. The conversion method can be expressed as follows:  
\[AOA=a_{0}+a_{1}\bigg{[}\frac{(AOA_{L}+AOA_{R})}{2}+\frac{L\dot{\theta}}{TAS} \bigg{]} \tag{15}\]  
where \(AOA_{L}\) is the left angle of attack and \(AOA_{R}\) is the right angle recorded in QAR. \(L\) is the length of the sensor arm, taking half of the fuselage length. \(\dot{\theta}\) is the pitch rate. Constants \(a_{0}\) and \(a_{1}\) are estimated using a linear least squares fit of \(\theta=a_{0}+a_{1}\frac{(AOA_{L}+AOA_{R})}{2}\) in the preprocessing step, i.e., \(a_{0}=3.19957\), \(a_{1}=0.602594\). (2) Build the Aircraft Response Function  
(1) Estimate the acceleration response coefficient  
The Fourier transform is performed on the vertical acceleration VRTG and the vertical wind \(w_{g}\) in the time T (takes 10 s, and recording frequency is 8 Hz), and converts it from the time domain to the frequency domain to obtain \(\ddot{z}(i\omega)\) and \(w_{g}(i\omega)\) of the nine recorded points in the current estimation window. Taking the value of \(\frac{\ddot{z}(i\omega)}{w_{g}(i\omega)}\) constitutes a new sequence of points. Since the equation form of \(\frac{\ddot{z}(i\omega)}{w_{g}(i\omega)}\) is unknown and indeterminate, we fit it here via a polynomial curve fitting method. Results show that the cubic polynomial fitting performs the best in terms of both accuracy and efficiency, in comparison to the linear, quadratic, and fourth-order polynomial fitting methods. It can be expressed as Equation (16):  
\[\frac{\ddot{z}(i\omega)}{w_{g}(i\omega)}\approx c_{0}+c_{1}\omega+c_{2}\omega^ {2}+c_{3}\omega^{3}. \tag{16}\]  
(2) Build the acceleration response function  
The integral of the acceleration response function \(I(\omega)\) can be written as Equation (17):  
\[I(\omega)=\int_{\omega_{1}}^{\omega_{2}}\bigg{[}\frac{\ddot{z}(i\omega)}{w_{g }(i\omega)}\bigg{]}^{2}\omega^{-\frac{5}{3}}d\omega=\int_{\omega_{1}}^{\omega _{2}}\Bigl{(}c_{0}+c_{1}\omega+c_{2}\omega^{2}+c_{3}\omega^{3}\Bigr{)}^{2} \omega^{-\frac{5}{3}}d\omega. \tag{17}\]  
Usually, take \(\omega_{1}=0.1\) HZ, \(\omega_{2}=0.8\) HZ to shield high-frequency and low-frequency noise. Figure 2: Attitude angles of an aircraft.