# Estimating Eddy Dissipation Rate with QAR Flight Big Data
## 2 Data and Methodology
### EDR Estimation
#### 2.3.1 The First Estimation Method  
Record the power spectral density of the input vertical gust velocity as \(\phi_{wg}(\omega)\), the vertical acceleration of the aircraft as \(\tilde{z}(\omega)\), input gust as \(w_{g}(\omega)\), then the power spectral density of the acceleration response can be estimated by analyzing the power spectrum and frequency response linear system of the aircraft's response to turbulence. As shown in Equation (5):  
\[\phi_{\tilde{z}}(\omega)=\left|\frac{\tilde{z}(i\omega)}{w_{g}(i\omega)} \right|^{2}\phi_{wg}(\omega). \tag{5}\]  
The estimation is conducted in the frequency domain, and the above equation is integrated between the frequency segments \(\omega_{1}\) and \(\omega_{2}\) to obtain the acceleration response energy between the frequency segments \(\partial_{\tilde{z}}^{2}\). As shown in Equation (6):  
\[\partial_{\tilde{z}}^{2}=\int_{\omega_{1}}^{\omega_{2}}\phi_{\tilde{z}}( \omega)d\omega=\int_{\omega_{1}}^{\omega_{2}}\left|\frac{\tilde{z}(i\omega)}{ w_{g}(i\omega)}\right|^{2}\phi_{wg}(\omega)d\omega. \tag{6}\]  
\begin{table}
\begin{tabular}{c c} \hline
**Turbulence Intensity** & **DEVG** \\ \hline
**steady flight** & \(DEVG<2.0\,\)m/s \\
**weak turbulence** & \(2.0\,\)m/s \(\leq DEVG<4.5\,\)m/s \\
**moderate turbulence** & \(4.5\,\)m/s \(\leq DEVG<9.0\,\)m/s \\
**strong turbulence** & \(DEVG>9.0\,\)m/s \\ \hline \end{tabular}
\end{table}
Table 5: Air turbulence intensity and the corresponding derived equivalent vertical gust velocity (DEVG).  
Introduce band pass filter \(H_{bp}(\omega_{1},\omega_{2},\omega)\), and Equation (6) can be approximated as Equation (7):  
\[\hat{\sigma}_{z}^{2}\approx\int_{0}^{\infty}\left|H_{bp}(\omega_{1},\omega_{2}, \omega)\right|^{2}\left|\frac{\tilde{z}(i\omega)}{w_{g}(i\omega)}\right|^{2} \phi_{avg}(\omega)d\omega. \tag{7}\]  
For the flight conditions of commercial transport aircraft, the frequency band in which the vertical acceleration response of most rigid-body aircraft occurs corresponds to the inertia sub-range of atmospheric turbulence. Therefore, according to [PERSON]\(\ddot{a}\)rm\(\ddot{m}\)n's energy spectrum theory [6], Equation (8) can be derived:  
\[\hat{\phi}_{avg}(\omega)=0.7V^{\frac{2}{3}}\varepsilon^{\frac{2}{3}}\omega^{- \frac{5}{3}} \tag{8}\]  
where \(V\) is true airspeed of the aircraft through the atmospheric turbulence, and \(\varepsilon^{1/3}\) is the EDR. Substituting Equation (8) into Equation (7), Equation (9) is derived:  
\[\hat{\sigma}_{z}^{2}=0.7V^{\frac{2}{3}}\varepsilon^{\frac{2}{3}}\int_{0}^{ \infty}\left|H_{bp}(\omega_{1},\omega_{2},\omega)\right|^{2}\left|\frac{ \tilde{z}(i\omega)}{w_{g}(i\omega)}\right|^{2}\omega^{-\frac{5}{3}}d\omega. \tag{9}\]  
Seen from Equation (9), the EDR value can be estimated when the aircraft response function and the acceleration response energy are known.  
The estimation of the aircraft response function can be obtained by constructing an equation of motion for the aircraft. However, parameters such as the wing and tail reference surface area, which are needed in the solution process, are not provided by QAR data.  
Estimating the acceleration response energy requires the following two steps:  
* Inverse Fourier transform of bandpass filter function, as show in Equation (10): \[h_{bp}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}H_{bp}(\omega_{1},\omega_{2}, \omega)e^{i\omega t}d\omega.\] (10)
* Use the Parseval theorem (assuming local stationary) and then use the convolution theorem to get its approximation, as show in Equation (11): \[\hat{\sigma}_{z}^{2}(t)\approx\frac{1}{2T}\int_{t-T}^{t+T}\left[\int_{0}^{t_{1 }}h_{bp}(t_{1},t_{2},\tau_{2})\tilde{z}(\tau_{1}-\tau_{2})d\tau_{2}\right]^{2 }d\tau_{1}.\] (11)