## 2 Vertical Acceleration and Turbulence Severity Estimation
### Vertical-Acceleration Based EDR Estimation  
The goal of the EDR algorithm, whether based on vertical acceleration or vertical wind, is to recover the same turbulence severity with acceptable accuracy. While the aircraft is flying through turbulence, the intensity and spatial frequency of turbulence are actually obtained by measuring the temporal frequency and acceleration response of aircraft. It is required to recover the objective EDR value which is independent of aircraft's airspeed, mass and vertical acceleration. As mentioned before, although the wind-based EDR estimation is much easier, there exist deviations in measured air data and empirically selected parameters. In this paper, a new acceleration-based EDR estimation algorithm is proposed. Given the power spectral density (PSD) of the input vertical turbulence component as \(\phi_{W_{z}}(\omega)\) and the frequency response function from turbulence input \(W_{z}(i\omega)\) to vertical acceleration \(a_{n}(i\omega)\), we get the PSD of the acceleration response by  
\[\phi_{a_{n}}(\omega)=\left|\frac{a_{n}(i\omega)}{W_{z}(i\omega)}\right|^{2} \phi_{W_{z}}(\omega) \tag{18}\]  
Equation (18) is built up in temporal-frequency domain since the measurements are made by aircraft flying through turbulence with airspeed \(V\). According to [PERSON]'s frozen-field hypothesis, the length scale of the vertical turbulence component \(\Omega\) can be obtained by \(\Omega=\omega/V\). The estimation is conducted in frequency domain, and the above equation is integrated between the frequency segments \(\omega_{1}\) and \(\omega_{2}\) to obtain the acceleration response energy:  
\[\partial_{a_{n}}^{2}=\int_{\omega_{1}}^{\omega_{2}}\phi_{a_{n}}(\omega)d \omega=\int_{\omega_{1}}^{\omega_{2}}\left|\frac{a_{n}(i\omega)}{W_{z}(i\omega )}\right|^{2}\phi_{W_{z}}(\omega)d\omega \tag{19}\]For flight conditions of civil aviation aircraft, a bandpass filter \(H_{bp}(\omega_{1},\omega_{2},\omega)\) should be cascaded to eliminate the influence of aircraft maneuvering and high-frequency structural vibration, where \(\omega_{1}\) and \(\omega_{2}\) correspond to the filter's half-power points. In practice, the lower cutoff frequency must be greater than the phugoid response frequencies, while high-end cutoff frequency is made to lie below the elastic response mode of the given aircraft. For the typical flight conditions of civil aviation aircraft, the frequency band within which most vertical acceleration response occurs corresponds to the inertial sub-range of atmospheric turbulence, and \(\omega_{1}\) and \(\omega_{2}\) are chosen as 0.1 and 1.0 Hz, respectively in Equation (19) [10]. The above equation is modified by  
\[\partial_{\alpha_{n}}^{2}\approx\int_{0}^{\infty}\left|H_{bp}(\omega_{1},\omega _{2},\omega)\right|^{2}\left|\frac{a_{n}(i\omega)}{W_{z}(i\omega)}\right|^{2} \phi_{W_{z}}(\omega)d\omega \tag{20}\]  
To estimate the response energy, the bandpass filter is transformed to time domain first, by Inverse Fourier transform (IFT) as follows:  
\[h_{bp}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}H_{bp}(\omega_{1},\omega_{2}, \omega)e^{i\omega t}d\omega \tag{21}\]  
After this, [PERSON]'s theorem (assuming local stationary) and the convolution theorem are adopted to yield the approximation [31]:  
\[\partial_{\alpha_{n}}^{2}(t_{k})\approx\frac{1}{2T}\int_{t_{k}-T}^{t_{k}+T} \left[\int_{0}^{\tau_{1}}h_{bp}(\tau_{2})a_{n}(\tau_{1}-\tau_{2})d\tau_{2} \right]^{2}d\tau_{1} \tag{22}\]  
where \(a_{n}(t)\) is a curve equation obtained by fitting the acceleration calculated by Equation (16). This states that the frequency-limited response mean-square value is obtained by computing a running mean-square value of the bandpass filtered acceleration sequence. Before been discretized, nonzero mean value must be removed in computing the outer integration in Equation (22). The averaging integral 2T (\(T=10\)s) is selected to satisfy the local-stationarity assumption and provide enough data values for statistical stability.