## 1 Introduction  
Atmospheric turbulence--a chaotic motion added to constant wind--is frequently encountered in flight. In particular, the vertical component of turbulence threatens aircraft stability and even flight safety. On one hand, in-flight turbulence reduces passenger riding comfort and even causes causalities. On the other hand, it also leads to the manipulation difficulty for pilots and induces fatigue on aircraft structure [1]. To avoid such hazard, it is therefore fundamental to explore the occurrence mechanism, detection and risk judgment method for turbulence. Atmospheric turbulence can be categorized into convective turbulence, low-altitude thermals, wake turbulence and clear air turbulence [2]. Clear air turbulence, though seriously threatening the flight safety of civil aviation aircrafts, is difficult for onboard weather radar to observe. A traditional method of assessing the turbulence hazard is by means of pilot weather report (PIREP). A major problem of the subjectively reported PIREP is that the turbulence severity experienced by the reporting aircraft may not be the same experienced by a following aircraft because of the different aircraft type and its configuration [3]. Scholars put forward the method of assessing turbulence severity based on vertical acceleration of aircraft [4]. However, the measured acceleration cannot reflect the turbulence severity because it is in fact the turbulence induced response and is also related to other flight parameters. Based on the measured acceleration value, a further derived equivalent vertical gust velocity (DEVG) method was developed, with airspeed and aircraft mass considered [5]. However, measuring the turbulence via vertical acceleration or DEVG could be sensitive to different aircrafts. We need to obtain an objective evaluation criterion of turbulence severity which is independent of aircraft. Eddy dissipation rate (EDR), a parameter to determine the amount of energy lost by the viscous forces in turbulence, has been widely used to describe the turbulence severity in recent years [6]. The National Center for Atmospheric Research (NCAR) proposed vertical wind-based and vertical acceleration-based EDR estimation method successively. A major advantage of the vertical wind-based algorithm is that only six parameters are required for computing EDR, including true airspeed, left and right vane angle of attack, inertial vertical speed, pitching and rolling angle [7]. The EDR value is obtained by frequency-domain-based maximum-likelihood estimation. The World Meteorological Organization (WMO) has added turbulence monitoring information in the aircraft meteorological data relay, in which vertical wind-based EDR value is reported [8]. However, this kind of EDR algorithm still must be improved because the air data measurements and derived vertical wind may not be accurate in turbulent flight. Furthermore, considering different aircraft, there are some empirically chosen parameters in the algorithm, including the selection of bandpass frequency and bias-correction term [9]. Vertical acceleration-based EDR algorithms, recovering the turbulence severity from an aircraft's vertical acceleration response, is a rigorous method with better accuracy theoretically [10]. The acceleration response, initially described by a polynomial fitting function, though lack of adequate accuracy, has been applied in modern EDR estimation [11]. The other way of describing the acceleration response is to build a linear-transfer function from vertical wind to aircraft vertical acceleration. Research shows that vertical turbulence mainly leads to aircraft plunge and pitching motion, inducing aircraft bumpiness. Two linear-transfer functions describing plunge and pitching motion--based on small-perturbation approximations--were built, with vertical turbulence as the only excitation. To eliminate the influence of maneuvering flight and high-frequency structural vibration, a bandpass filter must be cascaded to the transfer function.