## Abstract

Two indicators of turbulence—the eddy dissipation rate (EDR) and derived equivalent vertical gust velocity (DEVG)—are calculated using aircraft observations from Hong Kong–based airlines, whose aircraft included Boeing and Airbus models, for 39 months from February 2011 to April 2014. Characteristics of the two turbulence indicators that were calculated at 1-min intervals from the flight data are investigated. For Boeing and Airbus aircraft, there are large seasonal variations in the 90th and 99th percentiles of EDR and DEVG, whereas there are relatively small seasonal variations in the medians of EDR and DEVG. For the turbulence encounters estimated from EDR and DEVG, the authors compute their correlations for each level of turbulence and each type of aircraft. Strong correlations (larger than 0.7) occurred for all levels of turbulence encounters for Boeing aircraft, whereas relatively weak correlations (less than 0.5) occurred for Airbus aircraft. This difference is due to the different characteristics of recorded Boeing and Airbus aircraft data (the number of decimals and data sampling frequency). Based on correlation analyses, the authors construct the best-fit curves using mean EDR values for each DEVG bin and mean DEVG values for each EDR bin and obtain relationships between EDR and DEVG for Boeing and Airbus aircraft. The EDR and DEVG-derived EDR for moderate-or-greater-level turbulence are generally similar for Boeing aircraft.

## 1. Introduction

Turbulence is unexpected bumps that occur during flight with scales between 10 m and 1 km (Lester 1994). Significant risks are associated with turbulence, including fuel loss, aircraft damage, and human injuries. A number of efforts have been made to lessen and prevent such risks. Because observational data of turbulence are essential for improving such efforts and serve as the first step in both properly understanding turbulence and accurately predicting it, such data have been utilized in various case studies and statistical analyses, as well as in the construction and validation of a turbulence forecasting system (e.g., Clark et al. 2000; Sharman et al. 2006, 2012, 2014; Wolff and Sharman 2008; Trier and Sharman 2009; Trier et al. 2012; Kim and Chun 2010, 2011a, 2012; Kim et al. 2014, 2015).

In airborne observations of turbulence, there are the pilot’s verbal reports (PIREPs) and aircraft-based observations. PIREPs are widely used because of their wide coverage in space and time, following pilots’ voluntary and/or mandatory participation in reporting meteorological conditions during flight; thus, the climatology of turbulence encounters has been reported in several local regions (e.g., Wolff and Sharman 2008; Lane et al. 2009; Kim and Chun 2011a). However, because PIREPs provide an inevitably subjective measure of turbulence, a high degree of uncertainty exists regarding the intensity, timing, and locations of turbulence encounters (Schwartz 1996; Cornman et al. 2004). Therefore, the use of in situ aircraft data has been strongly demanded to obtain more accurate observational information regarding turbulence, especially for better forecasting of turbulence. Such flight data have recently been made available to the research community via collaborations with airline industries that produce and store flight data (e.g., Kim and Chun 2012; Lane et al. 2012; Sharman et al. 2012, 2014; Chan and Wong 2014; Gill 2014; Gill and Buchanan 2014; Hon and Chan 2014; Kim and Chun 2016; Meneguz et al. 2016).

There are three indicators of turbulence estimated from in situ aircraft data: the vertical acceleration (Sherman 1985), the cube root of the eddy dissipation rate (EDR) (MacCready 1964; Cornman et al. 1995, 2004; Haverdings and Chan 2010; Sharman et al. 2014) and the derived equivalent vertical gust velocity (DEVG) (Sherman 1985; Truscott 2000; Gill 2014; Kim and Chun 2016). Among these indicators, EDR and DEVG operationally provide objective measures of turbulence and have been used in the Aircraft Communication Addressing and Reporting System (ACARS) and Aircraft Meteorological Data Relay (AMDAR), which support weather forecasting, monitoring, and safety alerting purposes (Moninger et al. 2003).

Operationally, EDR can be estimated using two methods. The first is based on the proportional relationship between EDR and the root-mean-square (RMS) vertical acceleration from the aircraft accelerometer data (MacCready 1964; Cornman et al. 1995; Sharman et al. 2014). Because this relationship is a function of the aircraft response characteristics and varies with altitude, aircraft mass, and airspeed, deriving EDR from vertical accelerations requires substantial aircraft-dependent information. The vertical acceleration-based estimation method has been deployed on United Airlines aircraft (Sharman et al. 2014). The second method, which is less aircraft dependent, is based on computing EDR directly from vertical wind velocity (Cornman et al. 2004; Haverdings and Chan 2010; Sharman et al. 2014). In this method, there are two ways to estimate EDR. First, Cornman et al. (2004) estimated EDR from the power spectrum of the vertical wind estimates within a discrete frequency range and maximum likelihood calculation of Smalikho (1997). This method has been deployed on Delta Air Lines aircraft (Sharman et al. 2014). Second, Haverdings and Chan (2010) estimated EDR in a way similar to Cornman et al. (2004), although with different angle-of-attack calibration and sampling frequencies. Since MacCready (1964) mentioned that EDR can be used as a turbulence-reporting standard, owing to its simplicity, EDR has been used as a representative turbulence metric by the International Civil Aviation Organization (ICAO; ICAO 2001, 2010; Sharman et al. 2014). The DEVG is estimated from vertical acceleration, with consideration of aircraft information and changes of aircraft mass, altitude, airspeed, and air temperature (Sherman 1985; Truscott 2000; Gill 2014). The DEVG algorithm has been implemented on Qantas, South African, and British Airways aircraft (Sharman et al. 2014).

There have been several attempts to determine EDR and DEVG thresholds for each level of turbulence. For EDR, MacCready (1964) suggested EDR thresholds for four classes of turbulence [null (NIL), light (LGT), moderate (MOD), and severe (SEV) turbulence] based on comparisons of EDR values and PIREPs. Bohne (1985) and Lee et al. (1988) estimated EDR thresholds based on comparisons of EDR values obtained from aircraft data and radar data. Stickland (1998) suggested EDR thresholds that were based on DEVG obtained from the same aircraft data. Publicly, ICAO suggested EDR thresholds based on significant meteorological information and aircraft-reported meteorological information (ICAO 2010). Most recently, Sharman et al. (2014) estimated EDR thresholds using aircraft-observed EDR and PIREPs; their results showed that the ICAO (2010) recommendations are too large and need to be modified. For DEVG, Sherman (1985) proposed using 9 m s^{−1} as the criterion for SEV turbulence, without suggestions for other levels of turbulence. Truscott (2000) classified DEVG into four classes of turbulence (NIL, LGT, MOD, and SEV turbulence); Gill (2014) used those same thresholds to validate a global turbulence forecasting model.

In this study, we use aircraft measurements provided by airlines based in Hong Kong and estimate EDR and DEVG. Spatiotemporal variations in the turbulence estimated by EDR and DEVG will be presented. Moreover, direct comparison between EDR and DEVG will be performed first using a relatively long-term dataset, which is used to calculate these two turbulence indicators on a one-to-one basis. There is a study by Stickland (1998) that presented a direct comparison between EDR and DEVG time series from Qantas Airways B747-400 data over a 3-month period (from October to December 1997) and generated EDR criteria using 1575 EDR and DEVG pairs. When compared with Stickland (1998), which used limited turbulence cases and one aircraft type, our current study can provide more statistically meaningful results.

This paper is organized as follows. In section 2, a brief description of the aircraft data used in the present study is provided, and the algorithms used to calculate EDR and DEVG are given. In section 3, spatiotemporal characteristics in the turbulence encounters from both EDR and DEVG are presented, and the correlations between turbulence encounters from EDR and DEVG are examined for Airbus aircraft (A) and Boeing aircraft (B) separately. Based on the correlation, we construct the best-fit curves between EDR and DEVG. In section 4, discussion is provided, and the results of this study are summarized in the last section.

## 2. Data and methodology

Figure 1 shows the horizontal distributions of the number of data accumulated within horizontal grid boxes of 0.5° × 0.5° at 1-min intervals, from flight data recorders on Boeing and Airbus aircraft from February 2011 to April 2014. The data used in the present study are from 1819 flights, consisting of 266, 128, and 303 flights by B747-400, B777-200, and B777-300, respectively, and 350, 152, and 620 flights by A320-200, A321-200, and A330-300, respectively. Note that quality control procedures by Gill (2014) and Meneguz et al. (2016) were applied to the dataset to eliminate suspicious values. Although data sampling rates vary for different parameters and depend on aircraft type, most parameters are recorded at every second; vertical acceleration is recorded with sampling rates of 8 and 10 Hz (Table 1). Note that both EDR and DEVG are calculated through the offline postprocessing algorithms using the time series of several variables recorded in flight data recorders; the variables are listed in Table 1. As shown in Fig. 1, most of the flights are heading to Shanghai, China, and Taiwan. Figure 2 shows the vertical frequencies of Boeing and Airbus aircraft data, which are accumulated within each 3-kft (1 kft = 305 m) bin of flight levels (FL) for the whole period (from February 2011 to April 2014). The maximum number and more than 40% of the flight data are between 37 and 40 kft. In the present study, we consider observations above 28 kft. This lower limit of altitude (28 kft) is chosen based on the quality control procedures of Gill (2014) and Meneguz et al. (2016) to remove erroneous data associated with aircraft maneuvering.

The vertical wind-based EDR is calculated using an equation from Haverdings and Chan (2010) as

Here, *σ*_{w} is the running mean standard deviation of vertical wind over a 10-s time window, *V*_{a} is the true airspeed, and *ω*_{1} and *ω*_{2} are cutoff frequencies—which are set as 0.1 and 2 Hz, respectively. A calculation of the vertical wind is described in Haverdings and Chan (2010). Although the sampling rates vary for the different variables and each type of aircraft (Table 1), all variables are interpolated at 4 Hz to achieve the required sampling rate of vertical wind-based EDR estimation. Note that the data used in the present study are from both heavy-sized aircraft (A330-300, B747-400, and B777-300) that are more than 136 tons and medium-sized aircraft (A320-200 and A321-200) (e.g., http:/www.faa.gov/air_traffic/publications/media/aim.pdf). From the EDR time series, following each flight route, turbulence events are counted at 1-min intervals using their maximum values during that 1-min time window.

As another aircraft-independent measure of turbulence, we calculate DEVG using flight data based on the formulation of Truscott (2000) and Gill (2014) as

Here, *m* is the aircraft mass in metric tons, Δ*n* is the peak value of the deviation of vertical acceleration from 1 g, and *V* is the calibrated airspeed. Considering aircraft types, *A* can be approximated as

Here, *H* is the altitude, is a reference mass of the aircraft in metric tons, and *c*_{1}, *c*_{2}, …, *c*_{5} are empirical constants dependent on aircraft type. Table 2 shows the constants (*c*_{1}, *c*_{2}, …, *c*_{5}) for each type of aircraft considered in the present study, based on Truscott (2000). DEVG is sensitive to parameters of *c*_{1}, *c*_{2}, *c*_{3}, *c*_{4}, and *c*_{5}, which are obtained from approximately fitted curves by considering reference aircraft mass, airspeed, altitude, and Mach number for each type of aircraft (Sherman 1985; Truscott 2000). When we made a calculation of DEVG with the parameters increased by ±10%, for the data of a Boeing 777-300 flight as a sample, the resultant DEVG is increased within ±10% regularly (not shown). Uncertainties in values of *c*_{1}, *c*_{2}, *c*_{3}, *c*_{4}, and *c*_{5} remain to be determined in the future research. From the DEVG time series following each flight route, turbulence events are counted at 1-min intervals using their maximum values during that 1-min time window.

## 3. Results

According to previous studies (e.g., Nastrom and Gage 1985; Frehlich 1992; Frehlich and Sharman 2004; Sharman et al. 2014), the probability density functions (PDFs) of EDR are fairly well matched with lognormal distributions. To examine this for the current data, PDFs of EDR and DEVG are computed for Boeing and Airbus aircraft for the whole period. In constructing the PDFs, we use 0.01 m^{2/3} s^{−1} and 0.1 m s^{−1} intervals for EDR and DEVG, respectively. The lognormal PDFs are computed with the mean and standard deviation of the natural logarithms of EDR and DEVG (Wilks 1995).

Figures 3 and 4 show the PDFs and lognormal distributions of EDR and DEVG, respectively. For EDR, the PDFs for Boeing aircraft are concentrated in relatively narrow ranges, in comparison with those for Airbus aircraft. The PDFs for both Boeing and Airbus aircraft fit fairly well to the lognormal PDFs, within a range of 0–0.2 and 0–0.3 m^{2/3} s^{−1}, respectively. The median and 90th and 99th percentiles of EDR are summarized in Table 3. Generally, the median EDR values for Boeing aircraft are approximately half of those for Airbus aircraft. This difference is related to different numbers of decimals in the variables that determine EDR (Table 1). Similarly, for DEVG, the PDFs for most Boeing and Airbus aircraft fit fairly well to the lognormal PDFs within a range of 0–4 and 0–2 m s^{−1}, respectively. Unlike EDR, the median DEVG values are quite different for each type of aircraft. The median DEVG values for B777-200 and B777-300 are much lower than those for other types of aircraft (B747-400, A320-200, A321-200, and A330-300). However, the 90th and 99th percentiles of DEVG for most types of aircraft, except for B777-200, are approximately 1 and 2 m s^{−1}, respectively. Note that the 90th and 99th percentiles of EDR and DEVG can be considered LGT and moderate-or-greater (MOG) turbulence encounters, respectively. Being consistent with Figs. 3 and 4, seasonal PDFs of EDR and DEVG for Boeing and Airbus aircraft generally follow the lognormal distributions (not shown).

Figure 5 shows the median and 90th and 99th percentiles of EDR and DEVG for each season [December–February (DJF), March–May (MAM), June–August (JJA), and September–November (SON)] and those averaged over the whole period from February 2011 to April 2014 for Boeing and Airbus aircraft. The seasonal medians of EDR and DEVG are generally similar to the medians for the whole period. For Boeing aircraft, significant seasonal variations appear in both the 90th and 99th percentiles of the EDR and DEVG, while for Airbus aircraft, large seasonal variations appear only in the 99th percentiles of EDR and DEVG.

Figure 6 shows the median and 90th and 99th percentiles of EDR and DEVG for each 3-kft bin from 28 to 43 kft for Boeing and Airbus aircraft. The variations in the median and 90th and 99th percentiles of EDR are much larger than those of DEVG, and the vertical distributions of EDR and DEVG are different between Boeing and Airbus aircraft. For Boeing aircraft, the median and 90th and 99th percentiles of EDR are the largest at altitudes between 34 and 37 kft, whereas those of DEVG are between 28 and 31 kft. For Airbus aircraft, the median and 90th and 99th percentiles of EDR are the largest at altitudes above 40 kft, whereas those of DEVG are between 28 and 31 kft.

Figure 7 shows scatterplots of EDR and DEVG for each type of aircraft. Most turbulence observations are within relatively lower ranges (approximately 0–0.1 m^{2/3} s^{−1} and 0–2 m s^{−1} for EDR and DEVG, respectively), as also shown in Figs. 3 and 4. The turbulence reports of DEVG less than 2 m s^{−1} and EDR larger than 0.1 m^{2/3} s^{−1} account for only approximately 1% of all turbulence encounters from Boeing aircraft but 25% from Airbus aircraft. The Pearson correlation (Wilks 1995), which is a measure of association between two variables, is calculated and represented by *r* in each plot shown in Fig. 7. The probability density functions of EDR and DEVG revealed in Fig. 7 show that the EDR and DEVG terminate well above zero in all Airbus data, and a similar termination, near zero, is shown for B747-400. To understand this somewhat unexpected result, we checked the data quality and any artificial thresholds imposed by Airbus data systems. We found that there were no serious errors in the Airbus data, which passed the standard quality control procedure, and no artificial thresholds were imposed in the Airbus data system. Instead, we found that the number of decimals and sampling frequency of the variables used for the calculation of DEVG and EDR for each flight type were different (Table 1), which significantly influenced the scatterplots in Fig. 7.

It is found that the correlation is much higher for data from Boeing than from Airbus: 0.74 for all Boeing aircraft and 0.47 for all Airbus aircraft. This is also likely due to the number of decimals and sampling frequency of the variables in the data observed from Airbus aircraft, which are generally lower than those from Boeing aircraft. Among the Boeing aircraft, the lowest correlation is from B747-400, which has the lowest number of decimals and sampling frequency. This demonstrates that the precision in the variables observed from flight is crucial for accurately determining the in-flight turbulence indicators of EDR and DEVG and their correlations. Meanwhile, the average correlation for lighter aircraft (A320-200 and A321-200) is 0.44, while that for heavier aircraft (A330-300, B747-400, B777-200, and B777-300) is 0.67. It is interesting that the correlations for medium- and heavy-sized Airbus aircraft are very similar. In the previous study by Stickland (1998), a strong linear correlation (0.92) between EDR and DEVG was shown from Qantas Airways B747-400 data for three months (from October to December 1997). Note that the EDR calculated by Stickland (1998) was based on vertical acceleration, which is similar to the DEVG estimation, unlike the EDR used in the present study.

We compute correlations between EDR and peak vertical acceleration and between DEVG and peak vertical acceleration for Boeing and Airbus aircraft (not shown). The average correlations between EDR and peak vertical acceleration for Boeing and Airbus aircraft are 0.68 and 0.45, respectively, whereas those between DEVG and peak vertical acceleration are 0.75 and 0.65. These relatively low correlations between EDR and vertical acceleration are likely because EDR considered in the present study is based on RMS vertical wind. Considering the flight type, the correlations are lower for Airbus (0.45 and 0.65 for EDR and DEVG, respectively), especially for EDR, than for Boeing (0.68 and 0.75 for EDR and DEVG, respectively). This difference is partially due to the lower precision (as quantified by the number of decimals) for Airbus than for Boeing, as shown in Table 1.

Figure 8 shows scatterplots of EDR and DEVG, accumulated within each 0.5 m s^{−1} bin of DEVG for Boeing and Airbus aircraft. Relative to the result for Boeing aircraft, there are comparatively large spreads of EDR in the overall range of DEVG for Airbus aircraft. For each DEVG bin, mean EDR values are computed, and a best-fit curve is constructed from those mean values. In constructing the best-fit curve, we ignore the mean value if there are less than 10 events within the given DEVG bin. The coefficient of determination (Wilks 1995), which is a measure of goodness of fit between the curve and the mean EDR value, is larger than 0.9 for both Boeing and Airbus aircraft. The best-fit curve is more likely in exponential form for Boeing aircraft, while it is close to linear for Airbus aircraft. The converted EDR from DEVG (DEVG-derived EDR) for Airbus aircraft is similar to that for Boeing aircraft with the difference of approximately 0.07 m^{2/3} s^{−1}.

Conversely, Fig. 9 shows scatterplots of EDR and DEVG, accumulated within each 0.05 m^{2/3} s^{−1} bin of EDR for Boeing and Airbus aircraft. Through the same process of constructing Fig. 8, the best-fit curve, which converts EDR into DEVG, is constructed. Unlike Fig. 8, there are relatively large spreads of DEVG in the overall range of EDR for both Boeing and Airbus aircraft. For Boeing aircraft, the exponential curve is constructed because the negative gradient is obtained on the quadratic curve. The converted DEVG from EDR for Airbus aircraft is lower than that for Boeing aircraft. The DEVG difference between Boeing and Airbus aircraft is substantial at the higher EDR value, while the difference is less significant for the lower EDR value.

Figure 10 shows the locations of MOG turbulence by EDR and DEVG-derived EDR and the differences in magnitude between EDR and DEVG-derived EDR based on Fig. 8 for the Boeing and Airbus aircraft. The MOG turbulence is determined by EDR larger than 0.16 m^{2/3} s^{−1}, based on the averaged 99th percentile of EDR for six aircraft types. As shown in Figs. 3, 5, and 6, the number of MOG turbulence encounters by EDR and DEVG-derived EDR from Airbus aircraft is larger than that from Boeing aircraft. For Boeing and Airbus aircraft, the number of MOG turbulence encounters by EDR is larger than that by DEVG-derived EDR. The differences in magnitude between EDR and DEVG-derived EDR are mostly less than 0.05 m^{2/3} s^{−1} (Boeing: 98.8%; Airbus: 91.1%), while those in the range of 0.05–0.1 m^{2/3} s^{−1} account for 7.1% in Airbus flights. The cases with a large difference (~0.3 m^{2/3} s^{−1}) appear more for Airbus aircraft (0.06%) than for Boeing aircraft (0.003%) because of the low correlation between EDR and DEVG. Nevertheless, the locations of MOG turbulence encounters are reasonably well matched between EDR and DEVG-derived EDR. There is a relatively large frequency of MOG turbulence along the Taiwan and Fukuoka, Japan, routes for Boeing aircraft and the Shanghai, Philippines, and Indonesia routes for Airbus aircraft.

## 4. Discussion

In this study, we investigate the characteristics of two aircraft-independent turbulence indices, EDR and DEVG, from in situ flight data and examine the correlation between the two indices. DEVG is calculated using the peak vertical acceleration-based algorithm, while EDR is calculated using the RMS vertical wind-based algorithm. Although there are some uncertainties in DEVG, which assumes a simplified and restricted model of the atmosphere (Sharman et al. 2014) and does not consider the effect of the pitch damping due to autopilot, DEVG is considered in the present study because it is widely used as an important aviation turbulence index. This study is unique in the sense that direct comparison of the two major turbulence indices, EDR and DEVG, using the same in situ flight data for a relatively long period has not been reported previously and that the current results can provide invaluable turbulence information for the area considered, South Asia, where turbulent information from PIREPs are nearly unavailable (Kim and Chun 2011b; Gill 2014; Lee and Chun 2015; Kim and Chun 2016). Furthermore, we first examine the correlations between EDR and DEVG observed from two different aircraft types, Boeing and Airbus, separately.

The correlation between EDR (or DEVG) and peak vertical acceleration is larger for Boeing than for Airbus. The correlation between EDR and DEVG is also larger for Boeing than for Airbus. This is not simply related to the weight of aircraft, considering much weaker correlations between EDR and DEVG in Airbus aircraft with similar weight to Boeing aircraft. Rather, differences in the precision (number of decimals) of the vertical acceleration and the sampling frequency of variables, in particular, the inertial vertical velocity and vertical acceleration, between Boeing and Airbus aircraft lead to differences in the correlations. Among the three Boeing aircraft datasets, the B747-400 data have the smallest number of decimals and low sampling frequency, and the relatively low correlation between EDR and DEVG for B747-400 is due to the data precision.

There have been several attempts to determine the EDR criteria for each level of turbulence (e.g., MacCready 1964; Bohne 1985; Lee et al. 1988; Stickland 1998; Sharman et al. 2014). MacCready (1964) estimated severity thresholds of EDR through comparing aircraft data and pilots’ descriptions. The lower boundaries of EDR values for LGT, MOD, and SEV turbulence suggested by MacCready (1964) were 0.03, 0.07, and 0.16 m^{2/3} s^{−1}, respectively. Bohne (1985) and Lee et al. (1988) estimated severity thresholds of EDR by comparing Doppler weather radar observations and aircraft data during storm penetration. The lower boundaries of EDR values for MOD and SEV turbulence suggested by Bohne (1985) and Lee et al. (1988) were 0.07 and 0.16 m^{2/3} s^{−1}, respectively. Based on the strong linear correlation between EDR and DEVG from Qantas Airways B747-400 data for 3 months, Stickland (1998) suggested EDR values for each level of turbulence using the linear relationship between EDR and DEVG (DEVG = EDR × 10.529). The lower boundaries of EDR for LGT, MOD, and SEV turbulence by Stickland (1998) were 0.19, 0.43, and 0.85 m^{2/3} s^{−1}, relative to the DEVG criteria (2, 4.5, and 9 m s^{−1} for LGT, MOD, and SEV turbulence, respectively). Sharman et al. (2014) proposed EDR values based on in situ aircraft data from B737 and B757 compared with PIREPs in the United States from 2004 to 2013 (from 2008 to 2013 for B737 and from 2004 to 2013 for B757), which were 0.013, 0.118, and 0.33 m^{2/3} s^{−1} for LGT, MOD, and SEV turbulence, respectively. Sharman et al. (2014) also showed how to convert EDR values from one aircraft to another, considering turbulence intensity from PIREPs, aircraft type, flight altitude, and aircraft weight. For example, Sharman et al. (2014) generated EDR values for B737 and B747 using B757 quadratic fits, which were 0.22 and 0.27 m^{2/3} s^{−1}, respectively, for MOD turbulence and 0.49 and 0.61 m^{2/3} s^{−1}, respectively, for SEV turbulence.

The best-fit curves between EDR and DEVG obtained from the current study can be used when examining the severity thresholds of turbulence. The thresholds of LGT-, MOD-, and SEV-level turbulence based on the ICAO (2010) criteria of EDR (0.1, 0.4, and 0.7 m^{2/3} s^{−1}, respectively) and Gill’s criteria (2014) of DEVG (2, 4.5, and 9 m s^{−1}, respectively) are compared with those calculated from the best-fit curves. First, based on Fig. 8, the EDR values corresponding to the LGT, MOD, and SEV turbulence criteria of DEVG are 0.081, 0.203, and 0.520 m^{2/3} s^{−1}, respectively, for Boeing and 0.128, 0.258, and 0.586 m^{2/3} s^{−1}, respectively, for Airbus. When compared with the ICAO criteria for each level of turbulence, the current threshold of LGT is generally similar, but the current thresholds of MOD and SEV are much smaller for both Boeing and Airbus. However, these EDR thresholds are somewhat larger than those from the aforementioned previous studies (MacCready 1964; Bohne 1985; Lee et al. 1988; Sharman et al. 2014), which were based mostly on medium-sized aircraft. When we recalculate the fitting curve for medium-sized aircraft (not shown), the obtained EDR values are smaller than the current results and are similar to those of the previous studies. Second, based on Fig. 9, the obtained DEVG values corresponding to the LGT, MOD, and SEV turbulence criteria of EDR from ICAO are 1.542, 4.689, and 7.347 m s^{−1}, respectively, for Boeing aircraft and 0.925, 2.676, and 5.241 m s^{−1}, respectively, for Airbus aircraft. Relative to the criteria of Gill (2014), the obtained thresholds of LGT and SEV are smaller, especially for Airbus, while the threshold of MOD is similar for Boeing and smaller for Airbus. It is noteworthy that the examination of the currently used criteria for each level of turbulence from EDR and DEVG, utilizing the present results in the fitting curves between EDR and DEVG, is limited primarily because we do not know the true criteria of either EDR or DEVG. Nevertheless, determining the thresholds of each level of turbulence from in situ flight data is essential for many areas, such as constructing reliable turbulence climatology and developing–evaluating turbulence forecasting systems in local [e.g., Graphical Turbulence Guidance (GTG; Sharman et al. 2006) in the United States and Korea aviation turbulence guidance (KTG; Kim and Chun 2011b)] and global [e.g., World Area Forecast Centers’ clear air turbulence forecasts (Gill 2014) and Global-GTG system (Williams et al. 2009)] regions. Further studies with data with different times, locations, and EDR algorithms such as vertical acceleration-based EDR and vertical wind-based EDR by Cornman et al. (2004) may lead to a robust conclusion.

## 5. Summary

In this study, we calculate two turbulence indices, EDR and DEVG, on a one-to-one basis using in situ flight data provided by Hong Kong–based airlines for a relatively long period (February 2011 to April 2014; a total of 39 months). EDR and DEVG are calculated using Eqs. (1) and (2), respectively, using the time series of the required variables at 1-min intervals. The PDFs and lognormal distributions for EDR and DEVG for Boeing and Airbus aircraft are computed. Although the PDFs of EDR and DEVG from Boeing aircraft are more concentrated in narrow regions than those from Airbus aircraft, all PDFs fairly well follow the lognormal distribution for the whole period. From the time series for EDR and DEVG, seasonal PDFs and PDFs for every 3-kft bin are obtained. The seasonal variations in the medians of the EDR and DEVG are weaker than the variations in the 90th and 99th percentiles of EDR and DEVG for Boeing and Airbus aircraft. Vertically, the variations in EDR are larger than those for DEVG for Boeing and Airbus aircraft. The maximum median and 90th and 99th percentiles of EDR occur at altitudes between 34 and 37 kft (above 40 kft), while those of DEVG occur at altitudes between 28 and 31 kft (between 28 and 31 kft) for Boeing aircraft (Airbus aircraft).

To identify quantitative differences, we compute the correlations between EDR and DEVG for six different types of aircraft. Stronger correlations between EDR and DEVG appear for Boeing aircraft than for Airbus aircraft. The different sampling rates and precision in the variables observed from Boeing and Airbus aircraft (Table 1) likely affect the correlations. We obtain the relationships between EDR and DEVG from the best-fit curves for Boeing and Airbus aircraft, which are constructed using mean EDR values for each DEVG bin and mean DEVG values for each EDR bin. Although Boeing and Airbus aircraft have different weight and features of data (e.g., sampling frequency and precision), the best-fit curves for EDR are relatively similar, while those for DEVG are different. For both Boeing and Airbus aircraft, converted EDR is obtained from DEVG for Boeing and Airbus aircraft, and the spatial distributions of MOG turbulence by EDR and DEVG-derived EDR are fairly well matched despite the large discrepancies in the numbers of MOG turbulence encounters. The magnitudes of difference between EDR and DEVG-derived EDR are primarily less than 0.05 m^{2/3} s^{−1} for Boeing and Airbus aircraft (98.8% and 91.1%, respectively).

In the present study, we first attempt to directly compare the two well-known in-flight turbulence indicators, EDR and DEVG, on a one-to-one basis, using relatively long-term data (39 months) in Asia, where other turbulence observational data, such as PIREPs, are nearly unavailable. In particular, correlation between EDR and DEVG is calculated for six different flight types (three Boeing and three Airbus flights), separately, and best-fit curves for EDR with respect to DEVG (Fig. 8) and for DEVG with respect to EDR (Fig. 9) for Boeing and Airbus aircraft are obtained. The fitting curve formulations may be practically useful to convert any in-flight observational data in EDR (DEVG) to DEVG (EDR), which can be invaluable resources for development and validation of aviation turbulence forecasting systems. One reservation of the current results is that the precision of the data, such as number of decimals and sampling frequencies, is somewhat different from each flight type, and this makes a difference in the correlation between EDR and DEVG (and resultant fitting curve formulations). Therefore, any attempt to apply the best-fit curve between EDR and DEVG to other regions and other types of aircraft requires caution. A more robust fitting curve formulation between EDR and DEVG for each flight type remains to be accomplished in the future, after obtaining flight data with the same data precision.

## Acknowledgments

This work was funded by the Korea Meteorological Administration Research and Development Program under Grant KMIPA 2015-1080. The authors thank the anonymous reviewers for many helpful comments and suggestions.

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## Footnotes

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