# Characterization of Wind Guests: A Study Based on Meteorological Tower Observations
## 3 Results and Discussion  
### Distribution Fitting of Gust Description  
It is to be noted that, although there exist many previous studies that have fitted various probability distributions to the average wind speed and gust amplitude, e.g., Weibull distribution [42; 43; 44; 45] and Rayleigh distribution [46], few among them try to address which distribution form is most satisfactory. More importantly, the parent probability distribution that is closest to other gust characteristics distributions is seldom explored. Hence, four two-parameter distribution forms of positive-values (see Equations (9)-(12)) were employed in this study to fit the 12 gust descriptors by using maximum likelihood estimation (MLE). These distribution forms were selected mainly because all of the gust descriptive statistics were zero- bounded [5]. The data used herein were obtained at the observation level of 40 m high. The log-likelihood (LogL) value was adopted as an index to evaluate the performance of various distributions; the candidate distribution with the largest LogL value is considered to be the best fit [47]. Essentially, a likelihood method is a measure to describe how well a particular model fits the data. In statistics, the maximum likelihood estimation (MLE) is commonly used for estimating the parameters of an assumed probability distribution, given some observed data. When maximizing a likelihood function under the assumed statistical model, the observed data is most probable. Given that the logic of MLE is both intuitive and flexible, it has become a dominant means of statistical \begin{table}
\begin{tabular}{c c} \hline \hline
**Stability Class** & **Range of Monin–Obukhov Length (m)** \\ \hline Unstable (u) & \(-1000<L<-200\) \\ Very unstable (vu) & \(-200<L<0\) \\ Very stable (vs) & \(0<L<200\) \\ Stable (s) & \(200<L<1000\) \\ Neutral (n) & \(|L|>1000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification of atmospheric stability condition [41]. inference. Particularly, log-likelihood (LogL) is used mainly because it is computationally simpler and easier to optimize. \[\text{Weibull}: f(x|\,a,b)=\frac{b}{a}\,(\frac{x}{a})^{b-1}\exp{[-(\frac{x}{a})^{b}]} \tag{9}\]  
\[\text{Gamma}: f(x|\,a,b)=\frac{1}{b^{\intercal}(a)}\,x^{a-1}\exp{(-\frac{x}{b})} \tag{10}\]  
\[\text{Lognormal}: f(x|\,a,b)=\frac{1}{xb\sqrt{2\pi}}\exp{[\frac{-(\ln{(x)-a)}^{2}}{2b}]} \tag{11}\]  
\[\text{Log-logistic}: f(x|\,a,b)=\frac{\exp{[\frac{\ln{(x)-a}}{b}]}}{b\chi\left\{1+\exp{[ \frac{\ln{(x)-a}}{b}]}\right\}^{2}} \tag{12}\]  
Figure 3 and Table 4 summarize the distribution fitting of 12 gust descriptors considered in this study. The median gust magnitude and gust amplitude were, respectively, 8.49 m/s and 2.92 m/s. The results show that the Weibull distribution was the most suitable for the 10-min average wind speed, gust level and gust amplitude (Figure 3a-c), while the distribution of gust factor, peak factor and turbulence intensity (Figure 3d-f) could be most satisfactorily expressed as a logarithmic logic distribution. Although the Weibull distribution function is often used to fit the average wind speed and gust wind speed, Figure 3 clearly demonstrates that it is not appropriate for unitless parameters, such as gust factor, peak factor and turbulent intensity. The median values of rising time and elapsed time were estimated to be 19.8 s and 28.4 s, respectively; their distributions conformed most closely to the lognormal distribution and the gamma distribution, albeit with minor difference in the LogL values. As can be seen from Figure 3g,h, the distribution peak of rising time was more pronounced and skewed to the right than that of lapsing time. Hence, the consequent distribution of the gust asymmetric factor (Figure 3l) was also right-skewed, which can be most desirably described by the lognormal distribution. The median value of GAF was about 1.46.