## 2 Methodology for Calculation of Peak Factors and Extreme Values
\begin{array}{l}m_{3}=x^{3}\big{(}6h_{3}+36h_{3}h_{4}+8h_{3}^{2}+108h_ {3}h_{4}^{2}\big{)}\\ m_{4}=x^{4}\big{(}3+24h_{4}+60h_{3}^{2}+252h_{4}^{2}+576h_{3}^{2}h_{4}+1296h_{4 }^{3}+60h_{3}^{4}+2232h_{3}^{2}h_{4}^{2}+3348h_{4}^{4}\big{)}\end{array} \right.. \tag{13}\]  
Theoretically, as soon as the skewness \(m_{3}\) and kurtosis \(m_{4}\) are determined, \(h_{3}\) and \(h_{4}\) can be calculated by solving Equation (13). Nevertheless, Equation (13) is a system of non-linear simultaneous and implicit equations for \(h_{3}\) and \(h_{4}\). For calculation efficiency and application, an explicit expression for \(h_{3}\) and \(h_{4}\) is more favorable. To this end, an approximate explicit expression for \(h_{3}\) and \(h_{4}\) is proposed [31]:  
\[\left\{ \begin{array}{l}h_{3}=\frac{m_{3}}{6}\left[\frac{1-0.015m_{3}+0.3m_{4 }}{1+0.2(m_{4}-3)}\right]\\ h_{4}=h_{40}\Big{[}1-\frac{1.43m_{3}}{m_{4}-3}\Big{]},\quad h_{40}=\frac{\left[ 1+1.25(m_{4}-3)^{1/3}\right]_{-1}}{10}\end{array} \right.\left(3<m_{4}<15\right) \tag{14}\]  
However, the accuracy of Equation (14) cannot be guaranteed when process \(Z\) has strong non-Gaussian properties. Therefore, another approximate expression for \(h_{3}\) and \(h_{4}\) is proposed:  
\[\left\{ \begin{array}{l}h_{3}=&0.2m_{3}-0.02m_{3}m_{4}+0.02m_{3} ^{3}+7.4\times 10^{-4}m_{3}{m_{4}}^{2}\\ &-9.2\times 10^{-4}{m_{3}}^{3}m_{4}-1.4\times 10^{-5}m_{3}{m_{4}}^{3}+1.5\times 1 0^{-4}{m_{3}}^{5}\\ &+1.1\times 10^{-5}{m_{3}}^{3}{m_{4}}^{2}+8.8\times 10^{-8}{m_{3}}{m_{4}}^{4}\\ h_{4}=&-0.07+0.03m_{4}-0.03{m_{3}}^{2}-0.002m_{4}^{2}+0.002{m_{3}}^{2}m_{4}\\ &+6\times 10^{-5}{m_{4}}^{3}-6.3\times 10^{-4}{m_{4}}^{4}-6.4\times 10^{-5}{m_{3}}^{ 2}{m_{4}}^{2}-9.7\times 10^{-7}{m_{4}}^{4}\\ &+1.5\times 10^{-5}{m_{3}}^{4}m_{4}+5.5\times 10^{-7}{m_{3}}^{2}{m_{4}}^{3}+6 \times 10^{-9}{m_{4}}^{5}\end{array} \right. \tag{15}\]  
The above expression has been verified as accurate by wind tunnel test results in this study. It is applicable for strong non-Gaussian processes. Therefore, Equation (15) is adopted for calculations of shape parameters \(h_{3}\) and \(h_{4}\). After \(h_{3}\) and \(h_{4}\) are determined, the CDF and PDF for \(z_{\rm ex}\) in Equations (3) and (4) can be determined. Thus, a \(p\)-fractile of the extreme value for \(Z\) can be calculated from \(U\) according to the translation process method following:  
\[z_{\rm ex}=g(u_{\rm ex}),\ u_{\rm ex}=\sqrt{2\ln[v_{0}T/\ln(1/p)]} \tag{16}\]  
The expectation of \(z_{\rm ex}\) is the peak factor \(g_{\rm peak}\), which has a form of  
\[g_{\rm peak}=z_{\rm ex,mean}=\int_{0}^{\infty}z_{\rm ex}p_{\rm ex}(z)dz. \tag{17}\]  
\(g_{\rm peak}\) corresponding to a \(57\%\) percentile can be derived as:  
\[g_{\rm peak}=\kappa\bigg{\{}\beta+\frac{\gamma}{\beta}+h_{3}\bigg{(}\beta^{2}+2 \gamma-1+\frac{1.98}{\beta^{2}}\bigg{)}+h_{4}\bigg{[}\beta^{3}+3\beta(\gamma-1 )+\frac{3}{\beta}\bigg{[}\frac{\pi^{2}}{6}-\gamma+\gamma^{2}\bigg{]}+\frac{5. 44}{\beta^{3}}\bigg{]}\bigg{\}} \tag{18}\]  
where \(\beta=\sqrt{2\ln(v_{0}T)}\), and \(\gamma=0.5772\) is [PERSON]'s constant. With this, peak factors for non-Gaussian processes can be calculated by Equation (18). In addition, it should be noted that when \(m_{3}=0\) and \(m_{4}=3\), \(h_{3}\) and \(h_{4}\) are zero. Thus, Equation (18) degrades as:  
\[g_{\rm peak,G}=\beta+\frac{\gamma}{\beta}=\sqrt{2\ln(v_{0}T)}+\frac{\gamma}{ \sqrt{2\ln(v_{0}T)}}, \tag{19}\]which is accordingly the peak factor expression for the Gaussian process. The extreme value of a general non-Gaussian process \(X\) can be calculated as follows:  
\[x_{ex}=\mu\pm g_{\text{peak}}\sigma. \tag{20}\]  
It is worth mentioning that Equation (20) is only applicable for obtaining a positive peak factor; when the mean value of \(X(t)\) is negative, the time history of \(X(t)\) should be multiplied by \(-1\) to calculate the corresponding peak factor. The flowchart of the HPM is illustrated in Figure 2.