## Wind Gust Definition
(2016, 2017) applied the method to derive observations of wind speed maxima from instrumentation on board a research aircraft ([PERSON] et al., 1991) and from a Doppler lidar ([PERSON] et al., 1991). Both instrumentations have the potential to provide wind gust measurements from heights and regions yet unreachable by the traditional in situ anemometers. According to the Rice theory, the probability that the instantaneous horizontal wind speed \(u(t)=u\) performs one up-crossing (the time derivative \(\dot{u}>0\)) of the level \(U_{x}\) is:  
\[\int_{0}^{\infty}d\dot{u}\int_{U_{x}-\dot{u}\Delta t}^{U_{x}}P(u,\dot{u})du \approx\Delta t\int_{0}^{\infty}\dot{u}P(U_{x},\dot{u})d\dot{u}, \tag{3}\]  
where \(P(u,\dot{u})\) is the joint probability of the wind speed and its time derivative ([PERSON] et al., 1991). The covariance of \(u\) and \(\dot{u}\) is:  
\[\left\langle(u-U)\dot{u}\right\rangle=\frac{1}{2}\frac{d}{dt}\left\langle u^{ 2}\right\rangle, \tag{4}\]  
where the angle brackets denote the mean. If the variance of \(u\) is stationary (does not depend on time), i.e., \(\frac{d}{dt}\left\langle u^{2}\right\rangle=0\), \(u\) and \(\dot{u}\) will be uncorrelated and hence statistically independent:  
\[P(u,\dot{u})=P(u)P(\dot{u}). \tag{5}\]  
Following ([PERSON] et al., 1991), both probabilities, \(P(u)\) and \(P(\dot{u})\), are assumed Gaussian:  
\[P(u)=\frac{1}{\sqrt{2\pi}\sigma_{u}}e^{-\frac{(u-U)^{2}}{2\sigma _{u}^{2}}}\] \[P(\dot{u})=\frac{1}{\sqrt{2\pi}\sigma_{\dot{u}}}e^{-\frac{\dot{ u}^{2}}{2\sigma_{u}^{2}}}, \tag{6}\]  
although real atmospheric turbulence is typically non-Gaussian, because higher order statistical moments contribute to the probability density function (e.g., ([PERSON] et al., 1991)). To estimate those, quasi-Gaussian turbulence models have been developed (e.g., the Quasi-Normal Scale Elimination (QNSE) method by ([PERSON] et al., 1991, 1992)), which are, however, beyond the scope of this work. For strong gusts (for which \(U>>0\)), the two probabilities in Equation (6) are fairly good approximations. Following from Equations (3), (5) and (6), the average rate \(\eta\) of up-crossing of the level \(U_{x}\) is:  
\[\eta(U_{x})=P(U_{x})\int_{0}^{\infty}\dot{u}P(\dot{u})d\dot{u}=\frac{\sigma_{ \dot{u}}}{2\pi\sigma_{u}}e^{-\frac{(U_{x}-U)^{2}}{2\sigma_{u}^{2}}}. \tag{7}\]  
The expectation value for the number of upcrossings of the level \(U_{x}\) within a time series of length \(T\) is \(N(U_{x})=\eta(U_{x})T\). The probability for upcrossing the level \(U_{x}\)\(k\)-times within the time interval \(T\) can be estimated by using the Poisson distribution:  
\[P(k,N)=\frac{N(U_{x})^{k}}{k!}e^{-N(U_{x})}. \tag{8}\]As we are interested in estimating the maxima of the wind speed time series, we will set \(k=0\), which gives the probability of \(u\) not exceeding \(U_{x}\) in period \(T\):  
\[P(0,N)=\exp\left(-N(U_{x})\right). \tag{9}\]  
Now, combining Equations (7) and (9) yields:  
\[g_{x}=\frac{U_{max}-U}{\sigma_{u}}, \tag{2}\]0  
where \(\tau=\frac{\sigma_{u}}{\sigma_{u}}\) is the turbulent time scale. From Equation (10), we can estimate the peak factor as \(g_{x}\approx\frac{U_{x}-U}{\sigma_{u}}\) yielding:  
\[g_{x}=\frac{U_{max}-U}{\sigma_{u}}, \tag{2}\]1  
where \(P\) is the probability of a peak factor not exceeding the value \(g_{x}\) in an ensemble of samples. That is, \(P(u<U_{x})=P((u-U)<(U_{x}-U))=P(\frac{u-U}{\sigma_{u}}<\frac{U_{x}-U}{\sigma_ {u}})\). Thus, the median peak factor in an ensemble of samples corresponds to a probability \(P=0.5\). The turbulent time scale \(\tau\) in Equation (11) can be estimated using information on the (one-sided) turbulence power spectrum \(S(f)\), where \(f\) is the frequency (e.g., [57]). \[g_{x}=\frac{U_{max}-U}{\sigma_{u}}, \tag{2}\]2  
The advantage of Rice theory is that it provides the theoretical basis to estimate the relationship between the gust duration and the wind gust speed. Equation (12) provides the turbulent time scale of the continuous wind speed signal including all scales of turbulent motion beyond the sample length \(T\).