## 2 Wind Field Simulation by Spectral Representation Method
### Stationary Wind Fields  
Consider an \(n\)-variate vector-value stationary wind fluctuation process \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)]^{T}\), where \(T\) denotes transpose. Its spectral matrix is given by  
\[\mathbf{S}(\omega)= \begin{bmatrix}S_{11}(\omega)&S_{12}(\omega)&\cdots&S_{1n} (\omega)\\ S_{21}(\omega)&S_{22}(\omega)&\cdots&S_{2n}(\omega)\\ \vdots&\vdots&\ddots&\vdots\\ S_{n1}(\omega)&S_{n2}(\omega)&\cdots&S_{nn}(\omega)\end{bmatrix}  \tag{1}\]  
\[S_{jk}(\omega,t)=\sqrt{S_{jj}(\omega,t)S_{jk}(\omega,t)}\gamma_{jk}(\omega) \tag{2}\]  
where \(\mathbf{S}(\omega)=\) a Hermitian matrix with positive definite property; \(\omega=\) circle frequency; \(S_{jj}(\omega),j=1,2,\cdots,n=\) auto-PSD function of \(x_{j}(t)\) and a real and non-negative function of \(\omega\); \(S_{jk}(\omega),j=1,2,\cdots,n;j\
eq k=\) cross-PSD function between \(x_{j}(t)\) and \(x_{k}(t)\) and a complex function of \(\omega\); \(\gamma_{jk}(\omega)\) = coherence function between \(x_{j}(t)\) and \(x_{k}(t)\). In order to consider the wave passage effect, \(\gamma_{jk}(\omega)\) should be a complex number and can be expressed as  
\[\gamma_{jk}(\omega)=\Big{|}\gamma_{jk}(\omega)\Big{|}e^{i\theta_{jk}(\omega)} \tag{3}\]  
where \(i=\sqrt{-1}\) = imaginary unit; \(|\cdot|\) = modulus of a complex number; \(\theta_{jk}(\omega)\) = complex phase of \(\gamma_{jk}(\omega)\). According to Equations (1) and (2), the spectral matrix can be represented as  
\[\mathbf{S}(\omega)=\mathbf{D}(\omega)\mathbf{\Gamma}(\omega)\mathbf{D}(\omega) \tag{4}\]  
where \(\mathbf{D}(\omega)\) is a diagonal matrix composed of auto-PSDs, i.e.,  
\[\mathbf{D}(\omega)=diag\bigg{[}\sqrt{S_{11}(\omega)},\sqrt{S_{22}(\omega)}, \cdots,\sqrt{S_{nn}(\omega)}\bigg{]} \tag{5}\]  
and \(\mathbf{\Gamma}(\omega)\) is the coherence matrix composed of coherence functions, i.e.,  
\[\mathbf{\Gamma}(\omega)= \begin{bmatrix}1&\gamma_{12}(\omega)&\cdots&\gamma_{ 1n}(\omega)\\ \gamma_{21}(\omega)&1&\cdots&\gamma_{2n}(\omega)\\ \vdots&\vdots&\ddots&\vdots\\ \gamma_{n1}(\omega)&\gamma_{n2}(\omega)&\cdots&1\end{bmatrix}  \tag{6}\]  
Because \(\mathbf{\Gamma}(\omega)\) is also a Hermitian matrix with positive definite property, it can be decomposed by the following Cholesky decomposition  
\[\mathbf{\Gamma}(\omega)=\mathbf{B}(\omega)\mathbf{B}^{\mathbf{T}*}(\omega) \tag{7}\]  
\[\mathbf{B}(\omega)= \begin{bmatrix}\beta_{11}(\omega)&0&\cdots&0\\ \beta_{21}(\omega)&\beta_{22}(\omega)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ \beta_{n1}(\omega)&\beta_{n2}(\omega)&\cdots&\beta_{nn}(\omega)\end{bmatrix}  \tag{8}\]  
where \(*\) denotes complex conjugate; \(\mathbf{B}(\omega)\) = decomposed coherence matrix and a lower triangular matrix; \(\beta_{jj}(\omega),j=1,2,\cdots,n\) = a real and non-negative function of \(\omega\); \(\beta_{jk}(\omega),j=1,2,\cdots,n;j\
eq k\) = a complex function of \(\omega\); these complex functions can be written in polar form, i.e.,  
\[\beta_{jk}(\omega)=\Big{|}\beta_{jk}(\omega)\Big{|}e^{i\theta_{jk}(\omega)} \tag{9}\]  
where \(\theta_{jk}(\omega)\) = complex phase of \(\beta_{jk}(\omega)\). Then, the samples of \(x_{j}(t)\) can be generated by [35]  
\[x_{j}(t)=2\sqrt{\Delta\omega}\sum_{k=1}^{j}\sum_{l=1}^{N}\sqrt{S_{jj}(\omega_{ l})}\Big{|}\beta_{jk}(\omega_{l})\Big{|}\cos[\omega_{l}t-\theta_{jk}(\omega_{l})+ \phi_{kl}];j=1,2,\cdots,n \tag{10}\]  
where \(\omega_{l}=l\Delta\omega\) = discrete frequency; \(N\) = frequency discrete number; \(\Delta\omega=\omega_{u}/N\) = frequency increment; \(\omega_{u}\) = upper cutoff frequency; \(\phi_{kl}\) = independent random phase angle uniformly distributed in \([0,2\pi]\). Obviously, the deterministic phase \(\theta_{jk}(\omega)\) in the simulation Equation (10) is derived from the phase of coherence functions, reflecting the wave passage effect of the wind field.