## Theoretical Framework
### Wind Simulation
\begin{array}{cccc}\cos(\omega_{1}t_{1}+\theta_{1,1})&\cos(\omega_{1 }t_{2}+\theta_{1,1})&\dots&\cos(\omega_{1}t_{N_{s}}+\theta_{1,1})\\ \cos(\omega_{1}t_{1}+\theta_{1,2})&\cos(\omega_{1}t_{2}+\theta_{1,2})&\dots& \cos(\omega_{1}t_{N_{s}}+\theta_{1,2})\\ \vdots&\vdots&\ddots&\vdots\\ \cos\Big{(}\omega_{1}t_{1}+\theta_{1,N_{\overline{p}}}\Big{)}&\cos\Big{(} \omega_{1}t_{2}+\theta_{1,N_{\overline{p}}}\Big{)}&\dots&\cos\Big{(}\omega_{1 }t_{N_{s}}+\theta_{1,N_{\overline{p}}}\Big{)}\\ \cos(\omega_{2}t_{1}+\theta_{2,1})&\cos(\omega_{2}t_{2}+\theta_{2,1})&\dots& \cos(\omega_{2}t_{N_{s}}+\theta_{2,1})\\ \vdots&\vdots&\ddots&\vdots\\ \cos\Big{(}\omega_{2}t_{1}+\theta_{2,N_{\overline{p}}}\Big{)}&\cos\Big{(} \omega_{2}t_{2}+\theta_{2,N_{\overline{p}}}\Big{)}&\dots&\cos\Big{(}\omega_{2 }t_{N_{s}}+\theta_{2,N_{\overline{p}}}\Big{)}\\ \vdots&\vdots&\ddots&\vdots\\ \cos\Big{(}\omega_{N_{f}}t_{1}+\theta_{N_{f},1}\Big{)}&\cos\Big{(}\omega_{N_{ f}}t_{2}+\theta_{N_{f},1}\Big{)}&\dots&\cos\Big{(}\omega_{N_{f}}t_{N_{s}}+ \theta_{N_{f},1}\Big{)}\\ \vdots&\vdots&\ddots&\vdots\\ \cos\Big{(}\omega_{N_{f}}t_{1}+\theta_{N_{f},N_{\overline{p}}}\Big{)}&\cos \Big{(}\omega_{N_{f}}t_{2}+\theta_{N_{f},N_{\overline{p}}}\Big{)}&\dots&\cos \Big{(}\omega_{N_{f}}t_{N_{s}}+\theta_{N_{f},N_{\overline{p}}}\Big{)}\end{array} \right] \tag{31}\]  
The cross-spectral density matrix describes the influence of the turbulence components at two points at a given frequency. This influence is due to the spatial dimension of the vortices in the wind field. Considering Cholesky, if the cross spectral density matrix \(S^{0}(\omega)\) is symmetric positive definite, then \(S^{0}(\omega)\) can be decomposed as [1]:  
\[S^{0}(\omega)=\overline{\mathbf{H}}(\omega)\overline{\mathbf{H}}^{\mathrm{T}} (\omega), \tag{32}\]  
where \(S^{0}(\omega)\) is the cross spectral density matrix for angular frequency \(\omega\); and \(\overline{\mathbf{H}}(\omega)\) is a lower triangular matrix. The process of Cholesky factorization is as follows:\[\overline{\mathrm{H}}_{11}(\omega)=S_{11}^{0}(\omega)^{1/2} \tag{33}\] \[\overline{\mathrm{H}}_{21}(\omega)=S_{21}^{0}(\omega)/\overline{ \mathrm{H}}_{11}(\omega)\] \[\overline{\mathrm{H}}_{22}(\omega)=(S_{22}^{0}(\omega)-\overline{ \mathrm{H}}_{21}(\omega)^{2})^{1/2}\] \[\overline{\mathrm{H}}_{31}(\omega)=S_{31}^{0}(\omega)/\overline{ \mathrm{H}}_{11}(\omega)\] \[\vdots\] \[\overline{\mathrm{H}}_{jk}(\omega)=\left(S_{jk}^{0}(\omega)-\sum_ {l=1}^{k-1}\overline{\mathrm{H}}_{jl}(\omega)\overline{\mathrm{H}}_{kl}( \omega)\right)/\overline{\mathrm{H}}_{kk}(\omega)\] \[\overline{\mathrm{H}}_{kk}(\omega)=(S_{kk}^{0}(\omega)-\sum_{l=1 }^{k-1}\overline{\mathrm{H}}_{kl}(\omega)^{2})^{1/2}.\]  
For two points with vertical separation\(r_{y}\), the \(S_{ij}^{0}(\omega)\) corresponding elements are obtained by the cross-spectral density function \(S_{uu}\left(z_{i},z_{j},n\right)\):  
\[S_{uu}\left(z_{i},z_{j},n\right)=\sqrt{S_{u}(z_{i},n)S_{u}\left(z_{j},n\right) }\sqrt{\mathrm{coh}\left(r_{y},n\right)}, \tag{34}\]  
where \(S_{uu}\left(z_{i},z_{j},n\right)\) is the cross spectral density function for two longitudinal turbulent components at space points \(z_{1}\) and \(z_{2}\); \(S_{u}(z_{i},n)\) is the single power density spectrum at \(z_{i}\), defined in Equation (38); \(\sqrt{\mathrm{coh}\left(r_{y},n\right)}\) is the root-coherence function, defined in Equation (35); \(n\) is the frequency in Hz. The root-coherence function defines the static dependence between two turbulent components at two different points. This function tends to zero when the separation \(r_{y}\) increases; in other words, the cross-influence of the wind turbulent is inversely proportional to the separation.