## 4 Verification of Correlation Functions
### Stationary Process  
Based on the target PSD and coherence function, the target auto-/cross- correlation functions \(R^{0}_{jk}(\tau)\) of the stationary process can be calculated by  
\[R^{0}_{jk}(\tau)=\int_{-\infty}^{\infty}\sqrt{S_{jj}(\omega)S_{kk}(\omega)} \gamma_{jk}(\omega)e^{i\omega\tau}d\omega;j,k=1,2,\cdots,n \tag{22}\]In order to verify the effectiveness of the simulated nonergodic wind field, the ensemble auto-/cross-correlation functions \(R_{jk}(\tau)\) need to be calculated by the generated multiple numerical samples \(x_{j}(t)\), \(j=1,2,\cdots,n\), i.e.,  
\[R_{jk}(\tau)=E[x_{j}(t)x_{k}(t+\tau)] \tag{23}\]  
where \(E[\cdots]\) denotes mathematical expectation. For the ergodic wind field, the temporal auto-/cross-correlation function \(R_{jk}^{(i)}(\tau)\) need to be calculated by the generated numerical samples \(x_{j}(t)\), \(j=1,2,\cdots,n\), i.e.,  
\[R_{jk}^{(i)}(\tau)=\frac{1}{T}\int_{0}^{T}x_{j}(t)x_{k}(t+\tau)dt \tag{24}\]  
where \(T=\) time duration of ergodic samples. When the estimated correlation functions match the corresponding targets, the simulated wind field is considered to be effective.  
Before numerical examples, it is necessary to qualitatively study the influence of phase on simulation from the perspective of theoretical analysis, as follows.  
Based on the stationary simulation formulation Equation (10), the ensemble auto-/cross-correlation function of the simulated samples \(x_{j}(t)\), \(j=1,2,\cdots,n\) can be represented by  
\[\begin{array}{c}R_{jk}(\tau)=2\Delta\omega\sum\limits_{m=1-1}^{n}\sum \limits_{l=1-1}^{N}\sqrt{S_{jj}(\omega_{l})S_{lk}(\omega_{l})}|\beta_{jm}( \omega_{l})||\beta_{km}(\omega_{l})|\\ \times\cos[\omega_{l}\tau+\theta_{jm}(\omega_{l})-\theta_{km}(\omega_{l})];j,k =1,2,\cdots,n\end{array} \tag{25}\]  
which will converge to the corresponding target \(R_{jk}^{0}(\tau)\) as the frequency increment \(\Delta\omega\) decreases. Equation (25) presents the analytical correlation functions, which are theoretically identical to the numerical simulation results. Therefore, they can be utilized to qualitatively analyze the influence of phases on correlation functions. It can be seen from Equation (25) that the ensemble autocorrelation function of the simulated samples is independent on the phase, while the ensemble cross-correlation function of the simulated samples is related to the phase.  
For the ergodic stationary simulation, the single-index \(\omega_{l}\) in Equation (25) is replaced by the double-index \(\omega_{ml}\), and the ensemble auto-/cross-correlation function of the simulated samples \(x_{j}(t)\), \(j=1,2,\cdots,n\) can be expressed as [6]  
\[\begin{array}{c}R_{jk}(\tau)=2\Delta\omega\sum\limits_{m=1-1}^{n}\sum \limits_{l=1-1}^{N}\sqrt{S_{jj}(\omega_{ml})S_{lk}(\omega_{ml})}|\beta_{jm}( \omega_{ml})||\beta_{km}(\omega_{ml})|\\ \times\cos[\omega_{ml}\tau+\theta_{jm}(\omega_{ml})-\theta_{km}(\omega_{ml})] ;j,k=1,2,\cdots,n\end{array} \tag{26}\]  
When the time duration of the simulated sample \(T\) is equal to the period \(T_{0}\), the temporal auto-/cross-correlation function \(R_{jk}^{(i)}(\tau)\) of the sample function \(x_{j}(t)\), \(j=1,2,\cdots,n\) has the same expression, i.e.,  
\[R_{jk}^{(i)}(\tau)=R_{jk}(\tau)\text{ for }T=T_{0} \tag{27}\]  
\[T_{0}=n\frac{2\pi}{\Delta\omega} \tag{28}\]  
Obviously, the temporal autocorrelation function of the simulated ergodic samples is also independent on the phase, while the temporal cross-correlation function is related to the phase.