# A Spectrum Correction Method Based on Optimizing Turbulence Intensity
## 2 Materials and Methods
### Basic Algorithm for Simulating Fluctuating Wind Speed Time History  
Considering the simulation process at a certain point \(v_{j}(t),(j=1,2,3,\ldots,m)\) in the turbulent wind field in space, the basic algorithm for simulating and solving the one-dimensional \(M\)-variable turbulent wind field \(\left\{v_{j}(t)\right\}\) is described, and its algorithm flow is shown as follows:  
1. According to the sampling theorem [16], the simulation parameters are set in the frequency domain, including: * Sampling frequency \(f_{s}\), in order to ensure that the analog signal can be reconstructed accurately without aliasing, \(f_{s}\) is required to be greater than two times the value of \(F_{max}\), \(F_{max}\) indicates the upper limit of the analog frequency range, which denotes also the cut-off frequency in this paper; * The initial frequency \(F_{min}\) and cutoff frequency \(F_{max}\), which are selected by considering the dimensionless power spectral density function image and the influence of truncation error in the simulation frequency range on the simulation variance [17], the frequency step size is denoted by \(df=\frac{F_{max}-F_{min}}{N}\), where \(N\) represents the frequency sampling number.
2. Set simulation parameters in the time domain, including: * Sampling interval \(dt\), \(dt=1/f_{s}\); * Analog time points \(M\), generally \(M=2N\).
3. Set the basic parameters of the simulated wind field, including simulation points m, simulation height \(z\), and spacing between simulation points dr, etc. \[dr=\left(\frac{z}{m-1}\right)\cdot\sqrt{\left(x_{k}-x_{j}\right)^{2}+\left(y_{ k}-y_{j}\right)^{2}}\] (3) where, \(x_{k},x_{j},y_{k},y_{j}\) represent the abscissa and ordinate values of simulation point \(k\) and simulation point \(j\), respectively.
4. Calculate the cross-spectral density matrix of each frequency sampling point \(\left\{S(f_{n})\right\}\), \((n=1,2,3,\cdots,N)\): \[S(f_{n})=\left[\begin{array}{cccc}S_{11}&S_{12}&\cdots&S_{1m}\\ S_{21}&S_{22}&\cdots&S_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ S_{m1}&S_{m2}&\cdots&S_{mm}\end{array}\right]\] (4) where, \(S(f)\) denotes the power spectral density function, \(S_{jj}\) denotes the autocorrelation spectrum of corresponding points, and \(S_{kj}(k\
eq j)\) denotes the cross-correlation spectrum between two simulated points, \((k=1,2,3,\cdots,j)\), which can be calculated as: \[S_{kj}(f_{n})=\sqrt{S_{k}(f_{n})S_{j}(f_{n})}\cdot coh(k,j;f)\] (5)  
Since the cross-correlation function of the fluctuating wind speed in one-dimensional space does not involve the change of height, the cross-correlation spectrum between the simulated point j and other simulated points is consistent, and the cross-correlation spectrum of the fluctuating wind speed in one-dimensional space can be obtained by the following formula:  
\[S_{kj}(f_{n})=S_{j}(f_{n})\cdot coh(k,j;f) \tag{6}\]
5. Combined with the double-indexed frequency method, the Cholesky decomposition method is performed on the cross-spectral density matrix of sampling points at each frequency, and the decomposed lower triangular matrix is obtained, that is: \[S(f_{kn})=H(f_{kn})H^{T}(f_{kn})\] (7)\[f_{kn}=(n-1)df+(k/m)df \tag{8}\]  
where, \(f_{kn}\) denotes the double-indexed frequency; \(H(f_{kn})\) denotes the decomposed lower triangular matrix.  
Random phase obedience \(\phi_{kn}\) is introduced and subjected to independent random distribution among intervals \((0,2\pi)\), carrying out the Fast Fourier transformation, namely:  
\[G_{kn}(t)=FFT(B_{kn}) \tag{9}\] \[B_{kn}(ndf)=\left\{\begin{array}{c}H(ndf+(k/m)df)e^{i\phi_{kn}},(0 \leq n<N)\\ 0,(N\leq n<M)\end{array}\right. \tag{10}\]  
Employing harmonic superposition method to generate the wind speed time history of the simulated point \(j\) samples, and \(v_{j}(t)\) can be calculated by Equation (11):  
\[v_{j}(p\cdot dt)=\sqrt{2 df}Re(\sum_{k=1}^{j}G_{jk}exp(i\frac{k}{m}\cdot 2\pi df \cdot p\cdot dt) \tag{11}\]  
where, \(p=1,2,3,\cdots,(M\cdot m-1)\).