# Study on Fluctuating Wind Characteristics and Non-Stationarity at U-Shaped Canyon Bridge Site
## 4 DWT Reconstruction and Analysis
### Analysis of Non-Stationary Wind Data Using Optimal Decomposition Levels in DWT  
To compare and analyze the differences in turbulence intensity, turbulence integral scale, and turbulent power spectral density between pulsating wind extracted from the time-varying mean wind after wavelet decomposition and pulsating wind calculated under the stationary assumption, wind speed samples from the middle bridge deck height on August 6 were selected. These samples passed the ADF stationarity test at a significance level of 0.05 when the decomposition process reached level 9. The \"db10\" wavelet at the optimal decomposition level (level 5) identified in the previous study was selected to reconstruct the sample data. Figure 13 illustrates the reconstruction results for selected periods of the U-component wind speed sample. Although the definition and calculation method of the non-stationary wind speed model are consistent, the parameters correspond specifically to the non-stationary wind speed model. The calculation formula is given as  
\[I_{a}^{*}=\frac{\sigma_{a}^{*}}{U*}(\alpha=u,v,w) \tag{13}\]  
Figure 14: Time-varying mean wind from wavelet reconstruction and original wind speed. Figure 13: DWT reconstruction results for the sample at level 5.  
where \(I^{*}{}_{\alpha}\) is the turbulence intensity in three directions, and the subscript \(\alpha\) represents \(u\), \(v\), and \(w\), respectively. \(\sigma^{*}{}_{\alpha}\) denotes the standard deviation of wind speed fluctuations and \(U^{*}\) represents the time-varying mean wind speed. The turbulence integral scales of the stationary and non-stationary models are calculated as follows:  
\[L_{\alpha}=\frac{U}{\sigma_{\alpha}^{2}}\int_{0}^{\infty}R_{\alpha}(\tau)d\tau,(\alpha=u,v,w) \tag{14}\]  
\[L_{\alpha}^{*}=\frac{\overline{U}^{*}}{\sigma_{\alpha}^{2}}\int_{0}^{\infty}R _{\alpha}^{*}(\tau)d\tau,(\alpha=u,v,w) \tag{15}\]  
\(L_{\alpha}\) and \(L^{*}{}_{\alpha}\) denote the integral scales of stationary and non-stationary turbulence, respectively, while \(R_{\alpha}\) and \(R^{*}{}_{\alpha}\) represent the autocovariance functions of stationary and non-stationary fluctuations, respectively. To account for large errors caused by [PERSON]'s hypothesis when autocovariance is small, it is recommended to limit the integral to the first \(t\), where \(R_{\alpha}(\tau)=0.05\sigma^{2}{}_{\alpha}\) or \(R_{\alpha}(\tau)=0.05(\sigma^{*}{}_{\alpha})\)[25]. Based on the two different wind speed models, the turbulence intensity and integral scale in the \(u\), \(v\), and \(w\) directions at the 50 m bridge deck height within a 1 h sample are illustrated in Figure 15. The results indicate that turbulence intensity under the stationary wind speed model is higher than that under the non-stationary wind speed model, suggesting that the stationary wind speed model overestimates turbulence in non-stationary wind, consistent with findings by [PERSON] et al. [12]. However, the turbulence integral scale exhibits slight differences: the stationary wind speed model yields a lower turbulence integral scale compared to the non-stationary model. When trends or periodic changes in actual wind speed are present, the stationary model interprets these changes as random fluctuations, thereby increasing the standard deviation of velocity fluctuations and resulting in higher turbulence. An inverse relationship exists between the turbulence integral scale and turbulence intensity. This indicates that analyses based on the stationary assumption may misinterpret some large-scale structures as fluctuating components, thereby increasing the standard deviation and failing to accurately capture long-term correlations or large-scale structures. Figure 15: Analysis of different wind speed models: (**a**) turbulence intensity (**b**) turbulence integral scale. Figure 16 illustrates the reconstructed power spectral density of the sample under different wind speed models.