# Study on Fluctuating Wind Characteristics and Non-Stationarity at U-Shaped Canyon Bridge Site
## 4 DWT Reconstruction and Analysis  
### DWT Analysis Process and Theory  
Based on the above studies, it is evident that the non-stationary model influences the pulsating wind characteristics in the canyon area. To further evaluate its impact, the wavelet transform (WT) method, commonly applied in analyzing non-stationary signals, is introduced. Discrete wavelet transform (DWT), a specific form of wavelet transform, decomposes a signal into wavelet coefficients at different scales, capturing its local characteristics. The widely used db10 wavelet DWT was adopted in this study [12,22,23] to extract time-varying mean wind speed and analyze the effects of different decomposition levels. The decomposition process is illustrated in Figure 8.  
The DWT db10 decomposition method decomposes the signal using a set of low-pass filters h[n] and high-pass filters g[n]. The db10 wavelet has predefined values for the coefficients of both the low-pass and high-pass filters. Given a signal sequence x[n], the db10 wavelet decomposition can be mathematically expressed as follows:  
\[A_{j}[m]=\sum_{n}x[n]\cdot h[2m-n] \tag{6}\]  
Figure 8: Flowchart illustrating the wavelet decomposition and reconstruction steps. Figure 7: Relationship between gust factor and turbulence for different models. \[D_{j}[m]=\sum_{n}x[n]\cdot g[2m-n] \tag{7}\]  
\(A_{j}[m]\) is the approximation coefficients and \(D_{j}[m]\) is the detail coefficients; \(j\) represents the decomposition depth (level). To reconstruct and calculate the time-varying mean wind at different levels, it is necessary to convolve the low-frequency coefficients \(A_{j}[m]\) and high-frequency coefficients \(D_{j}[m]\) of that level with the corresponding filter coefficients. The reconstruction can be expressed as  
\[A_{j-1}[n]=\sum_{m}\bigl{(}A_{j}[m]\cdot h_{r}[2m-n]+D_{j}[m]\cdot g_{r}[2m-n] \bigr{)}. \tag{8}\]  
The symbol \(A_{j-1}[n]\) denotes the reconstruction outcome at the current decomposition level, accounting for boundary effects. In this study, the maximal decomposition depth \(N_{0}\) is established at 11, a parameter determined through Equation 9. This equation is derived from the principles of decomposition as described by the Mallat algorithm [24]. \[N_{0}=log_{2}(T\cdot s) \tag{9}\]  
\(N_{0}\) denotes the maximum decomposition level of the discrete wavelet transform (DWT); \(T\) represents the time interval; and \(s\) indicates the sampling frequency of the wind record. For the imported samples, those passing the ADF stationarity test at a significance level of 0.05 were directly averaged and analyzed based on the original stationarity hypothesis. For samples failing the test, the db10 wavelet basis was used to decompose the original wind speed data \(U(t)\), reconstruct the low-frequency component \(U^{\bullet}(t)\), and adjust the time-varying mean wind speed. The reconstructed component \(U^{\bullet}(t)\) is subtracted from \(U(t)\) to obtain the pulsating wind speed \(u^{\bullet}(t)\), which is then subjected to a stationarity test. If stationarity is not achieved, the process iterates with \(N-1\). To comprehensively analyze the time series characteristics of pulsating wind under different orders and verify the effectiveness of wavelet decomposition, this study employed the ACF and PACF methods to identify autocorrelation and partial autocorrelation in the pulsating wind sequence. This ensures that the extracted pulsating wind accurately represents the medium- and high-frequency fluctuations in the original signal, thereby providing a foundation for further predictions and analyses. The autocorrelation function (ACF) test is a method for analyzing time series data. It measures the correlation of a signal with itself over specified time delays. For time series data, the ACF reveals linear dependencies between data points, particularly the degree of correlation between adjacent points.