# Study on Fluctuating Wind Characteristics and Non-Stationarity at U-Shaped Canyon Bridge Site
## 3 Analysis of Fluctuating Wind Characteristics  
### Non-Stationarity Theory and Analysis Method  
In wind engineering, wind speed is typically divided into mean wind and fluctuating wind under the assumption of stationarity, mathematically expressed as follows:  
\[U(t)=\overline{U}(t)+u(t) \tag{1}\]  
\(U(t)\) represents the measured wind speed, \(U(t)\) represents the mean wind speed within the time period, and \(u(t)\) is the fluctuating wind speed.  
A wind speed series \(\{X_{t}\}\) is considered stationary if its statistical properties remain constant over time. The stationary model assumes that wind speed follows a stationary random process. This assumption is often valid in flat terrain. However, the stationary model is applicable only when wind speed characteristics do not exhibit statistically significant variations over time. In complex mountainous terrains, wind field changes often exhibit pronounced non-stationary characteristics.  
The non-stationary model decomposes wind speed into a time-varying mean wind and a fluctuating component, defined as the difference between the measured wind speed and the time-varying mean wind:  
\[U(t)=\overline{U}^{*}(t)+u^{*}(t) \tag{2}\]  
Figure 1: Layout of wind tower at U-shaped canyon bridge site.  
\(U(t)\) represents the measured wind speed, \(U^{*}(t)\) represents the time-varying mean wind, and \(u^{*}(t)\) is the fluctuating wind.  
Selecting an appropriate wind speed model is crucial for analyzing wind characteristics. Stationarity tests play a key role in selecting appropriate wind speed models. Common methods for assessing time series stationarity include the following:  
Unit Root Tests: The Augmented Dickey-Fuller (ADF) test extends the original Dickey-Fuller test and is suitable for higher-order autoregressive processes.  
KPSS Test: The Kwiatkowski-Phillips-Schmidt-Shin test assumes stationarity as the null hypothesis, contrasting with unit root tests.  
Runs Test: This detects random patterns in a time series to indirectly infer stationarity.  
Coefficient of Variation Test: This assesses stability by comparing the ratio of the standard deviation to the mean across different time periods.  
In this study, the ADF test is employed to assess stationarity. The ADF test is an extended form of the unit root test, which refers to a characteristic of a time series where shocks have permanent effects. In the context of the ADF test, if the time series has a unit root, it implies that the series follows a random walk, possibly with drift, does not revert to a long-term mean or trend line, and evaluates whether a unit root exists in a time series.  
The general form of the ADF test equation can be written as  
\[\Delta y_{t}=\alpha+\gamma y_{t-1}+\sum_{i=1}^{p-1}\delta_{i}\Delta y_{t-i}+ \epsilon_{t} \tag{3}\]  
where \(\Delta yt\) denotes the first-order difference in the time series, \(\alpha\) represents the constant term, and \(\gamma\) is the parameter under investigation, corresponding to the coefficient of the lagged value \(y_{t-1}\) of the original time series. \(\delta_{i}\) is the coefficient of the autoregressive term after differencing, and \(p\) denotes the number of lag periods. The default lag period is given by \(12\times(nobs/100)\)'\((1/4)\), where \(nobs\) refers to the number of observations. \(\epsilon_{t}\) represents the error term.  
The null hypothesis states that the time series contains a unit root and is non-stationary. If the \(p\)-value is below the chosen significance level of 0.05, the \(p\)-value is a measure of the evidence against a null hypothesis. It represents the probability of observing the test statistics or something more extreme, assuming the null hypothesis is true. In the context of the ADF test, a low \(p\)-value suggests strong evidence against the null hypothesis of a unit root. The significance level is a threshold set by the researcher to decide when to reject the null hypothesis. If the null hypothesis is rejected, this indicates that the time series is stationary.