# Coastal Wind Measurements Using a Single Scanning LiDAR
## 2 Materials and Methods
### Scanning Mode and the Velocity Volume Processing (VVP) Method
#### 2.3.2 Velocity Volume Processing (VVP)  
Figure 3 illustrates the schematic of the measurement positions and heights for Case 1. The small red square and the diagonal line passing through indicate the 100s and the coastline, respectively. The heading angle of the 100s and the along- and across-wind direction sectors are shown in the pie chart in the upper left corner. The radial wind speeds were measured along the line of the colored dots, with the measurement line moving to the next position at an interval of 1 s. For this case (i.e., Case 1), it takes roughly 20 s to complete a cycle; the device then repeats the operation. A set of radial wind speeds for an arc at the same radial distance was used to compute the horizontal wind speeds and directions in the process of wind vector retrieval using the VVP method. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Start Date** & **End Date** & **Duration** & \(\varphi_{\text{range}}\) & \(\omega\) & \(\theta\) & \multirow{2}{*}{\(U_{\text{ref}}\)} & **Height** \\  & **(LST)** & **(LST)** & **(Hours)** & (\(\Gamma\)) & (\(\Gamma\)) & (\(\Gamma\)) & (\(\Gamma\)) & **(\(\Gamma\))** \\ \hline
1 & 2019-02-20 & 2019-02-25 & 119.6 & 60 & 3 & 5.07 & V1 & 47 \\
2 & 2019-01-22 & 2019-01-26 & 97.5 & 45 & 3 & 5.07 & V1 & 47 \\
3 & 2019-01-26 & 2019-01-30 & 96.1 & 30 & 3 & 5.07 & V1 & 47 \\
4 & 2019-01-30 & 2019-02-04 & 112.5 & 15 & 3 & 5.07 & V1 & 47 \\
5 & 2019-02-13 & 2019-02-20 & 169.3 & 15 & 1 & 5.07 & V1 & 47 \\
6 & 2019-04-19 & 2019-04-23 & 100.2 & 45 & 3 & 14.11 & V1 & 107 \\
7 & 2019-02-04 & 2019-02-08 & 102.5 & 45 & 3 & 2.05 & V1 & 207 \\
8 & 2019-06-25 & 2019-09-22 & 2150.6 & 45 & 3 & 5.07 & V1 & 47 \\
9 & 2019-04-23 & 2019-06-25 & 1512.2 & 45 & 3 & 0.00 & SA & 7 \\ \hline \hline \end{tabular} \(\varphi_{\text{range}}\): azimuth angle range; \(\omega\): scan rate; \(\theta\): elevation angle; \(U_{\text{ref}}\): reference sensor for the validation; Height: validation height.  
\end{table} 
Table 2: The experimental period and scanning settings for each case. The Japanese Local Standard Time (LST) is +9 hours from UTC, sonic anemometer (SA). Figure 3: Measurement of horizontal and vertical positions of the radial wind speed for Case 1. The red square indicates the location of the 100s. The pie chart shows the definition of the along and across wind direction sectors. As noted earlier, the radial wind speeds obtained from the 100s were converted to horizontal wind speeds using the VVP method, which is discussed next. The radial wind speed can be expressed as the projection of the actual velocity vector along the radial direction, i.e.,  
\[Vr=u\cos\theta\sin\varphi+v\cos\theta\cos\varphi+w\cos\theta, \tag{2}\]  
where \(Vr\) is the radial wind speed, \(u\), \(v\), and \(w\) are the horizontal and vertical wind speed components, \(\theta\) is the elevation angle, and \(\varphi\) is the azimuth angle. In the original VVP, the wind speed is assumed to vary linearly about the analysis point, thus resulting in derivative terms of each wind speed component along all three spatial directions [26; 27]. Therefore, in total, there will be 12 parameters on the right-hand side of Equation (2). As it is unnecessary to have the full set of these parameters for wind resource assessment considered in this study, we used an assumption to simplify the equation, following previous studies [25]. If we assume that the wind field is homogeneous in time and space for one scan cycle, and that the vertical component \(w\) is negligibly small compared to the horizontal wind speed components, Equation (2) can be written as  
\[Vr=u\cos\theta\sin\varphi+v\cos\theta\cos\varphi. \tag{3}\]  
As \(Vr\), \(\theta\), and \(\varphi\) are obtained from the 100s, \(u\) and \(v\) can be simply computed using the linear-least-squares method.