## 2 Data and Methods
### Inversion Method  
In a horizontally homogeneous wind field, the variation of radial wind speed with azimuth for a VAD scan follows a sinusoidal curve [24] (refer to Figure 2). Thus, vertical profiles of horizontal wind vectors can be obtained by performing triangular transformation and sinusoidal fitting of radial wind speed with azimuth at each height [15]. The equations are as follows [24]:  
\[V_{0i}=a+b\cdot\cos(\theta_{i}-\theta_{max}), \tag{1}\]  
\[W_{s}=\frac{b}{\cos\varphi}, \tag{2}\]  
\[W_{d}= \begin{cases}\theta_{max}&if\ b<0\\ \theta_{max}+\pi&if\ b\geq 0^{\prime},\end{cases}  \tag{3}\]  
where \(V_{0i}\) is the radial wind speed (negative when coming to the Lidar) at \(i\) th direction (\(i=1,\ldots 24\)) (e.g., the squares in Figure 2), \(a\) is the vertical shift of the sinusoidal fitting curve, which can be used to derive the vertical velocity, \(b\) is the amplitude of the sinusoidal fitting curve, \(\theta_{max}\) is the phase shift (for the definition of \(a\), \(b\), and \(\theta_{max}\); please refer to Figure 2a), \(\theta\) is the azimuth, \(\varphi\) is the elevation angle, \(W_{s}\) is the horizontal wind speed, and \(W_{d}\) is the horizontal wind direction. Before using the above method to derive the horizontal wind speed and direction, some preprocessing is required on the original radial wind speed data from the Lidar. The raw data recorded by Lidar not only include instantaneous radial wind speed but also the Confidence Index (CI), which indicates the reliability of the data. To eliminate unreliable data, data with CI = 0 are discarded. Since precipitation could significantly affect the performance of Lidar, data with precipitation greater than 1 mm are discarded. Considering the uncertainty of Lidar radial wind speed at low wind speeds, data with radial wind speeds less than 2 m/s are discarded [25]. In practice, the carrier-to-noise ratio (\(CNR\)) of some points in a given VAD will have outliers due to the influence of clouds or precipitation as well as the fact that some of the radial directions may be completely or partially obstructed by obstacles such as wind turbines or wind towers (as shown by the circles in Figure 2a). To control the quality of the Lidar radial wind speed data, previous studies usually set a threshold of \(CNR\geq-27\) dB or higher as a cutoff and exclude data below the threshold [26; 27; 28]. This traditional method is simple, but due to the influence of environmental factors such as noise levels, atmospheric conditions, and reflection characteristics of the target, just using \(CNR\) to screen data may potentially remove some high-quality data. Therefore, in this study, we remove outliers using the standard deviation of \(CNR\) (\(\sigma_{CNR}\)) at each height in a full VAD scan. \(\sigma_{CNR}\) is calculated using the following equations:  
Figure 2: Fitting results for radial wind speed obtained at 08:00 on 4 April 2020 at 73 m AGL. (**a**) Traditional method (retain data with \(CNR\geq-27\) dB) and (**b**) \(\sigma_{CNR}\) filter. The blue squares represent the original radial velocity, and the gray ‘x’ s represent the corresponding \(CNR\) values. \[\overline{CNR}=\frac{1}{n}{\sum\limits_{i=1}^{n}CNR_{i}}, \tag{4}\]  
\[\sigma_{CNR}=\sqrt{\frac{1}{n}{\sum\limits_{i=1}^{n}}{\left(CNR_{i}-\overline{ CNR}\right)^{2}}}, \tag{5}\]  
where \(\overline{CNR}\) is the average of \(CNR\) for all azimuths at a certain range gate, \(CNR_{i}\) represents the \(CNR\) value at the \(i\) th direction, where \(i\) ranges from 1 to 24 (\(n=24\)). In this study, data points with \(\left|CNR_{i}-\overline{CNR}\right|>\)1.2\(\sigma_{CNR}\) are defined as outliers and removed. As seen in Figure 2b, this method can effectively remove data points with abnormal \(CNR\) values (indicated by the green circles in Figure 2a), and improve the fitting. The Goodness of Fit (GOF) increases from 0.938 (Figure 2a) to 0.948 (Figure 2b), and the relative absolute error (RAE) decreases from 0.232 to 0.205.