# An Inter-Comparison Study of Multi- and DBS Lidar Measurements in Complex Terrain
## Materials and Methods
### Second-Order Statistics of \(v_{r}\)  
One of the characteristics of wind-speed measurements from Doppler lidars is that, due to the measurement principle, \(v_{r}\) is retrieved from a measurement within a probe volume. The width of the laser beam is usually negligible compared to the length of the probe volume along the path. The effect of the measurement volume on \(v_{r\ lidar}\) can thus be simplified to line averaging along the direction of the lidar beam. This can be expressed as  
\[v_{r\ lidar}=\int\limits_{-\infty}^{\infty}\varphi\left(s\right)\mathbf{n}\cdot \mathbf{V}\left(\mathbf{r}+\mathbf{n}\mathbf{s}\right)ds, \tag{5}\]  
where \(\varphi\left(s\right)\) is the weighting function along the probe volume, \(\mathbf{r}\) is the centre of the probe volume and \(\mathbf{V}\) is the three-dimensional velocity field. Depending on the signal processing and shape of the emitted laser pulse, \(\varphi\left(s\right)\) will vary and different forms have been suggested (e.g., [23; 24]). Here we follow the concept that the weighting function can be estimated from a convolution of the form of the laser pulse \(\varphi_{p}\left(s\right)\) and the gate-weighting function \(\varphi_{g}\left(s\right)\), which results from the size and shape of the sampling gate--i.e., the internal signal processing [25]. Since the effects of line averaging depend on the scale of the turbulent eddies, it is convenient to express the problem in Fourier space. The Fourier transform of (5) can be written as [14; 26]:  
\[\hat{\varphi}_{r\ lidar}\left(\mathbf{k}\right)=\ \hat{\varphi}\left(\mathbf{k}\cdot \mathbf{n}\right)\mathbf{n}\cdot\mathbf{\hat{V}}\left(\mathbf{k}\right), \tag{6}\]  
Figure 1: Sketch of the multi-lidar (ML) and Doppler beam swinging (DBS) strategies next to a reference mast with a sonic as used in this study. Note that the angles and positions of the lidars differ from the instrument setup for reasons of illustrative clarity. where \(\mathbf{k}\) is the wave vector \(\left(k_{1},k_{2},k_{3}\right)\) and \(\hat{\ }\) denotes the Fourier transform. For the WLS 200s (a scanning lidar manufactured by Leosphere) used in this study, both weighting functions can be approximated by a Gaussian function [27]. \(\hat{\varphi}\left(\mathbf{k}\right)\) can thus be approximated by  
\[\hat{\varphi}\left(\mathbf{k}\right)=\hat{\varphi}_{p}\left(\mathbf{k}\right)\hat{ \varphi}_{g}\left(\mathbf{k}\right)=e^{-\mathbf{k}^{2}\sigma_{p}^{2}/2}e^{-\mathbf{k}^{2} \sigma_{g}^{2}/2}, \tag{7}\]  
where \(\sigma_{p}\) is the standard deviation of \(\varphi_{p}\left(s\right)\) and \(\sigma_{g}\) is the standard deviation of \(\varphi_{g}\left(s\right)\). In practice, pulsed lidars use an accumulation time to derive the radial wind speed. We thus need to account for the temporal averaging. This can be done by adding the additional term \(\xi\left(k_{1}\right)=sinc\left(k_{1}\cdot t\cdot u/2\right)\), where \(t\) is the averaging time [14]. Assuming a homogeneous turbulence field along the probe volume and [PERSON]'s hypothesis of frozen turbulence, we can express the line averaging in (6) on the measured turbulence spectrum, \(F_{\
u_{r}}\left(k_{1}\right)\), using the spectral density tensor \(\Phi_{ij}\left(\mathbf{k}\right)=\mathbf{\hat{V}}\left(\mathbf{k}\right)\mathbf{\hat{V}}^{*} \left(\mathbf{k}\right)\) as [14; 26]  
\[F_{\
u_{r}}\left(k_{1}\right)=\ n_{i}n_{j}\left|\xi\left(k_{1}\right)\right|^{ 2}\iint\hat{\varphi}\left(\mathbf{k}\cdot\mathbf{n}\right)\hat{\varphi}^{*}\left(\mathbf{k }\cdot\mathbf{n}\right)\Phi_{ij}\left(\mathbf{k}\right)dk_{2}dk_{3}\, \tag{8}\]  
where \({}^{*}\) denotes the complex conjugate. Summation over repeated indices is assumed.