# An Inter-Comparison Study of Multi- and DBS Lidar Measurements in Complex Terrain
## Materials and Methods
### Second-Order Statistics of \(v_{r}\)
In analogy to what [PERSON] and co-workers [26] suggested for sonic anemometers, we can now define a spectral transfer function between the spectrum measured by the lidar and a point measurement as  
\[T_{\
u_{r}}\left(k_{1}\right)=\frac{F_{\
u_{r},\,\,\text{l}i\,\,\text{l}i\, \,\text{l}i\,\,\text{l}i\,\,\text{l}i\,\,\text{l}i\,\,\text{l}i\,\,\text{l}i \,\,\text{  
(\(\mathbf{x_{0}}\)). In case of the conical scan of a profiling (DBS) lidar this would be the centre of the circle of the lidar scan:  
\[\mathbf{d}=\ \mathbf{x}-\mathbf{x_{0}}, \tag{11}\]  
where \(\mathbf{x}\) is the location of the centre of the averaging path. Adding the measurement location to (6) and including the temporal averaging, \(\hat{v}_{r\ lidar}\) can now be rewritten as [26]  
\[\hat{v}_{r\ lidar}\left(\mathbf{k},\mathbf{d}\right)=\ \hat{\zeta}\left(k_{1}\right)e^{i \mathbf{k}\cdot\mathbf{d}}\hat{\varphi}\left(\mathbf{k}\cdot\mathbf{n}\right)\mathbf{n}\cdot\hat{ \mathbf{V}}\left(\mathbf{k}\right). \tag{12}\]  
The term \(e^{i\mathbf{k}\cdot\mathbf{d}}\) basically decorrelates the wind field which is sensed at the different measurement locations for high wavenumbers by introducing a phase shift. By combining (3) and (12), a matrix \(\mathbf{N}\) can now be defined, which accounts for the effects of line averaging and path separation in the vector reconstruction. The entries of \(\mathbf{N}\) are  
\[N_{ab}\left(\mathbf{k}\right)=M_{ab}\hat{\zeta}\left(k_{1}\right)\ e^{i\mathbf{k} \cdot\mathbf{d_{b}}}\hat{\varphi}\left(\mathbf{k}\cdot\mathbf{n_{b}}\right) \tag{13}\]  
and its inverse \(\mathbf{J}=\mathbf{N}^{-1}\). The vector reconstruction in the Fourier domain then becomes  
\[\mathbf{\hat{v}}\left(\mathbf{k}\right)=\ \mathbf{J}\left(\mathbf{k}\right)\cdot\hat{\mathbf{V}} \left(\mathbf{k}\right) \tag{14}\]  
The spectrum of e.g., the \(u\) component derived from three lidar beams can be written as  
\[F_{u}\left(k_{1}\right)=\ \iint\sum\limits_{b=1}^{3}I_{1b}\left(\mathbf{k} \right)I_{1b}^{*}\left(\mathbf{k}\right)\Phi_{ij}\left(\mathbf{k}\right)dk_{2}dk_{3}\, \tag{15}\]  
where again the summation for repeating indices \(i\) and \(j\) is assumed. [PERSON] and co-workers call \(\mathbf{J}\) \"weighting factors,\" because they define how \(\Phi_{ij}\left(\mathbf{k}\right)\) is weighted in the vector reconstruction [9]. The effect of the beam separation on the time series-derived wind vector is twofold. The decorrelation leads to a damping of the contribution of the spectral tensor, \(\Phi_{ij}\left(\mathbf{k}\right)\), in the direction of the desired component (e.g., \(i=j=1\) for \(u\)) for high wavenumbers. However, the decorrelation effect is also present for the other components of \(\Phi_{ij}\left(\mathbf{k}\right)\). As a result, some of the spectral energy of these components is folded onto \(F_{u}\left(k_{1}\right)\). This effect is often referred to as cross-contamination in the literature [9; 26], because the desired spectrum is contaminated by the other terms of \(\Phi_{ij}\left(\mathbf{k}\right)\). As explained in Section 2.2, the effect of the probe volume averaging is mainly attenuation at small wavenumbers. Using the spectral tensor model of [18], this effect for two different conically scanning lidars was investigated [9; 10]. It was shown that the combined effects of beam separation and line averaging strongly depend on the beam configuration and the probe volume, which define \(\mathbf{J}\), but also the atmospheric conditions i.e., \(\Phi_{ij}\left(\mathbf{k}\right)\). Experimental evidence confirms these results and an underestimation as well as an overestimation of \(\overline{u^{\prime 2}}\) and \(\overline{v^{\prime 2}}\) has been reported [8; 9; 11; 12]. It should also be noted that, in complex terrain, the assumption of a statistically homogeneous turbulence field implicitly inherent in (15) is likely to be violated and additional errors might be introduced into the turbulence measurement.