# An Inter-Comparison Study of Multi- and DBS Lidar Measurements in Complex Terrain
## Materials and Methods  
### Wind Vector Reconstruction and Scanning Strategies  
Doppler lidars provide an estimate of reflectivity-weighted, radial velocities of aerosol particles. This velocity was applied as a good approximation for the radial wind velocity. In the context of wind profilers, the difference between Doppler velocity and real-wind velocity has been addressed by [21]. For Doppler lidars, the assumption seems justified as most of the backscattering for optical wavelengths is from particles which are small enough to be advected with the wind (e.g., [22]). Moreover, a homogeneous aerosol concentration within the probe volume needs to be assumed at this point. The measured radial velocity \((v_{r\;i})\) is thus a projection of the three-dimensional wind vector (\(\mathbf{v}\)) onto the lidar beam  
\[v_{r\;i}=\;\mathbf{v_{i}\cdot}\boldsymbol{n_{i}}, \tag{1}\]  
where \(\boldsymbol{n_{i}}\) is the unit vector in the direction of the lidar beam and \(i\) denotes the individual beam and its location. In meteorological applications, it is convenient to write this equation as:  
\[v_{r\;i}=\;u_{i}\text{cos}(\theta_{i})\text{cos}\left(\phi_{i}\right)+v_{i} \text{cos}\left(\theta_{i}\right)\text{sin}\left(\phi_{i}\right)+w_{i}\text{ sin}\left(\theta_{i}\right), \tag{2}\]  
where \(u\) is directed into the mean wind direction during the averaging interval, \(v\) is perpendicular to \(u\) in the horizontal and \(w\) is the vertical component of the wind vector. The projection onto the radial is defined by the elevation angle \(\theta\) in addition to the angle between the mean wind direction and the beam \(\phi\). The azimuth angle (\(\phi\)) or beam-pointing angle is often used instead of \(\phi\) in the literature to define (2). However, in this paper, the use of \(\phi\) makes the notation in the later chapters more consistent. Assuming homogeneous flow (i.e., \(\mathbf{v_{i}}=\mathbf{v}\)) between the measurement locations or--as is the case for ML configurations--the intersection of the beams in a single location, multiple measurements can be combined to derive \(\mathbf{v}\) using matrix inversion  
\[\mathbf{v}=\;\boldsymbol{M^{-1}\cdot}\boldsymbol{v_{r}}, \tag{3}\]  
where \(\boldsymbol{M}=\{\boldsymbol{n_{1}},\boldsymbol{n_{2}},\ldots\}\). For a reconstruction of all components of \(\mathbf{v}\), at least three beams need to be combined. The horizontal wind speed (\(V_{h}\)) is then defined as:  
\[V_{h}=\;\sqrt{u^{2}+v^{2}}. \tag{4}\]  
Classical profiling wind lidars usually rely on a scanning strategy, which employs a conical scan above the instrument and measures \(v_{r\;i}\) at a fixed height and at different azimuth angles. The wind profiler lidar (manufacturer Leosphere, type Windcube v2), which is used as a reference instrument in this study, scans in DBS mode and uses four beams (with \(\theta=62^{\circ}\)), which are arranged in two perpendicular planes (Figure 1). An additional vertical beam is used to measure \(w\) directly. The resulting separation between two opposing beams in this setup is roughly equal to the measurement height. Especially in complex terrain, the assumption of homogeneous flow across the different beams of a profiling lidar is often violated; significant, direction-dependent errors in the estimation of the mean can be introduced [4; 5]. The main reason for the observed errors is thought to be the difference in \(\overline{w}\) (the overbar denotes the mean) across the different beam locations [4; 5]. Differences in the other components will of course also introduce an error. For the site of the current experiment, a deviation to mast-bound cup anemometer measurements of approx. \(-4\%\) to \(+2\%\) in \(\overline{V_{h}}\) has previously been reported [6]. Also the scatter in the inter-comparison was higher than in comparable experiments in flat and homogeneous terrain [1].