# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach
## 1 Introduction
The line-of-sight velocity depends on the half opening angle \(\phi\), the azimuth angle \(\theta\) and the measuring distance or focal distance \(f_{d}\). The normal vector is calculated as \(\vec{n}(\theta,\,\phi)=[\cos(\theta)\sin(\phi),\,\sin(\theta)\sin(\phi),\,\cos (\phi)]=[n_{x},n_{y},n_{z}]=\vec{n}(x,\,y,\,z)\) where \(n_{i}\) is the component of the normal vector in \(i=x,y,z\) direction. The vector \(\vec{n}(\theta,\,\phi)\) can also be expressed in Cartesian coordinates, which are mainly used in this study. The line-of-sight velocity is given by:  
\[v_{los}=u\cdot n_{x}+v\cdot n_{y}+w\cdot n_{z} \tag{1}\]  
If \(v_{los}\) is measured from multiple azimuthal directions and elevations, the wind velocity components \(u,\,v,\,w\) can be calculated. If these measurements are made from a single point and encompass a volume of the atmosphere, it must be assumed that \(u,\,v,\) and \(w\) apply at all points in the wind field, i.e., that the flow is homogeneous. This leads to errors in the determination of wind speeds in heterogeneous, \"complex\" flow. [PERSON] [7] has shown with a simple mathematical model of a flow over a hill, that the error in the horizontal wind speed can be in the order of magnitude of 10%. In order to reduce the error during data collection, multi-lidar systems [8; 9; 10] can be used. The multi-lidar systems measure simultaneously \(v_{los}(x,\,y,\,z)\) at the same point. From this system of equations, the wind velocity components can be calculated without assuming the flow is homogeneous. The first simultaneous measurements with several lidar systems were carried out at the _Musketeer Experiment 2007_ in flat terrain [8]. Measurements in complex terrain were made during the _Kassel_Experiment_ 2014 [9] and during the _NEWA_ campaign in Perdigao [11]. However, this measurement configuration is very cost-intensive (greater than 200,000E per lidar device), which is not affordable for site assessments. Furthermore, a large effort has to be made for a successful operation of the multi-lidar configuration [10]. Another possibility to improve the measurements is to use wind field reconstruction models to better convert \(v_{los}\) data into a wind vector or wind field [5; 12; 13]. Finally, it is possible to use flow modelling to try to reduce differences between lidar measurement and traditional measurements when a campaign has already been completed. These typically focus on the use of commercial Computational Fluid Dynamics (CFD) programs to model flows and \"correct\" (in reality, \"adjust\") lidar measurements in complex terrain to better match point measurements from anemometers. Despite a continuing need to validate such tools in complex terrain, a 2015 study [14] concluded that with proper parametrization and flow modelling, flow simulations can be a useful tool in the post-processing of lidar measurements in complex terrain and this is likely to be an important approach in future [2]. In this study, the main focus is on the use of ground-based lidar systems with VAD scan patterns with additional vertical measuring points. This paper introduces several modified measurement strategies and adapted evaluation algorithms to show that it is possible to reduce uncertainties without having to forego the advantages of ground-based lidar systems. The results were also used to explore three questions about the effect of the scan design:  
1. Is the length of the trajectory circulation time important in the comparisons with the met mast? 2. Does the half opening-angle \(\phi\) have an influence on the accuracy of the correlations? 3. Is the local resolution of the measurement points important for the result? Section 2 gives an overview over the measurement campaign, the test site and the methods used for the wind field reconstruction. In Section 3 the results of the comparison between the results of the investigated methods and the reference in-situ measurements are presented. The discussion of the results is in Section 4 and conclusions are presented in Section 5.