# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach
## 1 Introduction  
Doppler wind lidar (light detection and ranging) systems emit laser pulses that are reflected by aerosols in the atmosphere. This backscattered light has a frequency shift (the Doppler effect) due to the movement of the aerosols. The wind speed in direction of the laser beam can be calculated from the frequency shift using the Doppler equation. The alignment of the laser can be described by the azimuth and elevation angle. Ground-based lidar systems can be operated with different operating modes, such as staring, velocity azimuth display (VAD), plan position indicator (PPI) or range height indicator (RHI). The operating modes differ in how the two angles to each other are moved. In PPI mode the elevation angle is fixed and only the azimuth angle is varied over a small sector range. In RHI mode the azimuth angle is fixed and the elevation angle is varied. Both angles are fixed when a lidar operates in a so-called staring mode. The most common mode for commercial wind profilers is the VAD mode. In VAD mode, the azimuth angle is varied over a circle with a fixed elevation angle. The numberof measuring points over the circle are specified by the manufacturer. Also, the backscattered light does not come from a single point but instead along a finite length. This probe length depends on the pulse duration and the measurement technology, for example the use of a pulsed or continuous wave (CW) light source [1]. The CW lidars measure the heights one after the other, while pulsed lidar systems measure several heights simultaneously. It is then possible to estimate a wind vector at one or more points in the lidar scan volume using the wind speed data and azimuth, elevation, and range information and by making assumptions about homogeneity and stationarity. This process is known as wind field reconstruction [1]. Ground-based doppler wind lidar systems have many advantages over wind measurements made with in-situ instruments mounted on meteorological masts (met masts). Commercial ground-based lidar devices can measure at heights up to 300 m, which is above the tip heights of modern wind turbines, while met masts frequently only reach 80-100 m. The lidar can also be reconfigured for different tasks [2], and moved easily so that several locations can be investigated with short measurement campaigns. Previous comparisons between lidar measurements and measurements made with cup anemometers on met masts show an almost perfect match between the two: [3] estimated a slope of 0.9558 and an offset of 0.1577 m\(\cdot\)s\({}^{-1}\) and a \(R^{2}\) of 0.9984 in flat terrain and homogeneous flow conditions, while [4] found a slope of 1.004, offset of \(-\)0.079 m\(\cdot\)s\({}^{-1}\) and an \(R^{2}\) of 0.996 based on 10 min averages. In complex terrain, the correlations between wind speed measurements with single points sensors on met masts and lidar systems show considerable differences. The reason for the large differences are the inhomogeneous flow conditions, which can be caused, for example, by the terrain (momentum-induced turbulence) or by thermal effects (buoyancy-induced turbulence). Further causes can be found in [5]. The magnitude of the differences depends on local conditions:  
* Fluctuations in the flow field: Changes in terrain roughness upstream of the measurements cause large variations and changes in wind speed with height lead to uncertainty in the wind speed [6]. * Lidar technology: The weighting function of the lidar signal processing also leads to over- or under estimation of the wind speed. In this study pulsed ground-based lidar systems were used, which performed modified VAD scans. In a VAD scan, the elevation angle is the half opening angle \(\phi\) of the cone. The lidar devices determine, at selected points, the line-of-sight velocity \(v_{los}(\theta,\,\phi,\,f_{d})\), which is the projection of the wind velocity vector \(\vec{v}(x,\,y,\,z)\) onto the normal vector \(\vec{n}(\theta,\,\phi)\) of the laser beam (Equation (1), the index indicates the axis direction).