## 5 Lidar Scan Pattern Optimization
### Lidar System Configurations  
The coordinate system and basic parameters used to describe the scan patterns analyzed in this paper are shown in Figure 1. A scan pattern may contain many different measurement points, but each point can be defined by its upstream preview distance \(d\) from the lidar unit in the \(-x\) direction, its radial distance \(r\) in the \(yz\) plane from the point \(d\) m directly upstream of the lidar unit, and the azimuth angle \(\psi\) in the \(yz\) plane, where \(\psi=0\) indicates a measurement at the top of the scan circle defined by \(d\) and \(r\). The measurement points can be equivalently described by their measurement distance \(F\) and cone angle \(\theta\). For simplicity, it is assumed that the lidar is located at the origin, which is defined as the rotor hub position. Both CW and pulsed lidars are examined here when analyzing scan patterns. Measurements from both types of lidars consist of wind speeds spatially averaged along the beam direction. However, the range weighting functions, which describe how heavily wind speeds at different radial distances are weighted, are different. The range weighting function for a CW lidar changes as the focus distance of the lidar increases; the full-width-at-half-maximum width of the weighting function is proportional to the square of the focus distance. Therefore, the farther a CW lidar is focused, the more spatial averaging occurs. The range weighting functions for pulsed lidars remain approximately constant regardless of the range gate distance. Figure 2 shows theoretical range weighting functions for a CW lidar with focus distances of 50 m, 100 m, and 150 m as well as for a pulsed lidar (shown without loss of generality at a measurement distance of 100 m), with parameters roughly based on the ZephIR 300 CW lidar [52; 53] and the WindCube WLS7 pulsed lidar [53; 54], respectively. More information about the range weighting functions can be found in [PERSON] et al. [36]. Figure 1: Scan pattern coordinate system and scan variables for a single lidar measurement point. Figure 2: Normalized range weighting functions for a continuous wave (CW) lidar with focus distances of 50 m, 100 m, and 150 m as well as a pulsed lidar. Several types of scan patterns are investigated in this paper for CW and pulsed lidars, all roughly based on lidar systems that have been developed commercially. The measurement quality achieved by each scan pattern is compared to a single point measurement upstream of the turbine at hub height, representing the most basic measurement scenario, as well as the longitudinal wind speed averaged over the rotor disk area upstream of the turbine, acting as an \"ideal\" preview measurement of the rotor effective wind speed. A preview distance of 50 m is used for the single-point and ideal measurements to match the closest range gate available for pulsed lidar measurements in this analysis. After the point measurement, the simplest lidar scan pattern investigated is the two-beam lidar, with both beams measuring at hub height. This configuration is analyzed for a CW lidar, as shown in Figure 3b, and a pulsed lidar, illustrated in Figure 3c. Note that all pulsed lidar scan patterns modeled in this paper contain 10 evenly spaced range gates beginning at a range of 50 m and ending at the farthest range (no more than 300 m), parameters achievable with the Avent lidar system [9; 10]. The next level of complexity is represented by a four-beam lidar, with CW and pulsed configurations shown in Figure 3d,e, respectively, based on the Windar lidar [7] and the four-beam version of the Avent system [10]. For the four-beam configuration, the beams are oriented at azimuth angles of \(\psi\) =45, 135, 225, and 315 degrees. The last realistic scan pattern investigated is the circular scan consisting of 50 points evenly spaced around a scan circle, based on the ZephIR DM lidar system [4], shown in Figure 3f.