## 5 Lidar Scan Pattern Optimization
### Lidar-Based Wind Parameter Estimation  
Line-of-sight velocity measurements from a lidar can be described as the opposite of the dot product between the unit vector in the direction that the beam is oriented (\(\ell=\left[\ell_{x},\ell_{y},\ell_{z}\right]\)) and the wind speed vector at the measurement point (\([u,v,w]\)):  
\[v_{LOS}=-\ell_{x}u-\ell_{y}v-\ell_{z}w. \tag{6}\]  
Note that the true velocity measured by a lidar is a spatial average of the LOS velocities along the lidar beam, modeled as the integral of the LOS velocities along the beam weighted by the range weighting function. For control purposes, wind parameters such as the rotor average wind speed, wind shear, and wind direction are more useful than the LOS velocities provided directly by a lidar. Therefore, the wind parameters of interest must be estimated from the LOS velocities.  
To estimate the wind parameters, an appropriate wind field model must first be defined. In general, it is not possible to distinguish between horizontal wind direction and horizontal shear (or vertical inflow angle and vertical shear) using LOS velocity measurements from a lidar scan pattern. Consequently, the two most common wind field models used for wind parameter estimation consist of (a) uniform wind speed across the rotor including horizontal and vertical wind direction, and (b) no transverse or vertical wind speeds, but linear horizontal and vertical shear [55].  
Regardless of which wind field model is assumed, for a scan pattern with \(N\) unique beam directions, \(\ell_{1}\ldots\ell_{N}\), the solution for the estimate of the rotor average wind speed perpendicular to the rotor, used for optimizing CW lidars in this paper, is given by  
\[\hat{u}_{eff}=-\frac{1}{N}\sum_{i=1}^{N}\frac{v_{LOS,i}}{\ell_{x,i}}. \tag{7}\]  
However, for pulsed lidars, measurements at different range gates can be combined more effectively by delaying the measured velocity signals at each scan point by the estimated amount of time it takes the wind to travel to the plane where the closest range gate is located, according to [PERSON]'s frozen turbulence hypothesis [32]. This effectively \"stacks\" the measurements from different range gates together before averaging by removing their relative time shifts (although the farther a range gate is from the rotor plane, the more the measurements will suffer from wind evolution). For pulsed lidars, an estimate of the rotor average wind speed, which is used for scan pattern optimization in this paper, can therefore be expressed as  
\[\hat{u}_{eff}\left(t\right)=-\frac{1}{N}\sum_{i=1}^{N}\frac{v_{LOS,i}\left(t- \frac{x_{0}-x_{i}}{U}\right)}{\ell_{x,i}}, \tag{8}\]  
where \(x_{i}\) is the longitudinal position of measurement point \(i\), \(x_{0}\) is the longitudinal position of the range gate closest to the rotor, and \(U\) is the mean rotor effective wind speed. Note that, in practice, the estimation of the rotor effective wind speed could be improved by delaying measurements at the different range gates according to their height-dependent mean wind speed, accounting for wind shear. In this work, however, it is assumed that the wind advection speed is the same at all heights.  
More information about wind parameter estimation using lidar measurements, including a discussion of nonlinear methods for estimating all five wind parameters described in this section simultaneously, can be found in [PERSON] et al. [55].