# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach
## 2 Materials and Methods
### Wind Field Reconstruction Methods
#### 2.2.3 Linear VAD Model  
This procedure is based on the idea of converting the turbulent wind vector \(\vec{v}(x,\ y,\ z)\) at the point \(x_{0},\ y_{0},\ z_{0}\) into a linear function through a Taylor polynomial up to the first order to represent the inhomogeneous flow conditions in complex terrain more accurately. The resulting Taylor polynomial for the wind velocity vector \(\vec{v}(x,\ y,\ z)\) is represented in Equation (7). This is made up of a stationary value \(\vec{v}(x_{0},\ y_{0},\ z_{0})\) and three directional derivative in space at the point of development (index 0). \[\vec{v}(x,y,z)=\vec{v}(x_{0},y_{0},z_{0})+\frac{\partial\vec{v}}{\partial x} (x-x_{0})+\frac{\partial\vec{v}}{\partial y}(y-y_{0})+\frac{\partial\vec{v}} {\partial z}(z-z_{0}) \tag{7}\]  
\[v_{los} =u_{0}u_{x}+u_{x}{{}^{\prime}}n_{x}(x-x_{0})+u_{y}{{}^{\prime}}n_{x}(y- y_{0})+u_{z}{{}^{\prime}}n_{x}(z-z_{0})\] \[+v_{0}u_{y}+v_{x}{{}^{\prime}}n_{y}(x-x_{0})+v_{y}{{}^{\prime}}n_ {y}(y-y_{0})+v_{z}{{}^{\prime}}n_{y}(z-z_{0}) \tag{8}\] \[+w_{0}n_{z}+w_{x}{{}^{\prime}}n_{z}(x-x_{0})+w_{y}{{}^{\prime}}n_ {z}(y-y_{0})+w_{z}{{}^{\prime}}n_{z}(z-z_{0})\]  
The linear approximation (Equation (8)) consists of twelve unknown variables. These are the three wind vector components (\(u,\ v,\ w\) with index 0) at the point \(x_{0},\ y_{0},\ z_{0}\) and the other nine variables are the partial derivatives (mark with a / in Equation (8)) of the three wind vector components in the Cartesian coordinate space (index \(x,\ y,\ z\)). This equation cannot be solved without further assumptions. Some terms have the same geometric identity and are linearly dependent: the components \(u_{y}{{}^{\prime}}\) and \(v_{x}{{}^{\prime}}\) and the terms \(u_{z}{{}^{\prime}}\) and \(w_{x}{{}^{\prime}}\) and the \(v_{z}{{}^{\prime}}\) and \(w_{y}{{}^{\prime}}\) terms had the same prefactors. Figure 6: Overview of the workflow to estimate the combination of data points the maximized the coefficient of determination \(R^{2}\), based on the input variables \(v_{los}\), the estimated wind velocity components \(u,\ v,\ w\), the model predicted \(\hat{v}_{los}\) and the total sum of squares (\(SS_{tot}\)) and the regression sum of squares \(SS_{reg}\). Each gradient pair has the same prefactors as shown in the example for \(u_{y}\) and \(v_{x}\): \(n_{x}y=xy/f_{d}=xn_{y}\) with \(n_{x}=x/f_{d}\) (\(f_{d}\) is the distance of the lidar measuring point). To solve the Equation (8), two important assumptions have to be made:  
1. The components are only evaluated along the vertical axis of the lidar device (coordinate origin), which means that \(x_{0}=y_{0}=0\). 2. The gradient of the vertical wind speed components in \(x\)- and \(y\)-direction \(w_{x}\) and \(w_{y}\) would be zero. CFD simulations and the results show very small changes in the inclination angle at this test site that support the second assumption [16]. There are no more assumptions possible for the components \(u_{y}\) and \(v_{x}\) to simplify the Equation (8) and so the two terms are not separable. A case to simplify Equation (8) is when all trajectory points are located on a horizontal plane with the height \(z_{0}=z\), then the partial derivatives in the direction of \(z\) would disappear. In this work, a lidar system is used which works with constant focal lengths \(f_{d}\) and thus places the measuring points along a sphere in space. With these assumptions it was possible to reduce the number of unknown variables from twelve to nine and a system of equations with these nine unknown variables is obtained. These nine variables consists of the three wind velocity components and the six simplified components of the strain tensor. In order to calculate the unknown components, at least nine linearly independent measuring points are required. The reduced system of equations is represented in Equation (9).