## 3 Application and Validation of MuLiWEA  
### Simulated Dual-Doppler Lidar Measurement Scenarios  
To test MuLiWEA, different lidar measurement scenarios were simulated considering a weighted average of the line-of-sight velocity over the considered sample volume. The sample volume was defined along a linear coordinate \(s\), which represents the line-of-sight distance from the lidar to the target point. We used a range gate length corresponding to a spatial extension \(\Delta p\) of 36 m and a Gaussian laser pulse with intensity \(I_{p}\), characterised by a Full Width Half Maximum \(\Delta r\) of 30 m [14]:  
\[I_{p}(s)=\frac{1}{\sqrt{\pi\Delta r}}e^{\left(-\frac{s^{2}}{\Delta r^{2}} \right)}. \tag{8}\]  
Figure 1: Illustration of a plan position indicator (PPI) scan with the azimuth angle \(\chi\) and elevation angle \(\delta\) indicated. Figure 2: Schematic view of a Cartesian grid with indicated measurement positions of two lidars in the vicinity with radius \(R\) of a grid point \(p\). The estimated line-of-sight velocity is then \(\partial_{LOS}(s)=\frac{1}{\Delta p}\int\frac{\frac{\Delta p}{2}}{2}\,v_{p}\left(s \right)ds\), where \(v_{p}(s)=\int_{-k_{p}\frac{\Delta t}{2}}^{k_{p}\frac{\Delta t}{2}}v_{r}\left(s ^{\prime}-s\right)I\left(s^{\prime}-s\right)ds^{\prime}\) with \(I=\frac{I_{p}(s)}{\int_{-\infty}^{+\infty}I_{p}(s)}\) and \(k_{p}=2.56\), where the primes denote the variable of integration along the line-of-sight distance \(s\). We simulated lidar measurements in the wake of a wind turbine with a rotor diameter of \(D=62\) m and a hub height of \(h_{h}=61\) m. To run the simulations, an unsteady wind field is calculated with the Parallelised Large-Eddy Simulation (LES) Model (PALM) [30]. The actuator line (ACL) approach [31] is included to simulate the wind turbine. For this experiment, a ten minute wind field is generated on a 4 m resolved grid with a temporal resolution of 0.4 s. The boundary conditions were chosen in order to simulate a neutral atmosphere (corresponding to a logarithmic vertical wind profile with roughness length \(z_{0}=0.05\) m and a friction velocity \(u_{*}=0.5\) m\(\cdot\)s\({}^{-1}\)), an average inflow wind speed at hub height of 9 m\(\cdot\)s\({}^{-1}\), a wind direction of 270\({}^{\circ}\) (West) and a turbulence intensity \(TI\) of 8% at hub height. In the lidar simulation used for testing MuLiWEA, the position of the laser beam was updated every 0.4 s. Within each range gate, fifteen equally spaced points of the corresponding sample volume were interpolated from the wind field. Finally, \(\partial_{LOS}\) was evaluated numerically. The validation procedure considers four different simulated lidar locations, two different PPI measurement scenarios and the effect of either enabling or disabling the continuity adjustment. The two following different PPI measurement scenarios were simulated and evaluated:  
1. A 'volume scan' based on six 30\({}^{\circ}\) azimuth sector PPI scans from each of the two lidars with elevations \(\delta\) ranging from 1.25\({}^{\circ}\) to 7.5\({}^{\circ}\). Note that this is not an actual volume that is spatially equally well covered in all directions, but a compromised way to gain volumetric information with a set of inclined PPI scans. Therefore, we refer to it as the 'volume scan' in the following. 2. A single elevation, 30\({}^{\circ}\) azimuth sector PPI scan from each of the two lidars with \(\delta=5^{\circ}\). This scenario will be referred to as the'single elevation scan' in the following. Each of the lidars measures 46 range gates spread equidistantly over distances between 400-850 m. The azimuth angle resolution is 0.5\({}^{\circ}\) and the scanning speed is 1.25\({}^{\circ}\cdot\)s\({}^{-1}\). Both scenarios aim at measuring a horizontal wind field at hub height between \(1D\) upstream and \(3D\) downstream of the turbine rotor with ground-based lidars.