## Methodology  
The MuLiWEA algorithm is based on the following assumptions:  
* The vertical wind speed component \(w\) is neglected because the elevation angles of the PPI scans are commonly small, and, therefore, it is difficult to estimate this component accurately from the measured line-of-sight velocity. This assumption limits the application of MuLiWEA to horizontal flows only, i.e., it might not be accurate in complex terrain sites, in convective atmosphere conditions, or in the near wake of a wind turbine. * As a consequence of the previous assumption, no vertical transport of mass is considered in the continuity equation, which is therefore applied in its two-dimensional form. * The flow is assumed to be stationary for the required time resolution because it generally takes a few minutes for the lidar scans to cover an area of interest, and it is useful to average a number of subsequent PPI scans to improve the data quality. * The flow is assumed to be incompressible. The measured line-of-sight velocity \(v_{LOS}\) is expressed in the Cartesian wind speed components in the \(x\)- and \(y\)-direction, namely \(u\) and \(v\), respectively, by Equation (1):  
\[\alpha_{k}u+\beta_{k}v=v_{LOS_{k}}, \tag{1}\]  
with \(\alpha=\sin(\chi)\cos(\delta)\) and \(\beta=\cos(\chi)\cos(\delta)\). Note that Equation (1) already incorporates that \(w=0\). The line-of-sight direction at a single measurement point \(k\) is defined by the azimuth angle \(\chi\) and elevation angle \(\delta\) of a lidar scanning beam (see Figure 1). The azimuth angle is expressed in the geographical reference system, i.e., \(\chi=0^{\circ}\) is directed to the North and increases clockwise. Since the angles are a known input with a certain pointing accuracy, the two remaining unknowns in Equation (1) are \(u\) and \(v\). To solve for these two wind speed components, at least two linearly independent \(v_{LOS}\) measurements are needed. Instead of fully synchronised measurements of two lidars at a single point in space at the same time (e.g., [18]), a concept to reconstruct 2D wind vector information from a larger set of PPI measurements from two lidars is used. In the worst case, the measurements are neither synchronised in time nor space. Therefore, the method has to analyse samples of \(K\) measurements from the two lidars around each grid point \(p\) (see Figure 2) over a longer time period of e.g., ten minutes. Measurements in the near vicinity of a grid point rather than exactly at a grid point have to be used and measurements at different heights above ground have to be combined. A suitable protocol for measurement point selection is found in the form of the Quad-Doppler wind synthesis, a step of the Extended Overdetermined dual-Doppler Formalism (EODD) [20], later also used by [PERSON] et al. [24]. The reconstruction of a wind field can be divided into five sequential steps:  
1. A temporal and spatial domain is defined. To ensure that an area of interest is covered with at least one PPI scan per lidar, and possibly multiple overlapping scans to enhance averaging, a time resolution of ten minutes is chosen. A Cartesian grid is generated on the area of interest covered by the overlapping PPI scans. The assumption of stationarity posed before is considered to be reasonable for this time scale. 2. For each grid point \(p\), the measurement set \(K\) that is located within its surrounding circle with radius \(R\), i.e., the radius of influence, is used to compute the local wind vector. The measurement set \(K\) contains contributions from all lidars (see Figure 2). Different lidars do not necessarily have to contribute an equal number of measurements for the same grid point. 3. For each grid point \(p\), a linear system based on Equation (1) can be established, which considers all measurements \(k=1,...,K\) per circle with radius \(R\)[20]: \[ \begin{bmatrix}\sum_{k=1}^{K}(\alpha_{k}^{2}w_{k})&\sum_{k=1}^{K}(\alpha_{k} \beta_{k}w_{k})\\ \sum_{k=1}^{K}(\alpha_{k}\beta_{k}w_{k})&\sum_{k=1}^{K}(\beta_{k}^{2}w_{k}) \end{bmatrix} _{p}