## 3 Preliminaries
### Construction of Snapshot Matrix  
In preparation for reconstruction, the establishment of an accurate snapshot matrix is critical. In the present paper, the 3D wind field to be measured is expressed by five stacked planes. Several boundary conditions are designed, and a series of speed and relative pressure distributions are calculated by CFD, which are combined into a matrix database. The continuity equation and the momentum equation are mainly used in the calculation process. In this study, wind flow is considered to be stable under a constant temperature. Thus the time term in the continuity equation can be ignored. Continuity equation:  
\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{ \partial(\rho w)}{\partial z}=0 \tag{12}\]  
Momentum equation:  
\[\left\{ \begin{array}{l}\rho(u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}+w\frac{\partial u}{\partial z})=\rho F_{x}+\frac{\partial p_{xx} }{\partial x}+\frac{\partial p_{yy}}{\partial y}+\frac{\partial p_{xz}}{ \partial z}\\ \rho(u\frac{\partial v}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{ \partial v}{\partial z})=\rho F_{y}+\frac{\partial p_{xx}}{\partial x}+\frac{ \partial p_{yy}}{\partial y}+\frac{\partial p_{xz}}{\partial z}\\ \rho(u\frac{\partial u}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{ \partial w}{\partial z})=\rho F_{z}+\frac{\partial p_{xx}}{\partial x}+\frac{ \partial p_{xy}}{\partial y}+\frac{\partial p_{xz}}{\partial z}\end{array} \right. \tag{13}\]  
where \(\rho\) is density; \(u\), \(v\) and \(w\) represent the velocity of the \(x\), \(y\) and \(z\) directions, respectively; \(p\) is pressure; and \(\rho F\) is the unit mass force. Based on the above equations, 28 boundary conditions are designed by changing the entrance speed and incident angle. Among them, the inlet velocity changes from 1 to 31 m/s with an interval of 5 m/s, and the incident angles are 30\({}^{\circ}\), 70\({}^{\circ}\), 110\({}^{\circ}\) and 150\({}^{\circ}\). Notably the inlet velocity is uniformly distributed along the horizontal and vertical directions. Both databases containing 28 wind speed and relative pressure distributions can be represented by \(V\), \(V\)\(\in\mathbb{R}^{n\times N}\), where \(n\) stands for the number of sampling points; \(N\) is the number of boundary conditions. In the simulation, 2500 sampling points are selected for each plane, and the dimension of the snapshot matrix is 12,500 \(\times\) 28; the following reconstruction calculation is based on this matrix. However, considering that the real airflow around the wind farm is much more complex, especially the effect of wind shear on the wind speed distribution, the atmospheric turbulence inflow of stable vertical wind shear using the Normal Wind Profile model is introduced. The inlet speed can determined by the wind profile power law [27]:  
\[V=V_{r}\bigg{(}\frac{z}{z_{r}}\bigg{)}^{\alpha} \tag{14}\]  
where \(V\)(\(m\)/s) is the wind speed at height \(z\) (\(m\)), and \(V_{r}\) is the known wind speed at a reference height \(z_{r}\). The exponent (\(\alpha\)) is an empirical coefficient, approximately 1/7, or 0.143 [27]. Furthermore, \(z_{r}\) in the construction of the database is taken at 0.8 m. The wind velocities at \(z_{r}\) change from 1 to 31 m/s with an interval of 5 m/s, and the incident angles are 30\({}^{\circ}\), 70\({}^{\circ}\), 110\({}^{\circ}\) and 150\({}^{\circ}\). The turbulence intensity and viscosity ratio are set to 5% and 10%, respectively. ANSYS ICEM is employed to generate the hexahedral structured mesh and boundary layer. The mesh is encrypted around the boundary, and \"o-block\" is performed near the three geometric models. A total of 16 boundary layers are set around the hemisphere model with a grid spacing of 1.16 cm, and 20 boundary layers are set around the sinusoidal model with a grid spacing of 1.44 cm.