# The Rapid Establishment of Large Wind Fields via an Inverse Process
## 2 Methods  
### Traditional POD and Gappy POD  
POD (proper orthogonal decomposition) is an effective way to reduce the data dimensions and rapidly reconstruct a large area. It already has many successful applications in data processing such as the derivation of the dynamic mode [10] and the stability analysis and the design of airfoil profiles [11]. POD satisfies the requirement of a low computing cost while in many circumstances having acceptable accuracies, which may be an effective tool to rapidly reconstruct a wind field with acceptable accuracy, where a large amount of data is processed [12]. POD uses \"snapshots\" to extract its basis vectors. The snapshots of the POD method can be obtained from either experimental measurements or from numerical simulations. The POD snapshots in this paper come from CFD simulations. For a given area, a CFD calculation is performed to generate the velocity distributions. A series of snapshots of the velocity distributions are then collected and denoted as \(\mathbf{U}^{I}\), where the superscript \(l\) indicates the _lth_ snapshot. Consider the collection of m snapshots, \(\left\{\mathbf{U}^{I}\right\}_{l=1}^{m}\). The correlation matrix \(\mathbf{R}\) is formed by computing the inner product for every pair of snapshots. \[\mathbf{R}_{ik}=\frac{1}{m}\Big{(}\mathbf{U}^{i},\mathbf{U}^{k}\Big{)} \tag{1}\]  
where \(\left(\mathbf{U}^{i},\mathbf{U}^{k}\right)\) denotes the inner product of \(\mathbf{U}^{i}\) and \(\mathbf{U}^{k}\). The eigenvalues \(\lambda_{i}\) and eigenvectors \(\mathbf{\psi}^{i}\) of \(\mathbf{R}\) are computed by singular value decomposition (SVD). The _jth_ POD basis vector \(\mathbf{\Phi}^{j}\) is given by a linear combination of snapshots. \[\mathbf{\Phi}^{j}=\sum_{i=1}^{m}\mathbf{\psi}^{i}\mathbf{U}^{i} \tag{2}\]  
where \(\mathbf{\psi}^{j}_{i}\) denotes the _ith_ element of the _jth_ eigenvector. The magnitude of the _jth_ eigenvalue describes the relative importance of the _jth_ POD basis vector. For each basis vector, the importance is quantified by defining the relative energy \(E_{j}\), which means the percentage of the _jth_ eigenvalue in the sum of the whole eigenvalues. The POD method can also be employed to reconstruct the missing or gappy data from the partial measurement data. To do this, the first step is to define a mask vector that describes a particular field vector to identify where the data are available and where the data are missing. For the solution \(\mathbf{U}^{k}\), the corresponding mask vector \(n_{i}^{k}\) is defined as \(n_{i}^{k}=0\) if \(\mathbf{U}_{i}^{k}\) is missing; \(n_{i}^{k}=1\) if \(\mathbf{U}_{i}^{k}\) is known, where \(\mathbf{U}_{i}^{k}\) denotes the _ith_ element of the vector \(\mathbf{U}^{k}\); the dot product is defined as \(\left(n^{k},\mathbf{U}^{k}\right)_{i}=n_{i}^{k}\mathbf{U}_{i}^{k}\); the induced norm is defined as \(\left(\left\|v\right\|_{n}\right)^{2}=\left(v,v\right)_{n}\).\(\mathbf{g}\) is set to be a solution vector that has some missing elements and is associated with the mask vector \(\mathbf{n}\), so the repaired vector \(\mathbf{\widetilde{g}}\), i.e., \(\mathbf{g}\), with all the missing data \"recovered,\" can be approximated by  
\[\mathbf{\widetilde{g}}=\sum_{i=1}^{p}b_{i}\mathbf{\Phi}^{i}. \tag{3}\]  
The coefficients \(b_{i}\) in the POD method can be acquired by solving the following minimization function:  
\[Error=\left\|\mathbf{g}-\mathbf{\widetilde{g}}\right\|^{2}=\left\|\mathbf{g }-\sum_{i=1}^{p}b_{i}\mathbf{\Phi}^{i}\right\|^{2} \tag{4}\]  
where \(\left\|\cdot\right\|\) defines  
grid points is very high, as it is the result of the large computation domain and fine grids/high-spatial resolution. This problem will be more pronounced in the cases of short-term wind forecasts because time is crucial for such forecasts. To solve the problem of extracting the basis vectors of the POD matrix, a subset of the wind field is defined in the procedure.