## 2 Reconstruction Method Based on PCA and Sparse Sensing  
Principal Component Analysis, abbreviated as PCA, is one of the most commonly used dimension reduction methods [23], which is called Proper Orthogonal Decomposition (POD) in mechanical engineering [24], Karhunen-Loeve (K-L) transformation in signal processing [25] and Hotelling transformation in image analysis [26].  
The core technical route of PCA is to express the high-dimensional vector \(x_{i}\in\mathbb{R}^{m}\) as a linear combination of multiple basis vectors \(\varphi_{k}\).  
\[x_{i}=\sum_{k=1}^{p}\alpha_{k}\varphi_{k} \tag{1}\]  
The basis vector \(\varphi_{k}\) and reconstruction coefficient \(\alpha_{k}\) can be obtained by Singular Value Decomposition (SVD). Given a snapshot matrix containing multiple state vectors \(\mathsf{X}=[x_{1},x_{2},\ldots,x_{n}]\), \(\mathsf{X}\in\mathbb{R}^{m\times n}\), the singular value decomposition of matrix \(\mathsf{X}\) yields:  
\[\mathsf{X}=\Psi\Sigma\Phi^{T} \tag{2}\]  
where \(\Psi\) and \(\Phi\) are left and right singular vectors of matrix \(\mathsf{X}\), respectively, \(\Psi\in\mathbb{R}^{m\times n}\), \(\Phi\in\mathbb{R}^{n\times n}\), \(\Sigma\in\mathbb{R}^{n\times m}\). All elements of matrix \(\Sigma\) are zero except those on the diagonal, and the diagonal elements are called singular values. When only the first \(p\) eigenvalues are taken, it is called truncated-PCA.  
\[\mathrm{X}\approx\widetilde{\mathrm{X}}=\Psi_{p}\Sigma_{p}\Phi_{p}^{T} \tag{3}\]  
The reconstruction coefficient vector \(\alpha\) can be obtained by minimizing the error between X and \(\widetilde{\mathrm{X}}\):  
\[\alpha=\Psi_{p}^{T}x \tag{4}\]  
Inspired by the truncated-PCA, the extracted features can be used to represent high-dimensional data, making it possible to reconstruct an unknown state with limited measurements in a wind field. Based on Equation (3), supposing there is a matrix of wind velocity \(V=\{v_{1},v_{2},\dots,v_{i},\dots,\)\(v_{N}\}\) (\(V\in\mathbb{R}^{n\times N}\)), which can be decomposed by SVD:  
\[V=\Psi\Sigma\Phi^{T} \tag{5}\]  
From Equation (1), PCA basis vectors can express high-dimensional states \(v\) as linear combinations of \(\varphi_{k}\) that define a low-dimensional embedding space.  
\[v=\sum_{k=1}^{p}\alpha_{k}\varphi_{k} \tag{6}\]  
The sparse sensing is introduced to solve \(a\), supposing only \(r\) (\(r\ll n\)) measurements are available, which enables the combination of sparse sensor placement and Equation (6) to determine the sparse variable \(y\) (\(y\in\mathbb{R}^{r}\)).  
\[y=Mv=M\sum_{k=1}^{p}\alpha_{k}\varphi_{k}=Ma\Psi \tag{7}\]  
where the measurement matrix \(M\in\mathbb{R}^{r\times n}\) represents measurement locations of \(v\). Point measurements require that the measurement matrix \(M\) can be structured in the following way:  
\[M=[e_{\lambda 1}\,e_{\lambda 2}\cdots\,e_{\lambda r}]^{T} \tag{8}\]  
where \(e_{j}\) are the canonical basis vectors for \(\mathbb{R}^{n}\) with a unit entry at index \(j\) and zeros elsewhere. The form of \(M\) can be shown as follows:  
\[M=\begin{bmatrix}1&0&\cdots&0&&&&\cdots&0\\ 0&\cdots&0&0&1&\cdots&&\cdots&0\\ 0&\cdots&1&0&\cdots&0&0&\cdots&0\\ \vdots&0&&&\cdots&0&1&0&\cdots&\vdots\\ 0&\cdots&&1&\cdots&0&0&0&\cdots&0\end{bmatrix} \tag{9}\]  
This linear system converts the solution of the unknown state \(v\) into the determination of the reconstruction coefficient \(a\) with the Moore-Penrose pseudoinverse. Subsequently, the reconstruction of \(v\) is obtained using:  
\[\hat{v}=a\Psi,\text{ where }a=\left\{\begin{array}{l}\Theta^{-1}y=(M\Psi)^ {-1}y,\,r=p\\ \Theta^{\intercal}y=(M\Psi)^{\intercal}y,\,r>p\end{array}\right. \tag{10}\]  
The accuracy of reconstruction results is measured by relative reconstruction error:  
\[RE=\frac{\|v-\hat{v}\|_{1}}{\|v\|_{1}}\times 100\% \tag{11}\]