## 2 Theory of Weighted Singular Value Decomposition  
Singular value decomposition (SVD) is a renowned and stable mathematical technique for data denoising as well as compression in the fields of imaging [26]. It has been effectively demonstrated for data reconstruction [27] and denoising [28] applications of the magnetic resonance imaging (MRI). SVD [29] decomposes the original signal into a superposition of the number of linearly independent and interpretable subspaces (or components), and then reconstructs the low-ranked approximate of the original signal using the small number of meaningful subspaces. Each of the components represent a unique energy contribution in the signal. The energies of the components vary in unique amplitude patterns over time, such as the slowly varying fluctuations, the coherent oscillations and vibrations, and the random noises (e.g., system noise produced by the optoelectronic instruments and background noise picked up by the sensing fiber). In the decomposed signal, the energies of the slowly varying fluctuations and the coherent oscillations spread over the small individual subsets of subspaces, while the energies of the random noise will be uniformly spread over the whole signal. Therefore,it is possible to separate the signal of interest from the noise present in the original signal by applying the SVD algorithm. The original signal of the phase-sensitive OTDR is made by the number of Rayleigh backscattering traces, which provides the spatial and the temporal information of the surrounding environment of the fiber-under-test (FUT). First, the spatial-temporal dataset is arranged into a rectangular matrix to create a 2D image. Assume \(A\) is the image matrix of size \(m\times n\) (\(m\leq n\)), where \(m\) is the number of Rayleigh backscattering traces and \(n\) is the number of sampling points representing the spatial-positions along the FUT in each trace. Thus, there exists a time series, \(x_{n}(t)\), made by the \(m\) number of amplitudes for each position. The matrix, \(A\), is normalized by dividing all the entries with the spectral norm (\(\|A\|_{2}\)). The normalized image matrix, \(A\), can be described using the SVD method according to the relation as:  
\[A=USV^{T} \tag{1}\]  
\[A=[u_{1}\ u_{2}\ \cdots\ \ u_{m}]\left[ \begin{array}{ccccc}s_{1}&0&\cdots&0& \cdots&0\\ 0&s_{2}&\cdots&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&0&\cdots&s_{m}&\cdots&0\end{array} \right]\left[ \begin{array}{c}v_{1}^{T} \\ v_{2}^{T}\\ \vdots\\ v_{n}^{T}\end{array} \right]=\sum_{i=1}^{m}u_{i}s_{i}v_{i}^{T} \tag{2}\]  
where \(U\) and \(V\) are \(m\times m\) and \(n\times n\) unitary orthogonal matrices, respectively; and \(S\) is a \(m\times n\) diagonal matrix; rank (\(S\)) = min (\(m\), \(n\)). The columns of \(U\), denoted by \(u_{i}\) of size \(m\times 1\), are called left singular vectors; and the rows of \(V^{T}\), denoted by \(v_{i}{}^{T}\) of size \(1\times n\), are called right singular vectors of matrix \(A\). Diagonal entries of matrix \(S\) are called the singular values of \(A\), and are represented by \(s_{i}\), where \(i\) = 1, 2, \(\ldots\), \(m\). The left and right singular vectors are the eigenvectors of \(AA^{\mathrm{T}}\) and \(A^{\mathrm{T}}A\), respectively; and \(s_{i}{}^{2}\) is the corresponding eigenvalues of A. The singular values describe information about the importance of their associated subspaces or vectors (left singular and right singular) in the image matrix \(A\) in the proposed denoising application for the phase-sensitive OTDR. In the matrix \(S\), the singular values are ordered from the highest to the lowest amplitudes, such that \(s_{1}\geq s_{2}\geq\ldots\geq s_{j}\geq\ldots\geq s_{k}\geq\ldots\geq s_{m}\geq 0\).