## 2 Theory of Weighted Singular Value Decomposition
The singular values with higher amplitudes (lowest indexes) represent the singular vectors of the components having high energy-amplitudes (e.g., slowly varying fluctuations, periodic oscillations, and background noises) in the original signal, and the singular values with the lower amplitudes (highest indexes) represent the singular vectors of the components having low energy-amplitudes (such as system noises). From Equation (2), by selecting the appropriate subset of the important singular values having larger amplitudes as well as the associated left and right singular vectors describing only the signal of interest (e.g., intrusion events or vibrations), we can reconstruct the image matrix referred to as, \(\tilde{A}\), without significant loss of information. Consequently, we can filter out the undesired energy components from the original signal's data, and could be able to reduce random noise [30]. The reconstructed image matrix, \(\tilde{A}\), can be written as:  
\[A= \begin{bmatrix}u_{j}&u_{j+1}&\cdots&u_{k}\end{bmatrix} \left[ \begin{array}{ cccc}s_{j}&0&\cdots&0\\ 0&s_{j+1}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&s_{k}\end{array} \right]\left[ \begin{array}{c}v_{j}^{T}\\ v_{j+1}^{T}\\ \vdots\\ v_{k}^{T}\end{array} \right]=\sum_{i=j}^{k}u_{i}s_{i}v_{i}^{T} \tag{3}\]  
where the size of the reconstructed matrix, \(\tilde{A}\), is \(m\times n\) (same as \(A\)), while the sizes of the matrices \(U\), \(S\), and \(V^{T}\) have been reduced to \(m\times r\), \(r\times r\), and \(r\times n\), respectively; \(r\) = \(k-j+1\); \(1<j<k<m\). Although the SVD denoising method eliminates most of the random noise and the (low-frequency) polarization-dependent interferometric amplitude fluctuations from the original signal, a small amount of background noise is present in the denoised signal. The removal of the low-frequency fluctuations and the background noise is essential for detection of a weak vibration event. Here, we propose using the weighted singular value decomposition method (WSVD)--a combination of SVD denoising and a weight factor of the position along the sensing fiber. WSVD is expected to enhance the SNR of both the location information and the periodic change in amplitude in the event of applied vibration. The weight factor is a scalar value and is calculated specifically for each position. The weight factor (\(w_{n}\)) at specific locations along the sensing fiber can be defined statistically as the energy of its time series, \(x_{n}(t)\), as:  
\[w\ \approx\ \int|x(t)|^{2}dt \tag{4}\]  
where \(dt\) is the resolution of the time series; \(dt\) = pulse duration. The reconstructed matrix in the Equation (3) can also be written as \(\tilde{A}=\ \left[ \begin{array}{cccc}\alpha_{i1}&\alpha_{i2}&\alpha_{i3}&\cdots& \alpha_{in}\end{array} \right]\) for \(i=1,2,\ldots,m\), where \(\alpha_{in}\) is the \(n\)th column vector of size \(m\times 1\) for the time series, \(x_{n}(t)\), at the \(n\)th position along the FUT. Multiplying the time series, \(\alpha_{in}\), with its weight factor, \(w_{n}\), further denoises the matrix, \(\tilde{A}\). Finally, the weighted reconstructed matrix, \(A_{uv}\), can be described as:  
\[A_{uv}=\ \left[ \begin{array}{cccc}w_{1}\alpha_{i1}&w_{2}\alpha_{i2}&w_{3} \alpha_{i3}&\cdots&w_{n}\alpha_{in}\end{array} \right] \tag{5}\]