## 3 Methodology
### Multiple Singular Spectrum Analysis Method Based on \(k\)-Means Clustering  
In accordance with MSSA theory, the initial step involves applying a Fourier transform to convert the multilevel wavelet coefficient profiles \(d_{{\ast}{\ast}_{k}}(x,\ t)\) into their frequency-domain counterparts \(\widetilde{d}_{{\ast}{\ast}}(x,f)\). The data \(\widetilde{d}_{{\ast}{\ast}}(x,f_{i})\) at a specific frequency \(f_{i}\) can be expressed as:  
\[\widetilde{d}_{{\ast}{\ast}}(x,f_{i})=[\widetilde{d}_{1},\widetilde{d}_{2}, \ldots,\widetilde{d}_{N_{x}}] \tag{4}\]  
Based on the Hankel transform [39; 40], the Hankel matrix of \(\widetilde{d}_{{\ast}{\ast}}(x,f_{i})\) can be represented as:  
\[\mathbf{H}= \begin{pmatrix}\widetilde{d}_{1}&\widetilde{d}_{2}&\cdots&\widetilde {d}_{p}\\ \widetilde{d}_{2}&\widetilde{d}_{3}&\cdots&\widetilde{d}_{p+1}\\ \vdots&\vdots&\ddots&\vdots\\ \widetilde{d}_{q}&\widetilde{d}_{q+1}&\cdots&\widetilde{d}_{N_{x}}\end{pmatrix}  \tag{5}\]  
where \(p\) and \(q\) represent two key parameters for constructing the Hankel matrix [41]. Typically, \(q\) is set as the integer value of \(N_{x}/2+1\), and \(p\) is defined as \(N_{x}-q+1\). The Hankel matrix can then be decomposed using SVD technique [40]:  
\[\mathbf{H}=\sum\limits_{i=1}^{r}\lambda_{i}\mathbf{U}_{i}\mathbf{V}_{i}^{T} \tag{6}\]where \(\mathbf{U}_{i}\) and \(\mathbf{V}_{i}\) represent the \(i\)-th eigenvector of \(\mathbf{HH}^{\mathsf{T}}\) and \(\mathbf{H}^{\mathsf{T}}\mathbf{H}\), respectively, while \(\lambda_{i}\) represents the \(i\)-th eigenvalue of \(\mathbf{H}\), satisfying the following:  
\[\left\{ \begin{array}{l}\mathbf{HH}^{\mathsf{T}}\mathbf{U}_{i}=\lambda_{i}^{2 }\mathbf{U}_{i}\\ \mathbf{H}^{\mathsf{T}}\mathbf{H}\mathbf{V}_{i}=\lambda_{i}^{2}\mathbf{V}_{i }\end{array} \right. \tag{7}\]  
where the matrix product form of Equation (6) is given by  
\[\mathbf{H}=\mathbf{U}\mathbf{\Lambda}\mathbf{V}^{\mathsf{T}} \tag{8}\]  
where \(\mathbf{\Lambda}\) represents the diagonal matrix composed of singular values, arranged in descending order along the diagonal. According to the composition characteristics of the GPR profile, the singular spectrum formed by the diagonal elements of the singular value matrix can be considered as primarily consisting of direct waves, reflection, diffraction, and noise [39]. In this work, we employ the \(k\)-means clustering algorithm to partition the singular values in the singular spectrum into \(k\) clusters based on their similarity, with the aim of defining the threshold between the singular values that correspond to different wave components. The following details describe this procedure. * Determine the number of clusters \(k\) and randomly select \(k\) singular values from the singular spectrum as the initial cluster centers. * Compute the Manhattan distance between each singular value in the singular spectrum and the cluster centers, as described by [28]. \[Q=|x_{i}-x_{k}|+|y_{i}-y_{k}|\] (9) where \(Q\) represents the calculated Manhattan distance. \(x_{i}\) and \(y_{i}\) represent the horizontal and vertical coordinates of the \(i\)-th singular value in the singular spectrum, respectively. \(x_{k}\) and \(y_{k}\) represent the horizontal and vertical coordinates of the \(k\)-th cluster center, respectively. A smaller value of \(Q\) indicates higher similarity, while a larger Q denotes lower similarity. In other words, singular values with similarity to the cluster center will be assigned to the same cluster. * Calculate the mean value of all singular values in each cluster and update the cluster center to this mean value. * Repeat second and third steps until the cluster centers stabilize or a predefined number of iterations is reached. Upon performing the mentioned calculations, the single values in the singular spectrum are disaggregated into \(k\) clusters based on their similarity. Utilizing the intrinsic characteristics of the GPR wavefield, the first cluster is thought to be represent direct waves, while several clusters with larger singular values are thought to be represent reflections.