## 2 Methodology
### Sparse Vector in Single-Trace Inversion  
According to Equation (1), taking 1D reflection coefficient inversion as an example, Figure 1 illustrates the results of the inversion operator \(\mathbf{G}\) (Figure 1a), noisy data (Figure 1b) with 10% noise, and the inversion reflectivity \(\mathbf{R}\) (Figure 1c) after 100 iterations. After 100 iterations, the inversion operator \(\mathbf{G}\) exhibits two horizontally distributed values at the positions corresponding to the reflectivity \(\mathbf{R}\) (at positions 33 and 66), while the values at other positions converge to a mean value. It can be observed that there is a certain correlation between the inversion operator \(\mathbf{G}\) and the positions of the reflection coefficients. According to Equation 6, as shown in Figure 2, the inversion operator \(\mathbf{G}\) (Figure 2c) is obtained by multiplying the inverse of \((\mathbf{W}\mathbf{W}^{\intercal}+\lambda\mathbf{Q})^{-1}\) (Figure 2a) with the transpose of the wavelet matrix \(\mathbf{W}^{\intercal}\) (Figure 2b). Since \(\mathbf{W}^{\intercal}\) is only related to the seismic wavelet, it can be inferred that the inverse of \((\mathbf{W}\mathbf{W}^{\intercal}+\lambda\mathbf{Q})\) determines the positions of the reflection coefficients. Figure 1: Damped least squares schematic with noise (\(\lambda=0.2\), \(p=0.8\), iteration \(=100\)). (**a**) Inversion matrix \(\mathbf{G}\); (**b**) noisy data with 10% noise; and (**c**) inversion reflectivity \(\mathbf{R}\). This figure validates the noise-robust interface localization capability of single-trace sparse inversion, illustrating the spatial correlation between operator \(\mathbf{G}\) and reflectivity \(\mathbf{R}\) during iterative optimization. For further analysis, \((\textbf{WW}^{\mathsf{T}}+\lambda\textbf{Q})\) is decomposed into \(\textbf{WW}^{\mathsf{T}}\) (Figure 3a) and \(\lambda\textbf{Q}\) (Figure 3b) diagonal elements (representing spatial constraint intensity), which significantly exceed those of \(\textbf{WW}^{\mathsf{T}}\) (reflecting wavelet autocorrelation energy). Since \((\lambda\textbf{Q})^{-1}\) dominates the matrix inversion process, the spatial resolution of inversion operator **G** is determined by **Q**'s constraint pattern: high diagonal elements in **Q** correspond to strong regularization zones (suppressing spurious reflections), while low elements indicate weak constraint regions (preserving true interfaces). Furthermore, the band-limited nature of \(\textbf{WW}^{\mathsf{T}}\) provides limited positional information, whereas **Q** encodes prior spatial constraints of reflectivity through its diagonal distribution. This mathematical representation of geological information establishes **Q** as the primary carrier of positional information. Consequently, the significant dominance of **Q**'s diagonal element magnitudes enables it to govern the positional information of reflectivity. We define the diagonal elements of the \(\lambda\textbf{Q}\) matrix as a vector \(\mathbf{q}\):  
\[\mathbf{q}=diag(\lambda\textbf{Q}). \tag{10}\]  
Clearly, the vector \(\mathbf{q}\) in the aforementioned equation is influenced by various factors, such as sparsity assumptions, the number of iterations, and others. Through multiple  
Figure 3: Schematic of the decomposition of the inverse matrix of \((\textbf{WW}^{\mathsf{T}}+\lambda\textbf{Q})\) (\(\lambda=0.2\), \(p=0.8\), iteration \(=100\)). **(a)** Matrix \(\textbf{WW}^{\mathsf{T}}\); **(b)** matrix \(\lambda\textbf{Q}\). The figure further reveals that the diagonal numerical superiority of the regularization sparse matrix \(\lambda\textbf{Q}\) dominates the spatial constraints of reflectivity coefficients. Figure 2: Inversion matrix **G** schematic (\(\lambda=0.2\), \(p=0.8\), iteration \(=100\)). **(a)** The inverse matrix of \((\textbf{WW}^{\mathsf{T}}+\lambda\textbf{Q})^{-1}\); **(b)** transposed wavelet matrix \(\textbf{W}^{\mathsf{T}}\); and **(c)** inversion matrix **G**.