## 2 Methodology
### Sparse Matrix in Multitrace Inversion  
In multitrace inversion, the sparse matrix \(\mathbf{Q}\) is composed of the sparse vectors \(q\) from each trace, representing the vertical sparse expression of seismic data:  
\[\mathbf{Q}=[q_{1},q_{2},q_{3},\cdots,q_{n}], \tag{11}\]  
Figure 4: Relationship between vector \(q\) and reflectivity position under different iteration counts (\(\lambda=0.2\), \(p=0.8\)). (**a**) Vector \(q\) after 1 iteration; (**b**) vector \(q\) after 10 iterations; and (**c**) vector \(q\) after 100 iterations. The figure illustrates the evolution of sparse vector \(q\) during iterative inversion: as iterations progress, \(q\) develops stable minima at reflectivity positions (e.g., samples 33 and 66), demonstrating the spatial constraint stability of geological sparse criteria.  
where \(n\) represents the number of seismic traces, and the sparse matrix \(\mathbf{Q}\) is jointly constrained by the seismic data and mathematical sparsity assumptions.  
To analyze the characteristics of the sparse matrix \(\mathbf{Q}\), this section designs a 2D wedge-shaped reflectivity model (Figure 5a). Seismic data are synthesized by convolving the model with a 30 Hz Ricker wavelet (Figure 5b), and 10% random Gaussian white noise is added to each trace (Figure 5c).  
Using the noisy seismic data for multitrace reflectivity inversion, and taking the Lp norm of 0.8 as an example, Figure 6 illustrates the changes in the sparse matrix \(\mathbf{Q}\) and the inversion results at different iteration counts. During the first iteration, the positional information in \(\mathbf{Q}\) (Figure 6a) resembles the boundaries of the seismic data, and the inversion result (Figure 6d) appears relatively blurred. By the 10 th iteration, \(\mathbf{Q}\) (Figure 6b) begins to exhibit wedge-shaped structural features, although the boundaries remain somewhat unclear, while the inversion result (Figure 6e) achieves a higher resolution. By the 100 th iteration, \(\mathbf{Q}\) (Figure 6c) demonstrates a sparse wedge-shaped structural characteristic, and as the precision of \(\mathbf{Q}\) improves, the accuracy and resolution of the inversion reflectivity (Figure 6f) are significantly enhanced.  
Through the analysis in this subsection, it is found that the two-dimensional sparse matrix \(\mathbf{Q}\) in multitrace inversion structurally aligns with the true reflectivity model, accurately resolving stratigraphic boundaries and lateral depositional continuity. The sparse vector \(q\) exhibits a direct spatial correspondence with reflectivity positions, where its minima systematically coincide with reflection interfaces. This positional relationship extends to the composite sparse matrix \(\mathbf{Q}\), which quantifies the subsurface architecture through vertical impedance contrasts and lateral geological patterns.  
Figure 5: The wedge model. (**a**) Reflectivity. (**b**) Synthetic seismic data. (**c**) Noisy seismic data with 10% random noise. The figure displays the designed wedge-shaped reflectivity model and its noisy seismic data, aiming to validate the adaptability of constraint operators to thin interbeds and noise under varying conditions.