## 2 Methodology
### Multitrace Reflectivity Inversion Based on Geologically Guided Sparse Criteria  
Building upon the aforementioned analysis, the iterative inversion framework demonstrates that under the sparsity assumption of seismic data, the sparse reflection coefficient matrix \(\mathbf{R}\) progressively converges toward the true subsurface reflectivity model. Leveraging this mechanism, we propose a multitrace reflectivity inversion method based on the geologically guided sparsity criterion, which integrates data-driven sparsity regularization with geological prior knowledge to enhance the fidelity of subsurface parameter estimation. The specific inversion equation is as follows:  
\[\mathbf{R}=(\mathbf{WW}^{\intercal}+\lambda\mathbf{L})^{-1}\mathbf{W}^{ \intercal}\mathbf{S}. \tag{15}\]  
Figure 8: Geological-guided sparse matrix **Geo** extracted from noisy data (\(j=5\)). **(a)** Structure-constrained matrix **Geo**; **(b) Geo** with added stratigraphic boundary information; and **(c) Geo** with added geological prior information. The figure illustrates the construction of the geologically guided sparse matrix by progressively incorporating structural constraints, stratigraphic boundaries, and internal geological body information. It demonstrates how geological priors optimize the spatial distribution of inversion results. The geologically guided sparse matrix \(\mathbf{L}\) is expressed as follows:  
\[\mathbf{L}=\mathbf{Q}\otimes\mathbf{Geo}, \tag{16}\]  
where \(\otimes\) is the Kronecker product, and \(\mathbf{Geo}\) is the constructed geological steering matrix, which is used to adjust the extreme value distribution of the column vectors in the sparse matrix \(\mathbf{Q}\). Figure 9 shows the inversion results \(\mathbf{R}\) of the reflectivity for the noisy synthetic seismic data based on the geological-guided sparsity criterion and the geological-guided sparse matrix \(\mathbf{L}\). Overall, \(\mathbf{Q}\) denotes the original sparse matrix constructed without geological constraints, and \(\mathbf{L}\) represents the geologically guided sparse matrix obtained by applying structural constraints through the Kronecker product with \(\mathbf{Geo}\). Figure 9: Inversion of noisy data using structure constraint extracted from noisy data (\(\lambda=0.002\), \(p=0.8\), \(j=5\), iteration \(=100\)). (**a**) Structure-constrained matrix \(\mathbf{L}\); (**b**) stratigraphic boundary structure-constrained matrix \(\mathbf{L}\); (**c**) geological prior information structure-constrained matrix \(\mathbf{L}\); (**d**) inversion using structure-constrained matrix; (**e**) inversion using stratigraphic boundary structure-constrained matrix; and (**f**) inversion using geological prior information structure-constrained matrix. The figure demonstrates the progressive construction of geologically guided sparse matrix \(\mathbf{L}\) through sequential incorporation of structural, stratigraphic, and geobody constraints. The corresponding inversion results validate the effectiveness of geological constraints in enhancing resolution and restoring subtle reservoir details. In Figure 9a, structural information is incorporated into the **L** matrix. In Figure 9b, =stratigraphic boundaries are added to the **L** matrix at the position \(\tau\) = 19, where the stratigraphic position consistently maintains low values. In Figure 9c, the internal features of the geological body are added to the **L** matrix, effectively removing anomalies around the stratigraphic position at \(\tau\) = 19. Comparing the geologically guided sparse matrix **L** in Figure 9a-c with the sparse matrix **Q**, it is evident that the wedge-shaped features are more distinctly characterized in **L**. Furthermore, as more specific prior geological information is added, the geological features in the **L** matrix become more deterministic and clearer.