## 2 Methodology
### Construction of the Geological-Guided Sparse Matrix  
The analysis in the previous section on the relationship between the sparse matrix \(\mathbf{Q}\) and the multitrace reflectivity inversion results provides guidance for optimizing these results. To this end, we gradually introduce geological structures, stratigraphic boundaries, and the internal information of geological bodies to construct a geological-guided sparse matrix \(\mathbf{Q}\). By integrating existing geological knowledge into the inversion process, the inversion results can be further improved. First, the geological structure constraint matrix is introduced. This matrix does not have sparse requirements in the vertical direction, and maintains good continuity in the  
Figure 6: Results of sparse matrix \(\mathbf{Q}\) and inversion under different iteration counts (\(\lambda=0.2\), \(p=0.8\)). (**a**) \(\mathbf{Q}\) after 1 iteration; (**b**) \(\mathbf{Q}\) after 10 iterations; (**c**) \(\mathbf{Q}\) after 100 iterations; (**d**) inversion after 1 iteration; (**e**) inversion after 10 iterations; and (**f**) inversion after 100 iterations. The figure validates the progressive enhancement of geological constraints on reflectivity spatial distribution and resolution through iterative optimization, demonstrated by structural evolution of sparse matrix \(\mathbf{Q}\) and corresponding inversion results. horizontal direction, which can reflect the approximate spatial distribution information of the subsurface [33]. The geological structure constraint matrix is as follows:  
\[\textbf{Geo}=f(\textbf{S},j)=\frac{\sum_{i=1}^{n}\lambda_{i}}{\lambda_{i}}, \tag{12}\]  
where **Geo** represents the geological structure constraint matrix, which reflects the basic structural morphology underground. \(f(\cdot)\) is the algorithm used to extract structural constraints, **S** is the input seismic data, and \(j\) is the length of the time window when extracting the structure. The algorithm for extracting structural constraints can determine the possible locations of reflectivity by inputting seismic data. The output range is larger than the actual range of reflectivity. The lower the value, the higher the likelihood of the presence of reflectivity. This is consistent with the relationship between the vector \(\mathbf{q}\) and the reflectivity position. The extracted structures are dependent on the extraction radius \(j\). When \(j\) is small, the wedge-shaped reflection structures exhibit strong discontinuities (Figure 7a); when \(j\) is appropriate, the extracted structural information aligns well with the boundaries of the seismic data (Figure 7b); and when \(j\) is excessively large, the structural information introduces artifacts at the boundaries, and the wedge-shaped reflection structures exhibit noticeable computational segmentation artifacts (Figure 7c). Subsequently, stratigraphic boundary information is further introduced to enrich the geological information characterized in the matrix. These stratigraphic boundaries refer to the interfaces between different geological layers, marking the division of sedimentary or metamorphic events throughout geological history. We use prior information from well logs and mud logs to identify the exact locations of these stratigraphic boundaries, which are then incorporated as geological constraints during the inversion process. Afterwards, stratigraphic boundary information is incorporated into the geological structure constraint matrix, as follows:  
\[\textbf{Geo}_{\tau,i}=0, \tag{13}\]  
Figure 7: Geological structure constraint matrix **Geo** extracted from noisy data. (**a**) \(j=1\); (**b**) \(j=5\); and (**c**) \(j=9\). The figure demonstrates the impact of different extraction radius parameters \(j\) on the structural characteristics of the constraint matrix **Geo**. where \(i\) represents the number of seismic traces, and \(\tau\) denotes the position of the stratigraphic boundaries.