## 4 Lateral Constraint Inversion Theory
### Optimization of Lateral and Vertical Weighting Factors
The lateral weighting factor was set to 0, while the vertical weighting factors were set to 0.1, 0.5, and 5. The impact of vertical weighting factors on inversion outcomes was assessed individually. As depicted in Figure 7, the inversion results generally capture the resistivity variations in the formation. Different vertical weighting factors exert varying degrees of influence on the inversion outcomes. As the vertical weighting factors increase, the vertical continuity of the inversion results gradually improves. However, significant deviations from the theoretical model are observed when the vertical weighting factor reaches 0.5, particularly in depicting interfaces and inclined surfaces beneath low-resistivity layers. When the vertical weighting factor reaches 5, the inversion results deviate from the theoretical model. A comprehensive evaluation indicates that excessively large vertical weighting factors should be avoided, possibly due to the smooth constraint underlying the OCCAM inversion, which favors enhanced vertical continuity of the inversion results. Therefore, a vertical weighting factor of 0.1 is recommended. Applying a lateral weighting factor of 0.5 and a vertical weighting factor of 0.1 to the low-resistivity fault valley model depicted in Figure 4, the inversion results, as illustrated in Figures 5 and 8, demonstrate a high degree of continuity in both lateral and vertical dimensions. These optimized weighting factors effectively capture intricate details of the model, including upper and lower interfaces and inclined planes. While there exists a slight  
Figure 7: Inversion results of low-resistivity fault valley model with different vertical weighting factors. (**a**) Vertical weighting factor of 0.1; (**b**) vertical weighting factor of 0.5; (**c**) vertical weighting factor of 5.  
disparity in interface depth between the inversion results and the theoretical model, overall consistency is observed. Figure 9 displays the corresponding decline curve in fitting errors. The rapid reduction in fitting error following the implementation of horizontal and vertical constraints, facilitated by the OCCAM inversion's speed, stability, and vertical smoothness, leads to near-ideal inversion outcomes typically achieved within three iterations. Model II: The schematic diagram of Model II is shown in Figure 10. This model depicts a structure where the middle layer exhibits high resistivity. The resistivity of this layer is 500 \(\Omega\)-m, while maintaining the same parameters as described previously. Refer to Figure 10 for the schematic representation of this model. The vertical weighting factor is set to 0, while the lateral weighting factors are 0.1, 0.5, and 5. The corresponding inversion results are depicted in Figure 11. The Figure  
Figure 8: Inversion results of low-resistivity fault valley model with lateral weighting factor of 0.5 and vertical weighting factor of 0.1. Figure 4.0 Schematic diagram of fault valley model with high resistivity in middle layer. Figure 4.1 Curve of fitting error decreasing with iteration times of low-resistivity fault valley model. illustrates that upon the application of lateral constraints, the inverted formation exhibits similarities to the inversion outcomes of a low-resistivity model concerning lateral continuity and interface morphology. As the lateral weighting factor increases, the smoothness and lateral continuity of the inversion results progressively improve, leading to a reduction in the discernibility of the transition characteristics of the lower interface within the moderately high-resistivity layer in the lateral direction. Notably, an optimal weighting factor is identified in this study, and a comprehensive comparison reveals that the inversion results with a lateral weighting factor of 0.5 align most closely with the design model, demonstrating enhanced lateral continuity and detailed characteristics.