Quadratic Reflectivity-Based Joint PP and PS Inversion for VTI Media Using the Stiffness Matrix

Shale is a critical hydrocarbon reservoir generally considered a typical transverse isotropy with a vertical axis of symmetry vertical transverse isotropic (VTI) media. Petrophysical experiments indicated that shales may exhibit moderate or strong anisotropy, exacerbating uncertainties in seismic reflection analysis and pre-stack seismic inversion. In addition, the significant difference in contribution between the elastic and anisotropic parameters to the reflection coefficients limits the stability of anisotropy predictions. A quadratic reflectivity-based joint PP and PS inversion for VTI media was proposed, which indirectly predicts the anisotropy using the stiffness matrix. First, based on the elastic impedance matrix, we derived the weak-contrast PP and PS reflectivities characterized by the stiffness matrix. The contribution analysis indicates that the PP wave is sensitive to the stiffness matrix C11, and the PS wave is sensitive to C13, which implies that the joint PP and PS reflection was more effective in analyzing the effects of anisotropic parameters. Then, to improve the accuracy of the reflectivity at the interface with high-contrast velocity, quadratic PP and PS reflectivities for isotropic media were derived. Tests on different typical models showed that the reflectivity with quadratic terms performed well in accuracy, even for strong anisotropy. Finally, a joint inversion constrained by the nonlinear relationship between the stiffness matrix and the anisotropic parameters was developed, which can overcome the ill-posedness of indirectly calculating the anisotropic parameters using the stiffness matrix. The model and field seismic data applications prove the feasibility and effectiveness of the proposed method.