A Linearized Alternating Direction Method of Multipliers Algorithm for Prestack Seismic Inversion in VTI Media Using the Quadratic PP-Reflectivity Approximation

Periodic horizontal thin interlayers, or formations developing horizontally layered fractures in a homogeneous background, are commonly studied as a transverse isotropy with a vertical axis of symmetry (VTI) media. Anisotropy needs to be considered in predicting such special reservoirs and can be inferred from the amplitude variation with offset (AVO) by the exact or linear reflectivity equations. However, the linear approximation is usually derived based on the assumptions of weak anisotropy and weak contrast, and the accuracy suffers when the anisotropy or the contrast of two adjacent media increases. In addition, due to the high nonlinearity of the Graebner equation, the stability and efficiency of the exact inversion are limited. To address these issues, we propose a robust prestack seismic inversion for VTI media using quadratic PP-reflectivity approximation. First, by deriving quadratic approximations for phase velocity and polarization, we analyze the advantage of the second-order terms at a higher contrast interface. Then, based on perturbation theory, we derive a quadratic PP-reflectivity approximation from the exact equation. It consists of first- and second-order terms with isotropic and anisotropic components, and the accuracy is significantly improved even with the enhanced anisotropy. Finally, we introduce the Hadamard product operator to linearly characterize the objective function and develop an improved alternating direction method of multipliers (ADMM) algorithm to handle the nonlinearity of the quadratic equations. The feasibility and stability of the proposed method are tested on two datasets, that is, the logging curves used as numerical experiments and a field dataset from a shale reservoir.