Nonlinear Pre-Stack Inversion Based on Exact VTI Medium Reflection Coefficient Equation

The assumption of transversely isotropic media with a vertical symmetry axis vertically transverse isotropic (VTI) has been widely utilized in the exploration of shale reservoirs, with its reflection response and pre-stack inversion attracting significant attention from geophysicists. However, accurately estimating anisotropic parameters using existing approximate reflection coefficient formulas for VTI media remains challenging. In theory, inversions based on exact reflection coefficients for VTI media could overcome these limitations. Nevertheless, such inversions present a highly nonlinear problem. The simultaneous inversion of five parameters further exacerbates the ill-posed nature of the inversion, making it difficult for conventional linearized algorithms to handle effectively. To address these challenges, this article proposes a nonlinear pre-stack inversion scheme based on the exact reflection coefficient equations for VTI media. Specifically, we establish an inversion objective function within a Bayesian framework and introduce a heuristic Newton-Raphson-based optimization (NRBO) algorithm for solving the nonlinear objective function. This approach mitigates the inaccuracies introduced by conventional linearized algorithms, enhancing the precision of the inversion results. Tests on 1-D single-well synthetic data, 2-D Hess VTI model synthetic data, and field data demonstrate that the proposed inversion scheme can stably and effectively estimate both elastic and anisotropic parameters simultaneously. Moreover, the inversion accuracy of the proposed approach surpasses that of inversions based on an approximate formula.