Time–Frequency Domain Joint Seismic Inversion for Fractured Porous Media Based on the Exact Graebner Equation

Accurate characterization of fractured porous media is essential for hydrocarbon exploration and coal mine safety, with traditional seismic methods often failing to provide reliable parameter estimates. Fractured porous media, characterized by horizontally aligned fractures embedded within a porous background, can be modeled as a transversely isotropic medium with a vertical symmetrical axis (VTI). A relaxation function was employed to establish quantitative links between porosity (ϕ), fracture weakness (δN), water saturation (Sw), and the full elastic stiffness tensor. While the linearized Rüger approximation was commonly adopted as the forward operator in prestack inversion for VTI media, its accuracy degraded significantly at moderate to large incidence angles, limiting reliable parameter estimation. To address this limitation, this study proposes an exact Graebner-based multi-domain inversion (EGMI) framework that leverages the exact Graebner equation for enhanced angular accuracy. A Bayesian inversion scheme was constructed using a Cauchy distribution as the prior, and joint time-frequency inversion was performed to exploit the noise robustness of time-domain data and the resolution enhancement from frequency-domain information. The inversion was implemented via a Levenberg–Marquardt optimization, with analytical Jacobians. Comparative experiments on both synthetic and field data showed that the EGMI method provides superior accuracy and resolution for fractured reservoir characterization, with significant implications for unconventional resource exploration and product optimization.