The study of thin-bed seismic phenomena is important in crustal, exploration and engineering seismology. Presently, seismic reflectivity theories based on single-interface assumption are widely used though they are only suitable for thick deposits. Thin-bed reflectivity theories are established on complex propagator matrices and are difficult to be applied to reveal thin-bed properties directly. Therefore, an approximation of thin-bed PP-wave reflection coefficients (R_PP) is derived in this paper. First, the relationship between thin-bed R_PP and incidence angles is analyzed through series expansion method. For PP-wave, its reflection coefficients are even power series functions of sine incidence angles. Then, for small incidence, R_PP of the thin bed is further simplified into a second-order series approximation with respect to the sine incidence angles. Simulations and accuracy analyses of the approximate formula show that approximation errors are smaller than 5% as the incidence angles smaller than 20 degrees. Based on this approximate formula, an approach is given for estimating thin-bed properties including P-wave impedance ratios and thickness. The estimation approach is applied in properties estimation of a thin bed model. Perfect performances of the model example show the future potentiality of the approximate formula in thin-bed Amplitude-Versus-Offset (AVO) analysis and inversion.

Subsurface rocks commonly contain fluids and exhibit elastic anisotropy. Fluid substitution in anisotropic media is therefore important in geophysical applications. The general fluid-substitution relation given by Brown and Korringa is expressed in terms of elastic compliance, whereas seismic wave propagation is formulated in terms of stiffness. In this study, a direct stiffness-domain formula for anisotropic fluid substitution is derived using the Sherman–Morrison formula. The undrained stiffness matrix is written as a rank-one update of the frame stiffness matrix. The derived formula is analytically equivalent to the conventional compliance-inversion form, and the singular condition is identified through determinant analysis. Numerical calculations for an anisotropic Berea sandstone model demonstrate that variations in the pore-fluid bulk modulus cause nonlinear and component-dependent stiffness changes. As the singular limit is approached, the undrained compliance matrix becomes increasingly ill-conditioned. The compliance inversion error reaches O (10⁻⁶), whereas the error of the direct rank-one expression remains on the order of 10⁻¹⁶ over the tested range. The proposed formula provides a direct stiffness-domain implementation of Brown–Korringa fluid substitution. It improves numerical accuracy over conventional compliance inversion near the singular limit.

“Thin-bed” reservoirs have become important targets of seismic exploration and exploitation. However, traditional amplitude versus offset/amplitude versus angle (AVO/AVA) technologies, for example, those based on Zoeppritz equations and their approximations for a single interface, are not sufficiently accurate for thin-bed stratigraphy. Analytic solutions of thin-bed reflectivity may become practical for thin-bed AVO analysis and inversion. Therefore, a linear analytic approximation of thin-bed P-wave reflectivity is developed under small-incidence and thin-bed assumptions. Numerical simulations show that the amplitude approximation errors are usually smaller than 10% for incidence angles less than 20 degrees, and the thin-bed thicknesses are less than one-tenth of the P-wave wavelength. Based on the least-squares approach, the inversion strategy is proposed using the approximate formula. A synthetic data test shows that the proposed inversion method can produce more accurate thin-bed properties than that based on the Zoeppritz equations, which reveals the potential of the inversion method based on the linear analytic approximate formula in the fine characterization of thin reservoirs.