Elastic waves in finely layered sediments: The equivalent medium and generalized O'Doherty-Anstey formulas

We study the influence of elastic 1-D inhomogeneous random media (e.g., finely layered media with variable density and shear and compressional velocities) on the kinematics and dynamics of the transmitted obliquely incident P- and SV-plane waves. Multiple scattering (resulting in localization and spatial dispersion of the elastic wavefield) is the main physical effect controlling the properties of the wavefield in such media. We analyze the wave propagation assuming the fluctuations of ve locities and density to be small (of the order of 20% or smaller), We obtain explicit analytic solutions for the attenuation coefficient and phase velocity of the transmitted waves. These solutions are valid for all frequencies. They agree very well with results of numerical modeling. Our theory shows that fine elastic multilayering is characterized by a frequency-dependent anistropy. At typical acquisition frequencies this anisotropy differs significantly from the low-frequency anisotropy described by the well-known Backus averaging. The increase of the phase velocity with frequency is quantified. It can partly explain the difference between well-log-derived velocities and lower frequency seismic velocities [e.g., vertical seismic profiling (VSP) velocities] in terms of localization. The low- and high-frequency asymptotical results for the phase velocity agree with those of Backus averaging and ray approximation, respectively. The theory describes the angle-dependent attenuation caused by multiple scattering. The proposed formulas are simple enough to be used in many practical applications as. e.g., in an amplitude variation with offset (AVO) analysis. They can be implemented for taking into account the angle dependence of transmission effects, or they can be used in an inversion for statistical parameters of sediments.