Reflectivityltransmissivity for one-dimensional inhomogeneous random elastic media: dynamic-equivalent-medium approach

We study the influence of 1-D randomly stratified media on both the transmissivity and reflectivity of obliquely incident elastic plane waves. For this, a combination of the invariant embedding method with the small-perturbation expansion is used. The resulting analytic expressions for the plane time-harmonic and transient P- and S I/-wave transmissivities and reflectivities can be used for both deterministically and statistically specified stratifications. They do not have any essential restrictions con-cerning the frequency range of validity. However, it is assumed that the fluctuations of the density and compressional and shear velocities are small in comparison with the average values of these parameters. The analytic expressions for the attenuation coefficients and phase velocities of the P and W transmissivities predict their smoothed behaviour well in one single realization of a 1-D random medium. Both expressions can be considered as fundamental wave attributes for a ‘dynamic-equivalent medium’. They can also be looked upon as a generalization of the well-known O’Doherty-Anstey formula, which can now account for oblique incidence and mode-conversion effects in an inhomogeneous 1-D medium with depth-dependent statistics. Likewise, the resulting frequency-dependent phase velocities can be considered as a broad-frequency-range generalization of their static approximations, obtained by what is known as Backus averaging. They postulate a frequency-dependent elastic transverse isotropy of the dynamic-equivalent medium, which replaces the random 1-D medium. We use the dynamic-equivalent-medium description to formulate some simple approximations of the P- and SV-wave reflectivities. Key words: anisotropy, attenuation, elastic-wave theory, layered media, scattering