Waves in stratified viscoelastic media with microstructure

An extension of the O’Doherty and Anstey theory [R. F. O’Doherty and N. A. Anstey, Geophys. Prosp. 19, 430–458 (1971)] is derived for waves in finely layered, anisotropic, viscoelastic media. Multiple scattering effects cause the direct wave to be delayed and broadened, subject to a deterministic integrodifferential delay equation. The kernel depends upon time‐domain autocorrelations of reflectivities and the relaxation functions defining the viscoelastic effects. The medium differs from a slowly varying medium by O(ε), and propagation over O(1/ε2) distances is considered. Both smoothly varying media and interfaces are considered simultaneously. A regular perturbation technique is used to show how an approximation to the field may be rapidly calculated. The signal delay predicted by the theory is identified with the retardation (drift) in the equivalent effective medium for the same interval. Numerical examples using synthetic fractal media show the approximate solutions to be in very good agreement with exact computations but 30–30 000 times faster.