BACKGROUND

This Short Note is part of a continuing effort to extend equivalent medium theory from the quasi-static and kinematic to the dynamic. In other words to extend its area of practical application. Backus (1962) taught us how to add fine-layered isotropic and transverse-isotropic materials, and Schoenberg & Muir (1989) extended this result to arbitrary anisotropy and fractures, and introduced a group theory and calculus to systematize the process. But however useful these results are, they do not speak to the problem of wavelet broadening, which was addressed so eloquently by O'Doherty & Anstey (1971).

There is a general understanding by workers in this area that a useful notion is to model the loss of energy from the coherent field to the scattered field (which does not place at 0 Hz) by replacing the elastic wave equation in Helmholtz form,

$$\nabla ^{T} C \nabla u + \omega ^{2} \rho u = 0,$$ (1)

by a standard form of the visco-elastic equation:

$$\nabla ^{T} C(i \omega ) \nabla u + \omega ^{2} \rho u = 0.$$ (2)

That is, the elastic matrix (stiffness tensor) C is replaced by a frequency-dependent form, $C(i \omega )$.