Spectral diffusion of elastic wave energy

Abstract

SUMMARY Elastic-wave propagation in strongly scattering media is described by an energy diffusion equation derived directly from the coupled mode equations. The physical model we assume consists of a stack of layers over a homogeneous, non-scattering half-space. The overlying stack contains the heterogeneities that cause the diffusion of energy by scattering and may have an arbitrary mixture of fluid and solid layers. The coupled mode equations provide a formally exact representation of the displacementstress field in 2-D heterogeneous media, so the resulting diffusion equation is physically based rather than phenomenological. By allowing the mode spacing to become very small, the diffusion equation is derived from a differential-difference coupled energy equation. All assumptions and approximations must be explicitly stated and implemented in the derivation of the energy diffusion equation. Strong forward scattering is assumed to dominate. The mean free path for multiple scattering must be large compared with the size of the medium fluctuations responsible for the scattering, and the spatial scale of the heterogeneities must be large compared with the wavelength. The energy diffusivity is a range- and frequency-dependent functional of the displacementstress field components and the horizontal gradients of the medium properties including anisotropy. The diffusivity functional is derived directly from the continuum limit of the mode-coupling matrix; it is essentially the spatial autospectrum of the coupling matrix weighted by a function describing the density of modes in spectral space. The approach presented in this paper is in contrast to energy diffusion equations derived from radiative transfer theory in which the diffusivity must be specified separately in an ad hoc manner. Although our energy diffusion equation is range-dependent, the computation of the diffusivity depends on local medium properties and field values for a plane-layered medium. An additional difference between the diffusion equation of this paper and previously published treatments of elastic-energy diffusion is that this paper describes energy diffusion in spectral space, for example slowness, as a function of range rather than diffusion in physical space as a function of time. The dimensions of the diffusivity in slowness space are [slowness2/length].