Random Heterogeneous Media Research Articles

Multiple scattering and energy transfer of seismic waves—Separation of scattering effect from intrinsic attenuation II. Application of the theory to Hindu Kush region

In order to separate the scattering effect from intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In paper I (Wu, 1985), radiative transfer theory is applied to seismic wave propagation and the energy density distribution (or the average intensity) in space for a point source is formulated in the frequency domain. It is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves. In this paper, the data from digital recordings in the Hindu Kush region are used as an example of application of the theory. We also discuss two approximate solutions of coda envelope in the time domain: the single scattering approximation and the diffusion approximation and discuss the relation with the frequency domain solution. We point out that in only two cases can the apparent attenuation be expressed as an exponential decay form. One is the dark medium case, i.e., when B o ≫ 0.5, where B o = ŋ s /(ŋ s + ŋ a ) is the seismic albedo, ŋ s is the scattering coefficient, ŋ a is the absorption coefficient. In this case the absorption is dominant, the apparent attenuation b can be approximated by the coherent wave attenuation b = ŋ s +ŋ a The other case is the diffuse scattering regime, i.e., when B o ≫ 0.5 (bright medium) and R≫ L s ,t ≫ τs, where R and t are the propagation distance and lapse time, L s and τ s are the scattering lengths (mean free path) and scattering time (mean free time), respectively. However, in this case the envelope decays with a rate close to the intrinsic attenuation, while the intensity decreases with distance with a coefficient b≈d 0 (ŋ a +ŋ s ) ≈ d s ŋ s where d o d s are the diffusion multipliers (0 1).

The incremental stiffness of an assembly of particles

The problem of fairly densily packed particles with rate independent though otherwise arbitrary interaction is investigated in two dimensions. It is shown that slow changes in a statistically uniform medium in the pre-failure deformation state can be described by a stiffness tensor. An approximate expression for this tensor is obtained from first principles. The connection with the perfectly random heterogeneous medium is established. A simple example for round rough particles in an isotropic packing of four contacts is given.

The effective permeability of a random medium

The authors investigate the effective permeability of a heterogeneous random medium by numerical simulation and by self-consistent perturbation theory for a class of models including the log-normal model. For a reasonable range of parameters they find close agreement between the simulation and the perturbative calculation. This suggests that both methods may be useful for appropriate models. However, the simulation approach is widely applicable in circumstances where a perturbation calculation is not possible.

Application of Volterra–Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. I. General Procedure

A general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed. It utilizes truncated Volterra-Wiener functional series, generated by the random conductivity field K(x), as classes of trial functions for the classical variational principles. The method simplifies, unifies and/or generalizes the earlier proposed variational techniques of Beran (3), Dederichs and Zeller (10), Hori (15), Kr6ner (17) and Prager (25). The general procedure is displayed in detail in the simplest case of interest, the construction of the optimal third-order bounds, that requires knowledge of the two- and three-point correlation functions for K(x). The evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions. A perturbation solution to this equation is given. For Miller's cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of Beran-Miller.

Static elastic and thermoelastic field fluctuations in multiphase composites

The spatially fluctuating strain and stress fields in a random heterogeneous thermoelastic medium are characterized by their mean values and square means in each phase. In the present paper the calculation of these quantities, which has been presented in a previous work for the case of a mechanical load only. is extended to include thermal expansion. Besides the derivation of some exact relations between the square means and the analytical properties of the effective thermal expansion coefficient the field fluctuations are calculated within an effective-medium procedure supposing an aggregate topology of the composite. Explicit results obtained for isotropic phases and spherical grain shapes are given in a more convenient representation than in our former work.

Multiple scattering and energy transfer of seismic waves -- separation of scattering effect from intrinsic attenuation -- I. Theoretical modelling

Summary. In order to separate the scattering effect from the intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In this paper, we apply radiative transfer theory to seismic wave propagation and formulate in the frequency domain the energy density distribution in space for a point source. We consider the cases of isotropic scattering and strong forward scattering. Some numerical examples are shown. It is seen that the energy density-distance curves have quite different shapes depending on the values of medium seismic albedo Bo = qs/(qs t q,), where vs is the scattering coefficient and va is the absorption coefficient of the medium. For a high albedo (B> 0.5) medium, the energy-distance curve is of arch shape and the position of the peak is a function of the extinction coefficient of the medium ve = qs + qa. Therefore it is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves.