Multiscale Media Research Articles

Ultrasonic velocity dispersion in bovine cortical bone

The evaluation of cortical bone quality has become possible in clinical practice, but the interaction between a broadband ultrasonic pulse and this complex multiscale medium remains poorly understood. Specifically, the frequency dependence of phase velocity has been sparsely investigated. This study aims at evaluating the determinants of the frequency dependence of phase velocity in bovine femoral cortical bone samples using an in vitro ultrasonic transmission device. Phase velocity is shown to vary quasi linearly in a 1 MHz restricted bandwidth around 4 MHz, which enables dispersion evaluation. Axial dispersion is significantly higher than radial and tangential dispersions. Significant differences in dispersion are obtained according to the anatomical location. The microstructure of each sample is determined using an optical microscope, which allows assessing the dependence of dispersion on the type of bone microstructure. Mostly positive, but also negative values of dispersion are measured. Negative dispersion is obtained mostly in samples constituted of mixed microstructure, which may be explained by phase cancellation effects due to the presence of different microstructures within the same sample. Dispersion is shown to be related to broadband ultrasonic attenuation values, especially in the radial direction. This dependence is compared with results derived from the local Kramers‐Kronig relationships.

Finite element modelling of transient elastic wave propagation in an inhomogeneous anisotropic fluid/solid multilayer medium: A time-domain method

The axial transmission technique is used clinically for cortical bone assessment. However, ultrasonic propagation in this multiscale medium remains unclear, in particular because of the heterogeneous nature of cortical bone. The aim of this work is to evaluate the effect of spatial gradients of elastic moduli on the ultrasonic response of the bone structure. Therefore, a 2D finite element time-domain method is developed to simulate transient wave propagation in a three-layer medium constituted of an inhomogeneous transverse isotropic solid layer sandwiched between two acoustic fluid layers and excited by an acoustic linear source located in one fluid layer delivering broadband ultrasonic pulses. The model couples the acoustic propagation in both fluid media with the elastodynamic response of the solid. The conditions of continuity are used to model the fluid-structure interaction. A constant spatial gradient of material properties in the direction perpendicular to the layer is considered in the solid structure. In the presence of a gradient, the first arriving signal (FAS) velocity depends on the average material properties when the thickness is smaller than the wavelength (guided wave modes) whereas the FAS velocity depends on the velocity at the surface when the thickness is larger than the wavelength (lateral wave).

Resistivity logging in a multi-scale medium with lognormal conductivity

We derived equations of effective conductivity coefficients to estimate mean current density and its variance in resistivity logging problems for isotropic multi-scale porous media with lognormal conductivity distribution. The equations have quite a simple form if the medium parameters satisfy refined Kolmogorov’s similarity hypothesis. The results of the refined perturbation theory obtained using subgrid modeling are compared with the results of numerical simulation and the conventional perturbation theory.

Multiple scattering of acoustic waves and porous absorbing media.

Porous media like air-saturated polymer foams with open cells, have a nontrivial frequency-dependent absorption that arises due to viscous and thermal effects at the scale of the rigid frame microstructure. In order to produce multiple scattering at ultrasonic frequencies, mesoscale scatterers are introduced in the porous medium host. The effective wave number of such a multiscale medium should take into account the peculiar absorption at the microscale and the multiple scattering at the mesoscale to describe precisely the propagation of a coherent acoustic wave. For this purpose, a simple model is developed. First, an equivalent fluid model, derived from a homogenization method, is used to describe the acoustic propagation in the host porous medium itself. Second, the scattering by the inclusions is described with a multiple scattering approximation (independent scattering approximation). This simple model allows to obtain the total effective wave number of the porous medium with mesoscale scatterers. After some validating results on the multiple scattering by an array of rigid cylinders in air, experiments on the multiple scattering by rigid cylinders embedded in a porous medium are presented and compared to the developed simple model. Incidentally, it appears that for the host medium itself, the equivalent fluid model is not capable to describe the high-frequency behavior whilst a multiple scattering approach with (thin) viscous and thermal boundary layers around the scatterers is accurate in the whole frequency range.

Wave scattering in a multiscale random inhomogeneous medium

In this paper, using the Fock method of the fifth parameter and weighted Fourier-transform with respect to the coordinates of the source and observer, an integral representation is obtained for the wave field in a randomly inhomogeneous medium without invoking the assumption about small-angle propagation. Random trajectory variations to a first approximation are taken into account in calculating the partial wave phase (the expression under the integral sign). The expressions for the field in a medium with different-scale irregularities and for the scintillation index, obtained using this integral representation, are compared with known results. The good agreement with results from the theory of single scattering in a medium with background irregularities, and with investigations of the scintillation index made in terms of Rytov’s method and path integrals, indicates that it is possible to use the approach developed in this study to describe the effects of simultaneous influence of different-scale irregularities.

Form of the Frequency Spectrum of Fluctuating Radiation in the Case of Interferometric Reception During Experiments on Radio Sounding of the Solar Corona

We study theoretically the profile of a spectral line of radiation which is received by an interferometer after propagation through a medium with developed turbulent structure. We derive a formula describing the resulting frequency spectrum of an initially monochromatic radiation after its propagation through a multi-scale medium with “strong” large-scale and “weak” small-scale irregularities. It is shown that the frequency spectrum observed in this case should comprise the main spectral line with Gaussian profile and “wings” decreasing according to a power law with index dependent on the orientation of the interferometer baseline with respect to the direction of regular drift of “frozen-in” irregularities of the medium. We analyze distortions of the spectral response of a very-long-baseline interferometer due to the effect of large-scale irregularities of a medium with developed turbulent structure on wideband radiation from extraterrestrial discrete radio sources. It is shown that if the amplitude fluctuations of the received radiation are weak, then the measured frequency spectrum contains information on the drift velocity and the index of the spatial spectrum of interplanetary plasma irregularities with scales from hundreds to thousands of kilometers.

Propagation in multiscale random media

Many studies consider media with microstructure, which has variations on some microscale, while the macroproperties are under investigation. Sometimes the medium has several microscales, all of them being much smaller than the macroscale. Sometimes the variations on the macroscale are also included, which are taken into account by some procedures, like WKB or geometric optics. What if the medium has variations on all scales from microscale to macroscale? This situation occurs in several practical problems. The talk is about such situations, in particular, passive tracer in a random velocity field, wave propagation in a random medium, Schrödinger equation with random potential. To treat such problems we have developed the statistical near-identity transformation. We find anomalous attenuation of the pulse propagating in a multiscale medium.

Applications of nonstationary stochastic theory to solute transport in multi-scale geological media

In this study, we make use of a nonstationary stochastic theory in studying solute flux through spatially nonstationary flows in porous media. The nonstationarity of flow stems from various sources, such as multi-scale, nonstationary medium features and complex hydraulic boundary conditions. These flow nonstationarities are beyond the applicable range of the ‘classical’ stochastic theory for stationary flow fields, but widely exist in natural media. In this study, the stochastic frames for flow and transport are developed through an analytical analysis while the solutions are obtained with a numerical method. This approach combines the stochastic concept with the flexibility of the numerical method in handling medium nonstationarity and boundary/initial conditions. It provides a practical way for applying stochastic theory to solute transport in complex groundwater environments. This approach is demonstrated through some synthetic cases of solute transport in multi-scale media as well as some hypothetical scenarios of solute transport in the groundwater below the Yucca Mountain project area. It is shown that the spatial variations of mean log-conductivity and correlation function significantly affect the mean and variance of solute flux. Even for a stationary medium, complex hydraulic boundary conditions may result in a nonstationary flow field. Flow nonstationarity and/or nonuniform distribution of initial plume (geometry and/or density) may lead to nonGaussian behaviors (with multiple peaks) for mean and variance of the solute flux. The calculated standard deviation of solute flux is generally larger than its mean value, which implies that real solute fluxes may significantly deviate from the mean predictions.

Acoustic Pulse Spreading in a Random Fractal

Fractal medium models are used to model, for instance, the heterogeneous earth and the turbulent atmosphere. A wave pulse propagating through such a medium will be affected by multiscale medium fluctuations. For a class of one-dimensional fractal random media defined in terms of fractional Brownian motion we show how the wave interacts with the medium fluctuations. The modification in the pulse shape depends on the roughness of the medium and can be described in a deterministic way when the pulse is observed at its random arrival time. For very rough media the coherent wave is confined to a surface layer.

Solute flux approach to transport through spatially nonstationary flow in porous media

A theoretical framework for solute flux through spatially nonstationary flows in porous media is presented. The flow nonstationarity may stem from medium nonstationarity (e.g., the presence of distinct geological layers, zones, or facies), finite domain boundaries, and/or fluid pumping and injecting. This work provides an approach for studying solute transport in multiscale media, where random heterogeneities exist at some small scale while deterministic geological structures and patterns can be prescribed at some larger scale. In such a flow field the solute flux depends on solute travel time and transverse displacement at a fixed control plane. The solute flux statistics (mean and variance) are derived using the Lagrangian framework and are expressed in terms of the probability density functions (PDFs) of the particle travel time and transverse displacement. These PDFs are given with the statistical moments derived based on nonstationary Eulerian velocity moments. The general approach is illustrated with some examples of conservative and reactive solute transport in stationary and nonstationary flow fields. It is found based on these examples that medium nonstationarities (or multiscale structures and heterogeneities) have a strong impact on predicting solute flux across a control plane and on the corresponding prediction uncertainty. In particular, the behavior of solute flux moments strongly depends on the configuration of nonstationary medium features and the source dimension and location. The developed nonstationary approach may result in non‐Gaussian (multiple modal) yet realistic behaviors for solute flux moments in the presence of flow nonstationarities, while these non‐Gaussian behaviors may not be reproduced with a traditional stationary approach.

An integral representation of the wave field in inhomogeneous media in terms of diffracting component waves

This paper gives a review of the advantages offered by an integral representation using diffracting component waves in solving wave propagation problems, deterministic as well as stochastic, in multiscale media. The component waves are constructed in such a way that they account for the diffraction on local inhomogeneities. As a result, the method presented here describes propagation effects which pertain to nonzero values of the wave parameter of the local inhomogeneities embedded in the smoothly inhomogeneous background medium. The technique involves in a natural way the concepts of complex rays and complex caustics. In random problems the method describes the influence of diffraction by local random inhomogeneities on the field also in the near‐caustic areas. The method given is particularly suitable for solving HF wave propagation problems in the disturbed ionosphere.

Sensing scaled scintillations

We review some of the ways in which the fractal concept has found application in wave-propagation contexts. The scaling properties of fractals in both geometrical and statistical situations are reviewed and the relation to inverse power laws discussed. The relationship among the self-similar scaling properties of fractals, Lévy distributions, and renormalized group theory is explored to provide a simple picture of wave propagation through multiscale media. Finally, the notion of using a wavelet transform in the processing of fractal time series is considered.

Optical beam propagation in a band-limited fractal medium

Fractal geometry is used to investigate the propagation of an optical beam in a weakly fluctuating multiscale medium characterized by a power-law spectrum. Of particular interest here is the evolution of the beam as it traverses the medium. We model this medium by using a series of phase screens, each described by a band-limited fractal function. The average intensity distribution is calculated by using a combination of Bragg-diffraction considerations and the Fresnel propagator. We display the beam evolution here.