Random Medium Model Research Articles

Polarimetric Remote Sensing of Geophysical Media with Layer Random Medium Model

Polarimetric Remote Sensing of Geophysical Media with Layer Random Medium Model

Porosity for the penetrable-concentric-shell model of two-phase disordered media: Computer simulation results

Computer-simulation results are reported for the porosity of a model of two-phase random media composed of identical D-dimensional spheres (D=2 or 3) distributed with an arbitrary degree of impenetrability λ, 0≤λ≤1; λ=0 corresponding to randomly centered or ‘‘fully penetrable’’ particles and λ=1 corresponding to totally impenetrable particles. We specifically consider the D-dimensional penetrable-concentric-shell model in which each sphere of diameter σ is composed of a mutually impenetrable core of diameter λσ, encompassed by a perfectly penetrable concentric shell of thickness (1−λ)σ/2. We develop two independent techniques to sample for the porosity. Simulation results agree with known exact results for the extreme limits of λ=0 and λ=1 up to three significant figures. The results for intermediate λ are new and compare favorably with approximate analytical expressions obtained by Rikvold and Stell.

Analytical solutions in differential form for the m-nth moment equations of waves in inhomogeneous random media

The series solutions of the m-nth moment equations are given in differential form for wave propagation in inhomogeneous random media when the small angle scattering approximation is valid. Three kinds of random media models are considered. In the first model, the material parameter in the medium fluctuates inhomogeneously in the direction of propagation and homogeneously in the direction transverse to the propagation path. In the second model, the fluctuation of the material parameter is homogeneous in the direction of the wave path and inhomogeneous in the direction transverse to the propagation path. In the third model, a monochromatic wave propagates in a medium and the statistical parameter of the medium varies both in longitudinal direction and transverse direction. In the first two cases, the solutions of the equations can be used to derive various moments of the wave with different wave numbers in different points. In the third case, the solutions can be used to study the moments of a monochromatic wave. The series solutions in differential form have some advantages over the series solutions in multiple convolution integral form in the study of certain problems related to the wave propagating in multiple scattering media.

Minimum wave-localization length in a one-dimensional random medium.

The frequency dependence of the localization length l for acoustic and electromagnetic waves in a one-dimensional randomly layered medium is studied both numerically and analytically. Through the consideration of different types of random-media models characterized by abrupt or continuous variation of the material parameters, it is shown that beyond the low-frequency behavior of l\ensuremath{\sim}${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}2}$, where \ensuremath{\omega} denotes the angular frequency, the localization length either approaches a constant or diverges at high frequencies. In all cases, the value of l for a given random medium is found to exhibit a well-defined lower bound whose value is generally several orders of magnitude times the correlation length of the inhomogeneities. The dependence of this minimum localization length on the amount of randomness, plus a comparison with the Schr\"odinger wave-localization length behavior, are presented and discussed.

D-dimensional interpenetrable-sphere models of random two-phase media: Microstructure and an application to chromatography

For a class of models of random two-phase media we derive and numerically evaluate expressions for the porosity (matrix volume fraction) φ and the specific surface (interface area per unit volume) s, as well as certain generalizations of these quantities. In these models the medium is considered as a suspension of mutually interpenetrable D-dimensional spheres of radius R, embedded in a uniform matrix. The models and quantities considered have applications to a variety of problems concerning transport, mechanical, electromagnetic, and chemical properties of composite media. Spatial dimensionality D=3 is appropriate for a wide range of composite media, whereas D = 2 describes certain fiber composites (parallel cylindrical inclusions), and D = 1 is appropriate for layered structures. The models considered here contain a hardness parameter ϵ, such that for ϵ = 1 they reduce to mutually impenetrable (hard) spheres, and for ϵ = 0 they reduce to randomly placed spheres. We consider three specific models in this class. The first of these, the permeable—sphere model, is considered in the Percus-Yevick approximation to obtain the two-point distribution function g 2(r 1, r 2). We obtain the n-point distribution functions g n (r 1, …, r n ) for n ⩾ 3 in the generalized superposition approximation. We also evaluate the lowest-order corrections to this approximation, finding that for all φ and ϵ it reliably gives φ as a function of inclusion number density. For the second model, the concentric-shell model, φ and s are evaluated in the scaled-particle approximation. The generalizations φ( R̂) and s( R̂), which are the volume fraction and specific surface available to a probe particle of finite radius R ̂ , are also derived and evaluated. For randomly placed spheres (ϵ = 0) these quantities are also considered for polydisperse media. An application to gel size-exclusion chromatography is considered. We find that varying the hardness parameter ϵ or the polydispersivity provides better agreement between theory and experiment than is obtained by using a monodisperse randomly placed sphere model of the gel. A comparison of the density expansions of φ and s for the permeable-sphere and concentric-shell models confirms that the considerable quantitative differences we find between these two conceptually distinct models are not simply the result of the approximation procedures that we use in evaluating their properties. Finally we introduce a new class of models, based on mixtures of spheres of different mutual interpenetrability. For one particular such model we obtain exact expressions for φ and s for general ϵ, as well as the two-point matrix function S 2(r) in the Percus—Yevick and Verlet—Weis approximations.

Active Microwave Remote Sensing of an Anisotropic Random Medium Layer

A two-layer anisotropic random medium model has been developed to study the active remote sensing of the Earth. The dyadic Green's function for a two-layer anisotropic medium is developed and used in conjunction with the first-order Born approximation to calculate the backscattering coefficients. It is shown that strong cross-polar-ization occurs in the single scattering process and is indispensable in the interpretation of radar measurements of sea ice at different frequencies, polarizations, and viewing angles. The effects of anisotropy on the angular responses of backscattering coefficients are also illustrated trated.

Porosity and specific surface for interpenetrable-sphere models of two-phase random media

We derive and numerically evaluate expressions for the porosity (matrix volume fraction) φ and the specific surface (interface area per unit volume) s for two specific models of a random two-phase medium in which the medium is considered as a suspension of interpenetrable spheres of radius R, embedded in a uniform matrix. The models and quantities considered have applications to a wide range of problems concerning transport, mechanical, and chemical properties of composite media. Both models contain a continuously variable hardness parameter ε, such that for ε=1 they reduce to mutually impenetrable spheres, and for ε=0 they reduce to fully penetrable spheres. The first of these models, the permeable-sphere model, has been defined only in the context of the Percus–Yevick approximation, which yields for it a unique pair distribution function g2(r1, r2). To find the associated gn(r1,...,rn) for n≳2, which are needed to evaluate φ and s we use the generalized superposition approximation. The second model, the concentric-shell model, can be fully defined without recourse to any particular approximation and proves to be isomorphic to the picture described by the scaled-particle theory of Reiss, Frisch, and Lebowitz. In this case we evaluate φ and s in the scaled-particle approximation introduced by those authors to implement the scaled-particle theory. For both models, we present numerical plots of φ vs dimensionless density, and of s vs φ. We also briefly discuss the relations between the results obtained in the two cases. Finally, we consider generalizations of φ and s that define the volume and surface available to a particle of finite size, rather than a point.

A Review of Tissue Characterization from Ultrasonic Scattering

Ultrasonic scattering is related to tissue architecture by a model which expresses the scattered wave pressure as the product of a frequency-dependent factor and the three-dimensional Fourier transform of a spatially windowed scattering function. The scattering function is the sum of the variations in compressibility and the variations in density, the latter being weighted by the cosine of the scattering angle. An analogous Fourier transform relation describes the average scattered intensity in terms of the correlation of the scattering function between points in a scattering volume. These Fourier transforms permit different scattering measurements to be related via trajectories in wavespace. Backscatter measurements made on blood, eye, liver, spleen, brain, and heart show the feasibility of differentiating tissues and determining spacing. Ultrasonic measurements of random medium model properties have demonstrated that average differential scattering cross sections per unit volume can be found accurately and precisely, and that scattering cross sections per unit volume can be used to make quantitative statements about changes in scattering properties. Measurements of calf liver ultrasonic differential and total scattering cross sections show that calf liver is a weak scatterer with nearly one-half of the total scattered power occurring between scattering angles of 25-45°. Measurements of scattering by fixed pig and human liver show qualitative agreement with predictions of scattering from architecture observed optically. Results presently accumulated indicate the promise of scattering measurements for tissue characterization, but extensive additional in vitro and in vivo development is required before their clinical utility for diagnosis is known.

Correlation function studies for snow and ice

In the interpretation of active and passive microwave remote sensing data, the random medium model has been used to characterize snow and ice fields. A correlation function is used to describe the random permittivity fluctuations with its associated mean and variance, and correlation lengths. Several snow and ice samples are studied in this paper, and the correlation functions are shown to be exponential in character with correlation lengths corresponding to the actual size of the ice particles in snow or air bubbles in ice.

Radiative transfer theory for passive microwave remote sensing of a two-layer random medium with cylindrical structures

In passive microwave remote sensing of earth terrain, the application of the radiative transfer theory to the model of random medium has been used in the interpretation of various experimental data. For vegetation fields with cylindrical structures, we model the vegetation layer as a two-layer random medium with a small correlation length lρ in the horizontal direction, and a large correlation length lz in the vertical direction. The closed form solutions for the brightness temperatures are obtained as lz approaches infinity. Using this model, the kernels in the scattering terms of the radiative transfer equations give rise to delta functions, indicating that forward scattering is dominant over all the other directions. For the case that lz is only a few wavelengths long, we find the first-order solution with an iterative method. The results are compared with that obtained using the Gaussian quadrature method to solve the radiative transfer equations numerically. We also apply this model to match experimental data collected from corn fields.

Computer simulation of ion scattering by polycrystals

Abstract The models of random media used at present in describing polycrystal scatteringcannot explain some experimental facts, such as the double-scattering peak in the energy spectra of reflected ions. Two models of simulation of ion interactions with polycrystals are proposed here, namely 1) the model of random filling of the coordinative spheres in which the polycrystal is represented by a set of singlecrystal coordinative spheres with arbitrary positions of atoms and the distances between atom may sometimes be smaller than those in single crystal, a fact that can be treated as the presence of an imbedded atom; 2) the model of arbitrary selection of the bombarded face in which the surface and incidence planes are selected randomly for each ion (or of a group of ions). In both models, the atoms are free, the Born-Mayer potential is used, and thermal vibrations of atom are included. A good agreement with experimental results has been obtained in both methods.

Active Remote Sensing of Layered Random Media

Analytical expressions for the backscattering cross sections have been derived for active remote sensing of an arbitrary number of random layers. These results are applicable to the interpretation of radar backscattering from vegetation and snow-ice fields. We illustrate the results by matching experimental data using the model of a three-layer random medium with air above and ground below.