Random Medium Model Research Articles

Application of Markov model of random medium for the interpretation of experimental results on muon intensity measurements at underground installations

Incomplete information on overburden rock composition is the source of systematic error arising in underground muon experiments. As was shown in our previous works this problem requires a probabilistical approach, namely the introduction of a statistical ensemble of media which includes all possible medium realizations. As an example of such an ensemble we proposed the Markov two-phase model of random medium and applied it to the KGF ground. In this paper we continue the discussion of this problem and consider further possibilities of application of this model for estimating the influence of ground properties uncertainties on experimental results.

Polymer dynamics in time-dependent Matheron-de Marsily flows: an exactly solvable model.

We introduce a model of random layered media, extending the Matheron-de Marsily model: Here we allow for the flows to change in time. For such layered structures, we solve exactly the equations of motion for single particles and also for polymers modelled as Rouse chains. The results show a rich variety of dynamical patterns.

Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Observations of seismic waves from earthquakes at depths between 100 and 200 km beneath Japan show that the initial portion of the S-wave arrival has greater duration than can be accounted for by earthquake source-time duration. The observed long duration of S waves has been explained as being caused by multiple forward scattering around the ray path between source and receiver. Array obser- vations of Lg waveforms have also shown that multiple forward scattering along the path between the source and receiver are important influences on the character of Lg waveforms and that the scattering cannot be explained only by vertical variations in velocity. Multiple forward scattering in 3D has been modeled using the Markov approximation for the parabolic-wave equation, which allows the calculation of seis- mogram envelopes in statistically characterized random media. To test the range of validity of the Markov approximation-derived solutions, we made 2D numerical calculations of wavefields in random media using approximations to the parabolic wave equation, which only models forward scattering, and by finite-difference so- lution of the scalar-wave equation, which gives complete wavefields. Media of back- ground velocity 4 km/sec are characterized using a Gaussian autocorrelation function with a5k mcorrelation distance and 5% rms fractional fluctuation. To compare with envelopes obtained from the Markov approximation, we calculated wavefields for source-receiver distances ranging from 50 to 200 km for several statistically identical realizations of random media. We obtain ensemble average envelopes by averaging envelopes from the realizations. We find a good agreement between ensemble- average envelopes obtained from the numerical solution of the parabolic-wave equa- tion and envelopes obtained from the Markov approximation. The later portion of the ensemble-average envelopes calculated using finite difference have larger am- plitudes than those from the Markov approximation, which is probably due to the late-arriving energy that has been scattered at wide angles from the global propa- gation direction between the source and the receiver. We observe that the variations among the numerically calculated envelopes of individual realizations of random media are well fit by a Rayleigh distribution, which describes the distribution of envelope amplitude when signals of one frequency but random phase are summed. Our results show that the Markov approximation provides reliable information about envelope shapes for forward-scattered wavefields but that the influence of wide-angle scattering and backscattering have some influences on envelope shapes and should be considered when analyzing data using random media models.

A hybrid experimental/theoretical scattering model for dense random media

The subject of scattering of electromagnetic waves by dense media has been one of intense interest in recent years. The present paper describes polarimetric backscatter measurements made at Ku-band on layers of a dense medium under very carefully controlled circumstances. The experiments have a dual purpose: 1) to evaluate the degree to which the experimental observations are predicted by theoretical, particle-based, random media models and 2) to test a proposed hybrid model by which the scattering and extinction properties of a dense medium are characterized experimentally, allowing future modeling of the polarimetric response for any arbitrary configuration of the medium. The hybrid model assumes that first-order vector radiative transfer (RT) is a suitable theoretical structure, providing that the extinction and phase matrix components are appropriately specified; the specification is accomplished through an inversion algorithm involving polarimetric backscatter measurements. The major conclusions of the study are the following: 1) hybrid model is an adequate description of the dense medium scattering behavior; 2) conventional RT appears to give a reasonable estimate of the observed radar response, but dense medium RT gives a very low estimate; 3) phase function of the effective volume scattering element of the medium, obtained via the hybrid model, suggests a larger effective scatterer than the physical ones.

Experimental Verification of a Random Medium Model for the Optical Behaviour of Ultrathin Crystalline Silicon Layers Grown on Porous Silicon

AbstractThe random medium model for porous silicon (PS) is applied to a porous silicon layer to be used as a back reflector in a thin film silicon solar cell realized in an ultrathin epitaxially grown Si layer on PS. Three-layer-structures (epi/PS/Si) have been fabricated by RPCVD of 150-1000 nm epitaxial silicon layers on the porous surface of silicon wafers. The light reflection has been measured in the 300-1000 nm wavelength range. An excellent agreement is found between the experimentally measured reflectance curves and those calculated using the random medium model. The analysis shows that the epitaxial growth has led to an appreciable reduction in the porosity in the intermediate PS layer from about 60% to 20-30%.

Lévy flight applied to random media problems

A random-medium model being the correlated distribution of points positioned in E-dimensional Euclidean space is considered. The construction of the medium starts from noncorrelated (Poisson) uniform distribution of parent particles; each of them initiates the finite Marcov chain of its descendants. The complete collection of correlation functions of all orders within the scope of the model has been obtained. The use of three-dimensional stable laws (Lévy laws) as transition probability density allows one to express the correlation functions of all orders in terms of two point ones only. Some numerical results are presented and discussed in connection with fractal structure simulation.

Local volume fraction fluctuations in random media

Although the volume fraction is a constant for a statistically homogeneous random medium, on a spatially local level it fluctuates. We study the full distribution of volume fraction within an observation window of finite size for models of random media. A formula due to Lu and Torquato for the standard deviation or “coarseness’’ associated with the local volume fraction ξ is extended for the nth moment of ξ for any n. The distribution function FL of the local volume fraction of five different model microstructures is evaluated using analytical and computer-simulation methods for a wide range of window sizes and overall volume fractions. On the line, we examine a system of fully penetrable rods and a system of totally impenetrable rods formed by random sequential addition (RSA). In the plane, we study RSA totally impenetrable disks and fully penetrable aligned squares. In three dimensions, we study fully penetrable aligned cubes. In the case of fully penetrable rods, we will also simplify and numerically invert a prior analytical result for the Laplace transform of FL. In all of these models, we show that, for sufficiently large window sizes, FL can be reasonably approximated by the normal distribution.

The exactly resolved nonlattice model of random media based on Markov walks with a stable law for jumps

A random-medium model which is a correlated distribution of points (particles) randomly positioned in the 3-dimensional space is considered. The construction of the medium starts from a noncorrelated (Poisson) distribution of parent particles, each of them initiates a finite Markov chain of its descendants. The complete collection of correlation functions of all orders within the scope of the model have been obtained. The use of the 3-dimensional stable law (Lévy law) as a transition probability allows us to present the correlation function in an explicit form.

Fractal nature of viscous fingering in three-dimensional pore-level models.

Using three-dimensional pore-level models of random porous media, we have studied viscous fingering in ``two-phase,'' miscible flow. When the viscosity ratio M (=${\mathrm{\ensuremath{\mu}}}_{\mathit{D}}$/${\mathrm{\ensuremath{\mu}}}_{\mathit{I}}$, the ratio of the viscosity of the displaced fluid to that of the injected fluid), is infinite, the flow is known to be modeled by diffusion-limited aggregation (DLA). We have observed and characterized the crossover from fractal flow at short times or large viscosity ratios to compact (i.e., Euclidean) flow at later times. In our pore-level model of three-dimensional flow in the limit of infinite capillary number (zero surface tension), the low viscosity fluid is injected at constant pressure on one face of the model porous medium. This modeling shows that this flow is fractal for large viscosity ratios (M=10 000), consistent with DLA. For realistic viscosities (M=30--1000), our modeling of the unstable flow shows that, although the flows are initially fractal, they become linear on a time scale \ensuremath{\tau}, increasing as \ensuremath{\tau}=${\mathrm{\ensuremath{\tau}}}_{0}$${\mathit{M}}^{0.16}$. This characteristic crossover time predicts that the flow become compact for patterns larger than a characteristic length, which increases with viscosity ratio as l=${\mathit{l}}_{0}$${\mathit{M}}_{\mathit{f}}^{0.16/(\mathit{D}}$-2), where ${\mathit{D}}_{\mathit{f}}$ is the fractal dimension. Once compact, the saturation front advances as x\ensuremath{\approxeq}${\mathit{v}}_{0}$${\mathit{M}}^{0.16}$t; the factor ${\mathit{M}}^{0.16}$ acts as a three-dimensional Koval factor. \textcopyright{} 1996 The American Physical Society.

Estimation of canopy water content in Konza Prairie grasslands using synthetic aperture radar measurements during FIFE

This paper presents the development of an algorithm to retrieve the canopy water content of natural grasslands and pasture from synthetic aperture radar (SAR) measurements. The development of this algorithm involves three interrelated steps: (1) calibration of SAR data for ground topographic variations, (2) development and validation of backscatter model for grass canopies, and (3) estimation of canopy water content by inverting a backscattered model for cross‐polarized ratio. The polarimetric radar data acquired by the Jet Propulsion Laboratory AIRSAR system during the 1989 First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) are used for this study. The SAR data have been calibrated and corrected for the topographical effects by using the digital elevation map of the study area. The backscattering coefficients obtained from the SAR data for each pixel are related to the canopy and soil parameters by employing a discrete random media model for vegetation. The model simulations indicate that biomass variations and surface treatments (burned and unburned) of grass canopies affect the C‐band backscatter signal but does not influence the L‐band signal. This model is then validated and adjusted over training areas where ground measurements were collected. An inversion technique is proposed to estimate the canopy water content by using the cross‐polarized and copolarized ratios of the SAR data at C band. The result of the inversion algorithm shows a good agreement with the grass biomass data collected during FIFE 1989 intensive field campaign.

Spectral analysis of randomly scattered signals using the wavelet transform

Estimation of power spectra is a central question in signal analysis and its applications such as Geophysics. This is classical for stationary signals for which the power spectral density, i.e. the Fourier transform of the covariance, is defined as a function of the frequency only. In the general case, signals under study are not stationary. This work has been motivated by signals obtained in the context of wave propagating in random media. A one-dimensional model of random media is given by a layered medium whose properties are random variables in each layer independent from one layer to another. It has been shown [M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, “Frequency content of randomly scattered signals”, SIAM Review 33, 526–629 (1991)], that if a pulse of typical wavelengths ϵ is sent in such a medium with small layers of order ϵ 2, the reflected signal is a non-stationary stochastic process which converges in distribution, as ϵ goes to 0, to a locally stationary Gaussian process. This convergence will be the key property in the estimation of the power spectrum for the reflected signal. We introduce an estimation of the local power spectrum using the wavelet transform. In our main theoretical result we identify the coefficients to be computed and we present the corresponding algorithm with numerical experiments. In the last section we show how this wavelet method enables us to get a fairly good estimation of the small parameter ϵ when this one is not known a priori.

Strong-fluctuation theory for a mean electromagnetic field in a statistically homogeneous random medium with arbitrary anisotropy of electrical and statistical properties

Most of the previously used strong-fluctuation theories exclusively deal with electrically isotropic random media. In this paper an extension of the method of changing the field variable is proposed to account for strong fluctuations of a statistically homogeneous random medium which possesses the most general anisotropy of the electrical and statistical properties. For the effective perturbation operator, a full perturbation series solution and a bilocal approximation in the long-wavelength limit are derived referring to a general anisotropic random medium. On this basis, the particular cases of random media characterized by ellipsoidal correlation functions with fixed axes of statistical symmetry as well as with randomly rotating symmetry axes and a random distribution of correlation lengths are considered. For a general model of random media, a coordinate-free solution to the mean-field Green's dyad problem in the spectral domain is described, and the nonlinear bilocal approximation for the effective perturbation operator is developed.

Solution of the kinetic equation for the deposited energy distribution in the power cross section model

A regular method for finding solutions of the equation for the distribution of energy deposited by atomic particles in elastic collisions is developed, using the model of an infinite random medium and power cross section. It is shown that in neglecting threshold energy the distribution derivative has a singularity at the target surface. The calculated distributions and other functions connected with the problem are given for the case of equal masses of the projectile and target atoms.

Random medium model approach to scattering from a random collection of discrete scatterers

The random medium model has been used quite successfully in the interpretation of active and passive microwave remote sensing data from earth terrain. In the random medium model, a correlation function with associated mean and variance is used to describe the random permittivity fluctuations. Most of the correlation functions used, thus far, in the random medium model have been of exponential form, or of other simple expressions derived from physical arguments. In this paper, we calculate the correlation function for a specific physical model of the random medium which is a random collection of discrete scatterers imbedded in a background medium of constant permittivity. Correlation functions are calculated for impenetrable and penetrable spherical scatterers. Once the correlation function is obtained, the strong fluctuation theory is used to calculate the effective permittivity. Then, the distorted Born approximation is used to calculate backscattering coefficients for a halfspace configuration. Theoretical results are illustrated by comparing effective permittivities and backscattering coefficients obtained from the penetrable and impenetrable models.

Born–Rytov controversy II Applications to nonlinear and stochastic scattering problems in one-dimensional half-space media

The Born and Rytov methods are applied to a study of the propagation and the scattering of waves in a one-dimensional (1-D) half-space medium with a permittivity either depending on the wave intensity or having a stochastic property. Both Born and Rytov series in the 1-D nonlinear half-space case are compared with their counterparts in a 1-D homogeneous dielectric half-space case. The multiple scattering in a 1-D random half-space, consisting of a host material and randomly distributed scatterers, is treated with a random medium model. We have derived the autocorrelation function for the case in which 1-D scatterers are randomly distributed in a host medium. We show that the autocorrelation function has an exponential-decay shape and depends on the fractional length, the average physical length, and the scattering strength of these scatterers. Under the bilocal approximation the effective permittivity for this 1-D random half-space medium is shown to have an oscillatory dependence on range. When it is applied to estimate the ensemble-averaged scattered fields in the 1-D random half-space case, the Rytov method is not as useful as the Born method because the Rytov method suffers a more serious divergence problem than the Born method.

Fluctuation analysis of elastic waves in random media via phase screen simulations

Phase screen simulations of vector wave propagation in elastic random media are applied to two independent studies of seismic wave scattering. In the first study, transmission fluctuations are analyzed for incident P waves, propagated through a two‐dimensional version of the two‐layer model of Flatté and Wu (1988) of the crust and upper mantle beneath NORSAR. Numerically simulated coherence functions for vector elastic waves are compared to analytic and simulated results based on the parabolic approximation to the scalar wave equation. The usefulness of phase screen simulations for analyzing fluctuations for moderately strong scattering (∼ 2%–5% perturbations), and for three‐component data, is demonstrated. In the second study, effects of multi‐scale random structure on the variances of the initial P wave peak‐to‐peak amplitude and the rms amplitude of the transverse component in the velocity window between the average P and S wave speeds are investigated. A large number of synthetic seismograms are generated for eight random media models. This study shows that the relative variance of the rms scattered phase is far less dependent on the model than that of the direct phase. The variance of the forward scattered P wave depends greatly on the strength of the large‐scale heterogeneities.

Experimental characterisation of UHF radiowave propagation through forests

An experimental measurements programme for characterisation of radiowave propagation at UHF band is discussed. Preliminary results on transmission loss, specific attenuation and delay spread are presented and discussed in terms of the biophysical and radio parameters that characterise the propagation path. The results are compared with predictions based on discrete random medium model.

Variance of Phase Fluctuations of Waves Propagating through a Random Medium

As an electromagnetic wave propagates through a random scattering medium, such as a forest, its energy is attenuated and random phase fluctuations are induced. The magnitude of the random phase fluctuations induced is important in estimating how well a Synthetic Aperture Radar (SAR) can image objects within the scattering medium. The two-layer random medium model, consisting of a scattering layer between free space and ground, is used to calculate the variance of the phase fluctuations induced between a transmitter located above the random medium and a receiver located below the random medium. The scattering properties of the random medium are characterized by a correlation function of the random permittivity fluctuations. The effective permittivity of the random medium is first calculated using the strong fluctuation theory, which accounts for large permittivity fluctuations of the scatterers. The distorted Born approximation is used to calculate the first-order scattered field. A perturbation series for the phase of the received field in the Rytov approximation is then introduced and the variance of the phase fluctuations is also calculated assuming that the transmitter and receiver are in the paraxial limit of the random medium, which allows an analytic solution to be obtained. Results are compared using the paraxial approximation, scalar Green's function formulation, and dyadic Green's function formulation. The effects studied are the dependence of the variance of the phase fluctuations on receiver location in lossy and lossless regions, medium thickness, correlation length and fractional volume of scatterers, depolarization of the incident wave, ground layer permittivity, angle of incidence, and polarization.

Nonlinear receiver compression effects on the amplitude distribution of backscattered ultrasonic signals

Nonlinear receiver compression effects on the amplitude distribution of backscattered ultrasonic signals are investigated by using digitized RF signals that have been compressed in a commercially made ultrasonic B-scan imaging instrument. Amplitude distributions of compressed RF and video signals were obtained from regions of B-scan images that correspond to approximately the same physical region in a random medium model with known backscatter amplitude characteristics. The amplitude distribution of the signal before compression was obtained by using a table constructed from measurements of the imaging instrument compression characteristics as a function of time gain compensation. While the results indicate the general form of the decompressed data agrees with single parameter model curves that are predicted by a widely employed Gaussian random process model, the signal-to-noise ratios of the decompressed envelope vary up to 20% from the 1.91 value predicted that model. This implies that effects such as nonlinearities, envelope smoothing, and noise which all may be present in varying degrees in practical ultrasonic imaging instrumentation can cause appreciable departures from theoretical data even under highly controlled conditions.

Laboratory measurements of spatial fluctuation and attenuation of elastic waves by scattering due to random heterogeneities

To study the effects of strong scattering on elastic waves, spatial fluctuation and scattering attenuation ofP waves were examined by laboratory experiments for 2-D models of random media approximately characterized by a triangular correlation function in the range of 2 10, both the intensity and the correlation distance of the amplitude fluctuation are approximate for any phase of theP-wave train. The correlation distance nearly agrees with the heterogeneity size. These fluctuation properties are quite consistent with the theoretical prediction by the forward-scattering approximation. (2) For 3 10 and 3 15° is lost from theP waves, while the energy scattered in the range of θ<15° is retained; and that the approximate theory overestimates by about three times the σ value of the model media used owing to the neglect of multiple scattering. (6) When the size of velocity heterogeneities responsible for forward scattering at 3<ka<6 is estimated from the θmin value of 15° on the basis of Wu's theory, it nearly agrees with the correlation distance for the initial phase of theP-wave train.