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We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto. [ABSTRACT FROM AUTHOR]

We present an analysis of enhanced wave transmission through random media with mirror symmetry about a reflecting barrier. The mathematical model is the acoustic wave equation, and we consider two setups, where the wave propagation is along a preferred direction: in a randomly layered medium and in a randomly perturbed waveguide. We use the asymptotic stochastic theory of wave propagation in random media to characterize the statistical moments of the frequency-dependent random transmission and reflection coefficients, which are scalar-valued in layered media and matrix-valued in waveguides. With these moments, we can quantify explicitly the enhancement of the net mean transmitted intensity, induced by wave interference near the barrier. [ABSTRACT FROM AUTHOR]

When a wave, such as sound or light, scatters within a densely packed particulate, it can be rescattered many times between the particles, which is called multiple scattering. Multiple scattering can be unavoidable when trying to use sound waves to measure a dense particulate, such as a composite with reinforcing fibres. Here, we solve from first principles multiple scattering of scalar waves, including acoustic, for any frequency from a set of two-dimensional particles confined in a circular area. This case has not been solved yet, and its solution is important to perform numerical validation, as particles within a cylinder require only a finite number of particles to perform direct numerical simulations. The method we use involves ensemble averaging over particle configurations, which leads us to deduce an effective T-matrix for the whole cylinder, which can be used to easily describe the scattering from any incident wave. In the specific case when the particles are monopole scatterers, the expression of this effective T-matrix simplifies and reduces to the T-matrix of a homogeneous cylinder with an effective wavenumber k⋆. To validate our theoretical predictions, we develop an efficient Monte Carlo method and conclude that our theoretical predictions are highly accurate for a broad range of frequencies. [ABSTRACT FROM AUTHOR]

Microwave remote sensing is significantly altered when passing through clouds or dense ice. This phenomenon is not unique to microwaves; for instance, ultrasound is also disrupted when traversing through heterogeneous tissues. Understanding the average transmission in particle-filled environments is central to improve data extraction or even to create materials that can selectively block or absorb certain wave frequencies. Most methods that calculate the average transmitted field assume that it satisfies a wave equation with a complex effective wavenumber. However, recent theoretical work has predicted more than one effective wave propagating even in a material which is statistically isotropic and for scalar waves. In this work we provide the first clear evidence of these predicted multiple effective waves by using high-fidelity Monte-Carlo simulations that do not make any statistical assumptions. To achieve this, it was necessary to fill in a missing link in the theory for particulate materials: we prove that the incident wave does not propagate within the material, which is usually taken as an assumption called the Ewald–Oseen extinction theorem. By proving this we conclude that the extinction length—the distance it takes for the incident wave to be extinct—is equal to the correlation length between the particles. [ABSTRACT FROM AUTHOR]

We theoretically investigate the scaling behavior of the localization length for s-polarized electromagnetic waves incident at a critical angle on stratified random media with short-range correlated disorder. By employing the invariant embedding method, extended to waves in correlated random media, and utilizing the Shapiro–Loginov formula of differentiation, we accurately compute the localization length ξ of s waves incident obliquely on stratified random media that exhibit short-range correlated dichotomous randomness in the dielectric permittivity. The random component of the permittivity is characterized by the disorder strength parameter σ2 and the disorder correlation length lc. Away from the critical angle, ξ depends on these parameters independently. However, precisely at the critical angle, we discover that for waves with wavenumber k, kξ depends on the single parameter klcσ2, satisfying a universal equation kξ≈1.3717klcσ2−1/3 across the entire range of parameter values. Additionally, we find that ξ scales as λ4/3 for the entire range of the wavelength λ, regardless of the values of σ2 and lc. We demonstrate that under sufficiently strong disorder, the scaling behavior of the localization length for all other incident angles converges to that for the critical incidence.

There has not been a satisfying numerical validation of the theory of effective waves in random particulate materials. Validation has been challenging because the theoretical methods for effective waves have been limited to random particulate media in infinite slabs or half-spaces, which require a very large number of particles to perform accurate numerical simulations. This paper offers a solution by providing, from first principles, a method to calculate effective waves for a sphere filled with particles for a spherically symmetric incident wave. We show that this case can excite exactly the same effective wavenumbers, which are the most important feature to validate for effective waves. To check correctness, we also deduce an integral equation method which does not assume the effective wave solution. Our methods are, in principal, valid for any frequency, particle volume fraction and disordered pair-correlation. With the methods we provide, it is now possible to validate, with a heavier Monte Carlo simulation, the predictions from effective wave theory. [ABSTRACT FROM AUTHOR]

chechan We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrm of a high-contrast random operator. We consider a family of elliptic operators Aε in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator Aε, we show that the point spectrum of A\ε converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator Ahom as. Furthermore, we prove that the eigenfunctions of A\ε decay exponentially at infinity uniformly for sufficiently small \varepsilon. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of hom. [ABSTRACT FROM AUTHOR]

Random granular media can exhibit characteristics that are often related to ordered media. In the present work, this feature is observed in the polarized Second Harmonic Generation (SHG) response from reduced iron‐doped lithium niobate (LN:Fe) powders, which is an unexpected effect due to multiple scattering. In addition, the subsisting‐order properties of the powders can be further controlled by magnetic induction to tailor the SHG response. The samples are characterized by X‐ray Diffraction (XRD), X‐ray Photoelectron Spectroscopy (XPS), and confocal Raman Spectroscopy. The SHG response in the absence and presence of an external static magnetic field is then studied as the fundamental beam focus is translated from air into the powder. The SHG intensity polarization state is studied as a function of the linear polarization of the fundamental beam at the focus depth position, where the maximum SHG recorded intensity is observed. These results demonstrate that the SHG response of LN:Fe powders can be modified by post‐thermal treatment in a reducing atmosphere for photonic applications. [ABSTRACT FROM AUTHOR]

A theory for the characterization of the fourth-order moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian–Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the Itô-Schrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth-order moment equations are considered. The general fourth-order moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the second-order wave moment. The fourth-order moment results for polarized waves are used in an application for imaging of partially coherent sources. [ABSTRACT FROM AUTHOR]

In this paper, we carried out a critical investigation of the strong fluctuation theory using a quantum field theoretic approach [Tatarskii VI. The effects of the turbulent atmosphere on wave propagation. Jerusalem: IPST; 1971]. We employed the Dyson and Bethe-Salpeter equations to derive the radiative transfer equation (RTE) for the Green's function of the radiant intensity. We proceeded to obtain estimates of the domain of validity of the RTE. The results thus obtained satisfy both the optical theorem for waves in random media and energy conservation. We also derived the RTE for our problem by employing a recently developed scale-separated asymptotic theory [Bal G., Komorowski T, Ryzhik L. Kinetic limits of waves in random media. Kinet Relat Models. 2010;3:529–644]. Although the RTE obtained by the two approaches are identical, we explain that the applicability regimes, and hence validity conditions are different. [ABSTRACT FROM AUTHOR]

Extensive research has been reported on MIMO systems for complex environments. It is noted that many communications studies make use of the channel matrix of transmitters and receivers, signal vectors, SNR, and eigenvalues. However, the channel matrix is often expressed in terms of channel modeling, and the detailed statistical characteristics of the channel in terms of physical characteristics of atmospheric turbulence, particulate matter, and rough surfaces and terrain are often not explicitly shown. This paper discusses the inclusion of expressions of physical characteristics of the random medium in the channel matrix. Therefore, the key point of this paper is that the channel matrix is explicitly given in terms of the actual physical parameters of the random environment using the mutual coherence function and stochastic Green's functions; we call this a 'physics-based' communications model. This paper presents a formulation of physics-based communications models in the presence of a random medium based on analytical statistical EM theory. We use the keyhole MIMO as an example to show the key points of this approach. It is hoped that this study may help extend channel modeling to include analytical statistical EM theory for atmospheric turbulence, ocean turbulence, and rough terrain. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of determining the field of a reflected wave in a multiscale inhomogeneous medium, which consists of a background deterministic large-scale irregularity and multiscale random irregularities with scales both larger and smaller than the wavelength. A layer with a linear permittivity profile is taken as the background irregularity. The small-scale irregularities are responsible for wide-angle scattering including backscattering. The first approximation of the perturbation theory is used to account for scattering from these irregularities. As the zero approximation and Green's function in determining the field backscattered by small-scale irregularities, we utilize the integral representation of the field, obtained in our earlier work by combining the method of double weighted Fourier transform (DWFT) and the Fock proper-time method. Asymptotic methods are used to reduce the sevenfold integral representation of a field to lower-order integrals. Conditions for validity of such representations are obtained. Formulas of the frequency coherence functions of waves reflected and backscattered from the turbulent plasma layer are given. The paper presents the results of the simulation of pulse sounding of a randomly inhomogeneous reflecting plasma layer demonstrating the effect of large-scale irregularities on backscattering by small-scale irregularities. [ABSTRACT FROM AUTHOR]

The Markov approximation is widely used in wave propagation in random media. This approximation is valid if the propagation path length is greater than the scale of the medium inhomogeneities affecting a particular statistical moment of a wave field and the moment changes insignificantly over this scale. These conditions might be violated for the variance of the phase fluctuations and other statistical moments of acoustic signals that have propagated through atmospheric turbulence: the scale of the largest eddies can be hundreds of meters, and fluctuations in the acoustic refractive index are relatively strong. In the current article, the phase variance of a spherical sound wave in statistically inhomogeneous turbulence is formulated without the Markov approximation. For propagation ranges smaller than the scale of the largest eddies, the phase variance without the Markov approximation is significantly smaller than when this approximation is employed. As the range increases, the difference between the two results tends toward a constant value (a 'memory' effect), which might be significant in many applications. The phase variance without the Markov approximation agrees better with the experimental data on sound propagation through the atmosphere, while the variance calculated with this approximation significantly overpredicts the data. [ABSTRACT FROM AUTHOR]

This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O'Doherty-Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers-Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations. [ABSTRACT FROM AUTHOR]

Abstract Random granular media can exhibit characteristics that are often related to ordered media. In the present work, this feature is observed in the polarized Second Harmonic Generation (SHG) response from reduced iron‐doped lithium niobate (LN:Fe) powders, which is an unexpected effect due to multiple scattering. In addition, the subsisting‐order properties of the powders can be further controlled by magnetic induction to tailor the SHG response. The samples are characterized by X‐ray Diffraction (XRD), X‐ray Photoelectron Spectroscopy (XPS), and confocal Raman Spectroscopy. The SHG response in the absence and presence of an external static magnetic field is then studied as the fundamental beam focus is translated from air into the powder. The SHG intensity polarization state is studied as a function of the linear polarization of the fundamental beam at the focus depth position, where the maximum SHG recorded intensity is observed. These results demonstrate that the SHG response of LN:Fe powders can be modified by post‐thermal treatment in a reducing atmosphere for photonic applications.

A comparative analysis of two algorithms ("by particles" and "by collisions") for estimation of the weighted mean particle flux is made on the basis of a test problem solving for a single-speed particle propagation process with isotropic scattering and multiplication in a random medium. It is shown that the first algorithm is preferable for a simple estimation of the mean flux and the second one fits better for assessment of the parameters of a possible superexponential growth of the flux. Two models of the random medium are considered: a chaotic "Voronoi mosaic" and a "spherically layered mosaic." For a fixed mean correlation radius, the superexponential growth turned out to be stronger for the layered mosaic. [ABSTRACT FROM AUTHOR]

Wave propagation in random media has broad applications in materials science and engineering. In this paper, we develop a quasi Monte Carlo (qMC)-based model reduction method for solving random Helmholtz equations. In the physical space, we construct multiscale reduced basis functions by using an optimization method together with the proper orthogonal decomposition method. Then, in the random space we employ the qMC method for discretization. Under mild conditions, we prove that the spatial grid size is only proportional to the wave number, and almost a first-order convergence rate is achieved in the random space with respect to the number of samples. Since the exact solution oscillates in both physical and random spaces, our approach provides an efficient strategy to find its numerical approximation. One significant advantage of our approach over existing methods is its applicability to generic random media which cannot be treated as random perturbations of homogeneous media. These are confirmed by a series of numerical examples. [ABSTRACT FROM AUTHOR]

This article presents an analysis of P-band radio signal propagation through forest based on 3D Numerical Method of Maxwell's equations (NMM3D). The simulation domain of the forested region covers an area of 35.8 × 35.8 m2 that includes 112–121 coniferous trees with a height in the range of 8–16 m. The modeled trees have primary and secondary branches to account for their impact on scattering and attenuation at P-band. The fast hybrid method (FHM) has been successfully developed to complete the NMM3D full wave simulations of trees with efficient use of CPU time and memory. The accuracy of the proposed FHM model is validated by comparing scattered fields to that of FEKO for the case with nine trees. To account for the seasonal variation of water content in trees and hence dielectric constant, the wave scattering model is applied to a range of dielectric constants. Analysis of the P-band signal propagation is performed by computing the average loss and correlation of the transmission fields. The results show that the transmissivities are about 0.9 for winter conditions (low dielectric constant) and about 0.7 for summer conditions (high dielectric constant). Simulations with randomly positioned trees (off the periodic grid) demonstrate that the transmissivity shows a maximum variation of 4.3% relative to the nonrandomly positioned trees. The correlation analysis of P-band radio signals with two different dielectric constants indicates a high correlation amplitude above 0.89 and a phase change of less than 6°. A sensitivity analysis is carried out for a range of tree heights and trunk radii.

Precipitation and in particular rain present a random medium with constituent particles having dimensions comparable to the wavelength at millimeter-wave frequencies. Due to considerable scattering of waves from such particles, significant signal attenuation and phase front aberration can take place which are important factors that must be considered in modern communication system or radar design. This paper presents a fast and accurate numerical method for phase coherent modeling of wave propagation and scattering in random media, like rain. In this approach, the random medium is divided into sufficiently large finite slabs and each slab is modeled as a network with multiple input/output ports where ports represent rays with different polarizations propagating in different directions entering and leaving the slabs. The cascaded networks of slabs can be used to find the response of the random medium to any arbitrary source. The proposed method considers multiple scattering among all scatterers and thus the entire physics of wave-particle interactions is accounted for. In addition, the method can model both the mean and the fluctuating parts of waves in the random medium. To demonstrate the applicability of the proposed method, it is used for the calculation of the specific attenuation in rain media as a function of rain rates and frequency within the millimeter-wave band. The results show a good match with the available rain attenuation data in the literature.

Microwave remote sensing is significantly altered when passing through clouds or dense ice. This phenomenon is not unique to microwaves; for instance, ultrasound is also disrupted when traversing through heterogeneous tissues. Understanding the average transmission in particle-filled environments is central to improve data extraction or even to create materials that can selectively block or absorb certain wave frequencies. Most methods that calculate the average transmitted field assume that it satisfies a wave equation with a complex effective wavenumber. However, recent theoretical work has predicted more than one effective wave propagating even in a material which is statistically isotropic and for scalar waves. In this work we provide the first clear evidence of these predicted multiple effective waves by using high-fidelity Monte-Carlo simulations that do not make any statistical assumptions. To achieve this, it was necessary to fill in a missing link in the theory for particulate materials: we prove that the incident wave does not propagate within the material, which is usually taken as an assumption called the Ewald–Oseen extinction theorem. By proving this we conclude that the extinction length—the distance it takes for the incident wave to be extinct—is equal to the correlation length between the particles.

The full-wave simulations of 3D Maxwell equations of forests at L-band are carried out with a two-step hybrid method. The tallest trees considered include one 8-meter tall trunk, 18 primary branches and 108 secondary branches. In the first step, the T-matrix of a single tree in the representation of vector cylindrical waves (VCW) is calculated by using the far-field scattering amplitudes obtained from the commercial solver FEKO. In the second step, the wave multiple scattering theory (W-MST) as represented by the Foldy-Lax equations is used to account for the wave interactions among different trees. The accuracy of the hybrid method is validated by showing that the computed solutions of three trees are in good agreement with those solved from FEKO. Next the hybrid method is used for Monte Carlo simulations of forest stands with up to 30 trees. The dependences of the solutions on tree heights and tree densities are illustrated. The results show that the full-wave based transmissivities are much larger than those obtained from the radiative transfer equation (RTE) model.

We study how atmospheric turbulence affects twisted space-time beams, which are non-stationary random optical fields whose space and time dimensions are coupled with a stochastic twist. Applying the extended Huygens–Fresnel principle, we derive the mutual coherence function of a twisted space-time beam after propagating a distance z through atmospheric turbulence of arbitrary strength. We specialize the result to derive the ensemble-averaged irradiance and discuss how turbulence affects the beam’s spatial size, pulse width, and space-time twist. Lastly, we generate, in simulation, twisted space-time beam field realizations and propagate them through atmospheric phase screens to validate our analysis.

The derivation of the probability density of irradiance for the received electromagnetic wave propagating through random inhomogeneities has applications in many diverse fields, such as optical wireless communications, radar, and imaging systems, to name few. The conventional approach in this regard is to combine distribution models, with each being found valid under different conditions, in a modulation process, e.g., the extended Rytov theory. However, in these techniques, the nature of the physical processes responsible for the randomness is not considered directly. Moreover, it is not clear how the model parameters are related to the physical environment. Recent research studies found that in random media, such as underwater, the exact physical processes in effect can significantly alter the distribution of the received signal intensity. In this article, a novel intensity distribution model is derived based on Rytov theory of propagation through weak turbulence. The physical parameters of the medium are directly included into the model parameters. Therefore, the resultant model is general enough to describe different physical environments. The utility of the model is validated by studying the effect of the medium parameters on the distribution through simulations. [ABSTRACT FROM AUTHOR]

Optical phase is a key information carrier in biomedical imaging and astronomical observation. However, it is often obscured by heterogeneous and scattering media, rendering optical phase imaging an utmost challenge. Limited by the memory effect or confinement, current methods have difficulties in retrieving phases at different depths. To address this challenge, a speckle three‐dimensional reconstruction network (STRN) is developed to solve the inverse problem of the scattering process. STRN is featured by stacked convolutional filters to distinguish the depth of each image from the spatially overlapped speckles. From single‐shot, reference‐free, and scanning‐free speckle patterns, STRN simultaneously extracts multiple phase objects at different depths with high fidelity. This new approach breaks the limitations of conventional approaches for phase reconstruction through scattering media and paves a novel avenue for optical communication and biological endoscopy applications. [ABSTRACT FROM AUTHOR]

Increasing the frequency range of physics-based predictions of earthquake ground motions requires to account for small-scale heterogeneities, which can only be described in a stochastic way. Although many studies have addressed the impact of random heterogeneities on ground motion intensity parameters obtained by numerical simulation, very few have verified the accuracy of their numerical solutions or controlled the scattering regime they were simulating. Here we present a comprehensive analysis of SH wave propagation in 2D random media which covers a broad range of propagation regimes from ballistic to diffusive. The coherent and incoherent parts of the wavefield are examined independently. The random media consist in correlated density and velocity fluctuations described by a von Kármán autocorrelation function with a Hurst coefficient of 0.25 and a correlation length a = 500 m. The Birch correlation coefficient which relates density to velocity fluctuations takes 4 possible values between 0.5 and 1, and the standard deviation of the perturbations is either 5% or 10%. Spectral element simulations of SH wave propagation excited by a plane wave are performed for normalized wavenumbers (ka) up to 5. Analysis of the decay of the coherent wave amplitude, obtained through different averaging procedures, allows for a direct measure of the scattering attenuation, which we successfully compare with the predictions of the Dyson mean field theory. We also present the comparison between the energy envelopes measured from the synthetics and their theoretical counterpart provided by the radiative transfer theory and the diffusion approximation. Excellent agreement is found between numerical simulations and theoretical predictions of radiative transfer theory for the mean intensity. The numerical study highlights the difference of attenuation length between the mean field and the mean intensity. In the forward scattering regime, the peak intensity appears to decay exponentially over a length scale known as the transport mean free path. Furthermore, the fluctuations of intensity in the ballistic peak exhibit a transition from Log-normal to Exponential statistics. This transition occurs for a propagation distance of the order of the mean free path, which offers an alternative method of estimating this parameter.

We consider a dilute lattice obtained from the usual Z 3 lattice by removing independently each of its columns with probability 1 − ρ. In the remaining dilute lattice independent Bernoulli bond percolation with parameter p is performed. Let ρ ↦ p c (ρ) be the critical curve which divides the subcritical and supercritical phases. We study the behavior of this curve near the disconnection threshold ρ c = p c site (Z 2) and prove that, uniformly over ρ it remains strictly below 1 / 2 (the critical point for bond percolation on the square lattice Z 2). [ABSTRACT FROM AUTHOR]

We investigate the accuracy of the recently proposed nonclassical transport equation. This equation contains an extra independent variable compared to the classical transport equation (the path length s), and models particle transport in homogenized random media in which the distance to collision of a particle is not exponentially distributed. To solve the nonclassical equation, one needs to know the s-dependent ensemble-averaged total cross section Σt(μ, s) or its corresponding path-length distribution function p(μ, s). We consider a one-dimensional (1-D) spatially periodic system consisting of alternating solid and void layers, randomly placed along the x-axis. We obtain an analytical expression for p(μ, s) and use this result to compute the corresponding Σt(μ, s). Then, we proceed to solve numerically the nonclassical equation for different test problems in rod geometry; that is, particles can move only in the directions μ = ±1. To assess the accuracy of these solutions, we produce benchmark results obtained by (i) generating a large number of physical realizations of the system, (ii) numerically solving the transport equation in each realization, and (iii) ensemble-averaging the solutions over all physical realizations. We show that the numerical results validate the nonclassical model; the solutions obtained with the nonclassical equation accurately estimate the ensemble-averaged scalar flux in this 1-D random periodic system, greatly outperforming the widely used atomic mix model in most problems.

Experimental data on the narrowing of fluorescence spectra of laser-pumped random media are interpreted within the framework of a probabilistic model of incoherent amplification of the stimulated component of fluorescence. R6G- and DCM-doped layers of dense-packed anatase and anatase-rutile particles and suspensions of DCM crystallites were examined under pulse-periodic laser pumping with various pump intensities at the wavelength of 532 nm. It was shown that, in the case of quasi-stationary fluorescence response at high pump intensities, amplification of stochastically propagating partial components of fluorescence field at various wavelengths is governed by the pump-independent factor of fluorescence amplification and the average number of stimulated emission acts in the pumped system. Analysis of the experimental data has shown that, despite certain simplifying assumptions, the considered model reasonably describes evolution of the fluorescence spectra of random fluorescent media with the increase in the pump intensity. A procedure of recovery of the fluorescence amplification factor from the sets of fluorescence spectra obtained at various pump intensities is considered. Features of the spectral shapes of the recovered amplification factors caused by the influence of the scattering matrices in pumped media are discussed. • Fluorescence enhancement in random media is analyzed applying a probabilistic model. • The key model parameter is the wavelength-dependent fluorescence amplification factor. • The amplification factor is highly sensitive to fluorophore-matrix interactions. • Enhancement is limited by the saturated average number of stimulated emission acts. [ABSTRACT FROM AUTHOR]

The uncertainty of fuel debris criticality can be estimated by Monte Carlo criticality calculations repeated over independent replicas of some properly modelled random medium. Incomplete randomized Weierstrass function (IRWF) is a modelling framework for such calculations. Under these approaches at hand, the theme of this paper is how to efficiently analyse extreme realizations of neutron effective multiplication factor (k eff) over IRWF random media replicas. To this end, a new bounded amplification (BA) technique is applied to IRWF. The numerical results clearly indicate that the BA-applied IRWF reduces a required number of random media replicas at least by an order of magnitude. To rigorously validate this efficiency gain, generalized extreme value (GEV) analysis is applied to a data set of k eff values obtained without applying BA. It turns out that the extreme values of these k eff values follow the Weibull distribution. Therefore, the theory of GEV guarantees the existence of an upper limit for these k eff values, and the actually computed upper limit is indeed smaller than or equal to the top two k eff values obtained from an order-of-magnitude reduced number of BA-applied IRWF random media replicas. This means that the efficiency gain brought by BA has been confirmed by the GEV methodology. In addition, a method of rejecting spurious estimation values due to statistical fluctuation is demonstrated concerning the extreme value index in the GEV analysis. • A new bounded amplification (BA) technique is introduced in random media criticality calculation by Monte Carlo. • BA increases computational efficiency by at least an order of magnitude. • This efficiency gain is confirmed by generalized extreme value analysis. [ABSTRACT FROM AUTHOR]

We theoretically investigate the scaling behavior of the localization length for s -polarized electromagnetic waves incident at a critical angle on stratified random media with short-range correlated disorder. By employing the invariant embedding method, extended to waves in correlated random media, and utilizing the Shapiro–Loginov formula of differentiation, we accurately compute the localization length ξ of s waves incident obliquely on stratified random media that exhibit short-range correlated dichotomous randomness in the dielectric permittivity. The random component of the permittivity is characterized by the disorder strength parameter σ 2 and the disorder correlation length l c. Away from the critical angle, ξ depends on these parameters independently. However, precisely at the critical angle, we discover that for waves with wavenumber k , k ξ depends on the single parameter k l c σ 2 , satisfying a universal equation k ξ ≈ 1. 3717 k l c σ 2 − 1 / 3 across the entire range of parameter values. Additionally, we find that ξ scales as λ 4 / 3 for the entire range of the wavelength λ , regardless of the values of σ 2 and l c. We demonstrate that under sufficiently strong disorder, the scaling behavior of the localization length for all other incident angles converges to that for the critical incidence. • The scaling behavior of the localization length for electromagnetic waves is studied. • Waves incident at a critical angle on stratified random media are considered. • Universal power-law scaling independent of microscopic parameters is obtained. • The universality of the scaling for all short-range correlated disorders arises. [ABSTRACT FROM AUTHOR]

Multiple optical scattering occurs when light propagates in a non-uniform medium. During the multiple scattering, images were distorted and the spatial information they carried became scrambled. However, the image information is not lost but presents in the form of speckle patterns (SPs). In this study, we built up an optical random scattering system based on an liquid crystal display (LCD) and an RGB laser source. We found that the image classification can be improved by the help of random scattering which is considered as a feedforward neural network to extracts features from image. Along with the ridge classification deployed on computer, we achieved excellent classification accuracy higher than 94%, for a variety of data sets covering medical, agricultural, environmental protection and other fields. In addition, the proposed optical scattering system has the advantages of high speed, low power consumption, and miniaturization, which is suitable for deploying in edge computing applications.

The propagation of longitudinal and shear elastic waves through a multi-phase material was studied and the effective elastic properties of the medium were evaluated. The distribution of the reinforcing inclusions was considered random throughout the matrix. The effective dynamic properties of the composites, including their effective bulk and shear moduli and effective densities, were examined along with the effective phase velocity and attenuation of the incident P and S waves. The Sabina-Willis model was employed to study the wave propagation behavior, and the model performance was analyzed through comparison with experimental data from the literature. The results indicated that wave propagation significantly depended on the physical and mechanical properties of inclusions relative to those of the matrix and the normalized wave number of the propagated elastic wave. Moreover, despite the fact that the elastic properties of the incidence in the P and S waves exhibited a similar trend, their values differed significantly. The results can serve as a design criterion for composite materials under dynamic loading. [ABSTRACT FROM AUTHOR]

In this article, we study wave propagation in vegetation at multiple frequencies using a hybrid method. First, the multiple scattering within a single plant is captured using the high-frequency structure simulator (HFSS) for extracting the T-matrix of the plant in vector cylindrical waves (VCW). Second, the Foldy–Lax equations (FLE) are applied with the extracted T-matrices to consider the multiple scattering among different plants. The accuracy of the hybrid method is verified by comparing the scattering results from nine wheat plants with direct HFSS simulations. The hybrid method is implemented with parallel computing using a physical iterative method to perform full-wave Monte Carlo simulations of a wheat field with up to 169 plants at $L$ -, $S$ -, and $C$ -bands. The impact of the gaps and the plant structure on the microwave propagations are studied by assessing the scattered fields and the resulting nonuniform transmission at different regions. The results are compared with modeling results using the classical radiative transfer equations (RTEs), and two major differences are observed: 1) the transmission calculated from the hybrid method is much larger than that of the RTE and 2) the full-wave simulation results show much weaker frequency dependence than RTE with saturation as frequency increases. [ABSTRACT FROM AUTHOR]

This article studies the existence and non-existence of transition fronts for a two species competition lattice system in random media, and explores the influence of randomness of the media on the wave profiles and wave speeds of such transition fronts. We first establish comparison principle for sub-solutions and super-solutions of the related cooperative system. Next, under some proper assumptions, we construct appropriate sub-solutions and super-solutions for the cooperative system. Finally, we show that random transition fronts exist if their least mean speed is greater than an explicit threshold and there is no random transition front with least mean speed less than the threshold.

Abstract In order to investigate the small-scale scattering heterogeneities underneath the northern Tien Shan, we analyze the P wavefield from teleseismic events. By using the teleseismic fluctuation method, we separate the total wavefield into coherent and fluctuating parts in the frequency band of 0.1–8.0 Hz. Subsequently, we investigate the scattering characteristics by analyzing the frequency-dependent intensities of the coherent and fluctuating wavefield between 0.3 and 2.5 Hz. We further constrain the velocity perturbations and correlation lengths by modeling the P-wave coda envelope with the Monte Carlo simulation. Strong scattering heterogeneities are revealed beneath the northern Tien Shan. The preferred scattering model can be described as a ~ 55- to 130-km-thick randomly heterogeneous layer with velocity perturbations of 6–9% and correlation lengths on the order of 0.4 km. We attribute these small-scale scatterers to isolated melt pockets from the upwelling hot mantle materials.

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. We consider the behavior of compressible two-phase flow in heterogeneous reservoirs with permeability and porosity being realizations of given statistically homogeneous random fields. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments. To our best knowledge, this is the first stochastic homogenization result in the case of compressible two-phase flow in random porous media. [ABSTRACT FROM AUTHOR]

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size ℓ ≫ 1 and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size L ≫ ℓ . More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of L ≫ ℓ ≫ 1 , using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and ℓ is optimal, where in this paper we focus on the case of d = 2 . This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor. [ABSTRACT FROM AUTHOR]

Numerical calculations of light propagation in random media demand the multiply scattered Stokes intensities to be written in a common fixed reference. In multiple-scattering schemes, a particularly useful way to perform automatically these basis transformations between reference frames is to write the scattered intensities in the Chandrasekhar-Sekera representation. The main drawback with this representation is the necessity of numerical tests to deal with the limiting situations of the small particle (Rayleigh) and forward/backward scattering. Here, a new set of basis functions is presented to describe the scattering of light by spherical particles (Mie scattering) in the Chandrasekhar-Sekera representation. These basis functions can be implemented in a new algorithm to calculate the Mie scattering amplitudes, which leads straightforwardly to all the scattering quantities. In contrast to the traditional implementation, this set of basis functions implies to natural numerical convergence to the above mentioned limiting cases, which are thoroughly discussed. [ABSTRACT FROM AUTHOR]

Single-well imaging plays an increasingly important role in the exploration and production of hydrocarbon in horizontal wells. Depending on the scattering strength, reflections from lateral sections below heterogeneous regions around the borehole suffer signal distortion and amplitude loss, which complicates the imaging of deeper structures. Thus, the understanding of scattering phenomena and their impact on single-well imaging is of significant interest for hydrocarbon exploration. This paper establishes a complex horizontal well numerical model with which the effects of heterogeneous overburden on deep reflection imaging are studied using dipole acoustic logging. Synthetic data were analyzed with respect to the observed reflectivity in the heterogeneous layer and along the deep reflection. The results show that strong velocity fluctuations increase the severity of the scattering effect when correlation distances are in the order of the dominant wavelength. The reflection images migrate and reveal additional structural details of the medium. Low frequency band imaging enables the extraction of undistorted real reflectors by using the Reflection Image Spectroscopy method. The results explain the influence of the heterogeneous overburden on single-well imaging results and provide theoretical guidance for logging interpretation. [ABSTRACT FROM AUTHOR]

Featured Application: The results obtained in the present research show the complex nature of the propagation of sound signals and noise in the conditions of the shallow waters of the Arctic basin shelf. Despite the rather simple hydrology of the water column, a high degree of diversity of the characteristics of the sub-bottom layer was observed here, and a statistical approach was applied to describe the influence of this layer. Analysis of the fluctuation nature of the signal indicates the need to search for new approaches to processing information received by acoustic measuring systems in the Arctic seas. The quantitative estimates obtained by the authors of the degree of influence of random bottom parameters in typical shallow-water Arctic regions can be used to predict propagation losses of low and medium-frequency sound signals, which is important in underwater communications, detection problems and environmental issues of reducing the impact of anthropogenic signals and noise on marine mammals. The statistical analysis performed in this paper is useful for predicting the presence of bottom areas with gas-saturated sediments. In addition, pronounced fluctuations in the parameters of the sub-bottom layer often make the known deterministic approaches to solving inverse problems of reconstructing the characteristics of the medium from the measured acoustic field inadequate in the Arctic seas and require the development of statistical methods. In this study, the problem of the influence of a horizontally inhomogeneous liquid bottom impedance, given by random Gaussian function of the speed of sound and by density, on the propagation of low-frequency sound in a shallow-water waveguide is considered. The model parameters are referenced to the conditions of sound propagation in the regions of the seas of the Russian Arctic shelf. By the example of statistical modeling of the sound field intensity, we show that sound speed fluctuations in the bottom lead to similar effects that were previously established for volumetric fluctuations of the speed of sound in the water layer. With the distance from the source, the decrease in the average intensity slows down in comparison with a deterministic medium in which there are no fluctuations. This deceleration of the decay of the intensity in a random waveguide can be significant already at short distances. Changes in the law of decay of intensity at a fixed frequency are mainly determined by the correlation radius of inhomogeneities and the average penetrability of the bottom, which leads to attenuation of sound propagating in the waveguide. [ABSTRACT FROM AUTHOR]

Grounded-source airborne time-domain electromagnetic (GATEM) method is an effective detection method of geological survey. The real geological media are rough, self-similar characteristics, and the diffusion process is anomalous diffusion. This paper primarily considers the GATEM responses of a grounded wire source in random media. Von K'arm'an function is used to establish three-dimensional (3D) random media model and the GATEM responses are realized based on 3D finite-difference time-domain (FDTD) method. The method is verified by homogeneous half-space model. The electromagnetic responses of random abnormal body model are analyzed, and the results show that the abnormal body can be clearly identified. The electromagnetic responses of a fractured model are analyzed, and the results show that the tilt angle of the fault can be reflected. [ABSTRACT FROM AUTHOR]

In this article, we develop a hybrid method for studying wave scatterings and propagation in a vegetation field. The coherent multiple scatterings among the plants and the gaps in the vegetation field are considered through a two-step hybrid method. First, the T-matrix of a single plant is calculated using the Huygens principle and the vector cylindrical wave (VCW) expansion based on the full-wave solutions from the Ansys High Frequency Structure Simulator (HFSS). Second, the multiple scatterings between different plants are considered through the Foldy–Lax (FL) multiple-scattering equations using the calculated T-matrix and the translational addition theorem. The convergence and the accuracy of the hybrid method are verified with the HFSS by comparing the solutions of scatterings from four wheat plants. The full-wave Monte Carlo simulations of a field with 169 wheat plants at the $L$ -band are subsequently performed using the hybrid method. Numerical results show that the transmission in the gap region is significantly higher than that of the inner region with the difference given quantitatively. The transmissivity of microwave through the wheat field with a large vegetation water content (VWC) can be significantly underestimated by the classical radiative transfer equation (RTE) and distorted Born approximation (DBA) model. [ABSTRACT FROM AUTHOR]

Multiple optical scattering occurs when light propagates in a non-uniform medium. During the multiple scattering, images were distorted and the spatial information they carried became scrambled. However, the image information is not lost but presents in the form of speckle patterns (SPs). In this study, we built up an optical random scattering system based on an liquid crystal display (LCD) and an RGB laser source. We found that the image classification can be improved by the help of random scattering which is considered as a feedforward neural network to extracts features from image. Along with the ridge classification deployed on computer, we achieved excellent classification accuracy higher than 94%, for a variety of data sets covering medical, agricultural, environmental protection and other fields. In addition, the proposed optical scattering system has the advantages of high speed, low power consumption, and miniaturization, which is suitable for deploying in edge computing applications. [ABSTRACT FROM AUTHOR]