Wave Propagation In Random Media Research Articles

Enhanced wave transmission in random media with mirror symmetry

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.

Paraxial Wave Propagation in Random Media with Long-Range Correlations

Paraxial Wave Propagation in Random Media with Long-Range Correlations

Wave propagation in random media: beyond Gaussian statistics

In this paper we review some aspects of wave propagation in random media. In the physics literature the picture seems simple: for large propagation distances, the wavefield has Gaussian statistics, mean zero, and second-order moments determined by radiative transfer theory. The results for the first two moments can be proved under general circumstances by multiscale analysis. The Gaussian conjecture for the statistical distribution of the wavefield can be proved in some propagation regimes, such as the white-noise paraxial regime that we address in the first part of this review. It may, however, be wrong in other regimes, such as in randomly perturbed open waveguides, that we address in the second part of this review. In the third and last part, we reconcile the two results by showing that the Gaussian conjecture is restored in randomly perturbed open waveguides in the high-frequency regime, when the number of propagating modes increases.

Multiscale model reduction for stochastic elasticity problems using ensemble variable-separated method

Elasticity is a fundamental model in mechanics and material sciences. In the article, we present an ensemble variable-separated multiscale method for elasticity problems in random media. A variable-separated method is utilized to provide a low-rank approximation for random fields. The variable-separated method is integrated with the Generalized Multiscale Finite Element Method (GMsFEM) to obtain an efficient reduced model. To this end, a few local problems are solved for snapshots to construct the multiscale basis functions in GMsFEM. An ensemble method is used to construct stochastic basis functions shared by a series of elasticity equations, and a residual decomposition is adopted to calculate the deterministic physical components for all ensemble members. The ensemble variable-separation is used to solve the local problems of the multiscale basis functions and achieves efficient online-computation for the basis. Then, under the Galerkin projection of the generalized finite element method, the solutions of stochastic elasticity problems are projected onto the low-dimensional space spanned by the online multiscale basis functions. This renders an effective coarse model. The ensemble variable-separated multiscale method is applied to steady-state elastic deformation and elastic wave propagation in random media. A few numerical results are carried out to demonstrate the accuracy and efficiency of the presented method.

Reflections on the Life of Oleg A. Tretyakov [In Memoriam

Dr. Oleg Alexandrovich Tretyakov, 83, passed away on Tuesday, 25 January 2022, in Istanbul, Turkey, after a short battle with lung cancer. He was an outstanding expert in electrodynamics and radiophysics. He was the chair of Commission B of the International Union of Radio Science (URSI) Committee of Ukraine since 1993, a Member of IEEE, and a professor emeritus of the Department of Theoretical Radiophysics, V.N. Karazin Kharkiv National University (KhNU), Ukraine. He had experience in solving diffraction problems by mathematically rigorous methods, wave scattering from rough surfaces, and wave propagation in random media, linear and nonlinear phenomena, and dynamic chaos in microwave oscillators and amplifiers. With him, we lost an internationally well-known and highly regarded leading figure in electromagnetic theory. His intellect, integrity, humor, professionalism, and humanity will be greatly missed.

Non-Markov behavior of acoustic phase variance in the atmospheric boundary layer

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.

Remarks on the principles of statistical fluid mechanics.

This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration, we review several functional integration approaches to the theory of turbulence. We note in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation. This article is part of the theme issue ''Scaling the turbulence edifice (part 1)'.

Petro-mineralogical controls on coda attenuation in volcanic rock samples

SUMMARYSeismic attenuation measurements, especially those obtained from coda decay analysis, are becoming a key data source for the characterization of the heterogeneous Earth due to their sensitivity to small-scale heterogeneities. However, the relation between the scattering attenuation measured from coda waves and physical rock properties is still unclear. The goal of this study is to identify the main petrophysical and mineralogical factors controlling coda attenuation in volcanic rocks at the laboratory scale, as a necessary step before modelling seismic waves in real volcanic media. Coda wave attenuation was estimated from ultrasonic S-wave waveforms. To quantify the heterogeneity of the rocks and link them with this attenuation parameter, we performed several categorizations of the pore and grain systems of volcanic samples. Considering that seismic attenuation in rock samples can be modelled using the framework of wave propagation in random media, a statistical analysis of shear wave velocity fluctuations was performed: this analysis gives correlation lengths ranging from 0.09 to 1.20 mm, which represents the length scale of heterogeneity in the samples. The individual evaluation of the pore space and mineral content revealed that the pores of the samples (characterized by large vesicles) have a bigger effect than the grains on the heterogeneity level. We have developed a framework where intrinsic properties of the host rocks drive seismic attenuation by correlating the petro-mineralogical characteristics obtained from image data processing and analysis, with the coda attenuation measured at ultrasonic frequencies. There is conclusive evidence that porosity alone is not the primary controller of coda attenuation: it is also changed by the alteration level (i.e. oxidation, coating of the vesicles, secondary minerals) and the size of grains and pores. Among all the parameters analysed, it appears that the pore space topology is the main contributor to scattering attenuation in the volcanic samples.

Valerian Tatarskii and Acoustic Wave Propagation in Random Media

Valerian Tatarskii and Acoustic Wave Propagation in Random Media

General formulation of source localization for multiple receivers with fully saturated scattered signals

A general formulation is provided for the problem of localizing a source from signals at multiple, arbitrarily separated receivers. The signals are assumed to be narrowband and fully saturated by scattering from homogeneous turbulence as they propagate along line-of-sight paths to the receivers. In this case, the complex Wishart distribution can be used to describe the joint distribution of the received signals. Previously derived results from the theory of wave propagation in random media can furthermore be used to model the covariance matrix between the received signals. Then, given a prior distribution on the source location, Bayes's theorem enables the maximum a posteriori (MAP) estimate for the source location to be formulated. This general approach incorporates source localization by beamforming and by signal amplitude as special cases. The behavior of the solution is illustrated with numerical examples.

Fractal catastrophes

We analyse the spatial inhomogeneities (‘spatial clustering’) in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes (‘caustics’) in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor (‘fractal catastrophes’) make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.

Effects of Soil Parameters Uncertainties on the Behaviour of Anisotropic Porous Media to Shear Waves

In the present study, the wave equations for shear waves propagating in anisotropic fluid-saturated porous media are established in order to obtain the solutions in terms of displacements and dispersion equation. The wave velocities in the vertical and horizontal directions are derived. The uncertainties of the soil parameters due to their spatial variability are taken into account via Monte Carlo Simulations. The results are restricted to the effects of the porosity and permeability uncertainties on the phase velocity and attenuation for SH wave in addition to the anisotropy for Love wave. Results show that the mean velocities are more sensitive to the random variations of the permeability than to that of the porosity, but both phase velocity and attenuation decrease as the uncertainties increase. On the other hand, the anisotropy level and the randomness significantly affect the dispersion of Love waves. The present approach which converts a deterministic solution in a probabilistic one may be used as an everyday tool for practical applications of shear wave propagation in random media.

A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for the Time-Harmonic Maxwell’s Equations with Random Coefficients

This paper develops an efficient Monte Carlo interior penalty discontinuous Galerkin (IP-DG) method for electromagnetic wave propagation in random media. This method is based on a multi-modes expansion of the solution to the time-harmonic random Maxwell equations. It is shown that each mode function satisfies a Maxwell system with random sources defined recursively. An unconditionally stable IP-DG method is employed to discretize the nearly deterministic Maxwell system and the Monte Carlo method combined with an efficient acceleration strategy is proposed for computing the mode functions and the statistics of the electromagnetic wave. A complete error analysis is established for the proposed multi-modes Monte Carlo IP-DG method. It is proved that the proposed method converges with an optimal order for each of three levels of approximations. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed numerical method and approach.

Correspondence between sound propagation in discrete and continuous random media

Wave propagation in random media is important in many applications such as sound propagation in a turbulent atmosphere and scattering by bubbles and microparticles in the ocean. Formulations for the statistical moments of the sound pressure field in continuous and discrete random media are usually done independently. In this presentation, it is demonstrated that the equations for the first two statistical moments in a continuous random medium have the same form as those for a discrete random medium if the scattering properties of the media are expressed in terms of the differential scattering cross section and total cross section. This analogy enables us to apply methods developed in wave propagation in continuous random media to discrete media, and vice versa. As an example, the existing theory of the interference of the direct and ground reflected waves in a turbulent atmosphere is used to study the effect of trees on the interference of these waves in a forest. The results obtained are compared with experimental data. The correspondence between wave propagation in discrete and continuous random media can also be used in other fields of physics.

Non-Markov character of sound propagation over relatively short ranges in the turbulent atmosphere

The Markov approximation is widely used in wave propagation in random media, including sound propagation in the turbulent atmosphere. This approximation, which significantly simplifies formulations for the statistical moments of the propagating field, is valid if the propagation range is greater than the outer length scale of random inhomogeneities. For sound propagation in the turbulent atmosphere, this length scale can be as large as 400–500 m (23% of the height of the atmospheric boundary layer), indicating that the Markov approximation might not be applicable for relatively short ranges. In this paper, the statistical characteristics of a sound field propagating in the turbulent atmosphere, such as the correlation functions of the log-amplitude and phase fluctuations, the mean sound field, and the mutual coherence function are calculated, without using the Markov approximation. The new results are then compared with those obtained in the Markov approximation. The difference between the two formulation...

High-order statistics for the random paraxial wave equation

We consider wave propagation in random media in the paraxial regime. We show how to solve the equations for the second- and fourth-order moment of the field in the regime where the correlation length of the medium is smaller than the initial beam width. We quantify the scintillation of the transmitted field and the statistical stability of the Wigner transform. We finally discuss a few applications to correlation-based communication and imaging methods.

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Ito–Schrodinger equation, that is, a Schrodinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.

A functional renormalization method for wave propagation in random media

We develop the exact renormalization group approach as a way to evaluate the effective speed of the propagation of a scalar wave in a medium with random inhomogeneities. We use the Martin–Siggia–Rose formalism to translate the problem into a non equilibrium field theory one, and then consider a sequence of models with a progressively lower infrared cutoff; in the limit where the cutoff is removed we recover the problem of interest. As a test of the formalism, we compute the effective dielectric constant of an homogeneous medium interspersed with randomly located, interpenetrating bubbles. A simple approximation to the renormalization group equations turns out to be equivalent to a self-consistent two-loops evaluation of the effective dielectric constant.

Experimental study of wave propagation in random media with different heterogeneity scales

Experimental study of wave propagation in random media with different heterogeneity scales

Monte Carlo Methods for Radiative Transfer with Singular Kernels

This work is devoted to Monte Carlo methods for radiative transfer equations with singular kernels, and is motivated by the study of wave propagation in random media with long-range dependence. As ...