Showing papers in "Waves in Random Media in 2004"

Quantum graphs: I. Some basic structures

Abstract: A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.

A critical survey of approximate scattering wave theories from random rough surfaces

Abstract: This review is intended to provide a critical and up-to-date survey of the analytical approximate methods that are encountered in scattering from random rough surfaces. The underlying principles of the different methods are evidenced and the functional form of the corresponding scattering amplitude or cross-section is given. The reader is referred to the original papers in order to obtain the explicit expressions of the coefficients and kernels. We have tried to identify the main strengths and weaknesses of the various theories. We provide synthetic tables of their respective performances, according to a dozen important requirements a valuable method should meet. Both scalar acoustic and vector electromagnetic theories are equally addressed.

A survey of ionospheric effects on space-based radar

Abstract: In this survey, we fully review almost all potential ionospheric effects on the performance of space-based radar systems (SBRs), which operate in the ambient ionosphere environment; in particular, we review the use of space-based synthetic aperture radar systems (SARs) for imaging. There are two families of effects involved. One is the effects of the background ionosphere (non-turbulent ionosphere), such as dispersion, group delay, refraction, Faraday rotation, and phase shift. The other is the effects due to ionospheric irregularities, such as refractive index fluctuation, phase perturbation, angle-of-arrival fluctuation, pulse broadening, clutter, and amplitude scintillation. These effects adversely affect SAR imaging in several respects, such as by causing image shift in the range, and degradations of the range resolution, azimuthal resolution, and/or the resolution in height (elevation). We also review ionospheric irregularity characteristics and descriptions, propagation channel statistics, ...

Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere

Abstract: In this paper, we study the effects of turbulent atmosphere on the degree of polarization of a partially coherent electromagnetic beam, which propagates through it. The beam is described by a 2 × 2 cross-spectral density matrix and is assumed to be generated by a planar, secondary, electromagnetic Gaussian Schell-model source. The analysis is based on a recently formulated unified theory of coherence and polarization and on the extended Huygens–Fresnel principle. We study the behaviour of the degree of polarization in the intermediate zone, i.e. in the region of space where coherence properties of the beam and the atmospheric turbulence are competing. We illustrate the analysis by numerical examples.

Coherent backscattering of light by complex random media of spherical scatterers: numerical solution

Abstract: Novel Monte Carlo techniques are described for the computation of reflection coefficient matrices for multiple scattering of light in plane-parallel random media of spherical scatterers. The present multiple scattering theory is composed of coherent backscattering and radiative transfer. In the radiative transfer part, the Stokes parameters of light escaping from the medium are updated at each scattering process in predefined angles of emergence. The scattering directions at each process are randomized using probability densities for the polar and azimuthal scattering angles: the former angle is generated using the single-scattering phase function, whereafter the latter follows from Kepler's equation. For spherical scatterers in the Rayleigh regime, randomization proceeds semi-analytically whereas, beyond that regime, cubic spline presentation of the scattering matrix is used for numerical computations. In the coherent backscattering part, the reciprocity of electromagnetic waves in the backscatt...

On the spectrum of the Laplacian on regular metric trees

Abstract: A metric tree r is a tree whose edges are viewed as non-degenerate line segments. The Laplacian Δ on such a tree is the operator of second order differentiation on each edge, complemented by the Kirchhoff matching conditions at the vertices. The spectrum of Δ can be quite varied, reflecting the geometry of a tree. We consider a special class of trees, namely the so-called regular metric trees. Any such tree r possesses a rich group of symmetries. As a result, the space L 2 (Γ) decomposes into the orthogonal sum of subspaces reducing the operator Δ. This leads to detailed spectral analysis of Δ. We survey recent results on this subject.

Modified spherical harmonics method for solving the radiative transport equation

Abstract: A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.

Target detection beneath foliage using polarimetric synthetic aperture radar interferometry

Abstract: In this paper, we demonstrate how the new technology of polarimetric synthetic aperture radar (SAR) interferometry can be used to enhance the detection of targets hidden beneath foliage. The key idea is to note that for random volume scattering, the interferometric coherence is invariant to changes in wave polarization. On the other hand, in the presence of a target the coherence changes with polarization. We show that under general symmetry constraints this change is linear in the complex coherence plane. These observations can be used to devise a filter to suppress the returns from foliage clutter while maintaining the signal from hidden targets. We illustrate the algorithm by applying it to coherent L-band SAR simulations of corner reflectors hidden in a forest. The simulations are performed using a voxel-based vector wave propagation and scattering code coupled to detailed structural models of tree architecture. In this way, the spatial statistics and radar signal fluctuations closely match t...

Nodal counting on quantum graphs

Abstract: We consider the real eigenfunctions of the Schrodinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds B. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.

Microwave radiometry of forests

Abstract: Microwave remote sensing observations provide all weather, day/night monitoring of the earth's surface and make it possible to probe forest vegetation at various depths by operating at different frequencies. Significant progress in microwave radiometry of land surfaces has been made by using advanced airborne and spaceborne instruments and by developing physical and statistical models needed for interpreting the data. At present, a new multi-frequency scanning radiometer, launched in 2002 is providing global observations of the earth's surface at a relatively high resolution, and collected data are currently under study. This paper provides a review of experimental and theoretical investigations carried out in recent years to study the relationships between microwave emission and forest features at regional and global scale. It is shown that, despite the relatively small amount of experimental data currently available, microwave radiometry has proved to be an efficient technique in monitoring for...

Telecommunication in a disordered environment with iterative time reversal

Abstract: We present a method to transmit digital information through a highly scattering medium in a MIMO-MU (multiple input multiple output multiple users) context. It is based on iterations of a time-reversal process, and permits us to focus short pulses, both spatially and temporally, from a base antenna to different users. This iterative technique is shown to be more efficient (lower inter-symbol interference and lower error rate) than classical time-reversal communication, while being computationally light and stable. Experiments are presented: digital information is conveyed from 15 transmitters to 15 receivers by ultrasonic waves propagating through a highly scattering slab. From a theoretical point of view, the iterative technique achieves the inverse filter of propagation in the subspace of non-null singular values of the timereversal operator. We also investigate the influence of external additive noise, and show that the number of iterations can be optimized to give the lowest error rate. (Some figures in this article are in colour only in the electronic version)

The calculated performance of forest structure and biomass estimates from interferometric radar

Abstract: Vertical structure and biomass are key characteristics of the forest random medium. This paper calculates the power and interferometric synthetic aperture radar (InSAR) sensitivity to tree height and vegetation density as it manifests in extinction, using a homogeneous, random-volume model of the forest medium, and accounting for speckle and thermal noise. Tree height and extinction are both related to biomass within the context of this simple model. Signal and noise calculations show that InSAR coherence and phase are more sensitive than radar power to structure and biomass to 10% variations in structure parameters over a wide range of medium to high density forests. For example, for extinctions of 0.2 db m−1 and other parameters as noted in the text, the sensitivity of InSAR coherence to 10% changes in tree height exceeds observation errors for trees shorter than about 37 m, as opposed to 14 m for radar power. InSAR phase sensitivity to 10% structural and associated biomass changes exceeds obse...

Irreversible quantum graphs

Abstract: Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the v ...

Weighted curvature approximation: numerical tests for 2D dielectric surfaces

Abstract: The weighted curvature approximation (WCA) was recently introduced by Elfouhaily et al [7] as a unifying scattering theory that reproduces formally both the tangent-plane and the small-perturbation model in the appropriate limits, and is structurally identical to the former approximation with some different slope-dependent kernel. Due to the simplicity of its formulation, the WCA is interesting from a numerical point of view and the aim of the present paper is to establish its accuracy on some representative test cases. We derive statistical formulae for the coherent field and the cross-section in the case of stationary Gaussian random surfaces. We then specialize to the case of isotropic Gaussian spectra and perform numerical comparisons against rigorous method of moments (MoM)-based results on 2D dielectric surfaces. We show that the WCA remains extremely accurate in a roughness range where other first-order classical approximations (small-slope and Kirchhoff) clearly fail, at the same computational cost.

On a differential operator appearing in the theory of irreversible quantum graphs

Abstract: A partial differential operator depending on the coupling parameter α≥0 is considered. The spectral properties of the operator strongly depend on α. The operator was suggested in Smilansky (2003 Waves Random Media 14 S143–53) as a model of an irreversible physical system.

On the validity of the inclusion of geometrical shadowing functions in the multiple-scatter Kirchhoff approximation

Abstract: The validity of the geometrical shadowing functions for use in calculations of the light distribution scattered from a rough surface is investigated. By the use of the multiple-scatter Kirchhoff approximation, single- and double-scattered contributions are calculated with and without the incident, intrasurface and scatter-shadowing functions. Explicit conditions are given for the validity of the geometrical shadowing functions.

Depolarization volume and correlation length in the homogenization of anisotropic dielectric composites

Abstract: In conventional approaches to the homogenization of random particulate composites, both the distribution and size of the component phase particles are often inadequately taken into account. Commonly, the spatial distributions are characterized by volume fraction alone, while the electromagnetic response of each component particle is represented as a vanishingly small depolarization volume. The strong-permittivity-fluctuation theory (SPFT) provides an alternative approach to homogenization wherein a comprehensive description of distributional statistics of the component phases is accommodated. The bilocally-approximated SPFT is presented here for the anisotropic homogenized composite which arises from component phases comprising ellipsoidal particles. The distribution of the component phases is characterized by a two-point correlation function and its associated correlation length. Each component phase particle is represented as an ellipsoidal depolarization region of nonzero volume. The effects o...

Application of a coherent model in simulating the backscattering coefficient of a mangrove forest

Abstract: In this paper, a single scattering mode li s present ed for a coherent forest scattering simulation. It is tested on the backscattering coefficient of mangrove forests, which are known to involve large coherent effects. Analysis of branches, leaves and ground contributions is done to understand the backscattering coefficient composition. Finally the sensitivity of the code is investigated. (Some figures in this article are in colour only in the electronic version)

The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory

Abstract: The Rytov perturbation method can be used to derive analytic expressions governing statistical quantities of an optical wave propagating through the Earth's atmosphere. It is generally accepted that the validity of these expressions is restricted to the weak fluctuation regime, and that the wave structure function for plane and spherical waves obtained via the Rytov method is valid in all fluctuation regimes, for sufficiently small separation distances. Data from experimental results for the wave structure function as a function of the fluctuation strength for a fixed value of the separation distance indicate that the Rytov method does not accurately model the behaviour of the wave structure function in moderate to strong fluctuation regimes. This is similar to what is observed for the scintillation index. Recently, however, it was shown that the integral definition of the scintillation index obtained via the Rytov perturbation yields analytic expressions that are valid in all fluctuation regimes when a f...

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

Abstract: The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.

P-wave velocity–porosity relations and homogeneity lengths in a realistic deposition model of sedimentary rock

Abstract: A realistic model of a porous rock is generated by simulating critical processes in sedimentary rock formation. The simulated processes include deposition of grains in different energy environments, compression with grain rearrangement effects due to the stratigraphic column pressure and cementation. It has been shown that such processes are fundamental in determining the appropriate correlations for the system to exhibit realistic electrical and hydraulic conductivities of sandstones. Here we analyse the theoretical model from the geometrical and acoustic standpoint, determining the relations of the homogenization volumes and sound speed to the resulting porosity of the model. For the quartz cemented contact model we find the correct velocity magnitudes and concavities in the velocity–porosity curves. We also show that using anisotropic rescaling as a compaction stage can lead to inadequate acoustic properties for the rock model, due to the importance of grain rearrangement.

Spectra of soft ring graphs

Abstract: We discuss a ring-shaped soft quantum wire modelled by δ interaction supported by the ring with a generally nonconstant coupling strength We derive the condition which determines the discrete spectrum of such systems, and analyse the dependence of the eigenvalues and eigenfunctions on the coupling and ring geometry In particular, we illustrate that a random component in the coupling leads to a localization The discrete spectrum is also investigated in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov–Bohm flux and the system exhibits persistent currents (Some figures in this article are in colour only in the electronic version)

Scattering from a layer of discrete random medium over a random interface: application to microwave backscattering from forests

Abstract: The distorted Born approximation is used to calculate the bistatic scattering coefficients from a layer of sparsely distributed discrete dielectric scatterers over a random interface. After specializing to the backscatter case, the scattering coefficient is determined as a sum of direct, direct reflected and interface scatter contributions. The direct reflected term contains contributions from the average interface and the interface fluctuations. These direct reflected terms include both incoherent and coherent or enhancement terms. The results are applied to backscattering from a mature hemlock forest over a roughened ground. The model results show that the direct reflected surface fluctuation terms give the dominant contribution to backscatter at P band and are equal in magnitude to the volume scatter at L band. Use of these new results brings the model predictions and experimental results into agreement.

Scattering of a plane wave by one-dimensional dielectric random rough surfaces—study with the curvilinear coordinate method

Abstract: We present a method giving the bi-static scattering coefficient of a one-dimensional dielectric random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. For each medium, this method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the diffraction amplitudes to be determined. The Monte Carlo technique is applied and the bi-static scattering coefficient is estimated by averaging the scattering amplitudes over several realizations. The results of a Gaussian random process with a Gaussian roughness spectrum are compared to published experimental and numerical data. Comparisons are conclusive.

A random necklace model

Abstract: We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.

Correlation length facilitates Voigt wave propagation

Abstract: Under certain circumstances, Voigt waves can propagate in a biaxial composite medium even though the component material phases individually do not support Voigt wave propagation. This phenomenon is considered within the context of the strong-permittivity-fluctuation theory. A generalized implementation of the theory is developed in order to explore the propagation of Voigt waves in any direction. It is shown that the correlation length—a parameter characterizing the distributional statistics of the component material phases—plays a crucial role in facilitating the propagation of Voigt waves in the homogenized composite medium. (Some figures in this article are in colour only in the electronic version)

Spectral theory and spectral gaps for periodic Schrödinger operators on product graphs

Abstract: Floquet theory and its applications to spectral theory are developed for periodic Schrodinger operators on product graphs 𝔾×ℤ, where 𝔾 is a finite graph. The resolvent and the spectrum have detailed descriptions which involve the eigenvalues and singularities of the meromorphic Floquet matrix function. Existence and size estimates for sequences of spectral gaps are established.

Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: I. Theoretical study

Abstract: In this paper the first- and second-order Kirchhoff approximation is applied to study the backscattering enhancement phenomenon, which appears when the surface rms slope is greater than 0.5. The formulation is reduced to the geometric optics approximation in which the second-order illumination function is taken into account. This study is developed for a two-dimensional (2D) anisotropic stationary rough dielectric surface and for any surface slope and height distributions assumed to be statistically even. Using the Weyl representation of the Green function (which introduces an absolute value over the surface elevation in the phase term), the incoherent scattering coefficient under the stationary phase assumption is expressed as the sum of three terms. The incoherent scattering coefficient then requires the numerical computation of a ten- dimensional integral. To reduce the number of numerical integrations, the geometric optics approximation is applied, which assumes that the correlation between two adjacent points is very strong. The model is then proportional to two surface slope probabilities, for which the slopes would specularly reflect the beams in the double scattering process. In addition, the slope distributions are related with each other by a propagating function, which accounts for the second-order illumination function. The companion paper is devoted to the simulation of this model and comparisons with an 'exact' numerical method.

Slowdown of the wave packages in finite slabs of periodic media

Abstract: A simple mathematical model is provided for a description of wave propagation through finite slabs of periodic media. The model concerns the devices of CROW (coupled resonators optical waveguide) type which are widely discussed in physical literature in the last several years. An incident pulse is considered, whose frequency is distributed in a small neighbourhood of a singular frequency for which the dispersion relation of the medium is very flat, and the group velocity (for the infinite medium) is small. It is proved that only a small part of the energy of the pulse propagates with the speed of light. Another part of the energy propagates with the group velocity. The price to pay for this slowdown is the reflection of the majority of the energy of the incident pulse.

Statistical properties of resonance widths for open quantum graphs

Abstract: We connect quantum compact graphs with infinite leads, and turn them into scattering systems. We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph. The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads. Finally, we analyse the statistical properties of the resonance widths and compare our results with the predictions of random matrix theory. Deviations appearing due to the dynamical nature of the system are pointed out and explained.