A study of the regions of validity for rough surface scattering models is conducted for surfaces with Gaussian and power law power spectra. Models included in the study are physical optics (PO), geometrical optics, small perturbation method and small slope approximation. The range of validity of the PO model is commonly described by a bound on the radius curvature of the surface relative to the electromagnetic wavelength. We show empirically that for backscattering the region of accuracy is more accurately described by a bound on surface slope. For surfaces with a Gaussian power spectrum, the PO model is accurate to within 2 dB for RMS surface slope values less than 0.59 cos(3) theta. For surfaces with a power law power spectral density, the PO model is accurate for significant slope values (RMS surface height/wavelength of the dominant spectral peak) less than 0.037 cos(3) theta. These conditions are valid up to approximately 30degrees. The regions of validity of other models in the study are also shown to be well approximated by bounds on surface slope.
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 forest environments, and in particular in separating forest types, estimating woody biomass and, in some cases, assessing soil surface properties.
This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourth-order moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second-order moments by using the circular complex Gaussian assumption. The stochastic Green's functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey-Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.
In this paper, a single scattering model is presented 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.
There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham-Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-,frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt in variance And reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.
We demonstrate how Mellin operator calculus can be used to study the mapping properties of the Dirichlet to Neumann map on graphs. These operators naturally arise in photonic crystal theory. The calculus adds rigour to some previously given heuristic arguments.
This paper describes the theoretical simulations carried out with a model of the backscattering coefficient of crops, where the leaf geometry is represented by a curved rectangular dielectric sheet. A general formulation is introduced for the bistatic scattering cross section of the curved sheet, and numerical results based on this approach are compared with those obtained by considering disc shaped leaves.
The spectral analysis of the Schrodinger operator on cubic lattice type graphs is developed. Similarly to the quantum mechanical tight-binding approximation, using the well known concept of the Dirichlet-to-Neumann map, asymptotic formulae for localized negative spectral bands of the Schrodinger operator on a periodic metric graph are established. The results are illustrated by numerical calculations.
We consider the real eigenfunctions of the Schr\"odinger 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.
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 backscattering direction allows the renormalization of the reversely propagating waves, whereafter the scattering characteristics are computed in other directions. High orders of scattering (similar to10000) can be treated because of the peculiar polarization characteristics of the reverse wave: after a number of scatterings, the polarization state of the reverse wave becomes independent of that of the incident wave, that is, it becomes fully dictated by the scatterings at the end of the reverse path. The coherent backscattering part depends on the single-scattering albedo in a non-monotonous way, the most pronounced signatures showing up for absorbing scatterers. The numerical results compare favourably to the literature results for nonabsorbing spherical scatterers both in and beyond the Rayleigh regime.
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 junction 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 filter function is applied to the atmospheric spectrum. The underlying physical theory is that as the wave propagates, intermediate refractive index scale sizes fail to refract or diffract the beam. Hence, these scale sizes do not contribute to the scintillation index. In this paper, we investigate the results of applying this concept to the wave structure function. Specifically, we apply a filter function to the atmospheric spectrum and develop analytic expressions for the wave structure function for plane, spherical and Gaussian beam waves using the Rytov perturbation method. It is shown that in weak fluctuations these expressions yield similar results to standard expressions obtained where no filter function is applied. However, in moderate to strong fluctuations, these new expressions predict a decrease in the value of the wave structure function as compared to the standard expressions, following the trend of the experimental data presented by Gurvich.
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 those observed for natural terrain. We demonstrate significant improvements in the detection of hidden targets, which suggests that this technology has great potential for future foliage penetration (FOPEN) applications.
In this paper, using the Fock method of the fifth parameter and weighted Fourier-transform with respect to the coordinates of the source and observer, an integral representation is obtained for the wave field in a randomly inhomogeneous medium without invoking the assumption about small-angle propagation. Random trajectory variations to a first approximation are taken into account in calculating the partial wave phase (the expression under the integral sign). The expressions for the field in a medium with different-scale irregularities and for the scintillation index, obtained using this integral representation, are compared with known results. The good agreement with results from the theory of single scattering in a medium with background irregularities, and with investigations of the scintillation index made in terms of Rytov's method and path integrals, indicates that it is possible to use the approach developed in this study to describe the effects of simultaneous influence of different-scale irregularities.
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 x 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.
Floquet theory and its applications to spectral theory are developed for periodic Schrodinger operators on product graphs G x 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.
We investigate the effect of a space-dependent random mass density field on small amplitude acoustic modes that are settled in a semi-infinite medium of a temperature growing linearly with depth. Using a perturbation method, the dispersion relation is derived in the form of Hill's determinant. Numerical solutions of this equation lead to the following conclusions: (a) a weak random field (with delta(eff) = 0.05) essentially affects long waves which experience attenuation and a frequency reduction; (b) for a stronger random field (with delta(eff) = 0.1), high-order sound modes behave as sound waves as they are attenuated and their frequencies are increased; (c) for a sufficiently strong random field (with delta(eff) = 0.2), mode coupling occurs, as a result of which the dispersive curves cross each other, the sound modes loose their identities, and some modes are amplified. Here delta(eff) denotes the effective strength of a random field.
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.
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.
This special section of Waves in Random Media is devoted to quantum graphs and their applications. We use the name quantum graph for a graph considered as one-dimensional (rather than just a purely combinatorial object) and equipped with a self-adjoint differential (or sometimes pseudo-differential) operator. This topic, rooted in numerous prior studies in different branches of physics, chemistry, and mathematics, has been actively developing and solidifying recently. We do not intend to provide in this introduction a complete historical description or a comprehensive bibliography on the subject (only token references will be given here). The reader is referred to the recent survey [24] and to the papers in this issue for a better historical account and references. We would just like to mention briefly that in applications quantum graphs occur usually in one of two ways: either as models of thin (mesoscopic) structures in the asymptotic limit when the width of the structure tends to zero, or as testing grounds where, due to the one-dimensional nature, it is hoped that the problems of interest will be resolved more easily than in more realistic models. Examples of the first type come from chemistry (free electron theory of conjugated molecules [18, 19, 30]), superconductivity (thin superconducting networks [1, 6, 29]), nanotechnology (quantum wires circuits [9]), optics (photonic crystals [11, 12, 13, 26, 27, 31]), scattering theory [16], averaging in dynamical systems [14, 15], and spectral theory of differential operators in singular domains [7, 8]. The second type of application of quantum graphs can be seen in modelling effects of electron propagation in non-simply-connected media [2, 3] and, most prominently recently, in quantum chaos [22, 23]. The papers collected in this issue represent some of the trends mentioned above. Articles by R Carlson [5], P Kuchment [25], and M Solomyak [33] address the mathematical background of quantum graphs. The basic definitions and results on spectra of quantum graphs are reviewed in [25], a detailed study of spectra of quantum graphs with a one-dimensional periodicity is conducted in [5], and analysis of spectra of quantum trees is given in [33]. Articles by G Berkolaiko, by S Gnutzmann, U Smilansky, and J Weber, and by T Kottos and H Schanz deal with quantum chaos problems. The paper [4] by G Berkolaiko continues the trend started in [22] of attacking quantum chaos problems in the quantum graph case. In particular, one tries to prove for the case of quantum graphs the Bohigas-Giannoni-Schmit conjecture on quantum signatures of classical chaos. The main result of [4], where the so called form factor is studied, provides one more step towards justification of the conjecture. Another contribution in the area of quantum chaos is the paper by S Gnutzmann, U Smilansky, and J Weber [17], where the statistics of nodal domains of the eigenfunctions of quantum graphs are studied. The nodal domains have been established as a tool in the search for the quantum signature of chaos and [17] supports further the role of quantum graphs as paradigms for quantum chaos. In the paper by T Kottos and H Schanz [21] a finite quantum graph is connected to several infinite leads and properties of such scattering systems are studied. In particular, the statistical properties of the resonance widths are compared with the predictions of random matrix theory. A quantum graph that consists of circular ‘beads’ of random lengths connected into the infinite chain by unit intervals is considered in the paper by V Kostrykin and R Schrader [20].
A linear spectral estimation technique, the PDFT algorithm, is used as part of a nonlinear iterative reconstruction scheme to obtain improved radar images. The iterative PDFT algorithm is used to address the limited resolution problem inherent to imaging objects buried in soil and hidden under foliage. This is achieved by subsequent application of two properties of the PDFT algorithm: the energy parameter of the PDFT algorithm is used to determine the target shape, while the shape information in turn is used to obtain super-resolved images. We describe algorithms able to exploit both properties automatically and without manual intervention. Two methods are investigated in particular, one iteratively optimizing the constraints by monitoring the energy parameter, the other method computing energy values for all points, from which a weighted prior function is determined. In addition, we discuss variants of both algorithm which provide an optimized trade-off between computation time and performance. Additional attention is given to situations, where a known target is embedded in an unknown background. Imaging results are presented for both synthetic and measured data.
