Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects

Abstract

This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder 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.     

Short pulse detection and imaging of objects behind obscuring random layers

This paper presents a theory of imaging objects behind layers of scattering media. The transmitter is a focused array or an aperture emitting a short pulse. The scattered pulse is received by a focused array or aperture. The received signal consists of two components: the pulse scattered from a random medium and from the target, and these two components can be distinguished by the use of ultra wide band (UWB) pulse. The second moment of the received signal includes the fourth-order moments of stochastic Green's functions, which are reduced to the second moments by the use of the circular complex Gaussian assumption, and of the generalized two-frequency mutual coherence function. This imaging theory is a generalization of optical coherence tomography (OCT), SAR and confocal imaging. It clarifies the relationships among resolution, coherence length, shower curtain effects and backscattering enhancement.     

Short pulse detection and imaging of objects behind obscuring random layers

This paper presents a theory of imaging objects behind layers of scattering media. The transmitter is a focused array or an aperture emitting a short pulse. The scattered pulse is received by a focused array or aperture. The received signal consists of two components: the pulse scattered from a random medium and from the target, and these two components can be distinguished by the use of ultra wide band (UWB) pulse. The second moment of the received signal includes the fourth-order moments of stochastic Green's functions, which are reduced to the second moments by the use of the circular complex Gaussian assumption, and of the generalized two-frequency mutual coherence function. This imaging theory is a generalization of optical coherence tomography (OCT), SAR and confocal imaging. It clarifies the relationships among resolution, coherence length, shower curtain effects and backscattering enhancement.     

Singular value distribution of the propagation matrix in random scattering media

The distribution of singular values of the propagation operator in a random medium is investigated, in a backscattering configuration. Experiments are carried out with pulsed ultrasonic waves around 3 MHz, using an array of 64 programmable transducers placed in front of a random scattering medium. The impulse responses between each pair of transducers are measured and form the response matrix. The evolution of its singular values with time and frequency is computed by means of a short-time Fourier analysis. The mean distribution of singular values exhibits a very different behaviour in the single and multiple scattering regimes. The results are compared with random matrix theory. Once the experimental matrix coefficients are renormalized, experimental results and theoretical predictions are found to be in a very good agreement. Two kinds of random media have been investigated: a highly scattering medium in which multiple scattering predominates and a weakly scattering medium. In both cases, residual correlations that may exist between matrix elements are shown to be a key parameter. Finally, the possibility of detecting a target embedded in a random scattering medium based on the statistical properties of the strongest singular value is discussed.     

Influence of ionospheric irregularities on the form of radio signal scattering spectrum in over-the-horizon radar sounding of the rough sea surface

Abstract

This paper is concerned with the backscattering of HF radio waves from the rough sea surface, which have propagated through the ionosphere with random large-scale irregularities.

For the sake of simplicity, it is assumed in calculations that the rough sea surface is a perfectly conducting surface with the known Philips power spectrum of irregularities. Ionospheric irregularities of a random medium that are isotropic and single-scale ones, with a Gaussian spectrum, are considered within the limits of the hypothesis of frozen-in irregularities.

Within the first approximation of perturbation theory, using, as the incident wave and the Green function, their geometrical-optics approximations, we obtained the expression for the backscattering spectrum of the ionospheric chirp radio signal with a Gaussian envelope. The expression involves the parameters of the receive–transmit antenna, the signal, the propagation medium, and of the scattering surface. Numerical simulation was used to investigate the influence of all the above-mentioned parameters on the backscattering spectrum. It is shown that travel of ionospheric irregularities has the largest influence on the scattering spectrum, the signal parameters mainly determine the size of the scattering area in the range, and the form of the coherent integration window determines the form of the received signal and can distort it.     

Influence of ionospheric irregularities on the form of radio signal scattering spectrum in over-the-horizon radar sounding of the rough sea surface

This paper is concerned with the backscattering of HF radio waves from the rough sea surface, which have propagated through the ionosphere with random large-scale irregularities.

For the sake of simplicity, it is assumed in calculations that the rough sea surface is a perfectly conducting surface with the known Philips power spectrum of irregularities. Ionospheric irregularities of a random medium that are isotropic and single-scale ones, with a Gaussian spectrum, are considered within the limits of the hypothesis of frozen-in irregularities.

Within the first approximation of perturbation theory, using, as the incident wave and the Green function, their geometrical-optics approximations, we obtained the expression for the backscattering spectrum of the ionospheric chirp radio signal with a Gaussian envelope. The expression involves the parameters of the receive-transmit antenna, the signal, the propagation medium, and of the scattering surface. Numerical simulation was used to investigate the influence of all the above-mentioned parameters on the backscattering spectrum. It is shown that travel of ionospheric irregularities has the largest influence on the scattering spectrum, the signal parameters mainly determine the size of the scattering area in the range, and the form of the coherent integration window determines the form of the received signal and can distort it.     

Influence of the Parameters of a Random Medium on the Characteristics of Transient Reflections

Based on the invariant imbedding method, we study numerically the statistical characteristics of the kernel of the backscattering operator in the case of normal incidence of a plane wave on a one-dimensional random medium with strong fluctuation intensities and various correlation radii of the irregularities. The local reflection coefficient of the medium is modelled by a centered Gaussian process with an exponential correlation function. The first eight one-point cumulants and the correlation functions of delta-pulse reflection are considered and the fluctuation phenomena are analyzed. The transition to the diffusion scattering regime is studded, and the numerical results are compared with the known analytical solutions.     

Effect of the illumination region of targets on waves scattering in random media with H-polarization

This paper addresses some parameters that have a significant effect on wave scattering in random media. These parameters are: target configuration, including size and curvature; random media strength, represented in the spatial coherence length; and incident wave polarization. Here, I present numerical calculations for the radar cross-section (RCS) of conducting targets and analyze the backscattering enhancement with different configurations. I postulate a concave illumination region and consider targets taking large sizes of about five wavelengths. In this aspect, waves scattering from targets are assumed to propagate in free space and a random medium with H-polarization. This polarization produces what is well known as creeping waves which in turn have an additional effect on the scattering waves that is absent in the case of E-polarization.     

Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator

The time-reversal effects on super-resolution in random scattering media are analysed using numerical finite-difference time-domain (FDTD) simulations. The analytical solutions and results have been presented previously in the literature, which provide confirmation of spot-size reduction and also explanations of the shower curtain effects and backscattering enhancement. However, the analytical solutions are based on several approximations. Thus, validation of the analytical results against realistic scattering events is necessary. Two-dimensional FDTD Monte Carlo simulations have been employed for this investigation to simulate wave propagation and scattering in a random medium. The scattering environments are created by randomly locating cylindrical rods in the background medium. The simulation process involves a point source emitting a Gaussian pulse wave that propagates through the scattering medium, gets time-reversed, and then back-propagated into the same scattering medium. The focusing behaviours including the location of the focal point and its spot-size as a function of its transverse position are analysed. The shower curtain, particle size, and time domain effects are also investigated. In comparison, the behaviours of focusing derived by numerical results are consistent with those of previously reported analytical results. However, there are some differences, which we speculate to be mainly because of the different phase functions.     

Boson peak, flickering noise, backscattering processes and radiative transfer in random media

The method of matrix Green’s functions in the classical theory of electromagnetic waves is stated. This method allows to obtain a closed equation system in the presence of the random media for the calculation both coherent, and incoherent (fluctuating) components of radiation. The density and heterogeneity of scattering media can be arbitrary. The coherent channel is calculated independently. The fluctuating radiation distribution in the medium is developed initially by an interference pattern generated by the coherent channel. The limitations of the processes speed are absent. The theory embraces such phenomena as the boson peak, flickering noise, memory effect, backscattering processes and also conventional radiative transfer equation and Fresnel’s formulae.     

Scattering of electromagnetic waves from a slightly random surface—reciprocal theorem, cross-polarization and backscattering enhancement

The present paper deals with the electromagnetic (EM) scattering from a perfectly conductive, random surface by means of the stochastic functional approach and aims to study the backscattering enhancement associated with co-polarized and cross-polarized scattering. The treatment is based on the stochastic functional theory where the random EM field is represented in terms of a Wiener-Hermite functional of the homogeneous Gaussian random surface. To obtain more precise solutions than the previous works (Nakayama J et al 1981 Radio Sci. 16 831-53), we first establish the reciprocal theorem for vector Wiener kernels which describe the stochastic functional representation of the EM field and, using this, we derive the reciprocal relations for the co-polarized and cross-polarized scattering distribution relative to TE and TM polarizations of incident wave. Solutions for the vector Wiener kernels up to the second are obtained so precisely as to satisfy the reciprocal relations and are expressed in terms of generating matrices, so that complex EM scattering processes described by the vector Wiener kernels are given dear physical interpretations. Compact operator representations are introduced to reformulate the hierarchical kernel equations, the mass operator equation and higher-order kernel solutions. It is shown that the second vector Wiener kernel physically describes a 'dressed double-scattering' process, similar to the scalar theory (Ogura H and Takahashi N 1995 Waves Random Media 5 223-42), and that the 'dressed double scattering', which involves anomalous scattering in the intermediate scattering processes, creates the backscattering enhancement in both co- and cross-polarized scattering for both TE and TM wave incidence.     

A Study of Optical Backscattering Enhancement from One-Dimensional Rough Surface Using Monte Carlo Method

In this paper, according to Kirchhoff approximation, the optical backscattering enhancement of one-dimensional random rough surface, which includes fractal rough surfaces and random rough surfaces with Gaussian and exponential correlation simulated by Monte Carlo method, is obtained. It is shown that backscattering enhancement of random rough surfaces will increase with increasing the rms height of rough surface for a given correlation length. The angle width of backscattering enhancement is directly proportional to incident wavelength and inverse proportional to correlation length of rough surface. Complex phase of scattering field from superposed rough surface is uniformly distributed, none of the directions is of more overweight. The backscattering enhancement is also studied by wavelet analysis. The numerical results show good consistent with that of the relative references.     

Scattering of an electromagnetic wave from a planar waveguide structure with a slightly 2D random surface

This paper deals with the scattering of an electromagnetic (EM) wave from a waveguide structure with a slightly rough surface. The waveguide structure is a dielectric film on a planar, perfectly conductive surface, and the top of the film is a two-dimensional (2D) homogeneous Gaussian random surface. The treatment is based on the stochastic functional theory where the random EM field is represented in terms of a Wiener - Hermite functional of the random surface. Numerical calculations show that enhanced backscattering and cross-polarization occur, but that no enhanced satellite peak appears for a 2D random surface, in contrast to the case of a 1D surface. The enhanced backscattering is caused by the interference of two double-scattering processes and is attributed to the existence of guided waves in the scattering structure.     

Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement

This paper Presents numerical simulations, theoretical analysis, and millimeter wave experiments for scattering from one-dimensional very rough surfaces. First, numerical simulations are used to investigate the effects of roughness spectrum, height variation, interface medium, polarization, and incident angle on the backscattering enhancement. The enhanced backscattering due to rough surface scattering is divided into two cases; the RMS height close to a wavelength and RMS slope close to unity, and RMS height much smaller than a wavelength with surface wave contributions. Results also show that the enhancement is sensitive to the roughness spectrum. Next, a theory based on the first- and second-order Kirchhoff approximation modified with angular and propagation shadowing is developed. The theoretical solutions provide a physical explanation of backscattering enhancement and agree well with the numerical results. In addition to the scattering of a monochromatic wave, the analytical results of the broadening and lateral spreading of a pulsed beam wave scattering from rough surfaces are also discussed. Finally, the existence of backscattering enhancement from one-dimensional very rough conducting surfaces with exact Gaussian statistics and Gaussian roughness spectrum is verified by a millimeter-wave experiment. Experimental results which show enhanced backscattering for both TE and TM polarizations for different angles of incidence are presented.     

Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation

This paper presents a theory of scattering from very rough metallic and dielectric surfaces using the first- and second-order Kirchhoff approximations (KA) modified with the angular and propagation shadowing. The shadowing functions limit the single and double scattered waves which are illuminated and not shadowed by the surface. The theoretical results are compared with the Monte Carlo simulations showing the range of validity of the theory. The theory is applicable to the range where the RMS height is close to a wavelength and the RMS slope is close to unity, and the second medium is lossy. The second-order scattering includes two waves travelling in opposite directions on the surface, giving a physical explanation of the enhanced backscattering.     

Interference effects in backscattering of light by a layer of a discrete random medium

Multiple backscattering of light by a layer of a discrete random medium is considered. A brief derivation of equations for describing the coherent and incoherent components of scattered light is presented. These equations are solved numerically in the approximation of doubled scattering of light by a semi-infinite medium of spherical scatterers having a size comparable with the wavelength in order to study the effect of the properties of particles on the angular dependence of interference effects. Calculations show that the half-width of the interference peak decreases upon an increase in lateral scattering by particles and that the degree of polarization has a complex angular dependence on the properties of the particles. For an optically thin layer of the medium, the relations defining the interference peak half-width and the scattering angle upon extreme linear polarization as functions of the effective refractive index are given.     

Numerical,analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement

Abstract

This paper Presents numerical simulations, theoretical analysis, and millimeter wave experiments for scattering from one-dimensional very rough surfaces. First, numerical simulations are used to investigate the effects of roughness spectrum, height variation, interface medium, polarization, and incident angle on the backscattering enhancement. The enhanced backscattering due to rough surface scattering is divided into two cases; the RMS height close to a wavelength and RMS slope close to unity, and RMS height much smaller than a wavelength with surface wave contributions. Results also show that the enhancement is sensitive to the roughness spectrum. Next, a theory based on the first- and second-order Kirchhoff approximation modified with angular and propagation shadowing is developed. The theoretical solutions provide a physical explanation of backscattering enhancement and agree well with the numerical results. In addition to the scattering of a monochromatic wave, the analytical results of the broadening and lateral spreading of a pulsed beam wave scattering from rough surfaces are also discussed. Finally, the existence of backscattering enhancement from one-dimensional very rough conducting surfaces with exact Gaussian statistics and Gaussian roughness spectrum is verified by a millimeter-wave experiment. Experimental results which show enhanced backscattering for both TE and TM polarizations for different angles of incidence are presented.     

Coherent backscattering of light by a layer of discrete random medium

We consider the problem of backscattering of light by a layer of discrete random medium illuminated by an obliquely incident plane electromagnetic wave. The multiply scattered reflected radiation is assumed to consist of incoherent and coherent parts, the coherent part being caused by the interference of multiply scattered waves. Formulas describing the characteristics of the reflected radiation are derived assuming that the scattering particles are spherical. The formula for the incoherent contribution reproduces the standard vector radiative transfer equation. The interference contribution is expressed in terms of a system of Fredholm integral equations with kernels containing Bessel functions. The special case of the backscattering direction is considered in detail. It is shown that the angular width of the backscattering interference peak depends on the polar angle of the incident wave and on the azimuth angle of the reflection direction.     

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

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 (~10 000) 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.     

Scattering of a TM plane wave from periodic random surfaces

This paper deals with the scattering of a TM plane wave from conductive periodic random surfaces. By means of the stochastic functional approach, the scattered field is expressed in terms of a harmonic series representation, in which the coefficients are homogeneous random functions and are given by Wiener-Hermite expansions. An approximate solution for the Wiener kernels is obtained up to the second order. Several anomalies appear in the angular distribution of the incoherent scattering because of combinations of scattering due to surface randomness and diffraction due to surface periodicity. These are incoherent Wood's anomalies associated with guided surface waves propagating along the surface, enhanced backscattering and diffracted backscattering enhancement. The physical reasons for these anomalies and numerical results are discussed.