A Fast Numerical Method for Electromagnetic Scattering From Dielectric Rough Surfaces

Based on the Propagation-Inside-Layer Expansion (PILE) and Forward-Backward method (FBM), the composite scattering from the target below a dielectric rough soil surface using the extended PILE (EPILE) combined with the Forward-Backward method (FBM) is studied. The accuracy and e-ciency of the EPILE + FBM for this speciflc type of composite scattering is researched by comparing with the method of moments (MOM), the in∞uences of the target size, target depth, target horizontal distance, the rms height, the correlation length, the incident angle and the soil moisture content, etc., to the bistatic scattering coe-cient (BSC) are also investigated.

A robust and accurate iterative algorithm is proposed for the solution of electromagnetic wave scattering from 1-D layered rough surfaces. The forward–backward method (FBM) has been applied extensively for solving electromagnetic scattering from single-layer rough surfaces due to its rapid convergence rate. Nevertheless, the performance of the FBM deteriorates, as it is applied to layered random rough surface problems. The proposed method greatly improves the convergence rate of the FBM by applying a residual minimization step. The computational cost of the minimization step and the forward–backward sweeps is expedited by using the spectral acceleration. In this communication, the proposed algorithm is applied as a solution to the problem of electromagnetic wave scattering from layered rough surfaces. Numerical results are presented to demonstrate the better convergence properties of the proposed method as compared with the standard FBM. In addition, the proposed method is shown to be more robust than the FBM, especially as the distance between interfaces decreases. The computational complexity of the proposed algorithm is also presented and discussed.

The Forward-Backward (FB) method and its accelerated version (SA-FB, Spectral Accelerated Forward-Backward) were developed to compute the electromagnetic scattering from ocean-like rough surfaces. The FB algorithm provides very accurate scattering results and a very fast convergence, but its computational cost is still O(N2) per iteration, being N the number of unknowns used to discretize the surface. The Spectral Acceleration Algorithm proposed later reduces this cost to O(N). Nevertheless, the SA-FB can only be applied to quasi-planar structures or slightly rough surfaces (such as the sea surface), but it cannot be used to solve problems involving very undulating surfaces such as terrain profiles. In this paper the SA-FB method is modified to enhance its scope of application to considerable undulating surfaces. It will be seen that significant changes with respect to the original formulation as well as modifications of the integration path are necessary to get accurate results when dealing with terrain profiles. The proposed method is presented for vertical and horizontal polarizations and for PEC and non-PEC surfaces.

In this paper the forward-backward (FB) method is used to solve the problem of scattering on a very rough surface for practical application to radio planning and site optimization for mobile radio systems. Novel spectral acceleration was developed for this method to reduce its computational cost to O(N), but the original implementation cannot be applied to solve the scattering on very rough surfaces. In this paper the spectral acceleration for the FB method is adapted to apply it on the calculation of scattering on considerable undulating surfaces.

One dimension band-limited fractal surfaces with different fractal dimension and different rms slope are created by Monte Carlo simulation. The curves of the single scattering shadowing function of the band-limited fractal surface with varying of the incident angle are obtained by utilizing a ray-tracing algorithm, the influence of different fractal dimension and different rms slope on the fractal surface shadowing function is analyzed. The curve of the shadowing function with varying of the incident angle for the fractal surface and Gaussian rough surface is compared with same rms slope. The results show that the curves of the shadowing function for the fractal surface with different fractal dimension and three Gaussian rough surface are basically same.

Numerous analytical radar-scattering laws have been published through the past decades to interpret planetary radar observations, such as Hagfors’ law, which has been commonly used for the Moon, and the cosine law, which is commonly used in the shape modeling of asteroids. Many of the laws have not been numerically validated in terms of their interpretation and limitations. This paper evaluates radar-scattering laws for self-affine fractal surfaces using a numerical approach. Traditionally, the autocorrelation function and, more recently, the Hurst exponent, which describes the self-affinity, have been used to quantify the height correlation. Here, hundreds of three-dimensional synthetic surfaces parameterized using a root-mean-square (rms) height and a Hurst exponent were generated, and their backscattering coefficient functions were computed to evaluate their consistency with selected analytical models. The numerical results were also compared to empirical models for roughness and radar-scattering measurements of Hawaii lava flows and found consistent. The Gaussian law performed best at predicting the rms slope regardless of the Hurst exponent. Consistent with the literature, it was found to be the most reliable radar-scattering law for the inverse modeling of the rms slopes and the Fresnel reflection coefficient from the quasi-specular backscattering peak, when homogeneous statistical properties and a ray-optics approach can be assumed. The contribution of multiple scattering in the backscattered power increases as a function of rms slope up to about 20% of the backscattered power at normal incidence when the rms slope angle is 46°.

The Fresnel corrected Kirchhoff approximation is applied to drive expressions for the temporal response and spatial coherence of high-frequency acoustic signals forward scattered from random rough surfaces. With the assumption of a Gaussian distribution of heights, the results depend only on the rms height and slope of the scattering surface. The expressions obtained are numerically evaluated for comparison with representative Arctic acoustic data. It is found that relatively low values of the rms height and slope must be assumed to match high-frequency temporal response data. These same slope and height values yield predictions of the transverse horizontal spatial coherence consistent with acoustic measurement for closely spaced receivers. [Work supported by NOARL.]

We present experimental results on the scattering of electromagnetic waves at millimeter‐wave frequencies from one‐dimensional very rough conducting surfaces with controlled surface roughness statistics. Very rough surfaces are defined as surfaces with rms height and correlation length of the order of a wavelength such that the rms slope is at least unity. It is expected that scattering experiments using these surfaces can provide useful insights since their statistics lie outside the range of validity of the present theories, namely, the Kirchhoff and perturbation theories. Strong backscattering enhancement at different incident angles, both in the transverse electric and transverse magnetic polarizations, are observed experimentally. Numerical calculations based on the exact integral equation method for cylindrical beam wave illumination compare favorably with the experimental results. The agreement between measurements and numerical calculations is good over a wide range of incident angles and for all scattering angles. The close agreement between the experimental results and numerical simulations indicates that this controlled experimental setup can be used to study scattering phenomena from one‐dimensional very rough surfaces with different roughness statistics as well as from two‐dimensional rough surfaces.

A theory of scattering from very rough surfaces is presented. Both the surface rms height and the correlation distance are close to a wavelength and the rms slope is close to unity. The theory is based on the first- and modified second-order Kirchhoff approximations. The second-order scattering includes the incident and scattering shadowing and the angular and propagation tapering shadowing. The angular shadowing limits the angular spectrum of the second-order scattering within those intercepted by the surface, while the propagation shadowing limits the propagation distance within those intercepted by the surface. The calculations are made for the rms height equal to 1.5, 1.0, and 0.5 wavelengths and the correlation distance equal to 1.8 wavelengths and show close agreement with the Monte Carlo simulation of the exact integral equation. The results show that the first-order Kirchhoff scattering is dominant for rms height equal to 0.5 wavelengths, but for rms heights equal to 1.0 and 1.5 wavelengths, the second-order scattering becomes comparable to the first-order and produces the enhanced backscattering. Therefore, this theory provides a possible physical explanation for the enhancement. The range of validity of the theory is also discussed. [Work supported by ARO and NSF.]

One goal of radar remote sensing is the extraction of terrain statistics and surface dielectric properties from backscatter data for some range of wavelengths, incidence angles, and polarizations. This paper addresses empirical approaches used to estimate terrain properties from radar data over a wider range of roughness than permitted by analytical models. Many empirical models assume, at least implicitly, that roughness parameters like rms height or correlation length are independent of the horizontal length scale over which they are measured, in contrast to recent surveys of natural terrain, which show that self-affine, or power-law scaling, between horizontal scale and roughness statistics is very common. The rms slope at the horizontal scale of the illuminating wavelength s(lambda) is directly related to the variogram or structure function of a self-affine surface, can be readily obtained from field-measured topography, and, when used in an empirical model, avoids the need for arbitrary wavelength-dependent terms. To facilitate comparison with earlier approaches, an expression that links the rms height at some profile length with the rms (Allan) deviation at an equivalent horizontal sampling interval is obtained from numerical simulations. An empirical model for polarimetric scattering as a function of s(lambda) at 35deg-60deg incidence from smooth to rugged lava surfaces is derived and compared with earlier models for backscatter from modestly rough soil surfaces. The asymptotic behavior of polarization ratios for the lava flows suggests that the depolarization of linear-polarized illuminating signals occurs as a first-order process, likely through single scattering by rock edges or other discontinuities, rather than as the solely multiple-scattering effect predicted by some analytical models. Efforts to fully understand radar scattering from geological surfaces need to incorporate wavelength-scale roughness, perhaps through computational simulations.

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.

An efficient method for simulating electromagnetic scattering from a dielectric rough surface over a frequency band is proposed in this paper. This method is based on Chebyshev series and Maehly approximation. The tapered plane wave is chosen as the incident wave to avoid the artificial edge diffraction due to the finite length of the simulated rough surface. The Gaussian rough surface is considered. Good agreements between the Maehly approximation and the exact results obtained by the method of moment (MoM) are exhibited. And the Maehly approximation is superior in the CPU time.

To numerically calculate bistatic scattering from a fractal rough surface with high permittivity, a comprehensive approach of the forward-backward method (FBM) with spectral acceleration algorithm (SAA) combined with the physics-based two-grid (PBTG) method is developed. Bistatic scattering from and transmitting through one-dimensional (1-D) fractal rough surface with high permittivity, under TE and TM tapered wave incidence, are numerically simulated. Angular reflectivity and transmittivity demonstrate the energy conservation and the functional dependences upon the surface parameters.

In this paper, an efficient numerical approach is developed for solving the baseline problem with interval uncertainties. Interval arithmetic is also utilized for developing the proposed method in the presence of uncertainties with only lower and upper bounds of parameters. In the proposed method, the equation of motion, performance index, and boundary conditions are first changed into some algebraic equations. This process converts the problem into an optimization problem. The presented technique is based on an interval extension of Chebyshev polynomials where its coefficients are achieved by world cup optimization algorithm, as a new optimization algorithm. The proposed method approximates the control and state variables as a function of time. The proposed solution is based on state parameterization, such that the state variable is approximated by the proposed interval Chebyshev polynomials with unknown interval coefficients. Finally, by solving the baseline problem in the presence of interval uncertainties, the reliability and effectiveness of the proposed method are demonstrated.

The Kirchhoff integral is a fundamental integral in scattering theory, appearing in both the Kirchhoff approximation and the small slope approximation. In this work, a functional Taylor series approximation to the Kirchhoff integral is presented, under the condition that the roughness covariance function follows either an exponential or Gaussian form-in both the one-dimensional and two-dimensional cases. Previous approximations to the Kirchhoff integral [Gragg, Wurmser, and Gauss (2001) J. Acoust. Soc. Am. 110(6), 2878-2901; Drumheller and Gragg (2001) J. Acoust. Soc. Am. 110(5), 2270-2275] assumed that the outer scale of the roughness was very large compared to the wavelength, whereas the proposed method can treat arbitrary outer scales. Assuming an infinite outer scale implies that the root mean square (rms) roughness is infinite. The proposed method can efficiently treat surfaces with finite outer scale and therefore finite rms height. This series is shown to converge independently of roughness or acoustic parameters and converges to within roundoff error with a reasonable number of terms for a wide variety of dimensionless roughness parameters. The series converges quickly when the dimensionless rms height is small and slowly when it is large.