Scattering from very rough surfaces based on the modified second-order Kirchhoff approximation with angular and propagation shadowing

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.

A new iterative solution to the problem of sea surface scattering is obtained which has the Kirchhoff approximation as its leading term. This solution is obtained by transforming an integral equation of the first kind for the surface potential to one of the second kind, and applying the method of successive approximations. The advantages of this method are that the conditions under which this solution converges may be readily established and the solution may be iterated to sufficiently high order in the interaction to examine the rate at which it converges. A numerical example of scattering from an echelette grating is considered, and the regime of convergence established. It is found that the new iterative solution converges only for such surfaces on which kσσ′<π><π where k is the wave number of the scattered radiation and σ and σ′ are, respectively, the rms surface height and slope. This finding raises doubts concerning the regime in which the Kirchhoff approximation is traditionally deemed applicable. Finally, numerical studies are presented to illustrate the convergence of the iterative solution.

Wave scattering from rough surface is important in many diverse areas, including radio communications, microwave remote sensing, underwater acoustics. The recent interest in bistatic radar, or the study of radio links above the sea surface, has led to the need for information concerning bistatic reflectivity of clutter viewed by such systems, particularly for near grazing incidence. The problem of developing a model for a rough surface is very difficult since at best /spl sigma/ is dependent upon (at least) radar frequency, geometrical parameters, physical parameters, incident and observation angles, and polarization. Therefore, it is necessary to restrict the areas of applicability of such models. A first major new consideration in this paper is to study the domain of validity of the Kirchhoff and small-perturbation rough surface scattering models (in the bistatic case). Comparisons between the numerical calculations and the models are presented for various surface rms height and correlation length both normalized to the incident wave number (denoted by k/spl sigma/ and kL, respectively). These two parameters form a two-dimensional space, into which the approximate regions of validity are established. The second new major consideration is the development of the theoretical composite model describing bistatic reflectivity. One application is implemented for the ocean surface case. Numerical results for the cross sections are compared to those found using Kirchhoff approximation (Physical Optics), small-perturbation model, and the phase perturbation technique.

The problem of electromagnetic scattering from a randomly rough surface is analyzed using the Kirchhoff approximation (stationary phase, scalar approximation), the small perturbation model and the two-scale model. The authors show that physical and geometrical optics approximations can be used for a rough sea surface. A small perturbation model gives a satisfactory solution. Unfortunately, it only applies to a smooth sea surface. Therefore, these methods have been shown to be limited in their regions of validity. Comparisons between the numerical calculations and the models are made for various surface rms heights and correlation lengths both normalized to the incident wave number (denoted by k/spl sigma/ and kL, respectively). The two-scale model proposed in this paper is the method which tries to conciliate the two previous aspects. One application is made in the ocean surface case. A set of experimental data, describing simultaneous bistatic radar cross section and forward radar cross section measurements of sea clutter taken with a moderately high resolution X-band scatterometer system operating near grazing incidence for both horizontal and vertical polarizations over a range of low grazing bistatic angles, is used but not presented in this paper to examine the applicability of bistatic scattering models.

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.

This paper presents radar-altimeter scattering models for each of the diagenetic zones of the Greenland ice sheet. AAFE radar-altimeter waveforms obtained during the 1991 and 1993 NASA multi-sensor airborne altimetry experiments over Greenland reveal that the Ku-band return pulse changes significantly with the different diagenetic zones. These changes are due to varying amounts of surface and volume scattering in the return waveform.In the ablation and soaked zones, where surface scattering dominates the AAFE return, geophysical parameters such as rms surface height and rms surface slope are obtained by fitting the waveforms to a surface-scattering model. Waveforms from the percolation zone show that sub-sruface ice features have a much more significant effect on the return pulse than the surrounding snowpack. Model percolation waveforms, created using a combined surface- and volume-scattering model and an ice-feature distribution obtained during the 1993 field season, agree well with actual AAFE waveforms taken in the same time period. Using a combined surface- and volume-scattering model for the dry-snow-zone return waveforms, the rms surface height and slope and the attenuation coefficient of the snowpack are obtained. These scattering models not only allow geophysical parameters of the ice sheet to he measured but also help in the understanding of satellite radar-altimeter data.

In this paper we study the validity of the small perturbation method for 1-D perfectly conductive random rough surfaces. We have obtained an expression for the diffuse component of the mean scattered intensity up to fourth order in powers of k0σ for S- waves (k0 = 2π/λ, a is the rms height). For P-waves it is not possible to obtain the fourth-order term. The perturbative expressions are derived from the Rayleigh hypothesis and also from the extinction theorem. In both cases the final expressions for the mean diffuse intensity are the same. Comparisons with exact numerical results and with those predicted by the Kirchhoff approximation are made. This permits us to establish some criteria on the validity of the perturbative solution, obtaining some new interesting facts. In particular, we show that the total incoherently scattered energy is the parameter that basically determines the applicability of the perturbative solution. Precise curves marking the zones of validity of the second-order perturbative intensity are drawn. Simultaneously, the shape of the diffuse halo as a function of the correlation length T, the angle of incidence θ0, and the polarization is studied at small k0σ.

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.]

Terrestrial light detection and ranging (LiDAR) offers significant advantages over conventional techniques for measuring the centimeter-scale surface roughness of natural surfaces, such as sea ice. However, the laser scanning technique is inherently limited, principally by the following: 1) the high inclination scanning angle of the sensor with respect to nadir; 2) the precision of the laser ranging estimate; and 3) the beam divergence of the laser. In this paper, we introduce a numerical model that has been designed to simulate the acquisition of LiDAR data over a regular rough surface. Results from the model compare well $(r^2 = 0.97) $ with LiDAR observations collected over two experimental surfaces of known roughness that were constructed from medium-density fibreboard using a computer numerical control three-axis router. The model demonstrates that surface roughness parameters are not sensitive to minor variations in the LiDAR sensor's range and laser beam divergence, but are slightly sensitive to the precision of the ranging estimate. The model also demonstrates that surface roughness parameters are particularly sensitive to the inclination angle of the LiDAR sensor. The surface RMS height is underestimated, and the correlation length is overestimated as either the inclination angle of the sensor or the true roughness of the surface increases. An isotropic surface is also increasingly observed as an anisotropic surface as either the inclination angle or the true surface roughness increases. Based on the model results, we propose a set of calibration functions that can be used to correct in situ LiDAR measurements of surface roughness.

The actual rough sea surface is described by Pierson-Moscowitz (PM) spectrum and Monte Carlo method; the composite electromagnetic scattering from the double rectangular crosssection conductor columns above the rough sea surface is investigated using the hybrid methodthe Method of Moment with the Kirchhoff approximation. Angular distribution curves of composite scattering coefficient are obtained; and this coefficient that varies with the parameter of sea surface, rectangular cross-section conductor columns and incident waves is calculated in detail. The characteristics of the composite scattering coefficient from the double rectangular cross-section conductor columns above the rough sea surface are also obtained. Results show that the hybrid methodthe Method of Moment with the Kirchhoff approximation, compared to the traditional Method of Moment, can obtain higher accuracy. Moreover, the hybrid method can effectively reduce the computation time and memory footprint, and the performance becomes more favorable with the increase of the sizes of rough surface.

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.

Compared with scattering from a rough surface only, composite scattering from a target above a rough surface has become so practical that it is a subject of great interest. At present, this problem has been solved by some numerical methods which will produce an enormous calculation amount. In order to overcome this shortcoming, the reciprocity theorem (RT) and the method of equivalent edge currents (MEC) are used in this paper. Due to the advantage of RT, the difficulty in computing the secondary scattered fields is reduced. Simultaneously, MEC, a high-frequency method with edge diffraction considered, is used to calculate the scattered field from the cone-cylinder target with a high accuracy and efficiency. The backscattered field and the polarization currents of the rough sea surface are evaluated by the Kirchhoff approximation (KA) method and physical optics (PO) method, respectively. The effects of the backscattering radar cross section (RCS) and the Doppler spectrum on the size of the target and the windspeed of the sea surface for different incident angles are analysed in detail.

One of the most popular methods for calculating the scattering of light from rough surfaces is the Kirchhoff approximation. A recent modification of the Kirchhoff approximation allows calculation of the scattering of electromagnetic radiation from rough surfaces with very high, or even infinite, slopes, which occurs frequently in, for example, imaging or testing of printed circuits or remote sensing applications. The modification involves a simple change in the way the surface normal is described which removes the explicit dependence of the resulting integral equations on the surface slope, and changing the sampling of the surface, removing the requirement of small slopes for application of the method. In this paper we present the application of the method to vector electromagnetic scattering, using the Stratton-Chu equation, from 2D rough metallic surfaces with infinite slope. The single and double scatter contributions to the total scattered intensity are calculated numerically with this new formulation. Geometrical shadowing is included by using ray tracing directly on the surface. The theoretical basis and preliminary numerical results of the application of this method to vector electromagnetic scattering from simple rough metallic surfaces (plane surfaces with single grooves or with a small number of identical grooves) with infinitely sloped structure are presented.

We study the electromagnetic scattering problem on a random rough surface when the height distribution of the profile belongs to the family of alpha-stable laws. This allows us to model peaks of very large amplitude that are not accounted for by the classical Gaussian scheme. For such probability distributions with infinite variance the usual roughness parameters such as the RMS height, the correlation length or the correlation function are irrelevant. We show, however, that these notions can be extended to the alpha-stable case and introduce a set of adapted roughness parameters that coincide with the classical quantities in the Gaussian case. Then we study the scattering problem on a stationary alpha-stable surface and compute the scattering coefficient under the first-order Kirchhoff and small-slope approximations. An analytical formula is given in the high-frequency limit, which generalizes the well known geometrical optics approximation. Some numerical results are given and discussed.

Predictions of the fourth-order small perturbation method (SPM) are examined for scattering from two rough surfaces in a layered geometry. Cross-polarized backscatter, in particular, is emphasized because use of the fourth-order SPM is required to obtain this quantity. The formulation of the SPM fields and incoherent ensemble-averaged normalized radar cross sections (NRCSs) up to the third and the fourth order in surface rms heights, respectively, are reviewed. It is shown that the fourth-order NRCS includes distinct contributions from upper and lower interface roughnesses, as well as an “interaction” term that couples the upper and lower interface roughnesses. A comparison with NRCS values computed using the “numerically exact” method of moments in the full bistatic scattering pattern is shown for verification, and NRCS values at the second and the fourth order are compared in order to assess the convergence of the SPM series. Although the number of parameters inherent in the two-layer rough surface scattering problem makes an exhaustive study of scattering effects difficult, several illustrative examples are presented to capture a range of scattering behaviors. The results emphasize the importance of interactions between the rough surfaces in producing cross-polarized backscattering and also indicate an increased significance of fourth-order contributions in the two-layer geometry as compared to the single-layer case.