First-order Small Slope Approximation Research Articles

Electromagnetic Scattering from Fractional Brownian Motion Surfaces via the Small Slope Approximation

Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful to evaluate the normalized radar cross section (NRCS) of fBm surfaces. This task has been accomplished in the past by using either the Kirchhoff approximation (KA) or the small perturbation method (SPM). However, KA and SPM have rather limited ranges of application in terms of surface roughness and incidence angle: a wider range of application is achieved by the small slope approximation (SSA), more recently developed, but the latter has not been applied yet to fBm surfaces. In this paper, the first-order SSA (SSA-1) is applied to the evaluation of scattering from fBm surfaces obtaining an analytical formulation of their NRCS. It is then shown that the obtained SSA-1 expression reduces to the KA and SPM ones at near-specular and far-from-specular scattering directions, respectively. Finally, the results of the proposed method are compared to experimental data available in the literature.

Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces

This paper presents new results of a simulation of radar backscatter from oil slick areas on a real three-dimensional sea surface, based on a physical hydrodynamic model of surface wave damping in the presence of oil films, the local equilibrium model (MLB). To solve this problem, the modelling was carried out by using the first-order small-slope approximation (SSA1) model. It presents the advantage of having a very good compromise between rapidity and accuracy of the calculation. The choice of the model is justified by solving the two-dimensional problem with several asymptotic methods and further comparing the results with a rigorous numerical method, based on the Method of Moments (MoM). Two approaches called “thin-layer” (TL) and “classical” were used to deal with the double layer (air/oil/sea) problem. The TL approach assumes that this double-layer problem can be seen locally as a Fabry–Pérot interferometer, which implies that the Kirchhoff-tangent plane approximation (KA) is valid. The classical approach consists in neglecting the presence of the oil layer for dealing with electromagnetic backscattering, which is valid for very thin oil films compared to the electromagnetic (EM) wavelength. It is shown that these two approaches have rather complementary validity domains: The TL approach is always valid for small observation angles, which makes it suitable for near nadir sensors such as altimeters, whereas the classical approach is valid for moderate observation angles, which makes it suitable for most satellite applications. The 3D modelling results are compared with C-band and X-band measured data (CSK experiment and OOW NOFO experiment) in VV polarization. The calculation takes into account that the oil film on the sea surface is mainly in an emulsion state. The results highlighted the relevance of the MLB hydrodynamic model, as well as the SSA1 EM model combined wit the classical approach, for quantifying NRCS in seas contaminated with marine oil or surfactants. The agreement is indeed very good in the X-band range.

An Analytical Formulation of the Correlation of GNSS-R Signals

We present an analytical formulation of the correlation coefficient of the electromagnetic fields scattered at near-specular direction by a rough or gently undulating surface and measured at two spatially separated positions occupied by a moving receiver at slightly different times. This allows us obtaining an explicit expression of the correlation time of the received signal in terms of radar and surface parameters. This work provides a contribution to the discussion, currently ongoing in the Global Navigation Satellite System Reflectometry (GNSS-R) scientific community, about the behavior of received signal fluctuations, especially when surface profile variations are such that the scattering is neither coherent nor completely incoherent. The scattering surface is here modeled as randomly rough, and the Kirchhoff Approximation (KA) or the first-order Small Slope Approximation (SSA1) is employed to compute the scattered field. In fact, the expression of the correlation coefficient is the same for both approximations. The obtained closed-form expression shows that, as the surface correlation length increases, the degree of coherence smoothly increases from the value obtained with the expression already available in literature for very rough surfaces to a value close to unity for gently undulating surfaces. The obtained behavior of correlation time as a function of surface parameters, system resolution and observation geometry is in agreement with numerical simulations available in literature. More in general, obtained analytical results are in agreement with the observed behavior of GNSS-R signals over flat land surfaces.

On the Convolutions of Sea Wave Spectrum in Radar Backscattering From Ocean Surfaces

By expanding the first-order small slope approximation scattering model in the polynomial terms containing convolutions of the sea wave spectrum, this letter quantitatively investigated how the convolutions of the sea wave spectrum affect the normalized ocean backscattering cross-section (NBCS) under various wind speeds, radar frequencies, and incident angles. First, numerical results show that only a portion of convolution powers of the sea wave spectrum contribute significantly to the NBCS, and such a portion of convolution powers are defined as the “effective convolutions”. Based on this, an empirical model related to the radar frequency, incident angle, and wind speed is established to determine the effective convolutions to reduce the computation cost. Finally, the NBCSs using the effective convolutions by the empirical model and the reference data calculated by the direct numerical integration are compared. The comparison results indicate that compared to the direct numerical integration, implementing the first-order small slope approximation with the effective convolutions has high accuracy and relatively low computational cost, notably with small Rayleigh parameters.

Quantitative Evaluation of the Effects of Large-Scale Sea Wave Components on Microwave Backscattering From Ocean Surface

This paper quantitatively examined the effect of these long-wave components with wavelength larger than the Bragg component on radar backscattering using the series-expanded first-order small slope approximation (SSA-1) and the Hwang sea wave spectrum model. Firstly, some roughness with various scales are constructed using different long-wave sea spectrum components and subsequently are cast into SSA-1 to obtain the corresponding normalized backscattering cross-section (NBCS) at different radar frequency, incidence angles, and wind speeds, respectively. Then, the effect of large-scale wave components on radar backscattering is revealed by analyzing the NBCSs with different roughness. Finally, the existing wavelength filtering theory explained the effect of large-scale waves on radar backscattering. It was also verified by another scattering model, i.e., the advanced integral equation model (AIEM) and radar measurements. Besides, we also investigate the effects of sea states and swell waves on the numerical simulation results. The effect of long-scale waves on NBCS clarifies the source of backscattering from the multiscale sea surface, which is helpful for both analytical and numerical scattering models to explore the sea surface scattering properties and better interpret the radar sensing of sea surface. Also, the examination of the effect of sea states and swell waves on radar backscatter is instructive for reducing the inversion errors of sea surface wind speed from radar measurements by means of configuring the appropriate radar parameters (e.g., high frequencies and large incidence angles).

Doppler characteristics of electromagnetic echoes from sinusoidal water waves illuminated by plane wave/Gaussian beam

The Doppler spectra of the scattering fields from sinusoidal water waves illuminated by a plane wave or Gaussian beam are numerically simulated with the method of moment (MoM). When the sinusoidal water surface is illuminated by a plane EM wave, the numerical results show that the Doppler spectrum of the scattering field is distorted seriously due to the frequency leakage. For this case, not only the resonance spectral peaks corresponding to the phase velocity of the water wave but also other equally spaced harmonic peaks appear on the spectrum curve. However, when the sinusoidal water surface is illuminated by a Gaussian beam with appropriate beam width, the harmonic peaks can be effectively suppressed. To better explain this phenomenon, the theoretical model for the spectrum of the scattering field from a sinusoidal water wave is derived based on the first-order small slope approximation theory (SSA-I). And this theoretical model proposed by us could give a reasonable explanation for the above phenomenon.

Scattering from a slightly rough chiral surface: analysis with the small perturbation method and the small slope approximation

We study the scattering of a linearly polarized electromagnetic plane wave by a two-dimensional random slightly rough surface separating the vacuum from a chiral medium. We implement the first-order small perturbation method (SPM) and the first-order small slope approximation (SSA) and determine the analytical expressions of the coherent and incoherent intensities. The effects of chirality on the polarization of the wave scattered within the vacuum are analyzed. The coherent intensity has a cross-polarized component as well as the incoherent intensity in the incidence plane. We show that there are configurations for which a total polarization coupling occurs with the co-polarized incoherent intensity equal to zero.

Surface reverberation based on small-slope approximation in deep water

For the reverberation experiments with a near surface source or receiver, surface reverberation arriving first after source signal is not influenced by bottom reverberation due to time-delay difference of acoustic multipath in deep water. A surface reverberation model is proposed in the paper. The Green’s function of sound propagation is described by the ray theory, and first-order small-slope approximation is employed for the surface scattering from full angle. The effect of bubbles scattering is also considered to get the surface reverberation theory. The reverberation model is verified by comparing the simulation results with the experimental data. The measured data show that the decaying rate of surface reverberation intensity decreases with the frequency increasing. Numerical calculation demonstrates that the frequency dependence is caused by the positive correlation between scattering strength and frequency, and the surface reverberation at low frequency for low sea state is dominated by the scattering from rough air-sea interface. Moreover, the reverberation data from experiment show that the surface reverberation is not sensitive to the change of receiving depth. A method of inversion for the two surface-wave spectral parameters in the model is achieved based on the reverberation model. The inversion results verify that the spectral parameters of rough surface can be obtained from surface reverberation data on the premise of the wind speed parameter. As a result, the scattering properties of rough interface will be obtained.

Investigation on THz EM Wave Scattering From Oil-Covered Sea Surface: Exploration for an Approach to Probe the Thickness of Oil Film

In this article, a theoretical study is carried out to seek an approach to remotely measure the thickness of the oil film. It is known that it is difficult to probe the oil thickness using a microwave radar due to the fact the dielectric constant of oil is very small compared with seawater for microwave frequencies. While in the range of terahertz (THz) electromagnetic (EM) wave frequency, the dielectric constants of oil and seawater are comparable, resulting in the THz EM fields scattered from the air–oil interface and those from the oil–water interface can produce constructive and destructive interference with the variation in the oil film thickness. Based on this principle, the sensitivity of the THz EM wave to the oil thickness is investigated in this work. First, the oil–seawater two-layer medium is equivalent to a single-layer medium by an equivalent dielectric constant model. Then, the THz EM scattering from the oil-covered sea surface is simulated using the first-order small slope approximation method (SSA-1) and the obtained equivalent dielectric constant. Meanwhile, the influences caused by the damping effect of the oil film, the reduction for friction velocity, and the change in equivalent permittivity on the normalized radar cross section (NRCS) are analyzed. The numerical results show that the NRCSs of THz frequencies are more sensitive to the change in oil thickness than that of microwave frequencies. This property makes the THz EM wave have the potential to probe the thickness of the oil film.

Effective approach to generate electrically large rough surfaces and the application to EM scattering problems

The small slope approximation (SSA) is a widely employed way to analyse the electromagnetic (EM) scattering problems from rough surfaces. However, it requires a surface sampling interval less than one-eighth of the wavelength, which makes it suffer from the problems of large memory consumption and low calculation efficiency for large rough surfaces, especially for higher microwave bands. In this study, a new way to generate the electrically large rough surfaces is proposed and its application to the calculation of EM scattering field is introduced on the base of first-order SSA. The calculation accuracy and consumed computer memory of the proposed method analysed by numerical comparisons show good performance.

EM Scattering From a Simple Water Surface Composed of Two Time-Varying Sinusoidal Waves

Some novel phenomena, which cannot be well explained by the traditional Bragg scattering theory, have been observed through the electromagnetic (EM) scattering from a water surface composed of two sinusoidal waves. According to the traditional Bragg scattering theory, the scattering intensity from water surface is only proportional to the spectral density of the Bragg resonant waves. However, the scattering field numerically simulated by the method of moment (MoM) method reveals that the resonant scattering field would also be affected by the amplitude of the non-resonant wave. In some special cases, despite the existence of the Bragg resonant waves, the Bragg resonant scattering field disappears when the amplitude of the non-resonant water wave is equal to some specific values. From the numerical results, another noticeable phenomenon is found that the Doppler spectrum of the scattering field is distorted seriously due to the frequency leakage. When the water surface with finite length is illuminated by a plane EM wave, not only the resonance spectral peaks corresponding to the phase velocity of the water wave but also other harmonic peaks appear on the spectrum curve. However, if a Gaussian beam is used instead of the plane EM wave, the harmonic peaks can be effectively suppressed. To better understand the phenomena, the theoretical model of the scattering field from the simple water surface is derived in the framework of the first-order small slope approximation method. And the empirical formulas for selecting the Gaussian beam width and water surface length are also proposed.

Monostatic and Bistatic Scattering Modeling of the Anisotropic Rough Soil

The electromagnetic scattering generated by agricultural tilled soils can be affected by a strong anisotropic component of the rough-surface profile. An accurate and reliable modeling of the normalized radar cross section, under both monostatic and bistatic geometries, is particularly important and desirable, especially for the correct estimation of the soil moisture content by means of satellite-based observations. In this paper, moving from the modeling so far proposed in the literature, we present and discuss a novel, more general (i.e., 2-D), spectral representation of an agricultural tilled soil, implementing a solution of the scattering based on the first-order small-slope approximation. Comparisons are given with a well-established model based on a 1-D representation of the soil correlation function, accounting for the radiation pattern of the illuminating antenna. The investigation gives new insight on the phenomenology of the bistatic scattering from the anisotropic soil, providing interesting information for the next generation of satellite missions, which foresees the possibility of launching companion satellites carrying aboard a passive receiver collecting the signal transmitted by active SAR-based platforms.

Investigation of EM Backscattering from Slick-Free and Slick-Covered Sea Surfaces Using the SSA-2 and SAR Images

This paper is devoted to investigating the electromagnetic (EM) backscattering from slick-free and slick-covered sea surfaces at various bands (L-band, C-band, X-band, and Ku-band) by using the second-order small slope approximation (SSA-2) and the measured synthetic aperture radar (SAR) data. It is known that the impact of slick on sea surface is mainly caused by two factors: the Marangoni damping effect and the reduction of friction velocity. In this work, the influences induced by these two factors on the sea curvature spectrum, the root mean square (RMS) height, the RMS slope, and the autocorrelation function of sea surfaces are studied in detail. Then, the slick-free and slick-covered sea surface profiles are simulated using the Elfouhaily spectrum and the Monte-Carlo model. The SSA-2 with the tapered incident wave is employed to simulate the normalized radar cross-sections (NRCSs) of sea surfaces. Furthermore, for slick-free sea surfaces, the NRCSs simulated with the SSA-2 at various bands are compared with those obtained by the first-order small slope approximation (SSA-1), the classic two-scale model (TSM), and the geophysical model functions (GMFs) at various bands, respectively. For slick-covered sea surfaces, the SSA-2-simulated NRCSs are compared with those obtained from C-band Radarsat-2 images and L-band uninhabited aerial vehicle synthetic aperture radar (UAVSAR) images, respectively. The numerical simulations illustrate that the SSA-2 can be used to study the EM backscattering from slick-free and slick-covered sea surfaces, and it has more advantages than the SSA-1 and the TSM. The works presented in this paper are helpful for understanding the EM scattering from the sea surface covered with slick, in theory.

Sea Surface Monostatic and Bistatic EM Scattering Using SSA-1 and UAVSAR Data: Numerical Evaluation and Comparison Using Different Sea Spectra

The microwave scatterometer is one of the most effective instruments in ocean remote sensing, which urges the need for some theoretical models to accurately estimate the scattering coefficient of the sea surface. For the simulation of the scattering from an ocean surface, the sea spectrum, or its inverse Fourier transform, autocorrelation function is essential. Currently, many sea spectral models have been proposed for describing sea waves. However, which spectrum should be adopted during electromagnetic (EM) computations? A systematic comparison of these models is needed to evaluate their accuracies. In this paper, we focus on numerical simulations of scattering from a rough sea surface in monostatic and bistatic configurations by using six different sea spectral models and the first-order small slope approximation (SSA-1). First, sea spectral models proposed by Elfouhaily et al., Hwang et al., Romeiser et al., Apel et al., Fung et al., and Pierson et al., are compared with each other from different points of view, e.g., the omnidirectional parts, the angular spreading functions, the autocorrelation functions, and the slope variances. We find that the spectra given by Elfouhaily and Hwang could reflect realistic wind sea waves more accurately. Then, the scattering coefficients are simulated in fully monostatic and bistatic configurations. Regarding the monostatic scattering, the results simulated using EM scattering models are compared with those obtained from the measured UAVSAR data in the L band and the empirical model CMOD5 in the C band. Comparisons are made for various incident angles, wind speeds, and wind directions. Meanwhile, special attention is paid to low to moderate incident angles. The comparisons show that, it is difficult to find one certain spectral model to simulate scattering coefficient accurately under all wind speeds or wind directions. Accurate estimations will be obtained using different methods according to different situations.

Doppler characteristics of EM scattering from sinusoidal or weakly nonlinear choppy waves

The spectra of electromagnetic (EM) scattering fields from sinusoidal or weakly nonlinear choppy waves are presented. In general, we always consider that the Doppler shift of EM scattering from a Bragg resonant wave is determined by the phase velocity of the water wave. However, the recent wave tank experiments and the numerical simulations demonstrate that not only the spectral peak corresponding to the phase velocity of water wave but also other equally spaced harmonic peaks can be found when a EM wave is illuminated on a sinusoidal or a simple weakly nonlinear choppy wave. The intensities of the harmonic peaks are affected by the amplitude, the frequency, and the hydrodynamic model of the water wave. To better explain these phenomena, a theoretical model for the spectra from the simple water waves are derived by the first-order small slope approximation theory (SSA-I). The comparisons with the numerical simulations illustrated that all the interesting phenomena could be successfully explained the theoretical spectral model.

Upwind–Downwind Asymmetry of the Sea Backscattering Normalized Radar Cross Section Versus the Skewness Function

From the first-order small slope approximation, this paper presents a closed-form expression of the sea backscattering normalized radar cross section (NRCS) versus the skewness function to predict the upwind-downwind asymmetry. This effect is produced by the fact that the surface height distribution slightly deviates from a Gaussian one. Introducing the azimuthal property of the sea height autocorrelation and skewness (spatial) functions, the integration over the azimuthal direction can be done analytically, whereas the integration over the radial distance can be done using the property that the skewness correlation distance is much smaller than that of the gravity waves. Then, the backscattering NRCS is expressed from the sea skewness spectrum at a wavenumber different of the Bragg wavenumber. In addition, a simple inversion algorithm is proposed to perform the parameters of the skewness function from experimental data in C and K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> bands.

Scattering From 2-D Perfect Electromagnetic Conductor Rough Surface: Analysis With the Small Perturbation Method and the Small-Slope Approximation

We study the scattering of electromagnetic wave from a 2-D random rough surface that separates the vacuum from a perfect electromagnetic conductor (PEMC). We implement the first-order perturbation method and the first-order small-slope approximation and we establish the analytical expressions of the coherent and incoherent intensities. In contrast with the two special cases, the perfect electric conductor and the perfect magnetic conductor, the coherent intensity reflected by the PEMC has a cross-component. We also show that there is a depolarization in the incidence plane and that there are configurations, where the wave depolarization is complete and the incoherent intensity is reduced to its cross-component.

An Improvement on SSA Method for EM Scattering From Electrically Large Rough Sea Surface

An improved facet model for the predication of the normalized radar cross section (NRCS) of electrically large rough sea surface was proposed based on the first-order small slope approximation (SSA-1) method, the Bragg scattering mechanism, and the specular scattering mechanism. The proposed method is able to evaluate both the complex reflective function and NRCS of electrically large sea surfaces from as low as ultrahigh frequency band to as high as Ka-band. The main idea is that a tilt sea facet can be regarded as the superposition of a planner facet and the microscopic profile; the latter is assumed to be a set of sinusoidal ripple patches. Thus, the integration kernel in SSA-1 over several small facets can be replaced by a large facet with a short wave modification. The efficiency increases because of the much larger mesh size than SSA. Then, both the backscattering and bistatic scattering NRCS results calculated by the proposed method were compared with those predicated by SSA-1, and all of the results show that the proposed method has the merits of high calculation efficiency as well as calculation accuracy.

Electromagnetic Scattering From 3D Layered Structures With Randomly Rough Interfaces: Analysis With the Small Perturbation Method and the Small Slope Approximation

We propose a theoretical study on the electromagnetic wave scattering from three-dimensional layered structures with an arbitrary number of rough interfaces by using the small perturbation method and the small slope approximation. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. They can be correlated or not, isotropic or not. The electromagnetic field in each medium is represented by a continuum of plane waves and a perturbation theory is used for solving the boundary value problem and determining the first-order scattering amplitudes by recurrence relations. The scattering amplitudes under the first-order small slope approximation are deduced from results derived from the first-order small perturbation method. We analyze with the small slope approximation model the combined influence of the anisotropy and cross-correlation upon the electromagnetic signature of a natural stratified structure.

High Frequency Roughness Scattering from Various Rough Surfaces: Theory and Laboratory Experiments

The scattering strength of isotropic and anisotropic rough surfaces was experimentally and theoretically investigated for high frequencies about 500 kHz. Emphasis was placed on studying the response from three two-dimensional rough surfaces which roughness was either isotropic (characterized by a Gaussian distribution) or anisotropic (characterized by a modified-sine surface). Theoretical predictions rely on the first-order small slope approximation either including a Gaussian structure function or a quasi-periodic structure function. The combination of true data and theoretical results indicates the importance of taking into account the anisotropy of a surface in a scattering prediction process. It is shown that the scattering strength varies a lot depending on the propagation plane. In the longitudinal direction of ripples, scattering strength is mostly in the specular direction, whereas in the transversal direction of the ripples, the scattering strength is spread in a very different way related to the particular features of the ripples, with several maxima and minima independent of the specular direction. Contrary to the isotropic surface, the scattering strength from an anisotropic rough surface is modified from one propagation plane to another, which explains why the entire rough surface should be taken into account without any simplification as it is often seen when dealing with scattering models. Compared to such a surface, positions of the emitter and of the receiver are naturally significant when measuring scattering strength.