Small Slope Approximation Method Research Articles

A scaled tank experiment for layered stochastic rough interface scattering reflection coefficient

The stochastic reflection coefficient of the seabed holds valuable acoustic information about the seabed, making it vital for seabed parameter detection. Therefore, it is crucial to achieve some improved results. In this study, we utilized the small-slope approximation method to obtain the fluctuation characteristics of the reflection coefficient for a layered medium with rough interface. Moreover, we addressed the changes in grazing angle by combining the stochastic rough interface scattering with the layered medium reflection. Initially, we present a simulation result, and subsequently, we conducted a scaled tank experiment to verify the variation characteristics of the reflection coefficient. These findings can serve as a theoretical reference for future seabed exploration.

The small slope approximation method applied to a two dimensional dielectric-topological insulator material rough interface

In this paper, the first order perturbation theory (PT) and the small slope approximation (SSA) are employed to analyze the scattering behavior of a two dimensional dielectric-topological insulator (TI) rough interface with Gaussian distribution. Zeroth and first order scattering amplitudes are utilized to obtain the analytic expressions for the coherent and incoherent intensities, respectively. For vertical incidence, an equation can be found whose numerical solution gives the Brewster angle at which the co polarized coherent intensity becomes zero. The scattering scenarios are analyzed where both polarization states of the incident field exhibit total depolarization and the co polarized incoherent scattering is zero.

Soil moisture sensor for agricultural applications inspired from state of art study of surfaces scattering models & semi-empirical soil moisture models

In this research work, models for the scattering from rough surfaces and semi-empirical soil moisture models are presented. Models using the “Bidirectional reflectance distribution function (BRDF)” are focussed in this study. Therefore the effect of the “surface roughness” on the moisture content estimation can be analyzed. “Kirchhoff Approximation Model,” “Small Perturbation Model,” “Integral Equation Model,” and “Small Slope Approximation Method” are mathematically investigated in this research work. Integral Equation Model and Small Slope Approximation Method are having the most extensive range of validity. The main disadvantage of these methods is that they follow a complex methodology and are pretty tricky to implement. Two semi-empirical soil moisture models, popularly known as the “Dubois model” and “Oh model,” are also presented and analyzed in this research work. These models assess the moisture content present in the soil depending upon several factors, i.e., incidence angle (θ), wavelength (λ), frequency (ν), scattering coefficients, etc. Scattering coefficients provide information in terms of polarization in “horizontal-horizontal (HH),” “vertical–vertical (VV),” “horizontal-vertical (HV),” or “vertical-horizontal (VH)” directions. Finally, a novel IoT-based resistive soil moisture sensor is developed and presented in this research work which provides voltage values corresponding to different moistured soil surfaces. Thus in this work, complex mathematics behind the scattering from rough surfaces is presented. Popular semi-empirical soil moisture models for moisture content estimation are presented. Finally, a prototype of the soil moisture sensor is developed to predict the moisture conditions for the different soil surfaces.

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.

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.

A Facet-Based Simulation of the Multipath Effect on the EM Scattering and Doppler Spectrum of a Low-Flying Target at Maritime Scene

A hybrid facet-based electromagnetic scheme is developed in this letter to investigate the multipath effect on the electromagnetic scattering and the Doppler spectrum of a low-flying target at maritime scene. In the scheme, the target and sea surface are both facetized. The facet-based small slope approximation method is developed to calculate the scattering upon the sea facet, which involves the composite scattering mechanisms upon both the long wave and short wave structures. The geometrical optics and physical optics method are combined with the fast ray-tracing technique in consideration of the multipath scattering. The equivalent edge current method is used to consider the scattering from the edged structures on the target. The tapered wave is used in the calculation process to eliminate the nonphysical edge diffractions. Numerical simulations present the multipath effect on the radar scattering characteristics of a low flying missile-like target above the sea surface. The Doppler spectra of the multipath scattering from the target flying at the time-evolving maritime scene are eventually simulated. The multipath effect is also investigated in the Doppler domain.

Scattering properties of a stratified air/snow/sea ice medium. Small slope approximation

The sea-ice thickness, a key parameter in Arctic studies, is derived from radar altimeter height measurements of the freeboard, taking into account not only snow load, but also the penetration depth of the electromagnetic waves inside the snow—this is the not generally the case. Within the framework of the small slope approximation method, we study in Ku-band (f = 13 GHz, λ = 2.31 cm in the air) the electromagnetic signature of an air/snow/sea ice rough layered medium. The snow is inhomogeneous and is represented as a stack of several layers with different relative permittivities. We show that the electromagnetic response is very sensitive to the isotropy factor of the air/snow interface and to the cross-correlation parameters of interfaces

A New Model for Sand-Ripple Scattering Based on SSA Method and Practical Ripple Profiles

In the former model for sand surface electromagnetic (EM) scattering, the sand ripples are generally viewed as a standard sine-shaped wave surface, which artificially deprives many inherent properties of natural sand ripples. For example, the ripples have two pretty distinct sides: Y-junctions and variable texture directions. For these reasons, these corresponding scattering models deserve doubts to final calculated results. On the other hand, the former models usually are statistical models, which are not favorable in target coupling scattering problems and synthetic aperture radar (SAR) imaging simulations. In this paper, based on the physical mechanism governing the formation of aeolian landforms, sand ripples of actual shape are generated under the discrete models. Furthermore, as the slope of the ripple surface is fully small, facet-based small-slope approximation (SSA) method is unquestionably utilized in the EM scattering studies of sand ripples. Finally, significant differences are shown between the results based on sine-shaped wave surface and practical ripple surface, which proves the necessity of the new model for EM scattering of sand ripples.

Ship motion estimation from polarized Doppler spectra from ship wakes on two-dimensional sea surfaces

The main purpose of this paper is to investigate the Doppler spectra from ship wakes on two-dimensional sea surfaces and further estimate the ship motion characteristics. The analysis of the ship wakes is helpful to detect the existence of ships on sea surface. And it will be an alternative method when the radar cross-section values are not competent to identify the ship target. In the study, Doppler spectra for different polarizations are compared with and without ship’s wakes based on the second-order small slope approximation method. As expected, there appears the second spectral peak when ship’s wake is considered. Moreover, the ship velocities, wind speed, and direction are also analyzed. As the results shown, there is a good linearity relation between the position of the second Doppler spectral peak and the ship velocity. Therefore, it is feasible to detect ship according the Doppler spectra.

Spectral Decomposition Modeling Method and Its Application to EM Scattering Calculation of Large Rough Surface With SSA Method

The small slope approximation (SSA) method is a practical method to calculate the electromagnetic (EM) scattering from rough surfaces. However, the SSA method requires that the interval for sampling surfaces must be small enough, such as less than one-tenth of incident wavelength. This constraint condition will cause the problem of huge memory consumption and insufficient memory when the EM scattering of large rough surfaces is calculated. Although the hard disk has large space to keep data and can solve the insufficient memory problem, its read/write speed is still too slow. In addition, massive data transmission will reduce the computational efficiency for the compute unified device architecture (CUDA) parallel computation under some conditions. In this paper, the main idea of the spectral decomposition modeling method is that the whole spectrum of rough surface is divided into several parts and these parts can be used to generate different-scale rough surfaces. Then, by analyzing the different-scale rough surfaces, the large rough surface can be achieved and applied to the calculation of EM scattering with the SSA method. Due to the small memory consumption of different-scale rough surfaces, it takes less time to translate data for the different-scale rough surfaces than that for the standard large surface. Thus, the spectral decomposition modeling method could readily be applied to CUDA parallel computation.

Scattering Studies for Two-Dimensional Exponential Correlation Textured Rough Surfaces Using Small-Slope Approximation Method

Two-dimensional exponential correlation rough surfaces characterized by textures are combined with the small-slope approximation (SSA) method to comparatively study electromagnetic (EM) scattering features of textured surfaces. The normalized copolarized radar cross section from 2-D exponential correlation rough surfaces characterized by stripe texture and block texture, respectively, is analyzed. Several numerical results show the effects of incident angle, texture angle, correlation length, and root-mean-square height on the copolarimetric scattering from the textured rough surface. The validity of the SSA method is verified by comparisons of theoretical value and measured data. Moreover, normalized amplitude distributions of backscattering fields from cells in a scene are studied through its statistical distribution and space correlation function, which are particularly useful for analysis and simulation of remote sensing images. Finally, based on the statistical distribution and space correlation function, the zero memory nonlinear transformation method is utilized to simulate EM scattering from very large scenes. The simulated scene coincides with the original one quite well.

Study on Doppler spectra of two-dimensional skewed sea surfaces

This paper deals with the effects of horizontal skewness on Doppler spectra. A.K. Fung et al. pointed out that the crests of the sea waves are tilted towards the wind direction, which is referred to as horizontal skewness. Although the nonlinear sea surface is more realistic than the linear one, it cannot embody the statistical front-back asymmetry of real sea waves due to the influence of the wind. Thus, the impact of horizontal skewness must be considered. The first-order modified Lagrange model for asymmetric ocean waves is combined with Monte Carlo method to produce the two-dimensional (2-D) time-varying skewed sea surfaces which are nonlinear and skewed simultaneously. Moreover, the second-order small-slope approximation method is used to calculate the scattering field and study the Doppler spectral characteristics from skewed sea surfaces. Horizontal skewness makes Doppler spectra broader in upwind case, but narrower in downwind case. In addition, the impact of horizontal skewness on Doppler spectra is dependent on wind direction and incident elevation angle.

Study of electromagnetic scattering from two-dimensional rough sea surface based on improved Wen’s spectrum

By combining the improved Wen's spectrum with the classic Monte Carlo algorithm and Donelan's experimental conclusion, a two-dimensional sea surface model suitable for different depths and different stages of wave growth is proposed in this paper. Based on the classic two-scale method for the calculation of sea backscattering coefficient, and the introduction of the simulation results obtained by the SSA (small slope approximation) method, the electromagnetic scattering calculation method is modified. Compared with the fitting result based on the backscattering coefficient model of GIT, it proves that the improved Wen's spectrum is suitable for Chinese coastal waves, and the proposed electromagnetic scattering theory is accurate and efficient.

SCATTERING ANALYSIS FOR SHIP KELVIN WAKES ON TWO-DIMENSIONAL LINEAR AND NONLINEAR SEA SURFACES

The ship Kelvin-wake models on two-dimensional (2-D) linear and nonlinear sea surfaces are combined with the second-order small-slope approximation method (SSA-II) to comparatively study the corresponding electromagnetic (EM) scattering characteristics. The nonlinear sea-surface models include the Choppy Wave Model (CWM) and the second-order Creamer model (Creamer (2)). Considering the limitations of using the ideal plane EM wave incident upon a rough sea surface of the limited size, the modifled tapered wave is utilized as the incident wave to derive the scattering waves. Due to the fact that the nonlinear efiects of Creamer (2) surfaces is obviously stronger than those of CWM surfaces, the bistatic normalized radar cross section (NRCS) calculated from Creamer (2) surfaces is signiflcantly greater than that of its linear and CWM surfaces for scattering angles departing from the specular direction, and the backscattering coe-cients from Creamer (2) surfaces are also the greatest except within quasi-specular (near vertical incidence) region. Moreover, for the linear and nonlinear sea surfaces, the in∞uences of difierent wind speeds and directions on scattering characteristics are also calculated and analyzed in detail. However, taking the ship Kelvin wakes into account, the corresponding scattering features are obviously distinct from those of the single linear and nonlinear sea surfaces. This helps to provide a basis to extract the related ship information through the scattering characteristics of ship Kelvin wakes. Also it shows that the small-slope approximation method is a very efiective analysis method to deal with the EM scattering from the rough sea surface.

Scattering and Doppler Spectral Analysis for Two-Dimensional Linear and Nonlinear Sea Surfaces

Two-dimensional linear and second-order Creamer [Creamer (2)] nonlinear sea-surface models are combined with the second-order small-slope approximation method to comparatively study the electromagnetic scattering and the Doppler spectral characteristics from sea surfaces. Due to nonlinear hydrodynamics, the bistatic normalized radar cross section (NRCS) calculated from Creamer (2) surfaces is slightly larger than its linear surface counterpart for scattering angles departing from the specular direction, and the Creamer (2) surface backscattering coefficient increases as well for wind direction angles around upwind and downwind, whereas, for the crosswind direction, it is interesting to note that the observations are contrary. However, as was pointed out by Toporkov in a 1-D surface case, the effect of the nonlinear surface model on the average NRCS is minute. In particular, the Doppler spectra of the backscattered echoes are compared for the linear and Creamer (2) surfaces at various incident angles. It is seen, as expected, that the Doppler shifts and spectral widths of 2-D Creamer (2) surfaces exhibit different features compared with those of the linear surfaces, agreeing with the 1-D cases. However, for larger incident angles, the Doppler spectral broadening for Creamer (2) surfaces is not as severe as was reported in previous 1-D studies that used full Creamer model. The reason for this discrepancy appears to be the reduced nature of the Creamer (2) approximation. Moreover, studies of the Doppler characteristics for different wind directions demonstrate that nonlinear effects become weaker as the wind direction varies from upwind to crosswind.

Model and measurement of high-frequency bottom backscattering intensity envelope.

High-frequency bottom scattering, generally, can be attributed to bottom interface roughness and volume inhomogeneities. The full grazing angle (0–90 deg) intensity envelope model of high-frequency bottom backscattering generated by these two kinds of mechanism was presented. Small-slope approximation method was used to explain bottom roughness backscattering. Bottom volume scattering coefficient was used to calculate bottom volume scattering assuming the volume scattering is homogeneous. The model incorporates the shape of transmitted pulse, directionality of transducer, scattering area, bottom characteristic, and so on. Tank experiment result shows the bottom roughness is the key factor to fine sand bottom backscattering. This backscattering intensity envelope model is useful for the application of bottom backscattering signal, such as sediment classification and geoacoustic inversion. [Work supported by the NSFC.]

THE SMALL-SLOPE APPROXIMATION METHOD APPLIED TO A THREE-DIMENSIONAL SLAB WITH ROUGH BOUNDARIES

In this paper we present new results on the small-slope approximation method. We consider different three-dimensional structures like a randomly rough surface separating two different media and a slab delimited by one or two rough surfaces. We extend the small-slope approximation to the fourth order terms of the perturbative development, and give the expression of the cross-sections for the different polarization states. Numerical examples are treated for the studied structures and a comparison with the small-perturbation is discussed.

Small-Slope Approximation Method: A Further Study of Vector Wave Scattering FROM Two-Dimensional Surfaces and Comparison with Experimental Data

This paper deals with the calculation of the scattering cross-section of polarized electromagnetic plane waves from 2-D metallic and dielectric randomly rough surfaces. The scattering cross- section of object is calculated by the Local Small Slope Approximation (SSA), the scattering cross-section is then compared with experimental data. In this paper, second order terms of the SSA method have been numerically implemented in order to obtain accurate results for a large range of slope. In this paper, we consider scattered and incident wave vectors in arbitrary directions, metallic and dielectric materials with complexpermittivity. Surfaces are considered with Gaussian probability density functions for surface heights and Gaussian or non-Gaussian correlation functions. The coherent and incoherent components of the electromagnetic intensity for cross- and co-polarization are calculated in the bistatic case and we give several comparisons of the theory with measured data.