Comparison of unified full-wave solutions for normal incidence microwave backscatter from sea with physical optics and hybrid solutions

Abstract

The unified full-wave solution for electromagnetic wave scatter from a rough surface is expressed as an integral similar in form to physical optics solutions. However it correctly predicts return for small and intermediate roughness scales where the physical optics approach fails. This full-wave solution is used here to evaluate microwave sea surface backscatter at normal incidence for both the like-and cross-polarized linear components. Surface heights and slopes are assumed to be Gaussian, the sea is characterized by its surface height spectral density function and both perfectly and finitely conducting surfaces are considered. Results from this complete full-wave evaluation are compared with approximations, involving single-scale (specular point) and two-scale models. For both of these models, however, it is necessary to assume a spectral wavenumber, ν_d, where spectral splitting between the large and the small scales of the rough surface occurs. The full-wave solution is used as a guide in the selection of ν_d and to study the accuracy and sensitivity of the different approximations to ν_d. It is also shown that results for cross-polarized backscatter based on the physical optics or two-scale models are totally inadequate.     

Calculations of radar backscattering coefficient of vegetation-covered soils

A model for simulating the measured radar backscattering coefficient of vegetation-covered soil surfaces is presented in this study. The model consists of two parts: the first is a soil surface model to describe the backscattered radar pulses from a rough soil surface, and the second part takes into account the effect of vegetation cover. The soil surface is characterized by two parameters, the surface height standard deviation σ and the horizontal correlation length l. The effect of vegetation canopy scattering is incorporated into the model by making the radar pulse subject to two-way attenuation and volume scattering when it passes through the vegetation layer. These processes are characterized by the two parameters, the canopy optical thickness τ and the volume scattering factor η. The model results agree well with the measured angular distributions of the radar backscattering coefficient for HH polarization at the 1.6 GHz and 4.75 GHz frequencies over grass-covered fields. These observations were made from an aircraft platform during six flights over a grass watershed in Oklahoma. It was found that the coherent scattering from soil surfaces is very important at angles near nadir, while the vegetation volume scattering is dominant at larger incident angles (> 30°). The results show that least-squares fits to scatterometer data can provide reliable estimates of the surface roughness parameters, particularly the surface height standard deviation σ. The range of values for σ for the six flights is consistent with a 2 or 3 dB uncertainty in the magnitude of the radar response.     

非高斯随机粗糙表面计算机仿真的研究

在摩擦学、光学等工程领域中,对于表面粗糙度的研究,总是以生成的随机粗糙表面为研究对象,且大多数研究都是建立在高斯随机表面的基础上,而实际的工程表面大多是非高斯随机表面.为此提出了一种基于快速傅里叶变换(FFT)、Johnson转换系统和自相关函数等理论仿真生成非高斯随机粗糙表面的方法,它可以生成具有给定偏斜度和峰度的随机粗糙表面.为了说明该方法的可行性和正确性,给出了不同偏斜度、峰度和自相关长度下的计算机仿真结果.结果表明:在一定的条件下用该方法仿真生成的非高斯随机粗糙表面,其输入的随机表面的统计参数与输出的统计参数吻合较好.     

Wetting of rough three-dimensional superhydrophobic surfaces

Wetting of rough three-dimensional periodic surfaces is studied. The contact angle of liquid with a rough surface (θ) is different from that with a smooth surface (θ₀) due to the difference in the contact area and effect of the air pockets. For non-wetting liquids (θ₀>π/2), the contact angle increases with roughness and may approach the value of π (superhydrophobic surface). For high θ₀, a homogeneous solid-liquid interface, as well as a composite solid-liquid-air interface with air pockets at the valleys of rough surface are possible. These two interfaces correspond to different states of equilibrium and result in different θ. A probability-based approach is introduced to handle the multiple states of equilibrium and to calculate θ. It is found also that increasing droplet size has the same effect as increasing period of roughness (size of asperities). For larger droplets and for larger asperities, the composite interface is less likely. For applications involving liquid’s transport near rough walls of a channel, an analogy between a droplet of non-wetting liquid and a gas bubble in wetting liquid is proposed. In order to increase bubbles mobility, the contact angle and the contact angle hysteresis should be minimized. Practical recommendations for design of superhydrophobic surfaces are formulated.     

A derivation of roughness correlation length for parameterizing radar backscatter models

Surface roughness is a key parameter of radar backscatter models designed to retrieve surface soil moisture (θ_S) information from radar images. This work offers a theory‐based approach for estimating a key roughness parameter, termed the roughness correlation length (L _c). The L _c is the length in centimetres from a point on the ground to a short distance for which the heights of a rough surface are correlated with each other. The approach is based on the relation between L _c and h _RMS as theorized by the Integral Equation Model (IEM). The h _RMS is another roughness parameter, which is the root mean squared height variation of a rough surface. The relation is calibrated for a given site based on the radar backscatter of the site under dry soil conditions. When this relation is supplemented with the site specific measurements of h _RMS, it is possible to produce estimates of L _c. The approach was validated with several radar images of the Walnut Gulch Experimental Watershed in southeast Arizona, USA. Results showed that the IEM performed well in reproducing satellite‐based radar backscatter when this new derivation of L _c was used as input. This was a substantial improvement over the use of field measurements of L _c. This new approach also has advantages over empirical formulations for the estimation of L _c because it does not require field measurements of θ_S for iterative calibration and it accounts for the very complex relation between L _c and h _RMS found in heterogeneous landscapes. Finally, this new approach opens up the possibility of determining both roughness parameters without ancillary data based on the radar backscatter difference measured for two different incident angles.     

A geometrical optics model based on the non-Gaussian probability density distribution of sea surface slopes for wind speed retrieval at low incidence angles

In this study, a large amount of data from precipitation radar (PR) and National Data Buoy Center (NDBC) buoys are collocated for the development and validation of a Geometrical Optics Model, in order to retrieve wind speed at small incidence angles. The omni-directional model is developed based on the combination of the quasi-specular scattering theory and non-Gaussian probability density distribution of ocean surface slope, and can be applied at incidence angles as high as 15°. There are four parameters included in the proposed model: the effective Fresnel reflection coefficient, the mean square slope, and the two coefficients associated with the kurtosis of the sea surface slope distribution. Using one half of the collocated data, the dependence of the four parameters on the in situ wind speed is acquired. The results show that the effective Fresnel reflection coefficient has a decrease relative to that obtained in previous studies. We combine the proposed model with the maximum-likelihood estimation (MLE) technique to retrieve the ocean surface wind speed at the 10 m height. The retrieved wind speeds are then validated against those measured by the NDBC buoys. The comparison shows that the root mean square error (RMSE) and bias between the model retrievals and buoy observations are 1.54 m s^–1 and 0.1 m s^–1, respectively, revealing high agreements in the wind speed estimations. The results of this study indicate that the proposed model and the PR measurements at low incidence angles can provide reasonably accurate estimates of the surface wind speeds within the range of 0–20 m s^–1.     

非高斯噪声下输出残差或状态估计误差的熵研究

基于非Gaussian噪声线性定常控制系统,通过控制滤波器输出残差或状态估计误差的条件概率密度函数形状来建立有效的滤波设计算法,创建滤波器输出残差或状态估计误差的条件概率密度函数的统一表现形式。利用复合概率密度函数的关系对残差或状态估计误差的条件概率密度函数的近似来实现非高斯残差的高斯化或相应的熵最小化。     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-Optimal Nonparametric Frontier Estimation via Linear Programming

A frontier estimation method for a set of points on a plane is proposed, being optimal in L₁-norm on a given class of β-Holder boundary functions under β ∈ (0, 1]. The estimator is defined as sufficiently regular linear combination of kernel functions centered in the sample points, which covers all these points and whose associated support is of minimal surface. The linear combination weights are calculated via solution of the related linear programming problem. The L₁-norm of the estimation error is demonstrated to be convergent to zero with probability one, with the optimal rate of convergence.     

Parameterizing the aerodynamic roughness length on a regional scale based on multi-source remote-sensing data

Aerodynamic roughness length (z₀) is one of those important biophysical parameters that influence energy exchange at the land–atmosphere interface, so it is significant to quantify the z₀ accurately. In this article, a scheme parameterizing land-surface z₀ at regional scale has been approached based on multi-resource remote-sensing data, including lidar and optical remote sensing. First, we retrieved the regional vegetation height from lidar data of Geoscience Laser Altimeter System (GLAS) on board the Ice, Cloud, and land Elevation Satellite (ICESat), and then the z₀ values of vegetated land surface were calculated using height data and canopy area index retrieved from remote-sensing data. Finally, the wall-to-wall map of z₀ in January and July 2008 were developed. The conclusions are as follows. (1) The vertical and horizontal structures of vegetation can be retrieved combining spaceborne lidar data and other optical remote-sensing data, so the vegetation characteristics and their intra-annual diversification of different land surfaces can be presented dynamically. The variation of z₀ with vegetation phenology can be quantified by modelling with vegetation height and multi-temporal leaf area index from multi-resource remote-sensing data. (2) The z₀ values of vegetated surface change significantly during leaf-on or leaf-off period in the year, but there are different features in the sparsely or densely vegetated surface. In the sparse vegetation areas, due to the relatively low leaf density in leaf-off season, the value of z₀ is also low. With the increase of leaf density in leaf-on season, the z₀ values will also increase. However, the relationship is complicated in the dense vegetation areas in leaf-on season; the z₀ values may or may not increase, but the zero-plane displacement heights will keep increasing continuously. This operational scheme to parameterize z₀ based on the vegetation height and canopy area index retrieved from multi-source remote-sensing data can be applied to quantify time serial z₀ at regional scale. Besides, it can also improve z₀ parameterization in land models or atmospheric models.     

Potential of ASAR/ENVISAT for the characterization of soil surface parameters over bare agricultural fields

The objective of this investigation is to analyze the sensitivity of ASAR (Advanced Synthetic Aperture Radar) data to soil surface parameters (surface roughness and soil moisture) over bare fields, at various polarizations (HH, HV, and VV) and incidence angles (20°-43°). The relationships between backscattering coefficients and soil parameters were examined by means of 16 ASAR images and several field campaigns. We have found that HH and HV polarizations are more sensitive than VV polarization to surface roughness. The results also show that the radar signal is more sensitive to surface roughness at high incidence angle (43°). However, the dynamics of the radar signal as a function of soil roughness are weak for root mean square (rms) surface heights between 0.5 cm and 3.56 cm (only 3 dB for HH polarization and 43° incidence angle). The estimation of soil moisture is optimal at low and medium incidence angles (20°-37°). The backscattering coefficient is more sensitive to volumetric soil moisture in HH polarization than in HV polarization. In fact, the results show that the depolarization ratio σ_HH⁰/σ_HV⁰ is weakly dependent on the roughness condition, whatever the radar incidence. On the other hand, we observe a linear relationship between the ratio σ_HH⁰/σ_HV⁰ and the soil moisture. The backscattering coefficient ratio between a low and a high incidence angle decreases with the rms surface height, and minimizes the effect of the soil moisture.     

The normal of a fractal surface

  We have discovered a class of fractal functions that are differentiable. Fractal interpolation functions have been used for over a decade to generate rough functions passing through a set of given points. The integral of a fractal interpolation function remains a fractal interpolation function, and this new fractal interpolation function is differentiable. Tensor products of pairs of these fractal functions form fractal surfaces with a well-defined tangent plane. We use this surface normal to shade fractal surfaces, and demonstrate its use with renderings of fractal mirrors.     

A new empirical model to retrieve soil moisture and roughness from C-band radar data

A new empirical model for the retrieval, at a field scale, of the bare soil moisture content and the surface roughness characteristics from radar measurements is proposed. The derivation of the algorithm is based on the results of three experimental radar campaigns conducted under natural conditions over agricultural areas. Radar data were acquired by means of several C-band space borne (SIR-C, RADARSAT) or helicopter borne (ERASME) sensors, operating in different configurations of polarization (HH or VV) and incidence angle. Simultaneously to radar acquisitions, a complete ground truth data base was built up with different surface condition measurements of the mean standard deviation (rms) height s, the correlation length l, and the volumetric surface moisture M_v. This algorithm is more specifically developed using the radar cross-section σ⁰ (HH polarization and 39° incidence angle off nadir), namely, σ_0HH,39, and the differential (HH polarization) radar cross-section Δσ₀=σ_0,23°−σ_0,39° in terms of an original roughness parameter, Z_s, namely Z_s=s²/l, and M_v. A good agreement is observed between model outputs and backscattering measurements over different test fields. Eventually, an inversion technique is proposed to retrieve Z_s and M_v from radar measurements.     

A novel Scan SAR imaging method for maritime surveillance via Lp regularization

In this article, a novel Scan mode synthetic aperture radar (SAR) imaging method for maritime surveillance is presented. Conventional Scan SAR is generally operated with severe azimuth resolution loss in order to cover a large area. The proposed imaging method changes the way Scan SAR illuminates sub-scenes and presents a new radar illuminating strategy based on ships’ spatial distribution in each sub-scene. To gain ships’ spatial distribution, a scene sensing algorithm based on radar range profiles together with a peak-seeking and clustering algorithm is introduced. After that, a Markov transfer-probability matrix is generated to make sure that radar illuminates each sub-scene randomly under the probability we calculated before. Finally, an imaging algorithm within the L_p (0 < p ≤ 1) regularization framework is utilized to reconstruct each sub-scene; the regularization problem is solved by an improved iterative thresholding algorithm. The whole wide swath image is joined by putting all the sub-scenes together. Experimental results support that the proposed imaging method can perform high-resolution wide swath SAR imaging effectively and efficiently without reducing the image resolution.     

A comparison between non-gaussian closure and statistical linearization techniques for random vibration of a nonlinear oscillator

The non-Gaussian closure technique is applied to a nonlinear single degree of freedom oscillator subjected to a stationary white noise excitation. The nonlinear restoring force in this system is a softening spring with a hyperbolic tangent behavior. This type of nonlinear system has practical applications in package cushioning. The governing differential equation of motion is used to generate relations between stationary response statistics. These relations are then employed to evaluate a corresponding number of unknown coefficients in a non-Gaussian probability distribution function. A truncated Gram-Charlier expansion is chosen for the probability density function. Up to the sixth order moments of the response process are obtained. The probability density functions predicted by this technique are then compared with density functions constructed by exact solution via the Fokker-Plank-Kolmogorov equation. A comparison is also made with the density function evaluated by statistical linearization for this system. Limitations of this non-Gaussian closure method are discussed.     

Geometrical uncertainty in turbomachinery: Tip gap and fillet radius

In turbomachinery, the prediction of component life is crucial. It has been found that the variability of gaps, fillets and small geometries, due to manufacturing process and in service operation, can be higher than expected. This has direct consequences on both component life and efficiency. The question arising is how to take into account the aleatory distribution due to manufacturing uncertainty during the design phase. The use of state of the art Computational Fluid Dynamics allows considering the uncertainties in the estimation of performance parameters. Uncertainty is treated including the statistical distribution of the variability in the geometrical models.In this work the methodology for the analysis of the impact of geometrical variations on components life and machine reliability is presented. A representative high pressure stage is used as a reference and fillet sizes and gaps are modified in order to generate the response surface to geometrical uncertainty. For that machine, the nominal rotor tip gap G₀ is 1.5% of the span.A preliminary analysis has been performed considering a value for the fillet radius r/G₀ = 50%, to underline the high impact of geometrical variations on the flow field. Then, five fillet radii at the rotor tip and three tip gaps have been studied (r/G₀ = 0.0, 0.03, 0.075, 0.15, 0.3 and G/G₀ = 1, 1.5, 2). A two dimensional Gaussian distribution for the fillet radius and tip gap is arbitrarily assumed. Starting from this distribution the probabilistic functions of corresponding mass-flow through the rotor gap and the heat loading of rotor tip are obtained.The overall variation of mass-flow can be more than 60% and the heat loading is modified by 15% if compared to nominal conditions. Moreover this work shows that by including the mean value without the probability distribution the discrepancy is about 6% for mass-flow and 7% of heat loading.     

Minimum distance between a canal surface and a simple surface

The computation of the minimum distance between two objects is an important problem in the applications such as haptic rendering, CAD/CAM, NC verification, robotics and computer graphics. This paper presents a method to compute the minimum distance between a canal surface and a simple surface (i.e. a plane, a natural quadric, or a torus) by finding roots of a function of a single parameter. We utilize the fact that the normals at the closest points between two surfaces are collinear. Given the spine curve C(t), t_min≤t≤t_max, and the radius function r(t) for a canal surface, a point on the spine curve uniquely determines a characteristic circle on the surface. Normals to the canal surface at points on form a cone with a vertex and an axis which is parallel to Then we construct a function of t which expresses the condition that the perpendicular from C(t) to a given simple surface is embedded in the cone of normals to the canal surface at points on K(t). By solving this equation, we find characteristic circles which contain the points of locally minimum distance from the simple surface. Based on these circles, we can compute the minimum distance between given surfaces.     

EO characteristics for a photoaligned VA‐1/4 π cell on a photopolymer layer

The generation of a liquid‐crystal (LC) tilt angle on a copolymer with chalconyl and cholesteryl moiety characteristics was performed, and the electro‐optical (EO) performance of the photoaligned vertical‐alignment (VA) 1/4 π cell by polarized UV exposure on a homeotropic photopolymer surfaces was studied. The LC tilt angles decreased as UV exposure time increased on the copolymer surfaces. A tilt angle of 87° in NLC was observed with an UV exposure of 3 min on the photoalignment‐2 surface. The LC tilt angle is attributed to increased chalcone moiety with increasing UV exposure time. Excellent voltage‐transmittance (V‐T) curves of the photoaligned VA 1/4 π cell by polarized UV exposure on the photopolymer surface for 3 min containing a cholesteryl moiety of 8% were obtained. The V‐T and response‐time characteristics can be improved by the presence of a cholesteryl moiety in the photopolymer.     

Review of: “ Télédétection spatiale: Université d' été européenne Toulouse 1990”. By MICHEL VAUZELLE (Cépadues-Éditions. Toulouse. 1992) [Pp. 309.] Price FF300.

In this paper, a simple model is proposed for measuring the vegetation cover over soil surfaces from radar signals acquired in semi-arid regions. In such regions, vegetation is characterized by the presence of clumps which partially cover the soil surface. The proposed model describes the relationship between the percentage of covered surface and the measured radar signal. Model simulations over Tunisian test areas, where ground parameters are controlled, are performed and compared with actual ERS2 radar measurements. A very good agreement is found. The model is then used to derive a map of the vegetation cover density for the whole studied site (in Tunisia). The approach used here is based upon supervised classification with classes defined by inverting the model and taking into account ERS calibration error. Each of the four classes thus defined exhibits a good classification rate, greater than 85%. Finally, two important applications for natural resources management are presented: vegetation monitoring and soil moisture monitoring.